ભાજપ
A 56915 4
سلام الله

!
* * ་ ་ * རཱ་“ -
1
General Library
OF
UNIVERSITY OF MICHIGAN.
PRESENTED BY
Author
Nov 18.
189..
1į
→ NOTICE.
Engin. Library
UF
870
.B31
THE Tables contained in my "Mathematical Treatise on the Motion of
Projectiles" (1873) were calculated before the law of the resistance of the
air to elongated projectiles had been completely determined.
In my
anxiety to make the work complete the Tables of Integrals for the cubic
law of resistance were carried to one place of decimals more than was
really necessary for the calculation of trajectories. Although the "Revised
Account" of all my experiments (1890) is accompanied with tables for the
Newtonian law, as well as a selection of tables from the work of (1873) for
the cubic law (so far as is required for the calculation of trajectories),
the original work (1873) is still of value, because it is not likely that such
extensive tables for the cubic law of resistance will be re-printed.
The price of the Treatise has been reduced to 5/-. The author will be
happy to present a copy to any scientific Library or Society.
F. BASHFORTH, MINTING VICARAGE, NEAR HORNCASTLE.
Recently Published by the same Author.
A Revised Account of the Experiments made with the Bashforth
Chronograph to find the Resistance of the Air to the Motion of Projectiles,
with the application of the results to the calculation of Trajectories
according to J. Bernoulli's Method. Price 12/-. 1890, Cambridge
University Press.
Re-print of" A Description of a Machine for finding the numerical roots
of Equations and tracing a variety of useful curves, &c." Communicated
to the British Association, 1845. Price 1/-. 1892, Cambridge,
Metcalfe & Co.
N.B. This machine was re-invented in 1872-3 and was then named the
Tide Predicter.
ALSO
An Attempt to test the Thecries of Capillary Action by comparing the
Theoretical and Measured Forms of Drops of Fluid. By F. Bashforth and
Professor Adams.
1883, Cambridge University Press.
Price £1 1s. Od.
፣
Recently published, by authority.
REPORTS ON
ON
EXPERIMENTS
MADE WITH THE
BASHFORTH CHRONOGRAPH,
To determine the Resistance of the Air to the Motion of Projectiles.
1865-1870. Pp. 170, price 1s.
London: W. CLOWES & SON; HARRISON & SONS; W. H. ALLEN & Co.;
W. MITCHELL; LONGMANS & Co.; TRUBNER & Co.
Edinburgh: A. & C. BLACK. Dublin: A. THом and E. PONSONBY.
Price 2s. 6d.
DESCRIPTION OF A CHRONOGRAPH,
Adapted for Measuring the varying Velocity of a Body in Motion
through the Air, and for other purposes.
BY
FRANCIS BASHFORTH, B.D.,
Professor of Applied Mathematics to the Advanced Class of Artillery Officers,
Woolwich, and late Fellow of St. John's College, Cambridge.
London: BELL & DALDY.
1866.
Price 2s. 6d.
TABLES OF THE REMAINING VELOCITY, TIME
OF FLIGHT, & ENERGY OF VARIOUS PROJECTILES,
Calculated from the Results of Experiments made with the
Bashforth Chronograph, 1865-1870.
London: E. and F. N SPON.
1871.
ON
Preparing for publication by the same Author,
CAPILLARY
ATTRACTION.
A FRAGMENT.
Also,
A PRACTICAL
ON THE
TREATISE
CONSTRUCTION OF OBLIQUE BRIDGES,
With Spiral and Equilibrated Courses.
1850.
ON THE MOTION OF PROJECTILES.
Z
a
D

E
S
S
Ζ'
M
Κ
A
H
THE CLOCK-CHRONOGRAPH.
University Library
A
MATHEMATICAL TREATISE
ON THE
68031
MOTION OF PROJECTILES,
FOUNDED CHIEFLY ON THE
RESULTS OF EXPERIMENTS
MADE WITH
THE AUTHOR'S CHRONOGRAPH.
BY
FRANCIS BASHFORTH, B.D.,
τ
PROFESSOR OF APPLIED MATHEMATICS TO THE ADVANCED ·CLASS OF
ROYAL ARTILLERY OFFICERS, WOOLWICH;
AND LATE FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE.
LONDON:
ASHER & CO.,
13, BEDFORD STREET, COVENT GARDEN,
BERLIN: 11, UNTER DEN LINDEN.
1873.
All rights reserved.
fchor
Rele
6-13-4)
Neclars
ΤΟ
MAJOR-GENERAL SIR FREDERICK ABBOTT, C.B.,
Member of the Late Council.of Military Education;
AND TO
LIEUTENANT-COLONEL C. F. YOUNG, R. A.,
Late Director of Artillery Studies,
This Work is Dedicated
AS SOME ACKNOWLEDGEMENT OF
THEIR JUDICIOUS EFFORTS TO PROMOTE
THE HIGHER SCIENTIFIC TRAINING
OF
OFFICERS OF THE ROYAL ARTILLERY,
AND ALSO OF
THEIR KIND ASSISTANCE AND ENCOURAGEMENT
AFFORDED TO
THE AUTHOR.
1872.
K
Extract from the Reports and Proceedings of the Ordnance Select
Committee for 1868, page 397, vol. vi.
"RESISTANCE OF AIR TO THE MOTION OF PROJECTILES.”
"Note.-Referring to Minute 23,351, p. 255, vol. v. :"
"The conclusions reported to the Ordnance Select Committee, with
"reference to the resistance of the air to the passage of projectiles within
"the limits of velocities of 1,500 to 1,100 feet, and recorded in § 2 of
the above minute,* are in accordance with the law and co-efficient
"previously derived by the Rev. F. Bashforth, B.D., Professor of Applied
"Mathematics to the Advanced Class of Royal Artillery Officers, from
experiments with his clock chronograph, and communicated in reports
"of December 1865 and October 1866."+
* The Engineer, Nov. 15, 1867.
+ Reports, &c., p. 1–17.
PREFAC E.
EARLY in the seventeenth century Galileo de-
termined the path of a projectile on the supposi-
tion that the horizontal motion would continue
the same as if there was no vertical motion, and
that the vertical motion would continue the same
as if there was no horizontal motion. But he
afterwards explained that a projectile would not
actually move in the parabolic path thus found,
partly because he had neglected to take account
of the resistance of the air, and partly because
gravity did not really act in parallel lines. Mathe-
maticians appear to have been content with this
solution until Newton gave a mathematical in-
vestigation of the path described by a projectile
moving under the action of gravity and the re-
sistance of the air supposed to vary as the velocity,
upon which, however, he remarked, "Cæterum,
"resistentiam corporum esse in ratione veloci-
"tatis, hypothesis est magis mathematica quam
naturalis." As Newton did not give a solution
in the case when the resistance of the air was
supposed to vary as the square of the velocity,
which he believed to be the law of nature, John
Bernoulli inferred that he was unable to solve
this problem. Although Bernoulli gave a solution
when the resistance of the air was supposed to
vary as any power of the velocity, that solution
was of no practical use in the state in which it
66
viii
PREFACE.
was left by its author. Moreover, as we now
know that the resistance of the air does not vary
even approximately, within practical limits, ac-
cording to any single power of the velocity, this
solution could not, under any circumstances, have
led to a correct knowledge of the laws of the re-
sistance of the air to the motion of projectiles.
Numerous mathematicians, as Euler, Legendre,
Lambert, Français and Poisson have attempted
to find the path of a projectile when the resistance
of the air was supposed to vary as the square of
the velocity. These solutions did not, however,
lead to any definite conclusion respecting the law
and amount of the resistance of the air. The real
advance made in the science of ballistics was due
to Robins, who first devised means for measuring,
in a satisfactory manner, the velocities of projec-
Afterwards Hutton, improving upon what
Robins had done, obtained very good results.
Numerous experiments have been made since his
time, the most notable of which were those exe-
cuted at Metz, which formed the basis of M.
Didion's Traité de Balistique. But the projectiles
used in those experiments were spherical shot
fired from smooth-bore guns, and the results ob-
tained did little more than confirm the previous
conclusions of Hutton.
tiles.
On the institution of the Advanced Class for
Officers of the Royal Artillery in 1864, there was
no satisfactory work on ballistics, and no experi-
ments made with elongated shot could be found
which were consistent among themselves, and
therefore deserving of confidence. The Navez
electro-ballistic instrument, at that time in use in
this country, was one of the most unphilosophical
of philosophical instruments. Feeling that the
satisfactory solution of any question in gunnery
depended upon the construction of a trustworthy
chronograph, it therefore became my duty to re-
PREFACE.
ix
commend that a proper instrument should be pro-
cured, and that a systematic course of experiments
should be undertaken to determine, in the first
instance, the resistance of the air to the motion
of projectiles. Although I had the hearty support
of the Council of Military Education in these re-
commendations, I met with no encouragement
from the Ordnance Select Committee and its asso-
ciates, who avowed the most perfect confidence in
their instrument and the results it afforded, which
were, however, of no use to me. Under these cir-
cumstances I undertook the construction of my
own chronograph, because it was desirable for me
to retain complete control over my own invention,
on account of the numerous modifications and im-
provements which the first example of such an in-
strument might naturally be expected to require.
However, this roughly constructed instrument far
exceeded my expectations on its first trial in No-
vember, 1865, with ten equi-distant screens.
The chronograph, after having thus passed its
first trial with success, was next used to determine
the resistance of the air to elongated projectiles
of the same diameter, but provided with various
forms of heads. The results of these experiments
shewed that it would not be necessary, for prac-
tical purposes, to employ more than one form of
head in future experiments. †
In the next place, an extended series of experi-
ments was made with a view to find according
to what function of the velocity the resistance of
the air varied, and also to determine whether the
resistance opposed to the motion of shot did ac-
curately vary according to the square of the dia-
meter. The form of head of shot selected for
these experiments was the ogival struck with a
* Reports, &c., p. 1–9.
† Reports, &c., p. 10—17, and Phil. Trans. for 1868, p. 417–441.
X
PREFACE.
radius of one diameter and a half. The diameters
of the shot varied from about three to nine inches.
Also in order to obtain great variations of velocity,
the charges of powder used in firing each kind of
shot were made to vary greatly. Afterwards ex-
periments were made with spherical shot fired
from the guns which had been used to fire the
elongated shot. In this way the resistance of the
air to service projectiles was found, in a conclusive
manner, for all attainable velocities above 900 feet
per second. As, however, the determination of
the law of resistance for velocities below 900 feet
per second was of little practical importance, and
as new arrangements of the screens would have
been required for that purpose, it was not thought
desirable to enter on that branch of the enquiry
without special authority for so doing. These
experiments were concluded early in 1869.*
Some few experiments were subsequently made
by me to determine the resistance of the air to
Whitworth flat-headed projectiles in consequence
of an application made to me from the Department
of the Director of Artillery and Stores, Woolwich.†
Immediately after these experiments were com-
pleted my chronograph was removed from Shoe-
buryness, where it had been four years; for, having
succeeded in perfecting my invention, and having
trained three officers of the Royal Artillery in the
use of my instrument and methods of reduction,
there seemed to be no reason for me to push en-
quiries further, unless the subject was taken up
officially and in earnest, as it was no part of my
duty as Professor or Referee to invent and provide
chronographs for the public service, or to make
experiments, because it was distinctly stated at the
first that the references themselves would always
be "the application of analysis to some practical
* Reports, &c., p. 18-152.
+ Reports, &c., p. 162–169.
PREFACE.
xi
"question." For evidence of the uniform success
of my instrument and methods of experimenting
I must refer to the collection of my Reports, pub-
lished by the authority of the Secretary of State
for War, which have been printed just as they
were written at the specified dates.
The original chronograph, called for distinction
the clock-chronograph, was constructed solely with
a view to determine the resistance of the air to the
motion of projectiles. The labour required to re-
duce each experiment would render this a most
inconvenient instrument for use when it was de-
sired to measure only the initial velocity of a
shot, as in the proof of gunpowder, &c. But
having received repeated applications from the
War Office respecting a simplified form of my
chronograph for common use, I succeeded, after
considerable trouble, in perfecting the gravity
chronograph, hereafter fully described. This in-
strument, constructed originally in a rough manner,
has been often used in the trial of small armı
ammunition, and has proved reliable and conve-
nient to use.
To complete the subject, I have given an ac-
count of the way in which my chronograph may
be used to measure extremely short intervals of
time, as I wished to explain clearly to the Com-
mittee on Explosives the right application of my
methods to such cases. I have also given a critical
examination of the crude form of my instrument
which was brought forward by Captain Andrew
Noble in 1866, and used by the Committee on Ex-
plosives, although it must be confessed that such
instruments, even in their best form, are not
adapted to determine the pressure of fired gun-
powder in the bore of a gun.
The consideration of the motion of a projectile
naturally divides itself into three parts-first, its
motion in the bore of the gun; second, its motion
xii
PREFACE.
through the air; and third, its motion during its
penetration into a solid substance. In the first and
third of these cases I have had no opportunity
of making any experiments whatever, and there-
fore I have nothing new to communicate on the
subject. The little that is known respecting the
laws of penetration of round shot into solid sub-
stances, is due to experiments made at Metz under
the direction of competent mathematicians. The
short chapter on the subject here given has been
adapted from M. Didion's Traité de Balistique. As
to the pressures exerted by the gas of fired gun-
powder, nothing for certain seems to be known.
If we are to accept the dicta of the Committee on
Explosives, the pressure of exploding pebble-powder
is liable to extreme variation from round to round,
even when fired under apparently the same con-
ditions. If this be really the case, further ex-
periments would be almost useless; but the methods.
of experimenting pursued by the committee are
far from leading to conclusive results, as is ex-
plained in Chapter I.
In the remaining case, the whole of the data
required for the calculation of the trajectories of
spherical and elongated projectiles used in this
treatise have been derived from recent experiments
made with my clock-chronograph. The mathe-
matical investigations are virtually the same as
those of John Bernoulli, published a century and
a half ago, but great labour has been required to
render them practically useful. If the resistance
of the air varied accurately as the cube of the
velocity, and if the force of gravity acted in strictly
parallel lines, it would now be easy to calculate
with precision the trajectory of a shot fired at any
inclination to the horizon. I mean that in no case
is it necessary to have recourse to the supposition
ds
that may be replaced by its mean value through
dx
PREFACE.
xiii
a moderate arc of the trajectory, as is done by
M. Didion* and M. Helié† and others. Whatever
may, therefore, be the length of the arc through
which the resistance can be supposed to vary as
the cube of the velocity, through that are the
motion of the projectile may be accurately cal-
culated by the methods and tables given in this
treatise, excepting any deviation due to the rota-
tion of the shot. It would be easy to take account
of the variation of the density of the air correspond-
ing to the height of any point in the trajectory. It
is believed that the Tables given are sufficiently
extended for all practical purposes; and when the
resistance of the air to the motion of shot for
velocities below 900 feet per second has been
determined, it will only be necessary to furnish
additional tables of coefficients corresponding to
tables I. and II.
Just now the results of my eight years' labours
have been arranged in a form adapted for the in-
struction of the Advanced Class of Royal Artillery
Officers, the class itself is threatened with extinc-
tion, inasmuch as a sufficient number of candidates
to form a new class did not, for some reason, offer
themselves for examination last spring. In con-
sequence of this a Committee was appointed to
consider "whether the Advanced Class is of so
"much benefit to the Service as to justify its con-
"tinuance, and whether any mode of educating the
"Royal Artillery Officers in the special branches
"of knowledge which the class was instituted to
"teach can be suggested in preference."
Committee drew up a circular letter containing
four questions, copies of which were forwarded
* Traité, &c., p. 88.
The four questions were as follows:
† Traité, &c., p. 244.
The
1. Whether the existence of that class, under the present regulations, is an
advantage to the Royal Artillery ?
2. Whether you can suggest any preferable way of imparting the necessary
Instruction to Candidates for appointments in the Scientific and Manufacturing
Departments?
xiv
PREFACE.
to thirty-four Officers of the Royal Artillery
varying in rank from General to Lieutenant. The
3. Whether you can account for the apparent unpopularity of the Class in
the Regiment?
4. Whether you can suggest any mode of inducing Candidates to come forward
in greater numbers?
Major-General F. Eardley Wilmot, R.A., replied to these questions in the
following terms:
"Sir,
"9a, Victoria Road, Clapham Common, 9th March, 1872.
"I have the honour to acknowledge the receipt of questions relative to the
"Advanced Class, Royal Artillery.
"Having already, as a Member of the Royal Commission of which Lord Dufferin
"L was Chairman, and also as a Member of the Committee of which Sir Edward
"Warde was President, very plainly expressed an opinion, after taking a con-
"siderable amount of evidence, I fear that as the reports of these two enquiries
are not considered as valuable on this point, I cannot state anything which shall
"add to their force.
"But considering that your request scarcely leaves room for any other course,
"I have the honour to append a few remarks in reply to the several questions.
"1. Very great. It offers the opportunity of instruction to officers of a
"character which can only be obtained by a temporary removal from the ordinary
"duties of the service. It provides a number of officers well qualified to take
a position among the scientific men of the day, and thus satisfies the public
"that there are in a professedly scientific corps a certain proportion of officers
"worthy of that character.
(6
LL
፡፡
"It furnishes, though only a few at present, a well-instructed body of officers
so far more fitted for the duties of the various departments in which they are
required to serve, substituting specific and actual knowledge for that general
"knowledge which is possessed by so many. It will eventually bring into the
departments more highly instructed officers, and providing for a continuous
supply will obviate one great hindrance to widely extended zeal, now resulting
"from the supposed necessity for the continued employment of any particular
officer beyond the period usually ordered for such service. It is a positive gain
"to the country to have officers capable of being utilized in any scientific duty
or enquiry, and of meeting those of other nations in this respect.
(C
"It forms a necessary complement to the education now given at the Royal
"Military Academy, especially in respect to mathematics; and it supplies a most
"valuable inducement to exertion by holding out the reward of employment as
'the well earned result, not of favour, but of ascertained qualification.
"
"2. I conclude this question does not apply to the particular 'course,' but
"the general arrangement under which the class is placed. I do not think that
any plan that does not involve an entire devotion to the special course of
"instruction, without attempting to carry it on with other duties, can satisfactorily
"effect the object. The constant interruption and consequent difficulty of making
steady progress must have been experienced by many officers who have tried
"various means for obtaining advanced education. The present system is based
on a sound principle, viz., a considerable mathematical foundation imparted
"by a highly qualified and devoted professor, accompanied by the systematic
"application of this to the practical duties of the service. Any departure from
"this general principle would, I think, be a mistake.
"C
"3. I conclude that 'unpopularity' is assumed as an apparent cause for there
"being comparatively few Candidates. We are only just settling down into the
"normal condition of peace, and consequently the effects of war and its rewards
"naturally still occupy the thoughts of officers as the surest road to advancement
"and employment. This helps to render 'science' for its own sake less 'popular'
"than it might be. There is a dislike among officers to throw their duties upon
'others, and to apply to enter the class when they know that inconvenience may
"be entailed upon the brigade or the battery by their absence. There is a feeling
"that employment is not heartily given to those that have qualified themselves
"for it in this way, and that those that are thus qualified are, ipso facto, considered
"to be disqualified for such duties as the Horse Artillery. The great majority of
PREFACE.
XV
Committee in their report (C. 589), presented to
both Houses of Parliament, recognised "the great
“interest taken in the subject by the officers gene-
"rally, and the importance which they (with
"hardly an exception) attach to the maintenance
"of some means of enabling officers of the corps
"to acquire that scientific knowledge which is
"so desirable in every respect." The Committee
practically recommended the continuance of the
class system, which provided at Woolwich instruc-
tion in Chemistry, Mechanics, Metallurgy, and
Mathematics specially adapted to the wants of an
"officers are not likely to select this particular line of preparation and employment,
"and would not care to have the latter, even without special preparation, hence
any prospect of 'popularity' is out of the question.
"C
4. By placing the officers on the seconded list, and sending others to do the
"duty which they vacate, the objection of those that did not care to compete
"would be met, and a real grievance which now tells strongly against the class
would be removed. I need scarcely add that a hearty encouragement from those
"in authority and comparative certainty of employment are essential. Under
"these circumstances, due notice being given, it is probable that candidates would
come forward in sufficient numbers.
(C
({
.66
"In conclusion, I need hardly refer to the evidence given to the Royal Com-
"mission and Sir E. Warde's Committee. In conversation with eminent foreign
"officers, the 'Advanced Class' and the 'Long Course' at Shoeburyness have
"always been referred to as among the most valuable arrangements, and most
"worthy to be copied by themselves. I hold the same opinion that was expressed
"35 years ago by General Lefroy and myself when urging the formation of a
Royal Artillery Institution, that it is impossible to maintain a reputation for
"being a scientific corps without some sound basis on which to ground it, over
"and above that given to Cadets.
"(
(C
"It is true that we have had men like Boxer and Dixon, who, without this
"special class instruction, have created departments which are the admiration of
"the world. But they each had a special training and views far in advance of
"the generality of officers, so well known that I need not comment on them.
"I am satisfied that either of them would have otherwise gladly hailed such
"opportunities as this class affords for preparing them for their work. Having
"been in charge of a department myself, it is difficult for me to conceive that any
officer can do otherwise than feel how enormous is the responsibility which he
"undertakes when he represents the special, scientific, and mechanical knowledge
"involved in the intelligent control and direction of so many foremen and others.
(C possessing in some cases the technical, and in some both the technical and the
"scientific knowledge of the work. Doubtless this alone is not sufficient to make
a good Head of Department, but there is nothing in this to hinder the fullest
development of that common sense fact, forbearance, and kindly feeling towards
"subordinates, which are necessary in every post of authority. To keep officers
"for a long time in specific employments until they have acquired a certain rule
"of thumb knowledge, and to reckon that as an equivalent for true scientific
'training, does not appear the wisest course, however highly they may value it,
“and undervalue the opposite system.
((
(C
"I need hardly say that I should deplore the abolition of the class as a very
serious national loss.
"I have, &c.,
"F. EARDLEY WILMOT, Major-General.
"To the President Committee on Advanced Class."
xvi
PREFACE.
Artillery Officer, at a total expense to government
of about £1200 per annum. Several Officers of the
Royal Artillery appearing to be distressed at see-
ing this large (?) sum of money "swallowed up by
"the Instructional Staff," proposed that a few
Artillery Officers should be selected by examina-
tion, and allowed to attend certain courses of lec-
tures at the School of Mines, King's College, and
University College, London. This formed the
"alternative scheme" of the Committee, which, if
adopted, would undoubtedly fail to lead to any
result beneficial to the public service.
If now the system of instruction tried during
the last eight years could have been pronounced
a failure, there would have been some reason for
its opponents to propose a change. But I main-
tain that the class system, instituted in 1864, has
been eminently successful in training for the public
service a sufficient number of Officers of great
ability and unwearied industry. In support of this
opinion, I refer to the official reports on the first,
second, and third classes-to the text books pre-
pared by Officers who have passed through the
advanced class-to the valuable papers contributed
to the Proceedings of the Royal Artillery Insti-
tution by former members of the advanced class;
and lastly, to the satisfactory manner in which
every Officer, who has received an appointment
after leaving the advanced class, has discharged
the duties of his office. There are certainly several
other Officers of the Royal Artillery who have
qualified by passing successfully the examinations
of the advanced class, but have not hitherto re-
ceived any appointment; while they have seen
prizes, which have been held out to them as in-
ducements to work, given to others whose scientific
attainments had never been tested. And this
goes
far to explain the existing dissatisfaction, and the
paucity of candidates for admission into the fifth
PREFACE.
xvii
advanced class. It, therefore, seems to me extremely
opportune that public attention should have been
so distinctly called at the present time to the higher
scientific education of Officers of the Royal Artillery,
and to some of the discouragements under which
professors and their pupils have so long laboured.
As none of the civilian lecturers pretend to have
even the most elementary knowledge of the e-
gimental duties of an Artillery Officer, it is plain
that their teaching is not likely to interfere with
the prospects of those Officers who confine their
attention to strictly professional duties. But just
now great changes are taking place in the arma-
ments of all nations, and it is a question who shall
have the direction of the changes in this country.
Shall selected Officers of the Royal Artillery re-
ceive, first, a high scientific training, and, when
found qualified, be appointed to some subordinate
posts, where they can carry on their education, and
avail themselves of the opportunity to apply to
practical purposes the instruction they have re-
ceived in Chemistry, Mathematics, &c., so as to
gradually fit themselves for the highest posts in the
manufacturing departments? Or, shall the heads
of departments be Officers who do not even pretend
to any
scientific knowledge, and who must, there
fore, in deciding most important questions, be de-
pendent for assistance upon subordinates, or in-
ventors, or scientific men called in for the occasion.
To me it appears that such duties would be best
performed by Officers of the Royal Artillery,
provided they possessed the necessary scientific
knowledge, because they would be likely to be
best acquainted with the wants of the service.
And even if they did not feel sufficient confidence
to deal single-handed with any important question,
they would be able to consult distinguished scien-
tific men, without being under any obligation to
follow implicitly their advice.
xviii
PREFACE.
The nation has suffered incredible losses of late
years from the want of proper scientific train-
ing in its advisers. Take the Armstrong breech-
loading system, a failure, upon which many millions
sterling have been wasted. The "grip" in front
of the seat of the shot was calculated to impede
the initial motion of the shot, and therefore to cause
the powder to exert its utmost destructive effect
on the gun. It ought, therefore, to have been fore-
seen that such a contrivance must fail.
But in dealing with the muzzle-loaders, the
very opposite course was taken. Some kind of
increasing twist of rifling was adopted with a view
to save the gun by facilitating the initial motion of
the shot. On what principle, it may be asked,
was the particular form of increasing twist chosen?
The failure of several very heavy guns in time
of peace seems to indicate that an error of a very
grave kind has been committed in adopting the
Woolwich system of rifling for those guns. Now
the proper form of rifling depends upon the nature
of the powder to be used. If the propelling force
acting upon the shot could be rendered perfectly
uniform, then the gun ought to be rifled with a
uniform twist. Consequently, if the use of pebble-
powder has a tendency to produce uniformity of
pressure of the gas propelling the shot, the proper
form of rifling must approximate to the uniform
twist.
Quite recently we have heard much of the great
power of a new 16-pounder field gun. When,
however, the gun was tried with its proper car-
riage, the firing of a few rounds sufficed to break
the axles of the gun-carriages. It need hardly
be stated, either that the strength of axles should
be perfectly understood, or that everything re-
lating to the recoil of a gun, so far as it depends
upon the weight and calibre of the gun, upon
the weight of the shot, or upon the kind of powder,
PREFACE.
xix
The present
ought to be perfectly well known.
difficulty may be got over by increasing the strength
of the axles, and so adding to the weight of the
gun, or by reducing the charge, and so diminishing
the boasted powers of the gun, but in either case
the gun ceases to be what it was intended to be.
The very numerous and costly experiments on
the penetration of iron plates throw little light
upon the subject, because they have been made
on no system, by the use of shot which generally
broke up in penetrating. Much valuable informa-
tion might be derived from experiments carefully
conducted on a small scale, by the help of a.
reliable chronograph, with shot made of the
Whitworth metal, or some other metal which did
not break up on penetrating.
If these and other questions of the same kind
were made the subjects of careful experimental
investigation under the direction of the best mathe-
maticians of the day, most instructive lessons would
be furnished to the Officers of the advanced class,
and the conclusions arrived at would prove of
the highest value to the nation. I here offer
the results of my eight years' labours, voluntarily
carried on under many discouragements and diffi-
culties, as my contribution towards an improved
state of things. The want of hearty official support
has led to a great waste of labour, because I have
had to do things for myself which ought to have
been done for me. So long as the educational
and the experimental departments do not work
well together, it must be to the great disadvantage
of the public service.
In carrying out the experiments which form the
foundation of the present treatise, I have been greatly
indebted to Major-General Sir F. Abbott, C.B.,
and to Colonel C. F. Young, R.A., as already
stated. At Shoeburyness, on all occasions, I re-
ceived every possible assistance in making the ex-
XX
PREFACE.
To
periments. My best thanks are, therefore, due to
Major-General F. Eardley Wilmot, R.A., to Major
Alderson, R.A., and to Captain Ellis, R.A.
Captain J. P. Morgan, R.A., Captain A. Ford,
R.A., and Captain J. Sladen, R.A., I am indebted
for most valuable assistance in making the ex-
periments and in the laborious work of their
reduction. I must also express my obligations
to the Officers of the Royal Artillery Institution for
the assistance derived from the use of their work-
shops in carrying out my plans.
For further information of a more practical
character, I must refer the reader to the excellent
treatise recently published by Lieutenant-Colonel
C. H. Owen, R.A., Professor of Artillery at the
Royal Military Academy, Woolwich, on the Prin-
ciples and Practice of Modern Artillery.
November, 1872.
CONTENTS.
CHAPTER I.
On the Measurement of the Elastic Force of Exploded Gunpowder
PAGE
1-18
CHAPTER II.
On the Motion of a Projectile in Vacuo
18-23
CHAPTER III.
On the Resistance of the Air
24-44
CHAPTER IV.
On the Motion of a Projectile in a Resisting Medium
45-66
CHAPTER V.
Description and Use of the General Tables
67-73
CHAPTER VI.
On the Law of Penetration of Projectiles
74-79
APPENDIX I.
Adaptation of the Chronograph shewn in the Frontispiece to Measure
a succession of extremely short intervals of time
80-84
APPENDIX II.
Description of a Gravity Chronograph
84-88
APPENDIX III.
On Interpolation and Quadratures
88-90
xxii
CONTENTS.
TABLE I.
Coefficients for the Cubic Law of Resistance. Elongated Projectiles
with ogival heads
PAGE
1
TABLE II.
Coefficients for the Cubic Law of Resistance. Spherical Projectiles
2-3
TABLE III.
3
Log P = log (3: tan & + tan³µ)
Po
TABLE IV.
4-38
Values of X, Y, and T for intervals of 0°.2
TABLE V.
38-39
РФ
P = 3 tan & + tan³ p
+
Values of X, Y, and T for intervals of 1°
TABLE. VI.
40-99
TABLE VII.
99
Table of values of hgt² = 193 14472 inches
TABLE VIII.
d2
A general table of values of
s for ogival-headed shot
100-102
W
TABLE IX,
d2
A general table of values of
t for ogival-headed shot
102-104
W
TABLE X.
A general table of values of
d2
s for spherical shot
105-107
พ
TABLE XI.
=
d2
A general table of values of
t for spherical shot
. 108—110
พ
TABLE XII.
Table of values of h = 1gt² = 16·0954ť² feet
111-112
ON THE MOTION OF PROJECTILES.
CHAPTER I.
INTRODUCTION.
ON THE MEASUREMENT OF THE ELASTIC FORCE OF
EXPLODED GUNPOWDER.
1. Robins made experiments about 1740 with a view to
determine the pressure exerted by fired gunpowder. He
came to the conclusion that when powder was exploded it
produced about 250 times its own volume of gas under the
atmospheric pressure. The elastic force of common air was
found to be increased four-fold when raised to a temperature
equal to that at which iron begins to be white hot. Robins
thence concluded that when powder filling the chamber in
which it is enclosed is exploded, it will exert a pressure equal
to 1000 times that of the atmosphere.* This result was far
below the truth.
2. The best experiments of this kind were made by
Rumford towards the close of last century. He measured
directly the pressure exerted by the gas at the instant of
explosion, and so avoided calculations based on mere hypo-
thetical views. He employed a cylindrical barrel B (fig. 1)
about a quarter of an inch in diameter and capable of
containing 25 grains of fine gunpowder. The barrel, which
was placed in a vertical position, was carefully closed by the
pressure of a heavy weight resting on the hemisphere E, and
acting in the direction indicated by the arrow. The ball
W when heated was employed to fire the charge. Various
charges were used, and by trial it was found what weight
* Robins' Tracts, p. 161.
B
2
ON THE MEASUREMENT OF THE ELASTIC
each charge was just capable of lifting.* The unit of volume
by which the quantity of powder was expressed was the one-
thousandth part of the volume of the barrel. Suppose x to
be the volume of the charge of powder so expressed, and
y the pressure in pounds per square inch excrted by the gas,
then
y=27·6151+0004
3. The above formula did not, however, correctly repre-
sent the experimental results for charges of powder more
than sufficient to fill three-fifths of the chamber. For higher
charges the elastic force of the gas appeared to be greater
than that given by the formula. These results of Rumford
must be considered highly satisfactory so far as they go, but
it is quite possible that, if a much larger chamber and higher
charges had been used, less heat would have been absorbed,
and consequently higher pressures would have been registered.
Rumford's papers are well worth a careful perusal, as his
methods are, in general, so direct and simple; but his cal-
culations respecting the power required to burst his gun are
fanciful and inconclusive. The results of Rumford, however,
do not enable us to calculate the motion of a shot propelled
in the bore of a gun by the explosion of powder, because, in
general, the shot is moving while the explosion of the powder
is in progress, and at no time is it possible to say what is
the ratio of the volume of powder actually exploded to the
volume occupied by its gas.
4. Captain Rodman, by the help of his pressure gauge,
carried out numerous experiments in America to determine
the pressure exerted by fired gunpowder, but his results were
not sufficiently consistent among themselves to merit any
great confidence. He gave a very partial trial to what he
called his velocimeter, by which he found the law of recoil of
the gun mounted in the gun pendulum. This was the first
step towards a satisfactory system of experimenting. But
the mass of the gun being too great, its initial motion was
too insignificant to give the law of pressure of the gas with
accuracy.
5. Experiments were made at Essen in 1867 by General
Mayevski, who took another step in the right direction, but
still he did not arrive at any satisfactory result. A Prussian
* Philosophical Transactions, 1797.
FORCE OF EXPLODED GUNPOWDER.
3
breech-loading gun of 4 was used, which fired an elongated
lead-coated projectile (fig. 3). A hole was bored through the
breech piece in the axis of the gun. Through this hole an
iron rod M was passed and screwed into the shot with which
the gun was loaded. Arrangements were made at the free
end of the rod M to break two wires a, a', connected with a
Boulengé Chronoscope, and placed in certain known positions.
In this way attempts were made to measure the times in
which different parts of the bore were described by the shot.
But, unfortunately, only one observation was made for each
round fired, and that by a chronoscope capable of measuring
only one interval of time. General Mayevski thus confesses
the unsatisfactory nature of his results:-"Les résultats
"obtenus font voir que la durée du trajet du premier pouce
"est très-longue et très-variée, comparativement à celles des
"déplacements suivants, de sorte que les durées que met le
projectile à parcourir différents trajets ne peuvent être
"déterminées avec précision qu'à partir d'un certain dé-
"placement de sa position initiale. C'est pour cette raison que
"l'origine des durées a été prise dans la plupart des expériences
“à 1 de pouce, et dans quelques-unes à 1 pouce de la position
"initiale du projectile."*
66
(6
6. Thus the most important point in the experiments was
given up. But it appears to me that if a suggestion pub-
lished by me in 1866 had been attended to, this difficulty
might, in a great degree, have been overcome. "The only
"method which appears to me to hold out any prospect of
success is one to which Captain Rodman gave a very partial
"trial. He mounted a gun in the old gun pendulum. A
66 cylinder was placed with its axis parallel to the bore of
"the gun. When the gun recoiled it drew back a tracing
"point, which marked a straight line on the surface of the
cylinder at rest. If, whilst the gun was stationary, the
"cylinder was made to rotate, the tracing point described a
"circle on the surface of the cylinder. If now the cylinder
"be made to rotate rapidly and left to itself, and if the gun
"be fired, the point will trace out a curve on the surface
"of the cylinder, the ordinate of which will represent the
"time occupied by the gun in recoiling through a space equal
"to the abscissa. The law of the work done to produce
"the recoil thus becomes known. By measuring the initial
"velocity of the shot, the total amount of work done by the
66
* Royal Belgian Academy, Oct. 10, 1868.
B2
4
ON THE MEASUREMENT OF THE ELASTIC
66
แ powder is found. The law of pressure of the gas would
"be found with greater exactness if the tracing point was
"connected with the projectile. This method of experimenting
seems to be the only one likely to give a satisfactory result,
"for if the rotation of the cylinder be uniform for the very
"short time the explosion is in progress, there can be no
"doubt that the coordinates of the diagram would give us
"a connection between the time and space; and knowing
"the weight and the velocity of the moving body at any
66 one point, we can tell the amount of pressure that was
"acting upon it at any instant, or for any position. From
"what has been said, it is evidently of great importance in
"the proof of gunpowder, that besides the trial for initial
"velocity, there should be some test of the stress thrown on
the gun.
992
7. Arrangements might be made for measuring the angular
velocity of the cylinder at the time of firing the gun. It
is evident that the proposed arrangement would connect time
and space in a satisfactory manner, provided (1) the angular
velocity of the cylinder did not change sensibly during the
progress of the explosion, and (2) that the rod attached to
the marking point did not stretch sensibly. If the curve was
one of continuous curvature then it would be plain that there
was no percussive action of the powder, and no "waves
"of pressure." When high charges were used, General
Mayevski found that the iron rod screwed into the shot was
torn asunder near the shot. Probably in any case a short
rod would have stood sufficiently long to give the initial
motion of the shot correctly. But in order to determine the
law of pressure throughout the whole length of the bore,
it would be necessary to use a rod tapering in substance from
the screwed end. In order to avoid a variable amount of
windage, it would be advisable to substitute for the solid rod
a cylinder, from which an internal conical cavity had been
bored.
8. It is evident that the mass of the gun is too great,
while the mass of the shot is too small, to admit of being
used with perfect success in determining the pressure of the
gas of exploding powder. It appears to me that the proposed
* Proceedings of the Royal Artillery Institution, 1866, p. 189, and Descrip-
tion of a Chronograph, &c., p. 29.
FORCE OF EXPLODED GUNPOWDER.
5
breech-loading gun of Captain Morgan, R.A.,* would enable
this difficulty to be completely overcome. In this case the
barrel is closed by a piston 4 (fig. 2) connected with a heavy
A
mass of metal. When the gun is fired the shot D is driven
forward, and the breech piece A is driven backward, while
the barrel remains nearly stationary. Here the recoiling
breech piece might be made 10 or 20 times the weight of
the shot, and there could be no difficulty in obtaining the law
of its recoil, from which might be deduced the law of the
propelling pressure. Such an experiment would enable us
to calculate the motion of the shot on the supposition that the
pressure per square inch exerted by the gas on the breech
piece and on the shot were equal at every instant. By com-
paring the calculated and measured initial velocity of the
shot the precision of the experiment would be tested. The
resistance due to rifling, or the friction of the lead coat, might
be simultaneously determined from the forward motion of
the barrel, just as the pressure of the gas was determined
from the recoil of the breech piece.
9. A Committee on explosive substances has been more
or less engaged in this country since 1866, but hitherto
nothing more than the most meagre statements of their
proceedings have been published. In the beginning of
their labours the Committee troubled themselves to little
purpose about chronoscopes. They desired to measure the
velocity at different points in the bore of the gun, and from
this they hoped to deduce the corresponding pressures of
the propelling gas. In the first place (Nov. 1866) they tried
the Leurs instrument then just received, but in vain.† From
the Pall Mall Gazette (July 15, 1868) it appears that the
Committee had then become embarrassed with riches, for
they had obtained a Schultz, Chronoscope, and were daily
expecting a Noble Chronoscope. A confident hope was
expressed that, through the instrumentality of these inventions,
"some of the most important problems relating to gunnery'
would be solved in a satisfactory manner. Fabulous stories
were told about the performances of these instruments. The
measurement of time with an error so great as the 5,000,000th
of a second was easily accounted for, although in both these
instruments the spark from a secondary coil was used as the
""
* Journal of the United Service Institution for 1870, and Proceedings of the
Royal Artillery Institution, Woolwich, Feb. 1871, p. 145.
† Proceedings of the Ordnance Select Committee, 1866, p. 350.
6
ON THE MEASUREMENT OF THE ELASTIC
66
recording agent, which, to compare small things with great,
strikes as directly or as capriciously as a discharge of forked
lightning. Thus Gloesener remarks that "l'etincelle ne
parâit pas éclater normalment de la pointe à la plaque;
"elle suit le chemin le plus facile ou qui oppose le moins de
"résistance, et non pas le plus court. La direction de son
"trajet dépend de différentes causes; de la nature, de
l'hétérogénéité, de l'etat hygrométrique, et de la con-
"ductibilité du papier; la forme, la conductibilité de la
"pointe et l'état de l'air qui l'entoure influent sur le trajet
qui suit l'etincelle."*
66
10. These two chronoscopes also agree in another point,
their cylinders are driven by toothed wheels. In the Noble
Chronoscope "there are four toothed wheels, each travelling
"five times as fast as the one behind it," which are used to
impart and maintain the velocity of the short cylinders
upon whose surfaces the records are received. The chatter-
ing noise made when the instrument is at work plainly shews
that the cylinders are in a state of excessive vibration. Up
to the present time I am not aware that the Committee have
published full particulars of a single experiment made with
this instrument in a precise form intelligible to the scientific
world.
11. It is manifest that little reliance can be placed on
the results given by chronoscopes where the records are
received on the surfaces of cylinders which are subject to
more or less vibration in consequence of toothed wheels
being used to drive them. But there are other equally strong
objections to the use of all chronographs whatever in the
determination of the pressure of fired gunpowder, by means
of observations made at six or eight points first at one end
and then at the other end of the bore of the gun. In order
to determine a rapidly varying velocity it is necessary to
measure with extreme accuracy the short time St in which.
Es
a short space ds is described, and then the ratio may be
St.
taken as the velocity at the middle point of the space ds.
But in order to determine the rapidly varying pressure of
the gas which is propelling the shot at any moment, it is
necessary to determine the increment of velocity dv which
* Traité Général des Applications de l' Electricité, t. I. p. 449.
FORCE OF EXPLODED GUNPOWDER.
7
Sv
δέ
takes place in time St, and then the ratio will represent
66
Now when
the accelerating force acting upon the shot.
the spark discharged from the secondary coil is used as the
recording agent, accuracy in space can only be secured by
cutting the primary wires precisely at predetermined points.
The Committee on explosives met with this practical diffi-
culty of securing accuracy in space in the very simplest
case when they attempted to break six primary circuits
simultaneously, for they remark that "great difficulties were,
"however, experienced in securing a simultaneous rupture
"of the primary wires, and the only method found perfectly
satisfactory was to place the whole of them on a small screen
"close to the muzzle of a rifle; on firing a flat-headed bullet
"the whole of the wires may be cut nearly simultaneously,
"and the action of the instrument examined." If so great
a difficulty was encountered in such a simple case, it may
be asked what means were used by the Committee to secure,
in actual experiment, the rupture of the primary wires by
the passage of the shot exactly at the appointed positions.
The contrivance, shewn in fig. 4, was used by the Committee
for this purpose. The elongated shot A rubbed against the
inclined plane of the shearing instrument D, which was half
an inch in length. The primary wire to be sheared is ee,
and f is a steady pin.* It is manifestly impossible to say
what would be the exact positions of the shot in each case,
when the primary wires were ruptured by such a contrivance.
12. But suppose this first and greatest difficulty sur-
mounted, and that accuracy in space could be secured, we
must consider the means used by the Committee to secure
accuracy in the records made by the spark on the rotating
cylinders. The Committee on explosives confess that, "It
"is obvious that the results furnished by an instrument
"intended to measure the very small intervals of time
"6 necessary for the investigations in which the Committee
"have been engaged, would be liable to be received with great
"suspicion were there no means of testing the accuracy of
"the instrument; accordingly, one of the chief objects in its
design was to arrange that the accuracy of the indications
"could at any time be tested with facility." The cylinders
66
* Preliminary Report, 1870, p. 6.
† Ibid.
8
ON THE MEASUREMENT OF THE ELASTIC
were accordingly put in rapid rotation, the primary wires
were cut by firing at them a flat-headed bullet, as nearly
simultaneously as possible, and the records were then ex-
amined. Now in this test it is to be observed that the
rupture of all the primary wires was nearly instantaneous,
and consequently nothing whatever depended upon the known
uniform and steady motion of the cylinders on which the
records were made, or upon the satisfactory action of the
stop-clock. The test employed did not test the instrument
as used in any experiment If the experiment tested any-
thing it merely tested the value of the secondary spark as
a recording agent, the value of which was well known before.
Although it has often been proposed to employ the spark
from a secondary coil in similar cases, I am not acquainted
with any results obtained by its aid in any single case which
are of the slightest scientific value. Still it is difficult to
find any other more satisfactory means of making records if
chronoscopes are to be used in such experiments, and it
therefore becomes necessary in the first place to consider
whether the deviations of the spark from the normal line
are such as would materially impair the value of the proposed
experiments. If I had used the secondary spark, as origi-
nally proposed, the errors caused by the known´ deviation
of the spark would have been sufficient to vitiate all my
experiments made to determine the resistance of the air to
the motion of projectiles. It is, however, extremely probable
that the errors in space due to the shearing apparatus would
be far more injurious than the errors in registration by the
secondary spark in the mode of experimenting pursued by
the Committee on explosives.
This
13. But the chief instrument employed by the Committee
was the Rodman pressure gauge, where the indenting knife
was replaced by a copper cylinder to be crushed by the
pressure of the gas acting on a piston of known area.
method has the great merit of being simple and direct, and,
provided it gave consistent results, there would be no reason
to dispute its value, so far as relative results were sufficient.
But the Committee have noticed a surprising difference
between the recorded pressures when the cylinder to be
compressed was placed near the bore of the gun and when
placed at a distance from it, which does not appear to admit
of a satisfactory explanation. The Committee remark that
"In experimenting with the 8-inch gun, the copper cylinders
"by which the pressure is measured at the front of the
FORCE OF EXPLODED GUNPOWDER.
9
(l
66
แ
66
cartridge have been placed occasionally in the same position
as the Rodman knives, which were employed for that
purpose during the early stages of the Committee's ex-
periments, that is to say, they were put at the outside
"surface of the gun, instead of being situated close to the
"bore.
In all five cases when this was done an enormous
"increase of compression resulted, &c." It is quite necessary
that this anomaly should be explained satisfactorily before
any confidence can be placed in the indications of the crusher
gauge.
แ
66
66
66
66
AC
14. It is natural to enquire what means were used in
order to deduce from a given compression of a copper cylinder
the pressure to which it had been subjected. The Committee
state that "A series of experiments having been made by
means of a testing machine to determine the pressure
required to produce a definite amount of compression in
copper cylinders corresponding to those used in the instru-
ment, the tabulated results furnished a means whereby the
amount of compression produced in the 'crusher' becomes
a direct indication of the pressure at that part of the bore
"where the plug is inserted." This was certainly a very
ready, but it can hardly be considered a satisfactory, way of
determining the pressure to which the cylinders had been
subjected. In the first place the modes of action of the
pressure of the gas and of the testing machine are quite
different. It cannot be safely assumed that copper cylinders
can be manufactured of such uniform quality that all will
shew equal compressions corresponding to equal great pres-
sures. The metal of which some of the cylinders were
composed appears to have been too soft, regard being had
to the pressures they were intended to measure, for one case
is mentioned where a cylinder was compressed from 0·500 inch
to 0.285 inch in length.
15. The Committee state that "the pressures have also
"been calculated, as before, from the various rates of move-
"ment in the bore indicated on the Noble Chronoscope, and it
"has been highly satisfactory to find that all the observations
"of pressure and velocity thus independently taken have cor-
"roborated and confirmed one another in a remarkable degree,
"with the exception of certain discrepancies as regards pres-
"sure, which have been confined to quick-burning powders,
† Preliminary Report, p. 8.
* Progress Report, p. 7.
ΤΟ
ON THE MEASUREMENT OF THE ELASTIC
"and which the Committee hope to explain in their detailed
“report.”*
"Now it is difficult to imagine how this mutual
confirmation is possible, except in cases where the maximum
pressure of the gas is attained before the shot moves sensibly
from its seat. For instance, it is quite impossible for the
crusher to confirm the pressures exhibited by the pressure
curves of Plate IX. of the preliminary report. It is quite im-
possible for a chronoscope to reveal the variation of pressure
said to exist in different parts of the powder chamber. Thus
there were three crusher gauges; namely, A placed in the
axis of the gun and at the bottom of the bore, B in the bore
opposite the centre of the charge, and C approximately in
front of the charge. Three rounds of Waltham Abbey
pebble powder, of density 1.78, were fired from an 8-inch
gun, loaded with a spherical shot weighing 69 lbs., with the
following results:†
Charge.
16 lbs:
Pressure at ¡
4 tons.
Pressure at B.
31 tons.
18
gg
20
212 99
3.
21 99
5
99%
99
""
Pressure at C.
2 tons.
3.
"2
21",
But we
For these low pressures we should expect "the crusher
gauge" to give particularly consistent results.
cannot be satisfied that the pressures indicated at A, or at
B, or at C, follow any law corresponding to the charges
of powder which produced them. It is unfortunate that
charges of 14, 12, 10, &c. lbs. were not also tried. Another
good test of the value of "the crusher gauge" would have
been obtained if, for every round fired, crusher plugs had
been inserted at equal intervals along the bore of the gun.
16. There is another peculiarity in the experiments made
by the Committee on explosives. It appears that, from the
nature of the shearing arrangement (fig. 4) in connection with
the Noble Chronoscope, it was found necessary to employ a
mechanically fitting projectile. For instance, a 7.995 in. bolt
was fired from an 8-inch gun. Mention is also made of a
windage of 0.02 in., while the service windage is generally
9:08 in. Now it is impossible to insist too strongly on the
necessity of carrying out experiments of this kind under
strictly practical conditions. The usual amount of windage
may afford great relief to the gun, by allowing part of the gas
at a high tension to escape, and so act as a safety valve.
* Quarterly Proceedings, &c., 1870, p. 280.
† Progress Report, p. 6.
FORCE OF EXPLODED. GUNPOWDER.
11
แ
66
66
17. After numerous experiments made with the 8-inch and
10-inch guns, the Committee on explosives were employed
to measure the pressures in the bore of the 35-ton gun.
When the gun had a bore of 11.6 in., 35 rounds were fired
with shot of 700 lbs., and charges of pebble-powder varying
from 75 lbs. to 130 lbs. Afterwards, with a bore of 12 in.,
33 rounds were fired with shot of 700 lbs. and charges of
pebble-powder varying from 110 lbs. to 120 lbs. After this
an incipient crack was found in the steel lining of the
gun. The initial velocity reached its maximum as an
11.6-inch gun, with 120 lb. charges, when it was 1370 ft.
per second, with a pressure on the powder chamber of
66 47 tons per square inch; whilst it fell off to 1346 ft. with
"a 130 lb. charge, though it gave 63 tons pressure on the
"base of the projectile, showing that the extra powder was
unconsumed, or that the exit of the shot was impeded."*
An explanation given in The Standard† may apply in this
case. "When the gun was tried with 130 lbs. of powder
"there was an idea that it failed to burn all the charge,
inasmuch as the velocity appeared to fall off. But the gun
was only once fired with this excessive charge, and it is.
66 now understood that the velocity taken on that occasion was.
"fallacious, the electric screen nearest the gun being broken-
"by the blast from the muzzle before the projectile reached
"it. It has been found necessary to take great care to prevent
a recurrence of this disturbing cause. This is just what
might have been expected from the very defective and un-
trustworthy chronometric instruments used, which are quite
insufficient for the performance of exact experiments.
แ
66
66
66
66
66
99
18. It is further stated that "the combined pressures
were 47 tons on the square inch with 120 lb. charges in
"the 11.6-inch bore, but these were reduced to about 35 tons
"in the 12-inch bore. Suddenly, at the eighteenth 120 lb.
"charge, the extraordinary internal pressure of 66 tons per
square inch was registered in the powder chamber; and
"the steel tube being calculated to withstand only 55 tons,
"an incipient crack took place in the bottom groove near
"the seat of the shot. I have not seen all the registered:
'pressures, but those of the last four 120 lb. charges varied
"from 40 to 66 tons, the initial velocity with 46·8 tons pressure,
“being six feet more than when the pressure was 66 tons. The
'
* Captain W. Dawson, R.N., Royal United Service Institution, Feb., 1872.
† Nov. 15, 1871.
12
ELASTIC FORCE OF EXPLODED GUNPOWDER.
"muzzle velocity and maximum pressures given by the last
"four 120 lb. pebble-powder charges in the 12-inch bore,
"with 700 lb. projectiles, were:-
Maximum Powder-Pressures at
Velocity.
Total Work.
Axis of Gun.
1366 feet. 9059 foot-tons.
46'8 tons.
Vent.
35.2 tons.
Projectile.
45.4 tons.
1360
8973
66.0
42.6
53.2
99
99
""
"2
""
1334
8639-
44.6
34.2
46.8
,,
"2
,,
""
1334
8639
37.6
27.0
39.6
99%
93.
19
19.
""
66
"The muzzle-velocity and pressures with smaller charges of
"pebble-powder are said not to present the same variety;
probably, because the effort of the shot to escape from
"them is more successful......Returning to the last four
120 lb. charges and the crack, let us ask, why did the
greatest pressure give the lesser velocity, if the exit of
"the shot was not impeded? Why this variation in the
"powder-pressure at all?"
66
66
'
19. Further, it has been stated that Captain A. Noble,
a member of the Committee on explosives "has fired by
electricity as much as 4 lb. of powder confined in cylinders
"of steel tempered in oil. The cylinders, which were 2 in.
"internal diameter and 1 calibres thick, usually expanded
".002" or '003", and one as much as '02", external diameter.
"......Some crusher gauges' were placed inside, and from
"them, I believe, Captain Noble deduced a maximum
66 pressure of 40 tons. I do not know, however, what
"reliance is to be placed on their indication under such
extreme pressures as they must have been subjected to;
"nor can I reconcile their indications with the expanding
"of the cylinders, which is more in harmony with Rumford's
"results."*
AC
This pressure of 40 tons per square inch is only about
two-thirds of that said to have been subsequently registered
in the proof of the 35-ton gun. It is undoubtedly a most
difficult matter to measure the pressure of the gas of fired
gunpowder. The same powder may, under apparently the
same circumstances, at different times exert a pressure liable
to extreme variations. This is, however, at present little
more than an unsupported theory adopted to explain
discordant results, which may in the end turn out to be due
to the manner in which the experiments have been conducted.
* Captain Morgan, R.A., in Proceedings of the Royal Artillery Institution,
1871, p. 420.
CALCULATION OF INITIAL VELOCITIES OF PROJECTILES.
13
And, regard being had to the striking performances of guns
at long ranges, which depend so much upon the uniform
action of the gunpowder, most thoughtful people will be
disposed to question the results arrived at by the Committee
on explosives. See a valuable paper by Captain Morgan, R.A.,
on the Determination of the Explosive Force of Gunpowder.*
ON THE CALCULATION OF INITIAL VELOCITIES OF
PROJECTILES.
20. The motion of a shot in the bore of a gun has been
usually calculated on the supposition (1) that all the powder
was converted into gas before the shot moved; (2) that there
was no escape of gas by windage; and (3) that the pressure
of the gas varied inversely as its volume for every position
of the shot. The last of these conditions would probably
hold in no case, but the first two might be approximately
satisfied by the Armstrong breech-loading gun, where,
however, there is an enormous and unknown amount of
friction. Sometimes it has been assumed that the work done
on the shot by the powder varied as the weight of the
powder. But this is not approximately true even for
moderate variations of the charge, the gun and shot re-
maining unchanged. For instance, it was found by firing
a hollow elongated studded shot, weighing 23.84 lbs., from
a 5-inch rifled gun, that the average useful effect per pound
of powder used was 147,921 foot pounds, for a charge of
the weight of the shot; 177,268 for a charge of 1; 187,032
for a charge of; 170,964 for a charge of; and 159,875
for a charge of In this case it appears that the useful
effect of each pound of powder attained a maximum when
the weight of the charge was about of that of the shot.
10
Again, when a solid shot weighing 473 lbs., and of the same
external form as the hollow shot, was fired from the same
gun, the useful effect of each pound of powder was 170,534
foot-pounds for a charge of the weight of the shot; 192,071
for a charge of; 199,242 for a charge of; and 200,834
for a charge of 12. Although the shot for our experiments
with the 3, 5, 7, and 9-inch guns were very carefully manu-
factured with a view to determine the resistance of the air,
the guns from which they were fired would afford varying
amounts of windage depending upon their different degrees
* Proceedings of the Royal Artillery Institution, 1871, p. 413.
† Reports, &c., p. 51.
3
14
ON THE CALCULATION OF
of wear and tear. Also the lengths of the bores of the guns
did not follow any law dependent on the calibres of the guns.
Consequently the useful effect per pound of powder obtained.
from one gun did not admit of direct comparison with that
obtained from another gun.
21. It was observed by Rumford* that when charges of
coarse grained powder were used, unconsumed grains were
frequently blown out of the gun, while some grains shewed
clear indications of having been once ignited and of having
been afterwards extinguished. It must be manifest, therefore,
from the above examples of the behaviour of powder, that
its explosion is not instantaneous, and unfortunately we have
no knowledge of the law of its explosion, depending as it
does upon its composition, upon the mode of its manufacture,
upon the size and form of the cartridge, upon the point of
its ignition, upon the state of the atmosphere, and upon
the pressure under which the explosion takes place. Powder
will ignite, but it will not explode in vacuo. Hence it
appears to be undesirable to attempt to make any mathe-
matical investigations respecting initial velocity so far as
it depends upon the charge and weight of projectile, especially
as the law which governs it may be easily determined by a
short course of careful experiments when the service powder
has been decided upon.†
22. But in obtaining a given initial velocity of the shot,
it is necessary to consider the stress thrown upon the gun.
The more uniform the pressure of the gas can be rendered,
so as to distribute the work done on the shot along the
bore, the better it is for the gun. A detonating mixture
would explode so suddenly and violently that it would destroy
the gun and perhaps fracture the shot without imparting
to it a high initial velocity. Gun cotton ignites so rapidly
that all attempts to retard its combustion have hitherto failed
to make it useful in large charges. If the powder ignites
quickly the whole of the charge will be exploded during
the initial motion of the shot, and the tension of the gas
at first will be very high, but it will generally fall quickly
as the volume of the gas is increased. Such a powder would
try severely the endurance of the gun. But when the
powder burns slowly, as the shot moves the explosion will
be going on in a manner calculated to sustain the pressure
*Philosophical Transactions, 1797. † Reports, &c., pp. 46, 47.
INITIAL VELOCITIES OF PROJECTILES.
15
of the propelling gas throughout the whole length of the
bore. În this way it is found that a slow burning powder
may give even a much higher initial velocity to a shot than
a quick burning powder, and at the same time exert a much
less destructive effect on the gun.
23. When quick burning powder was used for large guns
it was most important to consider the initial motion of the
shot, so as to provide, as far as possible, for an early increase
of space for the gas to occupy, for the motion of the shot
must at first be comparatively slow, and there was a danger
that the gas, being confined in a limited space, might burst
the gun. It seems, therefore, that the law laid down by
Professor Hélie was in that case a sound one; namely, that
every obstruction to the initial motion of the shot ought to
be removed.* This consideration gave rise to the rifling of
guns with an increasing twist. Now it is plain that modern
rifled guns firing elongated shot have a much greater strain
thrown upon them than the old smooth bore service guns of
equal calibre. But it does not yet clearly appear whether
this is due chiefly to the increased weight of elongated shot
and the diminution of the escape of gas by windage, or
whether the rifling has any perceptible destructive effect beyond
that due to the cutting away of the metal of which the
gun is composed. If two elongated shot were fired from a
gun rifled with a uniform twist, one studded and the other
without studs, under otherwise identical conditions, and if
their initial motions were accurately determined, it would
be clearly shewn what effect the uniform twist had upon
the initial pressure exerted by the gas. It is known that
the total "work done" in giving rotation to a shot is quite
insignificant when compared with that required to give the
usual velocity of translation, but it is not known what effect
a uniform twist of the rifling has upon the initial motion
of the shot. This is a subject worthy of careful experimental
investigation, for if the increasing twist does not sensibly
relieve the stress upon the gun by facilitating the initial
forward motion of the shot, it fails to effect that saving of
the gun for which it was designed. The more rapidly the
powder explodes the greater is the apparent necessity for
an increasing twist in the rifling; and, on the other hand,
the slowly burning powder recently introduced into the
service must tend to to diminish the practical importance.
*Hélie Traité, p. 400.
16
ON THE CALCULATION OF
of the increasing twist, and perhaps render it entirely un-
necessary.
24. It is undoubtedly right in principle to free the
initial motion of the shot, so far as possible, from all needless
obstructions. The increasing twist has, as we have seen,
been tried in deference to this principle, only it is at present
doubtful whether the advantages secured by its adoption
counterbalance the practical disadvantages of its employment.
But the Armstrong system of breech-loading ignored all
considerations of this kind, for a "grip" in the bore was
placed just in front of the seat of the shot, while the shot
was covered with a thin coating of lead to take the rifling.
An experiment was made at Woolwich in 1865 to determine
the statical pressure required to force a 12-pound shot along
the bore of an Armstrong breech-loading gun.* It was
found that a pressure of from 16.25 tons to 20 tons was
necessary to force a cold 12 lb. shot through the "grip" in
front of the shot chamber, and a pressure of from 3 to 5.5 tons
to force it along the bore. It is plain that the shot could
not move forward before the tension of the exploding powder
was sufficient to exert a pressure of 16 tons on the base
of a 3-inch shot, and even after that the shot would move
slowly. This would cause so great a loss of time that pro-
bably the whole of the powder would be exploded before
the shot had moved sensibly from its seat, and consequently
the gas would exert its greatest possible destructive force upon
the gun.
This error in principle, combined with the use of
the destructive powder formerly in use in this country,
termed poudre brutale or poudre brisante on the continent,
seems to be quite sufficient to account for the failure of the
Armstrong system. The 12-pounders firing shot 3 inches in
diameter and the 40-pounders firing shot of 4 inches in
diameter appear to have been so far encouraging, in the
first instance, that the English Government proceeded with
the manufacture of 110-pounders firing lead-coated pro-
jectiles 7 inches in diameter. General Peel has stated that in
1860-1, £2,830,625, and in 1861-2, £3,006,049, were spent
in the manufacture of warlike and miscellaneous stores, "a
"great portion of which was for the 110-pounder Armstrong
guns which had been adopted" without any sufficient trial.
Repeated failures merely led the late Ordnance Select Com-
66
* Quarterly Proceedings of the Ordnance Select Committee, 1865, p. 23.
INITIAL VELOCITIES OF PROJECTILES.
17
mittee to search for a more enduring metal of which to form
their breech pieces.
25. The failure of the 7-inch Armstrong breech-loaders
led to the adoption of the muzzle-loading system for 7-inch
guns and for those of larger calibre. As it was impossible to
use lead-coated projectiles for muzzle-loading guns, it became
necessary to try other kinds which have been found to give
equally good results. After the large muzzle-loading guns
had proved successful, the same system was adopted even
for field-guns, and thus the Armstrong system was completely
superseded. Still it appears that the question of breech
versus muzzle-loading for large guns has not been finally
settled. The recently introduced slow burning powder
requires a long bore for its useful consumption, while the
limited space in forts and ships restricts very much the
lengths of muzzle-loading guns.
4
26. Inasmuch as Krupp has manufactured breech-loading
guns of 9 inches bore firing lead-coated shot of 300 lbs., which
successfully competed at Tegel with the Woolwich or muzzle-
loading gun of 9 inches bore, it is plain that the failure of
the Armstrong system for 110-pounders cannot be simply
ascribed to the use of lead-coated projectiles. Neither can
it be supposed that Krupp's success is due to the superiority
of the metal of which he constructs his guns. It may be
asked, does he contract the bore of his gun just in front
of the shot, so as greatly to impede its initial motion?
27. Sir William Armstrong, in his address as President
of the Institution of Mechanical Engineers in 1869, when
speaking of gunpowder, confessed that a new light had just
"dawned upon the subject," which altered the prospect. A
comparison of dates leads to the conclusion that this new
light was derived by him from the results of the Tegel
experiments, during the progress of which it was proved
that a slow burning powder might be found, which would
throw less stress upon the gun and give a higher initial
velocity to the shot than the quick burning powder in use
in this country. The different effects produced by different
explosive agents long ago suggested that there might be
different degrees of violence exerted by powders which gave
the same initial velocity to the shot. Shortly after I was
appointed referee to the Ordnance Select Committee in 1864,
I in vain suggested that a few inexpensive experiments
C
18 CALCULATION OF INITIAL VELOCITIES OF PROJECTILES.
66
should be made to determine the law of explosion of
gunpowder. As no attention whatever was paid to my
recommendation, when I was writing a description of my
chronograph in 1866, I took the opportunity to record my
opinion as follows:-"From what has been said, it is
evidently of great importance in the proof of gunpowder
66 that, besides the trial for initial velocity, there should be
66 some test of the stress thrown on the gun. "At present, 1872,
two Boulengé chronoscopes are used to measure the initial
velocity of the proof shot, and the crusher gauge is used to
measure the maximum pressure of pebble-powder. If such
a system, giving reliable results, had been introduced ten
years ago it might have saved many millions of money and
many vain attempts to transgress the laws of nature.
28. If, as seems probable, the progress of events should
render it necessary to reconsider the system of breech-loading,
it is to be hoped that lead-coated projectiles will be abandoned
and that the same kind of shot will be used for one bore,
be the gun loaded at the muzzle or the breech. The lead-
coated projectiles are said to have the good property of
preventing the scoring of the guns, while the objections
urged against their use are numerous and important.
the same shot was made suitable for firing from a breech
or a muzzle-loading gun, the two systems might be in use
at the same time without causing any confusion, and that
system might finally be adopted which seemed to be best
adapted to the wants of the service.
If
* Proceedings of the Royal Artillery Institution, 1866, p. 190, and Description
of a Chronograph, p. 30
( 19 )
CHAPTER II.
ON THE MOTION OF A PROJECTILE IN VACUO.
29. In our investigations we shall suppose gravity to act
in parallel lines and to exert a constant accelerating force, g.
The projectile will therefore move in the vertical plane in
which it is projected. Let the point of projection be taken
as the origin, and let x, y be the coordinates of the centre
of gravity of the projectile at the time t, the axis of y
being vertical and that of x being the horizontal line in
which the vertical plane of projection intersects the horizontal
plane through the origin. Let also V be the velocity and
a the angle of projection.
The equations of motion are
Integrating, we have
Also
dx
d²x
dt²
d²y
It²
dt
=
const. V cos a
(1).
dy
const.-gt V sina - gt
sina-gt
dt
x = Vt
cosa, and y =
Vt sina - gt...…………….(2),
because when t=0, x=0, and y=0.
These two equations (2) give the coordinates of the pro-
jectile at any given time t. Also the velocity v may be
found from equations (1), for
dy
v2
(de)² = (da)² + (din)
dt
dt
=V² cos²a + {'V² sin³a − 2gt V sina+g²t²}
=V²- 2g (tV sina - gt) V-2gy... (3).
–
€ 2
20
ON THE MOTION OF A
Eliminating t between equations (2), we have
9
y =
=x tana
x²...
2 V² cos² a
.(4),
the equation to the path of the projectile, which is a parabola
with its axis vertical.
30. The equation to a parabola with its axis parallel to
the axis of y, its latus rectum 7, and its vertex A at the
point hk, is
or
-
(x − h)² = 1 (k − y), since PN² = 1 × AN (fig. 5);
*2
x² - 2xh+h² = lkly.
= 0
If this curve passes through the origin, the condition x=
2h 1
х
when y=0 gives h=lk, leaving y=7
7. Compar-
ing this with (4) and equating coefficients of like powers
of x, we have
from which it appears that
2h
т
and/
=tana, and
1
9
212 cos² a'
1/2 sin 2a
V" sin" a
2 V² cos² a
h
k
and l=
2g
29
g
Or, the equation (4) may be put under the form
V2 sin 2a\2 2 V² cos² a
XC
2g
9
(-3
V2 sin² a
·Y+
2g
1/2 sin 2α
So that the coordinates of the vertex are
and
2g
V2 sin² a
2 V2 cos² a
and the latus rectum equals
"
2g
9
The coordinates of the focus of the parabola are
V2
瓦
​sin 2α,
2g
and
V2 sin² a
V² cos² a
V2
V² cos 2a
k-
(sin³a – cos³a) =
4
2g
2g
2g
2g
The height of the directrix above the horizontal plane
72
through the point of projection = k+
HX.
4 2g
PROJECTILE IN VACUO.
21
V2
If x, y be the coordinates of p then np=XH- mp= ―y.
2g
Now the velocity acquired by a body falling freely from
rest through a height np
= ×
2g
= √(29 × np) = √ √ {29 (17-
2g
Y = √(V' —
√( Vª — 2gy)
equals the velocity which the projectile would have in its
path at the point p, by equation (3) of Art. 29.
31. To find the range on a horizontal plane passing
through the point of projection we have y = 0, and equation
(4) becomes
2
0 = X tana
9X*
2172 cos² a'
2172
which gives x=0 or X=
sina cosa OR (fig. 5).
g
In order to find the time of flight we observe that the
horizontal velocity
dic
=
dt
V cosa is constant, and therefore
the time in which the range x is described
X
V cosa
2 V sin a
g
And generally when the shot
has reached a point whose
abscissa is x, the time of flight t=
=
X
V cos a
by equation (2).
32. If it be required to find the range and time of flight
on a horizontal plane at the distance
projection we have +x tana-
above the point of
2
gx²
2 V² cos² a
an equation
which gives
V2 sina cosa
±
g
V2 sin² a
2g
If k' be + and <
(-x+ V² ain'α)}.
this equation gives two real values
of x, On and On', corresponding to the ascending and descend-
ing branches (fig. 6).
2 V2 cos² a
sin²
-'
9
2g
If k' be + and
V" sin² a
then the two values of x are
2g
2
equal, or the shot just touches the plane at A.
"
ON THE MOTION OF A
22
If ' be + and >
V² sin' a
2g
then both values of x are im-
possible and the projectile never reaches the plane.
If k' be negative, or if the plane be below the point of
projection, there are always two real values of x, as Om, Om';
Om giving that point in which the trajectory would intersect
the given plane if it was produced backwards in Op.
33. In the same manner we must proceed if we wish
to find where a given projectile will strike a given object
represented by the equation y = f(x), for we have two.
gx2
equations y=x tana
and y=f(x), given to find
2172 cos² a
x and y.
We have found the range on the horizontal plane
172
X = sin 2a, where for a given value of V it is plain that
x will be a maximum when sin2a=1=sinπ or when a=
9
a=1.
34. The inclination 0, of the path of the projectile to
the horizon corresponding to the time t, is given by equations
(2), for
dy
dy dt
tan
dx
མི་ཆེ་
V sina - gt
V cosa
in terms of t and known quantities; or differentiating equation
(4) we obtain
tan 0
dy
dx
tan a
gx
V2 cosa
in terms of x and known quantities.
35. It may be useful to treat the motion of a projectile.
in vacuo in the same manner as we shall treat the motion
of a projectile in a resisting medium. The equations of
motion are
d²x
dt2
d²y
0;
dt2
=-9;
dx
dy
giving
dt
V cosa, It
V sina - gt.
dt
PROJECTILE IN VACUO.
23
Let
dy
dy
dt
tan 0 = p
=p=
dx
dx
dt
then
dx dy
&&&
dt
dp
dx
dy d³x d²y
dt dt
19-19 f f --
dt
dx
therefore
dp
and
X =
Also since
V2 cos² a
9
V2 cos²α
g
V2 cos² a
g
dť²
dx\
dx
dt
dt
Jap=
α
0
dy-p
dx
y= [pdx=-V" cos'a [pdp
2
V² cos² a
9
For the time we have
dt dx
α
9
V² cos²a
g
*S* (tan@ + tan'0) de
1
√2 cos² a
;
•A
"S"
(1+tan²0) d0
V" cos² a
g
α
«Y®.
0
V² cos² a
V cosa
dt
;
X
dp
dx' dp
V cosa
V
cosa
therefore
t =
9
Sdp=
V cosa
g
9
a
g
*S* (1+tan³0) do
V cosa
g
aTo.
a
0
י 70
α 6
α
The numerical values of XX-X°, for the ascend-
ing branch; and for the descending branch X = «X° + °Xº.
In the same manner the numerical values of Y and
may be found in the Table y=0·00, p. 40.
x Ө
.0
• and "T
( 24 )
CHAPTER III.
ON THE RESISTANCE OF THE AIR:
36. THE received theory of the resistance of the air to
the motion of bodies is unsatisfactory in the extreme. It
is based on the erroneous supposition that the moving body
disturbs those particles of air alone which lie in its way, and
no account is taken of the condensation of air in front of
the body, or of the lateral disturbance, or of the disturbance
propagated through the air when the velocity of the dis-
turbing body is below the velocity of the sound wave.
Although the form of the rear of the moving body is known
to modify the resistance opposed to its motion by the air,
no account is taken of that matter in the commonly received
theory. It is further probable that, the amount of the re-
sistance of the air to the motion of a body, at a given
velocity, will in some measure depend upon whether the
motion of the body is being accelerated or retarded. In con-
sequence of these difficulties, it has long been felt that actual
experiment would alone determine the resistance of the air
in each particular case with satisfactory accuracy. The im
portant effect of the air in modifying the motion of projectiles
has caused numerous experiments to be made with a view
to determine the amount of its resistance to the motion of
spherical balls, and, more recently, to the motion of elongated,
shot furnished with various forms of heads.
•
37. The earliest of these experiments of any value were
made by Robins (1742), and by Hutton (1783-91). They
were remarkably successful, considering the inherent defects
of the ballistic pendulum, which was used by them in the
measurement of velocities. In order to determine the resis-
tance of the air to a projectile, it is necessary to find the
velocity lost by it in moving through a given distance. But
as the ballistic pendulum allowed only one velocity to be
measured for each round, it was necessary to fire a number
of rounds with the pendulum placed at one distance from the
gun, and afterwards, having changed the distance between.
ON THE RESISTANCE OF THE AIR.
25
the gun and the pendulum, to fire another set of rounds
under as nearly as possible the same conditions. The dis-
tance between the gun and pendulum in Hutton's experi-
ments varied from 30 to 430 feet. The weights of his sphe-
rical balls varied from about 1 to 6 lbs., and their diameters
from 2 to 3 inches. In the course of these experiments,
the weight of the pendulum was increased from about 500
to 1800 lbs. Although a gun may be charged to all appear-
ance in the same manner each time, there will in general be
found to be considerable variations in the initial velocities
of the shot, which are important when exact experiments
are being attempted with the aid of the ballistic pendulum.
As the distance between the gun and the pendulum is in-
creased, the difficulty of making the shot strike the receiver
properly is also increased. The ballistic pendulum also limits
the weights of the shot which it is possible to use in ex-
periments of this kind..
38. In 1839 and 1840, experiments on the resistance of
the air to the motion of spherical shot were carried on at
Metz, by order of the French Minister for War, by a com-
mission of which MM. Piobert, Morin, and Didion were
members. The weights of the projectiles used in these
experiments varied from about 11 to 50 lbs., and their dia-
meters from 4.0 to 8.7 inches. The distance between the
gun and the ballistic pendulum varied from about 50 to
330 feet. The mean weight of the receiver was nearly
six tons.
39. M. Didion has given a comparison between the
results obtained at Metz and those derived from the pre-
vious experiments of Robins and Hutton by a careful re-
calculation of their observations. Didion's experiments gave
p' =0·02694 (1+0.0023v) when the shot moved with a velocity
of v metres per second. The experiments of Hutton made
with one pound balls gave p'= 0·02693 (1. + 0·002587v), and
those of Robins, made with small-arm bullets 2 inch in
diameter moving with low velocities, gave the same result
as the Metz experiments. But the experiments of Hutton
made with balls of three and of six pounds, as well as those
of Robins made with small-arm bullets moving at high
velocities, appeared to give higher values of p'. M. Didion,
*Didion, Lois de la Résistance de l'Air, p. 22.
4
26
ON THE RESISTANCE OF THE AIR.
therefore made the first term in the value of p' constant in
all cases and = 0.027. Then putting p' = 0·027 (1 + xv), and
giving p' its experimental value, and substituting for v the
mean velocity of the shot during the experiment, the nume-
rical value of x was determined in each case. In this way it
was found from Hutton's experiments with balls of one pound
moving with a mean velocity of 400 metres per second, that
a=0.00257; with balls of three pounds moving with a mean
velocity of 430 metres per second, that x=0.00274; and with
balls of six pounds moving with a mean velocity of 450
metres per second, that x= = 0.00272. The experiments of
Robins made with small arms gave, for a mean velocity of
455 metres per second, x=0.00451. In conclusion, M. Didion
adopted the formula p' = 0·027 (1+0·0023v) as the result of
the most recent experiments made at Metz, by the use
of service projectiles, by the aid of instruments of the
greatest precision. Applying now the formula of M. Didion,
p' = 0·027 (1+0.0023v), and also the new formula derived
from the mean of Hutton's experiments with three and six
pound balls, p' = 0·027 (1 + 0·00273v), to calculate the re-
sistance of the air to a spherical shot 10 inches in diameter
moving with a velocity of 1400 feet, we find that M. Didion's
formula gives 1088 lbs., and that the formula derived by
him from Hutton's experiments gives 1189 lbs. Recent ex-
periments have given 1204 lbs., shewing that Hutton's
experiments were more exact than those of Didion, which
fact leads to the conclusion that Hutton had reached the
limiting weight of shot which could with advantage be
employed in conjunction with the ballistic pendulum. Not-
withstanding this, very large ballistic and gun pendulums
were constructed at the Elswick engine works in 1855 for the
experimental establishment under the Ordnance Select Com-
mittee at Shoeburyness, which were "designed to afford
"data for calulating the initial velocity of cannon shot."
But these monsters have never been used for the purpose
for which they are said to have been designed.†
is probably the most costly ballistic pendulum that ever was
constructed. Elaborate models of these unused instruments
may be seen in the Rotunda at Woolwich, which are said
to have cost £800.
This
*Reports, &c., p. 121.
† Since the above was written I have been informed that these ballistic
pendulums have been taken down, as the sheds were wanted for manufacturing
purposes.
ON THE RESISTANCE OF THE AIR.
27
40. The first proposal to use electricity in ballistic experi-
ments seems to have been made by Professor Wheatstone,
in 1840, who shortly afterwards caused instruments to be
constructed for that purpose. About this time Colonel
Konstantinoff, of the Russian Artillery, consulted Professor
Wheatstone respecting a chronoscope which he desired to
obtain. He afterwards (March, 1843) employed M. Breguet,
of Paris, to construct an instrument for him. The problem
proposed by Colonel Konstantinoff was very clearly stated
as follows: "Disposer un instrument qui pût indiquer et
"conserver trente ou quarante observations successives, faites
"dans des espaces de temps très rapprochés, d'un phénomène
se passant plus ou moins loin de l'endroit où se trouve
"placé l'instrument d'observation."* Great credit is due
to Colonel Konstantinoff for thus clearly stating the problem
to be solved. The construction of the instrument was com-
menced in June, 1843, and the work was completed in May,
1844. In June, 1845, M. Breguet was expecting to hear
the result of a trial of his instrument at St. Petersburg, but
I am not aware that any experiments made with his instru-
ment have ever been published. M. Morin has remarked.
that the problem was not solved in a manner completely
satisfactory.t
66
66
41. Another instrument of the same kind was brought
before the English scientific world by General Sabine, in
his address as President of the Royal Society (Nov. 1866).
He said: "A chronoscope recently devised by Captain
"Schultz of the French Artillery appears, as far as can
"be judged previous to a practical trial, to correspond to
"these conditions, and to be likely to supply a means of
surmounting the difficulty; it is not improbable that it
"will be tried in the present year." Although the Com-
mittee on gun cotton wisely abstained from taking any
further steps in the matter, General Lefroy, R.A., F.R.S.,
as President of the Ordnance Select Committee, applied to
have a Schultz Chronoscope procured for the use of the
Committee, in consequence of the favourable terms in which
the President of the Royal Society had spoken of that
invention. This chronoscope was really brought forward
in 1859, and was somewhat ostentatiously exhibited at Paris
in 1867. It was noticed in The Engineer,‡ and in The
* Moigno, Traité de Télégraphie, 1849, p. 95.
† Du Moncel, Exposé, fc., Vol. II., p. 340.
The Engineer for Dec. 1867, p. 509.
28
ON THE RESISTANCE OF THE AIR.
Practical Mechanic's Journal, where it was stated that
"Captain Schultz, in fact, finds that he can observe and
(6 register time to the ten-millionth of a second."* Far more
satisfactory would it have been if evidence had been produced
of the satisfactory use of this chronoscope in some actual
experiment. In due time the Schultz instrument arrived
at Woolwich (July, 1868), but I am not aware that it
has ever been employed in this or any other country in any
experiment where more than two screens were used. A
few instances of its employment in the measurement of
single velocities were given. by Colonel Benet, which
results are of no scientific value.
42. The instrument most extensively used in this country
down to 1866, was the Navez Electro-Ballistic Pendulum,
which was introduced from Belgium by Colonel Younghusband,
R.A., F.R.S., and for some time a member of the late
Ordnance Select Committee. Perhaps no instrument has had
so much money wasted upon it in the way of experiment
as that of Navez, and yet it gave no results of any value. In
confirmation of this opinion I refer to the Ballistic Reports
of the late Ordnance Select Committee, 80, V. 114, pp. 34
(1862); 84, V. 132, pp. 130 (1863); and to 84, V. 132, pp.
284 (1865). All these labours with the Navez instrument
led to the adoption in England of Didion's methods of cal-
culation, and the reprinting of a selection of his. tables in
1865, even after these things had been discarded at Metz! It
has been pointed out that Hutton's experiments gave better
results than those of Didion, shewing clearly that the science
of ballistics had made no progress in the interval 1800—1865.
Now the instruments of Breguet and Schultz appear to have
been such decided failures, that nothing has ever been heard
of any important experiments having even been attempted by
their aid, so that in such cases the loss of the money paid for
the instruments was about the total loss. But the case was
different with the Navez instrument; it afforded no means
of testing the accuracy of its indications; it was not sufficiently
bad to secure its immediate rejection by the late Ordnance
Select Committee. At one time, it gave very fair results, and
at another, it gave extremely bad results. As an instance of
this I will refer to the experiments made by the help of two
Navez Pendulums, under the direction of General Lefroy, R.A.,
* Practical Mechanic's Journal, for Oct., 1867.
+ Electro-ballistic Machines, 1866.
ON THE RESISTANCE OF THE AIR.
29
to determine the loss of velocity of 12 lb. and 70 lb.
elongated shot, for the information and guidance of the
Armstrong and Whitworth Committee. Each result was a
mean obtained from five good rounds, and the means did
not differ notably from the individual quantities. We now
know that the experiments with the 70 lb. shot were very
fair, and that the experiments with the 12 lb. shot were
extremely bad.
43. Such was the state of our knowledge of the resistance
of the air to the motion of projectiles, when, in 1864, it
was decided to form an Advanced Class for the scientific
instruction of Officers in the Royal Artillery. On that
occasion the council of Military Education sought my as-
sistance in the department of Applied Mathematics, and
afterwards the duties of Referee to the late Ordnance Select
Committee in Mathematical questions were imposed upon
me. A very short examination of the experiments, which had
then been made under their direction, shewed me that, in
general, they had been made on no systematic plan, or, where
there was some approach to system, the results were dis-
cordant. I felt it to be my duty therefore to recommend
the construction of a chronometric instrument, which should
be capable of giving the times occupied by a shot in passing
over a series of successive equal spaces. This recommenda-
tion was made by me before I was aware that Colonel
Konstantinoff had felt and expressed the same want. The
Committee had, however, lavished such extravagant praise
upon their Navez, that they could not listen to any proposals
for the introduction of a rival chronoscope, and in the end
I was driven to the hard necessity of inventing and con-
structing my own instrument. Late in 1865 the new
Chronograph was tried in Woolwich Marshes with 10
screens with perfect success. It was afterwards removed to
Shoeburyness, where it remained from 1866-1870, during
which time the experiments were made which are described
in the "Reports on experiments made with the Bashforth
Chronograph, 1865-1870. The Council of Military Educa-
tion took great interest in these questions, and it was through
their influence that authority was obtained for carrying out
experiments in the interest of the technical education of
officers in the Royal Artillery.
* Reference to this publication is made under the short title, Reports, &c.
London, published by W. Clowes and Sons; Harrison and Sons; W. H. Állen
and Co.; Longman and Co.; and Trübner and Co. Price, 1s. pp. 170.
30
ON THE RESISTANCE OF THE AIR.
44. After the Chronograph had passed its first trial with
success in December 1865, it appeared desirable to make
use of it, in the first instance, to determine the resistance
of the air to elongated projectiles of a given diameter and
weight, but varying in the forms of their heads. The forms
selected were the hemispherical, the prolate hemispheroidal
having axes in the ratio 1 to 2, and two ogivals struck
with radii of one, and of two diameters of the shot. Ten
solid shot were provided of each kind, but unfortunately
sufficient care was not taken to secure equality of weight.
In addition twenty hollow ogival-headed shot, ten of each
kind, were provided. All the rounds were fired with a
charge of 5 lbs. of powder, so that the hollow shot had a
much higher initial velocity and consequently a much higher
velocity of rotation than the solid shot. The diameter of
the shot usually denoted by d was 4.7 inches. The distance
between successive screens was 150 feet denoted by l, and 2b
was the coefficient of resistance. The following is a state-
ment of the average results:
Mean Mean
Weight of Initial
Mean
Gun. Form of Head. Rounds.
sphere
40 Pr.
M. L. į spheroid
No. of Shot-w. Velocity
lbs. f-s.
value
10
20006
of bl².
Date.
5
39.33 1202 ·000840 0001329 Oct.,
4
Ogival (1 diam) 10
Ogival (2 diam) 8
38-69 1196 ·000669 ·0001043 1866. ·
21.81 1541 ·001245 ·0001097
21.94 1545 001181 0001042
The range of velocities obtained in these experiments was
not sufficient to indicate a departure from the cubic law of
resistance. It is apparent from the above values of bw÷d"
that the resistance to the hemispherical head was decidedly
greater than that opposed to the remaining three forms.
The resistance of the air to the hemispheroidal and the
ogival heads varied so little that it was plain that any of
these forms most serviceable in other respects might be safely
adopted. The slight variations in the resistances to the three
latter forms lead to the conclusion that the amount of re-
sistance offered by the air to the motion of elongated shot
is little affected by the more or less pointed apex, but depends
chiefly upon the form of the head near its junction with the
cylindrical body of the shot. In this neighbourhood the
forms of the hemispheroidal head and the ogival head struck
with a radius of two diameters are the same, and the re-
sistances are little different. For a detailed account of these
ON THE RESISTANCE OF THE AIR.
31
experiments see the Philosophical Transactions for 1868,* and
the published reports.†
45. It was still necessary to seek for a satisfactory answer
to the two questions: (1) Does the resistance of the air vary
as the cube of the velocity for all practical velocities of the
projectile? and (2) does the resistance of the air vary exactly
as the square of the diameter? With a view to solve these
questions the following officers were associated with me to
carry out an extended series of experiments with guns of
3, 5, 7, and 9-inch calibres :
Lieut.-Colonel C. F. Young, R.A., Director of Artillery
Studies.
Lieut.-Colonel C. H. Owen, R.A., Professor of Artillery,
R.M.A.
Captain J. P. Morgan, R.A., now Assistant Superintendent
of the Royal Gunpowder Factory, Waltham.
Captain A. Ford, R.A., now Assistant Director of Artillery
Studies.
Captain J. Sladen, R.A., now Assistant Instructor Royal
Laboratory.
The last three officers had then recently completed a two
years' course of instruction in the Advanced Class. In turn
they superintended the firing of the gun and assisted in
the chronograph room, which was about a quarter of a mile
from the range. Most of the reductions of the experiments
were made by the same three officers in the most satisfactory
manner. As the following treatise is founded entirely upon
the results of experiments made with my chronograph,
1865–1870, it will be proper to give a brief description of
the instrument, and shew in what manner the reductions
of the observations have been made, referring the reader
for fuller information to the published description of the
chronograph.‡
46. The chronograph was designed with a view to secure
the following conditions:
(1) The time to be measured by a clock going uniformly.
(2) The instrument to be capable of measuring the times
* Phil. Trans., 1868, p. 417.
† Reports, &c., p. 10.
Bell and Daldy, 1866; Proceedings of the Royal Artillery Institution, 1866,
p. 161; and Revue de Technologie Militaire, t. VI., p. 199.
32
ON THE RESISTANCE OF THE AIR.
occupied by a cannon ball in passing over any number of
successive equal spaces.
(3) The instrument to be capable of measuring the longest
known time of flight of a shot or shell.
(4) Every beat of the clock to be recorded by the inter-
ruption of the same galvanic current, and under precisely the
same conditions.
(5) The time of passing each screen to be recorded by
the momentary interruption of a second galvanic current,
and under precisely the same conditions.
(6) Provision to be made for keeping all the strings or
wires of the screens in an uniform state of tension.
47. In contriving my instrument I at once abandoned
all attempts to drive the cylinder at a known uniform angular
velocity. The frontispiece gives a general view of the in-
strument as constructed and used in experiment. A is a
fly wheel mounted on a vertical axis. K is a cylinder
intended to be covered with paper prepared to receive the
records made by the two markers m, m'. When the fly
wheel is spun by hand, the toothed wheel B allows the string
C to unwind from the drum M, and so lower the stage S,
which descends smoothly by its own weight along the slide L.
Two electro-magnets E, E are fixed to the stage S. These
magnets, by means of arms a, a', work levers b, b', which
give motion to the markers m, m'. A lever h acts upon the
springs s, and so raises the markers m, m', or keeps them
gently pressed against the paper mounted on the cylinder K.
If we suppose the markers depressed, and the fly wheel spun,
as a gyroscope, the markers m, m' will trace two parallel
spiral lines upon the cylinder. But if we cause a galvanic
current to circulate round the lower electro-magnet E', and
if the galvanic circuit be so arranged that the pendulum of
a half-seconds clock shall interrupt and restore the current
once every double swing, then we shall have upon the spiral
traced by m' records made every second. The forces tending
to destroy the motion of the fly wheel are the resistance of
the air and the friction at the bearings, and at the marking
points m, m'. The only force tending to drive the heavy
fly wheel is that derived from the tension of the string C
acting through the toothed wheel B, about three-quarters
of an inch in radius. The cylinder K and the fly wheel
ON THE RESISTANCE OF THE AIR.
33
A are respectively 4 and 18 inches in diameter. The object of
these arrangements was to remove all causes which might
be likely to disturb the perfect smoothness of the rotation
of the cylinder. As I knew that it was quite impossible
to secure sufficient uniformity of angular velocity of the
cylinder, I merely endeavoured to secure a uniformity of loss
of angular velocity. The regular beating of the clock gave
a diminishing scale of time on the cylinder. By means of
necessary corrections and interpolations this scale was further
divided into a correct scale of tenths of a second.
It was
not necessary to carry the subdivisions of the scale further
than tenths of a second, because it was found that intervals
of time less than the twentieth of a second could always
be treated by the ordinary method of proportional parts.
48. Suppose now a second galvanic current to circulate
about the other electro-magnet E, and also to pass through
ten or more equidistant screens, so contrived that the passage
of a shot through any screen shall cause a momentary inter-
ruption of the galvanic current at that screen, but allow the
current to be restored again before the shot can arrive at
the next screen. Each interruption of the current will be
recorded by the marker m upon the rotating cylinder K.
Afterwards the clock and screen records are measured off,
starting from some line drawn on the surface of the cylinder
parallel to its axis.
49. It is well known that there will always be some loss
of time between the interruption of a current and the regis-
tration of the corresponding record. As it is impossible
entirely to remove this difficulty care must be taken that
each marker shall make its own records under precisely
the same circumstances, so that we may be permitted to
assume that the loss of time is the same for every record
made by the same marker. The clock records present no
difficulty because the pendulum goes on swinging and
breaking contact once a second, hour after hour, and we have
no reason to suppose that the loss of time in making the
records will vary in the course of 10 or 20 seconds. But
precautions are required to secure a like condition for the
screen records. If the galvanic current was allowed to
circulate for a few minutes about the electro-magnet E,
and then was interrupted in rapid succession 10 or 15 times
in a second, it might be expected that the remaining mag-
D
34
ON THE RESISTANCE OF THE AIR.
netism would have an effect in delaying the records which
would gradually decrease with each successive interruption.
This would be a very injurious kind of error, for the varying
loss of time would follow a law, and therefore it would be
impossible to eliminate it. The following arrangement was
made with a view to obviate this difficulty. The action
of the lever h, when used to raise the markers m, m' from
the paper, simultaneously breaks the galvanic current be-
longing to the electro-magnet E, and restores the circuit
when the markers are depressed preparatory to the firing
of the gun. The manner in which this is effected is shewn
by fig. 15. The screen circuit is joined at A and B. When
the switch 7 stands as represented in the figure, the current
is broken. While the screens are being mended 7 is brought
in contact with m, and thus the ringing of the bell indicates.
the proceedings on the range. And when 7 is brought in
contact with n the galvanic current circulates about the
electro-magnet E. But between the depressing of the
markers and the firing of the gun there is in general a
little delay which requires further precautions. A self-acting
contact breaker (fig. 14) is placed in the circuit and near the
gun. When the firing party are ready, a lever ab is raised
and a pin ƒ is inserted, which turns the galvanic current
through the contact breaker D. The same motion of the
hand that pulls the lanyard to fire the gun withdraws the
above-mentioned pin f, and allows the lever ab to be brought
down into the position ac by a spring, and so shutting out the
contact breaker D, completes the circuit just before the shot
reaches the first screen. Further, it is not important for the
losses of time to be absolutely equal for records made by the
markers m and m', provided they be small for each marker.
50. The most troublesome part of my undertaking was
the invention of a proper screen, which would allow the same
galvanic circuit to be used for all the screens on the range,
the records being made by the momentary interruption of
this single circuit. Provision had to be made for closing
the circuit interrupted at one screen before the shot arrived
at the next screen. This was satisfactorily accomplished
in the manner shewn in fig. 12. Springs b, d, f,
formed of hard brass wire, and were fixed in wood as re-
presented. The ends of the springs passed through holes
in copper plates as a, c, e, &c. Wood being a non-conductor,
the current which entered at a passed through the spring b,
were
ON THE RESISTANCE OF THE AIR.
35
into the plate of copper c; thence by the spring d it entered
the plate of copper e, and so on along the whole length
of the top of the screen. The tops of all the screens and
the chronograph were included in one circuit, as shewn
in fig. 13. The holes in the plates of copper through which
the springs projected were about half an inch in length,
and the springs, when not held down by weights in contact
with the lower edges, rose up by their own elasticity and
remained in contact with the upper edges of their holes. For
an experiment the springs were held down by equal weights
suspended by strong sewing cotton so as to form the screens.
These springs were about two inches apart for experiments
with bores of 3 inches and upwards. The shot in its course.
broke one or more threads as it passed through each screen,
and so, releasing one or more springs, caused the marker m
to register a corresponding record.
51. When every precaution had been taken it was found
that there were still little irregularities in the readings of
several successive records. The corrections required were
most simply performed by plotting on an enlarged scale, and
then drawing a curve. They may also be performed by
differencing the readings and applying small corrections
required to render the second or third orders of differences
regular. It has been suggested that the method of least
squares might have been employed to obtain the most pro-
bable results from our experiments. But I very much doubt
whether any other method of dealing with our observations
would have given results so accurate and certain as the one
adopted. We had no supposition to make respecting the law
connecting space and time which we were in search of, we
merely assumed that there was a law. By inspection we
could see whether one mode of correction gave the sum of
the squares of the errors greater or less than another, when
there was any doubt whatever. But in such cases the
practical difference of the two systems of correction would
have been insensible. When the corrected screen readings
were converted into time, and again differenced, we were
able, almost by inspection, to obtain corresponding values of
the velocity of the shot and of the resistance of the air,
just at the instant when the shot passed any of the screens.
In this manner we obtained several corresponding values of
velocity and resistance for each good round fired. As it
was found impossible to express the resistance of the air
D 2
36
ON THE RESISTANCE OF THE AIR.
in terms of the velocity by any simple formula, the resistance
has been expressed in terms of the cube of the velocity in
connection with a variable coefficient. In considering the
motion of a shot it will be necessary from time to time to
change the coefficient so as to keep it in accordance with
the velocity of the shot. The law of the cube of the velocity
has been chosen on account of the simplicity of the formulæ
to which it gives rise.
52. The experiments were carried out with guns of 3,
5, 7, and 9-inch calibres, for which the following elongated
shot were manufactured with great care in the Royal Arsenal,
Woolwich; namely, shot 2.92 inches in diameter, weighing
12, 9, and 6 lbs. for the 3-inch gun; shot 4.92 inches in
diameter, weighing 47.68 and 23.84 lbs. for the 5-inch gun;
shot 6.92 inches in diameter weighing 122.3 and 61.15 lbs.
for the 7-inch gun; and shot 8.92 inches in diameter weighing
250 and 125 lbs. for the 9-inch gun. The charges used
in each gun were generally the one-sixth, the one-eighth,
and the one-twelfth of the weight of the solid shot. The
shot had similar external forms, the heads being of the
ogival form struck with radii of one diameter and a half.
The resistance of the air was found to vary as the square
of the diameter of the projectile, but when the resistance
of the air to a given projectile was expressed under the form
26 × (velocity)³, it was found that 26 attained a maximum
value for a velocity of about 1200 feet per second, and con-
sequently decreased for higher and lower velocities. The
numerical values of b were determined experimentally for
ogival-headed elongated shot and other kindred forms, for all
velocities between 900 and 1700 feet per second.* These
coefficients in a convenient form will be found in Table I.
Afterwards guns of the same calibres were used to fire
solid and hollow spherical shot, in order to determine
the coefficients for the cubic law of resistance to spherical
shot corresponding to all velocities between 850 and 2150
feet per second. These coefficients are given in a convenient
form in Table II.
53. The following examples are given as illustrations of
the great utility of differencing when systematic experiments
or independent calculations are being made. Suppose that
* Reports, &c., p. 152.
† Reports, &c., p. 114.
ON THE RESISTANCE OF THE AIR.
37
the following logarithms of numbers have been independently
calculated, and that the calculator wished to test the accuracy
of his work.
Calculated
Corrected
Nos. logarithms.
logarithms.
1280 107210
+ 3380
A₂
107210
Δι
Correc-
+ 3380
Δη
1290 110590
- 27 tions.
110590
27
+ 3353
+ 3353
1300 113943
· 35 + 10
113943
- 25
+ 3318
1310 117261
+3328
- 05
20
117271
- 25
+ 3313
1320 120574
- 35+ 10
+ 3303
120574
- 25
+ 3278
1330 123852
- 24
+ 3278
1
123852
+ 3254
1340
127106
+ 3253
– 25
- 26+
2
127105
+ 3228
+ 3229
- 24
1350
130334
-
23 - 1
130334
+ 3205
+ 3205
- 24
1360
133539
- 23
133539
- 23
+ 3182
+3182
1370 136721
136721
The difference of type indicates the erroneous figures in one
place and their corrections in the other. Tables of the
squares, cubes, &c., of numbers have respectively their second,
third, &c. differences constant, thus
No. Squares.
Cubes.
1
1
2
4
+
3
9
+
4
16
5
+
9
1:00 17 O
Δη
1
+2
8
+2
27
7
+2
64
25
+2
125
++++
A₂
7
+ 12
A3
+19
+6
+18
+ 37
+6
+24
61
+ 6
+11
+ 30
6
36
+ 91
+2
216
+6
+13
+36
7
49
+2
+ 127
343
+6
+ 15
+ 42
8
64
+ 169
+2
512
+6
9
+17
+48
81
+ 217
+ 2
729
+6
+19
+ 54
10
100
+ 271
+2
1000
+6
11
+21
+ 60
121
+ 331
+2
1331
+23
+66
+6
12 144
+2
+ 397
1728
+6
&c.
&c.
As the final differences are here constant, we could go on
and calculate complete tables of squares and cubes of integers.
In the former case adding 2 to 17 we have 19, and adding
19 to 81 we get 100, the square of 10 and so on; and in the
latter case adding 6 to 48 we get 54, and 54 to 217 we get
271, and adding 271 to 729 we get 1000, the cube of 10.
Difference machines have been constructed to work on this
principle. In the same way we may calculate formulæ of the
form u= a + bx + cx², where the first order of differences A,,
38
ON THE RESISTANCE OF THE AIR.
3
would be b+2cx, and the second, A,, would be constant and
=2c. If u = a + bx + cx² + dx³, then the column of the third
order of differences 4, would be constant and + 6d. In
the generality of tables, no order of differences would be
absolutely constant, but the final order used would in general
change very slowly, as in the above example taken from
a table of logarithms.
54. We now proceed to shew how the first and second
differential coefficients may be expressed in terms of the
successive differences.
By finite differences we have
or wx+1
u = u₂+ Aux
ux
Au = u
x+1
ის ე.
D.
Ux+2} = U x +¿+AU x+1=U₂+AU+A (u +▲u )=u +2▲u + A² u
U
3 Δ Δ 201
¹ x + B1 = U x + 3 AU₂+ з▲² u¸ + ▲³µ»
X
29
&c.
And generally
n.(n-1)
n.(n−1).(n − 2)
Ux+ni
ux + n Aux +
▲²u +
A³u₂+ &c.,
*
1.2
1.2.3
6
=u₂+n{Au¸—‡▲³u„+}A³u¸—‡A*u„+‡A³u„−‡A®u„+&c.},
13
+n³ {{A³u¬†Ã³u„+ 11A¹u¸ ¬¦¦A³u +‡‡ZA®u„+&c.},
+
&c.
360
Expanding now unt by Taylor's Theorem, we have
x+n?
du nl d² u n০t²
3
d³u n³ ³
7³
U
=U₂+
x+nl
Ux
+
dx 1
x
XC
+
dx² 1.2 dx³ 1.2.3
3
+&c.,.
and equating the coefficients of the first and second powers
of n in the two expansions of u+n' we have
7 du
doc
d'u
20
{A*u%+‡A³u¸
= Au„ ‡A³u„+
▲u — — A³µ¿ + } A³u„ − ‡A*u„ +‡A³u¸ ¬ £A°u„ + &c.,
ľ² = ▲³u - ▲³µ₂+}}^ *u - 184° u + 1}}A®u - &c.
dx²
20
3
20
20
ON THE RESISTANCE OF THE AIR.
39
Ux-41
Au
Ux-31
+аи 2х-46
+ Au
x-31
И
+ A²ux-31
The following scheme will explain the meanings of the
several terms:
A³u
+ A² U x-4² + A³ U x-4?
u
+ A*Ux-4!
x-21
+ AU x-27
+ D³ux-32
+ D³ux-4?
U x-l
A³u
+4² U x-22
+A+Ux-31
+ AⓇux-st
'x-41
+ Aux-1
+
A³ux-21
+ A³u 'x-31
+ A²ux-l
+ A*ux-22
+A®Ux-3?
น
+ Aux
+ A³u
+ A²ux
+ A*ux-l
A&u,
น
x+1
+ Au
+ A³u₁₂
Δ
x+1
+ A²ux+?
+ A*ux
+4x-21
+ A³ux-b
+ AⓇu
x-21
+ A® U¸-l
Ux+21
Au
+AU x+22
+ A³ux+!
+1³um
A² u
Ux+31
+Aux+81
+ 422 x+22
+ A*ux+l
+ A Ux+2²
A³ux+2²
and
น
'x+41
+AU x+1¿
Ux+58
Now
therefore
3
- A³u
Δ
+ A² U x+3?
ندلله
W
x-1
+ Aux-I
Δυ
Au
A²ux-
A³u
c-l
+호스 ​u
= + 4 + u
+ A³ux-i
A&u
x-1
+1 1 4 5 ~ 2-1
c-l
•
- 1945 u
12
-} } A³u
x-l
- 1 § 4°
u
U-t
7
Δ
+ 137A®u=+1&Z A ® u „_ ¿ + 1 § Z A ® u x-1
80
&c.
Taking the sum we find
x-1
&c.
で
​d²u
2
dx²
2
A²u x-1
1
x-l
1
X-l
13
180
11³▲³ux-¿+ &c.,
A&u
&c.
x-l
x-21
x-31
Let s be the distance the ball has moved from some fixed
point to a screen at a time t,, 7 the distance between successive
screens, and t.„, to_19 to, tai ti &c. the observed times of the
shot passing successive screens; then we shall have
dts
ds
-219
- At¸ — §Ã³t, + {▲³ts − 14³ts + ‡Ã³t, — ¿A°t,+ &c.,
40
ON THE RESISTANCE OF THE AIR.
or
d's.
dt.
= V,
velocity which the ball had at the distance s,
7
6
At¸− {ót,+{A³t,— {A*t,+ ‡Ã³t,— ¿Ãºt¸+ &c.
...(1),
supposing v, to denote the velocity, and f, the retarding force
at distance s,
で
​d²ts
&c.
-l
5-21.
$-37
ds
But
ds
d2s
d²ts
f₁ = d; = -1; (d)"
f=
dt,
8
ds²
1/2" (A³t¸_1 — 1½ Aˆts-21 + 1&▲°t¸-„¿¬&c.)………. (2).
The use of formulæ (1) and (2) will be best understood from
their application to numerical examples.
55. Hollow elongated shot weighing 23.84 lbs. 5-inch
gun. Diameter of shot 4.92 inches.
The readings of the seconds and their corrections were
as follows:
Correction
Corrected
Seconds. Reading. applied.
Reading.
1 25.90 - 001 =
25.899
A₁₂
+ 23.893
Δ2
2
49.79002
49.792
48
+ 23.845
3
3
73:64
73:64 — 003
73.637
51
4
97.43+001
97.431
+ 23.794
+ 23.740
3
54
5
121:17 + ·001
121-171
It was usual to read off a few seconds beyond the limits
of the screen records, in order to determine the exact motion
of the cylinder. In this case the screen records extended
from 74 to 100. Hence it was sufficient to form a complete
time-table extending from 3.0 to 4.5 seconds. By interpolation
we can now find where the half-second beats would have been
marked, just as well as if the clock had given a. record every
half second. The result is
Seconds.
Divisions.
3.0
73.637
A₁
+ 11.904
Δ.
3.5
85.541
14
+ 11.890
4.0
97.431
13
+ 11.877
4.5
109:308
14
+ 11.863
5.0
121.171
ON THE RESISTANCE OF THE AIR.
41
By a further interpolation, we find the places of every
division of the seconds into tenths, so far as is required for
the reduction of the experiment.
Seconds.
Divisions.
Seconds.
Divisions.
3.0
73.637
Δι
3.8
92.677
Δι
+ 2·382
+ 2·377
3.1
76.019
3.9
95.054
+ 2·381
+ 2·377
3.2
78.400
4.0
97.431
+ 2.381
+ 2.377
3.3
80.781
4.1
99.808
+ 2·380
+ 2.376
3.4
83.161
4.2
102.184
+ 2·380
3.5
85.541
+ 2·375
4.3
104.559
+ 2·379
+ 2.375
3.6
87.920
4.4
106.934
+ 2·379
+ 2·374
3.7
90.299
4.5
109.308
2.378
1
Here it is plain from the column A, which gives the lengths
of the scale corresponding to successive tenths of seconds,
that for fractions of a second less than one-tenth of a second,
the common rule of proportional parts will apply. This is
the time-table to be used in converting the screen readings,
after the application of slight corrections, into time.
Correc- Corrected
6
४
123 LOCO 78
Screens. Reading. tion.
74.89-001 =
2 77:48 + ·005 =
Reading.
74-889
Δι
+ 2.596
Δε
77.485
+ 57
+ 2.653
-
80·14 — 002
•002 =
80·138
+ 57
4
82.85 ·002 =
+ 2.710
82.848
+ 58
5
+ 2.768
85.61 + ·006
85.616
+ 58
+ 2.826
88.44 +002
88.442
+ 57
91.33 - 005 =
+ 2.883
91.325
+ 58
+ 2.941
94.26 + ·006
94.266
+ 58
+ 2.999
9.
97.27-005 =
97.265
+ 57
+ 3·056
10 100.32 + ·001 = 100·321
The reading for the first screen was 74.889
The reading for 30 seconds was
Difference
73.637
1.252
The length of the tenth of a second was 2.382, so that the
1.252
10 2.382
shot passed the first screen at 3·0+ 1¹‰
= 3·0526 seconds.
And in the same manner, the other corrected screen readings
42
ON THE RESISTANCE OF THE AIR.
were converted into seconds. The result will be found
to be
Screen, passed at,
seconds.
6
7:
8
123 LOCO → ∞
1
3.0526
Δι
+ 1090 Δη
3.1616
+24
+ 1114
3.2730
+24
+ 1138
4
3.3868
+ 25.
+ 1163
5
3.5031
+25
+ 1188
3.6219
+ 25
+ 1213
3.7432
+24
+ 1337
3.8669
+24
+ 1261
9
3.9930
+ 25
+ 1286
10
4.1216
The screens were 150 feet apart = 7. The velocity v at the
fifth screen
150
At - A³t
150
∙1188 – 10025
300
=
1276·1f-s.
•2351
Also the retarding force at the fifth screen
3
V
5
= ƒ = -2/½ (A³t, — √ Aˆ†)
12
⚫0025
V.
- 2bv³.
5
(150)2
And putting it under the form used in Table I.,
W
.0025 23.84
K¸=2b (1000)³ = (1000)³• (150)** (4.92)*
d2
109.4.
But this experiment was made when the density of the air
was 1.0076, so reducing 109-4 accordingly, we get
W
2b (1000)³ d²
109.4―0·8=108.6.
The tabular number corresponding to a velocity 1275 f-s
is given 108.4 in Table I. It is manifest that, as the column
A is nearly constant in this experiment, the value of the
coefficient 26 of the cube of the velocity is also nearly con-
stant; or, for the range of velocity obtained in this experiment,
ON THE RESISTANCE OF THE AIR.
43
the retarding force varies very nearly as the cube of the
velocity.
56. From what has been said in Art. 53 it appears that
when a function of the form a'+as+bs² is tabulated for
successive integral values of s, the second differences of those
quantities will be constant and equal to 26.
The experiments of 1865 and 1866 gave the second dif-
ferences of the times of passing the equi-distant screens At
nearly constant, as in the numerical example just given,
because they were made with velocities near those which
gave a maximum value of 2b. It was therefore inferred
that if the space s was described in the time t, then t might
be expressed in terms of s as follows:
or
t=as+bs²,
ds
1
and differentiating
dt
a + 2bs'
d2s
26
and
dť
=f=
ds
(a+2bs)² * dt
2bv³ x v³,
or the resistance of the air varied approximately as the cube
of the velocity for the particular velocities obtained in these
experiments.
57. We will now pass to an experiment made with a
6.92-inch spherical shot, weighing 44.094 lbs., fired from
a 7-inch gun.
The process for finding by experiment the
exact time at which the shot passed each screen was precisely
the same as before.
Screens, passed at,
seconds.
1 2.90068
2 2.98499
Δι
+ 8431 Δ2
+ 306
Аз
+
8737
حماد
3
3.07236
+10
+ 316
4
+ 9053
3.16289
+ 10
+ 326
5
+ 9379
3.25668
+ 10
6 3.35383
7 3.45444
8 3.55861
9 3.66645
10 3.77807
+ 9715
+ 10061
+ 10417
+ 10784
+ 11162
+ 336
+10
+ 346
+10
+356
+ 11
+367
+11
+ 378
44
ON THE RESISTANCE OF THE AIR,
Here
V3
=
7
▲t, — {A³t, +}▲³t₂
150
·09053 — 1·00326 + 1·00010.
· 1686 7 f-s.
=
3
V
ƒ₁ = − 2/3 (4°t, — ▲'t,)
f
-
therefore
го
K₁ = (1000)³ 2b
d2
3
V
3
(150)2
(0031600010)
2bv,3;
(1000)³×·00306
44.094
1
X
X
(150)²
(6·92)* 1.0441
120'0,
=
including a correction for the density of the air. The tabular
value of K, for a spherical shot corresponding to a velocity
of 1690f-s is 121.4. In the last example the column.
headed ▲² is far from being constant. Several coefficients
corresponding to velocities between 1700f-s, and 1400f-s
were found from this single round.
2
The reductions of the coefficients of resistance given in
Tables I. and II. were made on the supposition that the
weight of a cubic foot of dry air was 5306 grains, which
corresponds to a height of 30 inches of the barometer and
a temperature of 65°.5 F., and not of 62° F. as stated at page
21 of Reports, &c.
( 45 )
CHAPTER IV.
ON THE MOTION OF A PROJECTILE IN A RESISTING MEDIUM.
After
58. THE history of the mathematical investigation of the
path described by a projectile in a resisting medium, which
has been made use of in this treatise, is somewhat curious.
About the time when Newton flourished it was common for
the leading mathematicians in England and on the Continent
to propose difficult problems by way of challenge to one
another. It was of course supposed that any mathematician,
who proposed a problem, knew how to solve it.
this manner Keill proposed the following question to John
Bernoulli in 1718: "Invenire curvam, quam projectile describit
"in aëre, pro simplicissima suppositione gravitatis atque
"medii densitatis uniformis, resistentiæ vero in duplicata
"ratione velocitatis."* Bernoulli received the challenge
early in February, 1718, and in a short time obtained a
solution, not only of the problem where the resistance varies
as the square of the velocity, which was proposed, but also for
a resistance varying according to any power of the velocity.
Before, however, he published his solution, he deemed it
right and proper to propose the same problem to his opponent
Keill. This was done in the following May. Bernoulli
gave Keill till September to declare whether he could solve
his own problem, adding that, if at the expiration of the
specified time no solution was received, silence would be
taken as a tacit confession of incompetency. At the in-
tercession of the friend through whom these communications
passed, the time given was extended to November, but no
solution was furnished by Keill. Bernoulli goes on to argue
that Keill in his difficulty would undoubtedly seek assistance
from all the English mathematicians. He therefore felt
persuaded that if any one in England had been able to
solve the problem, a solution would have been produced
within the prescribed time. After this, Bernoulli furnished
the following statement of his result to a friend, in which
the resistance of the air is supposed to vary as the 2n' power
* Opera, II., p. 396.
th
46
ON THE MOTION OF A PROJECTILE
of the velocity: "Assumta indeterminata z construatur area
“Л(aa + zz)"-¹ dz quæ vocetur Z; sint autem coordinatæ curvæ
"quæsitæ x et y. Fiat x=[(zdzZ-¹:") et y = f(adzZ¹:").
"Dico curvam quæ inde oritur esse quæsitam."*
Bernoulli published his analysis in 1721, which he de-
scribed as no more than a chain of equalities deduced from
the general formula for the determination of resistances which
he had published in 1713. The following is a sketch of his
method of treating the problem.‡
Let the plane xy be the vertical plane in which the
trajectory is supposed to lie, the axis of a being in the in-
tersection of this plane and the horizontal plane through the
point of projection O, fig. 7. The force of gravity g is sup-
posed to be constant and always to act in a direction parallel
to the axis of y, which is vertical. The point of projection
is the origin. If t be the time and R the retarding force of
the air acting in a tangent to the trajectory at the point x, y,
(dy 1 dq)
then Bernoulli begins with the equation R-g
Ids 2 ds
where 2q is the vertical chord of curvature at the point x,y,
a demonstration of which he had published in 1713.§ This
may be shewn to be true as follows, p denoting the radius
of curvature:
+
-
22
ds\
$2
g cose, or
dt
=др
gp cos 0=gq,
P
d's
dť²
2 ds
and differentiating
g dq = also – R− g sin 0 = -R-9%
-
g
dy
ds
therefore
dy
R=-9
ds
g dq
2 ds
dy
9
+
1 dq
dq
ds
2 ds
1+
dy
dx
2
Now
d'y
dzy
dx²
dx2
d³y
dq
dy
dy
2
+
1+
dx
dx
dx
(税​別​)
da
´d²y
dx²
27
* Opera, II., p. 399.
†
Opera, II., p. 513.
† Acta. Erudit. Lips., 1721, Maj., p. 228.
§ Acta. Erudit. Lips., 1713, p. 77; Opera, I., 533.
IN A RESISTING MEDIUM.
47
dq
or
ds
1
2
dy
d³y
3
ds dx³
+
ds dx
dy
2 (24/
+
dy
2
2
dx²
dq
2 ds ds
ds d³y
therefore
R =
g
dx dx³
3
2 (d²y
dx²
2
But as R varies as the 2nth power of the velocity, we have
R=kv² = k(gq)” = k
2 N
ds
dx
- gd³y
dx
And equating the two values of R, we find
or
+
ds d³y
g dx dx
2 (d'y
dx²
ds
3
2n-1
ds
2
dac
k
-9
d²y
dx²
n-2
d³
k (− 9 )** (da) ** = 1 (dy)** dy
g)"-1
2dx²
dy 2
1
2n-1
a
(a² + 2²) 2017.
3
Let now
dx
ds\ 2n-1
then
dx
α
d²y
1 dz
Also
and
d³y
3
1 d²z
dx²
2
a dx
2
doc³
a dx²
2n-1 dz
By substitution, we have
1
2k (-g)"-1 (a² + z²)
2n-1
an-
α
and putting Z for ſ(a² +2²)
2
dx'
and integrating, we have
dz
2nk (− 1)" Z = − 9 (da)" ;
(-)
2n-1 dz
1
dz
n-1 d2%
2
dx
N-1
α dx
dx²
29
48
ON THE MOTION OF A PROJECTILE
~S
therefore
dx
or
X
Also
dy=
therefore
!
a
• (1) *
g k
α
$(9)*
g
2218
α
dx
dz
(— 2nZ)
dz
(— 2nZ)
1
(
His
g
1
zdz
y =
g
(- 2nZ)
zdz
(— 2nZ)
59. Suppose we apply these results to the particular case
where the resistance varies as the cube of the velocity, we
have 2n = 3. Let u。 be the velocity of the shot at the vertex
of the trajectory. At this point=0; therefore z = 0, and
d'y 1 dz
dx²
But
a dx
Ο
dy
u2=g× radius of curvature at the vertex
α
9 dz'
dx
dz
or
&&
ga
at the vertex.
dx
2
u
And
And since
222-1
Z= √(a² + 2²) 272 ¹ dz
= f(a² + z²) dz
=C+ a²z+
n
Z=
23
3
dz
dx }
dz
2nk (-2) "Z- -g (de)",
9
we have 3 (-2)* (C++) — — 9 (d);
α
3k
い
​3
g
ga\
and at the vertex 3% (-
3k (-2)*0=-3(-94)*,
or
да
3
C
아
​30--%2 which determines C.
u
31
IN A RESISTING MEDIUM.
49
Alse
and
dy-tane = p ==
dz = adp;
a
,
therefore since 3k ( − 2)* ( C+ a²z +
3k (− 2)* ( C + a²z + - ) = − 9 (ad),
or
И
dx
Ο
9
Xx
g
dp
ku 3
9
Ο
ku
g
3
0
3
(3p+ p³)
dp
(3p+p³)
pdp
dx


and
0
Y
9
ku
3
(3p+p²)
9
66
Respecting this solution Montucla has remarked, "Mais
"la solution de ce grand géomètre ne pouvoit être utile pour
"la pratique, parce que l'aire de la courbe ne pouvant la
"plupart du temps être exprimée en termes finis, il est
comme impossible, au moins d'après l'analyse actuelle,
"d'en tirer la valeur des inconnues; si l'or y parvenoit on
"n'arriveroit qu'à des valeurs si compliquées qu'elles ne
"seroient d'acun usage.
The case where 2n3 is per-
haps the simplest of all, and yet the only way to render
Bernoulli's results practically useful, was to calculate by
quadratures tables of the values of the above integrals for
11
all possible practical values of p and
ku
3
Ο
9
or y, so numerous
that all intermediate values might be obtained from the
tables either by a simple interpolation or by proportional
parts.
* Professor Adams communicated to me the above expressions for x and
Y
and an additional one for the time in a letter dated Nov. 13, 1866, when the few
ballistic experiments I had then made seemed to indicate a cubic law of resistance.
Professor Adams at the same time remarked that the "form of solution was
"worked out by my own investigations, but on looking into the history of the
"problem, I find that an equivalent process was given long ago by John Bernoulli.”
E
50
ON THE MOTION OF A PROJECTILE
60. Explanation of symbols used.
g denotes the accelerating force of gravity and equals
32.191 feet per second in the latitude of Greenwich,
w the weight of the shot in pounds,
d the diameter of the shot in inches,
2bv³ the retarding effect of the air for a velocity v feet
per second.
K=26 ×
го
d²
× (1000)³.
x, y are the horizontal and vertical coordinates of the
centre of gravity of the shot at time t, the origin being at the
point of projection.
v denotes the velocity, and u the horizontal velocity of the
shot at the point (x, y).
is the inclination to the horizon of the tangent to the
trajectory at the point (x, y).
vo denotes the velocity of the shot in the ascending branch
when moving in a direction inclined at an angle to the
horizon, and up the corresponding horizontal velocity, so that
Up=v¿ cosp. v' and u' denote corresponding quantities in
the descending branch.
Иф
Иф
$
3
P3 tano tan³ p.
Ᏸ
ap and ays denote the differences of the coordinates of
two points where the trajectory is inclined at angles a, B
to the horizon; thus in fig. 9, axe An, and ays = Bn; and
in like manner ats would denote the time of describing AB.
3
61. Suppose a body projected in a direction inclined at
an angle a to the horizon, to be acted upon by the force
of gravity g acting in parallel lines, and by a retarding force
26 × (velocity) acting at every point in the direction of the
tangent to the trajectory of the body at that point, then there
will be no force tending to draw the projectile out of the
vertical plane of projection. Fig. 8, let O the point of pro-
jection be the origin, and let the axes of x and y respectively
be horizontal and vertical in the plane of the projectile's
motion. Also let x, y be the coordinates of the centre of
gravity of the shot at time t.
The equations of motion are
3
d2x
ds\³ dx
2b
26
dt²
dt ds
ds\² dx
dt dt
and
d²y
3
26
dt²
ds\³ dy
dt ds
-9=-2b
ds
2
26 (da) 241-9;
dt dt
g;
dy
IN A RESISTING MEDIUM.
51
therefore
Let
dx
d²x
d'y de- dy d'--g de
dt
dt
at
dy
P
dx
ཝེཊྛི མྨེ་
dx d'y
dp
dt
then
dt
dy d³x
dt
19-39-4
dx
dt
or
dp dx
dt' dt
9
62. Again
dt
(1).
3
d*x
'ds" dx
- 26
261+
dt2
dt
dt
(()
dx
= −2b (1+p²)
dt
therefore
drx
dt2
dx
dt
therefore
or
2 (1+2)主​一​(1+2) 串
​1
dt
26
- 1 =0+ 3 (p+3).
3 /dx
dt
At the vertex let the horizontal velocity
dx
dt
to
when p=0;
dp
dt
and
1 1
3 /dx
dt
1
1
3
=C,
3 κ
1
3
3u
+
26
9
(2 + 2),
3
}
E 2
52 MOTION OF A PROJECTILE IN A RESISTING MEDIUM.
or if u =
dx
dt
=
the horizontal velocity of the projectile at the
point (x, y), then
1
- 100%
||
1
26
(3p + p³)
g
(2).
Substituting for
or
or
1
น
dt
or in the last equation its value
И dx
de given by (1), we have
1 dp
9 dt
3
dp 1 26
dt
3
(3p+p³),
U
g
2bu 3
0
dp = - 2 {1 - 20x (3p + p³)}',
dt
น
U
Ο
t =
9
2bu³
2b Mu 3
if y
9
Mg
9
dp
{1 − y (3p+p³)}}
(3),
resistance of the air at the vertex of trajectory
By equation (1)
weight of the shot
dp dx
dt dt
9, or
dp (dx 2
dx dt
g;
dx
1 /dx
2
1
1
2b
therefore
dp
(3p+p³)
9
}}
9
2 P rp
И
0
or
X
9 dt
2
9
3
{1 − y (3p + p³)}-}
dp
9 {1 − y (3p + p³)}:
{
dy
Again,
dr-P; therefore
dx =p;
dy
dx
=p
dp
P Jp
dp
or
3 - - T
น.
g
(4).
2
pdp
g {1− y(3p+p³)}
pdp
(5).
{1 −y (3p+ p³)}3
MÉTHODS OF FINDING THE VALUES OF X, Y, ANd T.
53
63. If tan &=p and tan o'p', then
dp = sec² pdp = (1 + p³) dø,
and the above equations (3), (4) and (5) take the forms
t
uo
Ø
9
Ꮳ
9
g
2
j7+
2. Φ Φι
2:0
(1+p²) do
{1 − y (3p+p³)};
(1+p²) dp
{1 − y (3p+ p³)}
*/*
(p+p³) do
{1 − y (3p+p³)}}
11
U
g
2
T'.... (6),
γ
u₂² + X‚³' ... (7),
2
g
и
'Y½º¹... (8).
g
64. Inasmuch as it was impossible to find the values of
x, y and t by direct integration, it has been necessary to
calculate by quadratures the values of X, Y and T for all
practical values of + not greater than 60°, and of - not
greater than 60° or 45°, for values of y=0·00, 0·01, 0·02…..0·18,
0.19, 02, 03, 0·4...4.9, 5.0. The value of do generally used
was the circular measure of 1°, but when 1-y (3p+p³) be-
came small, the successive values of
1
were
1 − y (3p+ p³)
subject to rapid variation. In such cases intervals of one-
fifth of a degree were used, and the results have been given
in preliminary Tables (pages 4 to 38). The calculations of
the values of X, Y and I were in general carried to one
decimal place beyond those given in the printed Tables.
By the ordinary rule of proportional parts, or, where great
accuracy is required, by interpolation, it will not be difficult
to find the values of X, Y and T for values of y and
intermediate to those given in the Tables.
EXAMPLES OF THE METHODS OF FINDING THE
NUMERICAL VALUES OF X, Y, and T.
65. From the Tables
10 X 5-10 X 0-5X204300934811082,
0.7
0
4Y 100 Y 100 Y=013448-002299011149,
0.7
5 T
3.2
0
0.7
=11027, °T='07836.
54
EXAMPLES OF THE METHODS OF FINDING THE
Suppose it was required to find the value of 3Y741,
7.
2-41 = 3 Y₁₂+ (Y.-Y.27) × 0*41,
3. Y 7·41 =³Y
3.2
3.2
3.2 7
=*004059+ (·006548 — ·005196) ×.0·41,
··004059+ ·001352 × 0.41
='004059+000554
='004613.
9
In the same way X
7.41 and T.
3
7.41
3.2
may be found.
3.2
Suppose now it was required to find the value of 0 in
the descending branch which satisfies the equation
зхо
·
0
³X.
o 3X,.,, when, is given =500 feet,
9
2.2
u = 1000 f-s, and g=32.191.
We have
3.X
2.2
0
9
32.191
2.
зхо
× 500 =
16.096
1000
=*01610
u (1000)²
· (·06129 — ·04731) +·00212
4
=³X*+*00212.
2.2
Now the difference of the values of X... for 4° and 5° is
•07457 — •06129 =·01328.
•00212
0=4°+
= 4°.1596.
•01328
2.2
Hence by proportional parts
Making use of this value of 0 we can now find the
corresponding values of y, and to.
Ө
We have supposed hitherto that we had a table calculated
for the required value of y. Suppose, however, we require
the numerical value of 3 Y
3.237
4.41
It has been shewn above that
3Y 7·41
⚫004613
3.2
and in precisely the same manner we find Y
3
7.41. •004567
3.8
}}
-'000046.
Shewing that an increment of 0.1 in
gives a decrement in Y
Hence, by proportional parts, an incre-
ment of 0.037 gives a decrement in
the value of Y
-·000017
NUMERICAL VALUES OF X, Y, and T.
55
Hence
3Y
7.41
3
32.37
7.41
Y -·37 × ·000046
3.2
='004613 — ·000017·004596.
66. In cases where we can suppose the resistance of the
air to vary as the cube of the velocity throughout the range,
it is easy to find the path and time of flight of a projectile
and its velocity at any point, supposing the initial velocity
and initial direction of the shot to be accurately known.
Let V be the initial velocity, a the angle of projection,
and 26 the coefficient of resistance, then u = V cosa is known,
and equation (2) gives u, for
1
3
1
+
26
(3p+p³),
9
or
(1000)*
(1000)
3
26 (1000)³
+
(3 tana+tan³a).
9
2bu 3
and referring to the table
Hence we know y=
2
g
for this particular value of y we obtain, (fig. 8)
2
a
0
2
Ο
OM="""X,, MA="""Y",
0
"X, MA= "Y, and time in OA= *Tz°.
9
Now for the descending branch, we have, for the point
where the motion of the shot is inclined to the horizon at
an angle - ẞ at P',
(1000) * = (1000) *
3
+
26 (1000)
9
3
(3 tan B+tan³ß),
3
which gives the value of u'ß, and therefore of v'e=u'ß secß.
2
2
AN' °X‚³‚ N'P'="₂²°Y‚³‚
g
,
I
and time in AP' – 。
=
20 Tf.
g
If we wish to find the range on the horizontal plane we
must have AM=N'L, or
U
0
9
а
Yy'
น
9
2
"Y
Y₁^, or "Y" = "Yy®.
56
EXAMPLES OF THE METHODS OF FINDING
By the help of the tables can be found, and this value
of ẞ must be used in calculating Mp, and the time in Ap.
67. Suppose it was required to find the height at which
the shot would strike a vertical target placed at distance OL,
and the time of flight. Here we have
or
u
ML=LO-OM= 0
9
2
°X7€ = (LO — OM) 2
27
И
Ο
7.
which gives ẞ by the help of the tables. The value of
so found must be used to find N'P' and the time in AP'.
We must proceed in the same way if it be required to find
where the shot will be at a given time.
68. It is manifestly easy to calculate the horizontal, and
thence the actual velocity, at any given inclination of the
trajectory to the horizon. Suppose us is the horizontal
velocity when the direction of motion is inclined at an
angle + to the horizon, that is in the ascending branch OA,
and us that when the inclination is in the descending
branch AP', we have
and
"
1:
Иф
И
$
3
1.
u
3
3
+
26
g
26
g
(3 tan+tan³),
(3 tan + tan³)..
69. Examples. The 16-pounder muzzle-loading gun fires.
an ogival-headed shot 16 lbs. in weight, and 3.54 inches in
diameter. If the angle of projection be 2°, and the initial
velocity 1358 f-s, find the trajectory and time of flight.
Here u₂ = cosa=1358
V cosa = 1358 cos 2° = 1357·2;
d².
w=16 lbs., d=3·54 in., and ='7832.
W
The mean coefficient of resistance used is that due to a
velocity of 1270f-s, and therefore K=108.5 and log
(1000) = (1000)
2
='67666,
8
K
=5276,
J
K d²
g W
+(1000)³ 20 P; - (1000) + 4 d P.
26
9
2
THE TRAJECTORIES OF PROJECTILES..
57
and therefore
u1139'0 f-s..
འ༦o
3
y=
2bu * = (1000) *
3
26
K d²
3
9
(1000)
9 го
(1000)
3.90,
u2
0.
g.
2
u
0
2
= 30.67 feet;.
3.9
g
g
"X.04118; "Y.=000761; 2T03788.
× 2X = 1659·7 feet; %=
0
3·9.
འvo
‚t = "/½ × ²T.° = 1″·340.
to
g
For the descending branch, suppose the mean coefficient of
resistance to be that due to a velocity of 1080f-s, and
therefore K=103.4 and log=5068. Taking u, as the
K
9
velocity of projection in a horizontal direction, we have
3
10003
100)* = (1000) * + (1000) *
26
3
P₁₁ = (1000)⁰
K d²
+
P
2.4.
น
го
2.4.
therefore
=99318;
u'
2.4
=
g
24.
1002·3 f-s..
3
K d²
น
y=
9. w
(1000)
= 3:718.
0X2.7
2.4.
3:7.
2.1
=
8-7
*03670; °Y, 24-000737; °T. •03920..
3.7
。°2.4 = 1479-2 feet; 0.429.70 feet; 。t=1"·3871.
=
0
Let x, y be the coordinates of the centre of gravity of the
projectile, and t the time when the shot is moving in the
descending branch of its trajectory in a direction inclined to
the horizon at an angle of 2°-4 or 2°24'.
x=2x+0x24=3138.9 feet,
Y = 2YQ+0Y2·4.
0.97 feet,
and the velocity
=
t = ₂to + ot2.4 = 2"-727,
20
0
· V'2.4 = '2.4 sec 2°·•4 = 1002.3 sec 2°41003·2 f-s.
The range on the horizontal plane
=
(31389 0.97 cot 2°24') feet.
Similar calculations have been made for a gun of 3.3 inches
firing 16 lbs. shot of 3-24 inches in diameter. Here V = 1307,
58
EXAMPLES OF THE METHODS OF FINDING
d2
w= 16 lbs., d=3.24 inches, and
'6561. The following is
го
a statement of the results.
u=1136·5 ƒ-s, x=3101 feet, y=-1·09 feet, t=2″•715,.
u'2.410186 f-s, and v'. 1019.5 f-s.
2.4
Hence it appears that if the 3.3 inch gun was fired at an
elevation sufficient to give the same range as the 3.6 inch
gun when fired at an elevation of 2°, the shot fired from
the former gun would have a higher striking velocity, but
would then have nearly the same inclination to the horizon.
As the elevation was increased, the advantage would be
more and more in favour of the 3.3 inch gun, both in striking
velocity and flatness of trajectory.
70. A spherical shot weighing 84 lbs. is fired from the
100-pounder smooth-bored gun with an initial velocity of
1564 feet per second. The angle of projection is supposed 5º.
Here V=1564f-s, u=1564 cos 5° = 1558;
d²
d²
d=8.9 in.; w = 94 lbs. ;
='8427; log = 9.92565.
W
W
The mean coefficient for the ascending branch is that due to
a velocity of 1300 f-s, therefore K=147·8, and log
(Table II),
3
(1000) *
(1000)
+
K
= '6619.
g
K d2
g
༤|ལུ
P
G
5
= 1.28236;
therefore
u = 920·5 f-s,
K d²
Y
g น
(10000)
= 3.017.
By the help of proportional parts, the following values of
X, Y, and Tare found from the Tables
*X3.017^=•13525; ³Y,
2
U
0
5
0
3.017
g
=3559.7 feet; Yo
0
5Y017-181.9 feet ;
3.017
=
3.017
·006 910; T
0
3.017
=·10760;
u
0
9
0
× 5T
0
= 3" 077.
5.0
3.017
9
U
X
THE TRAJECTORIES OF PROJECTILES.
59
For the descending branch we suppose the shot projected
horizontally with an initial velocity u920-5 f-s, and to be
acted upon by the resistance of the air varying as the velocity
cubed, the mean coefficient used being that due to a velocity
K
of 900 f-s. Here K=138.2, and log
='6329,
g
3
(1000)* = (1000) *
= 2.81811;
K d²
+
P
9 го
8.
therefore
8
2 =7080f-s
K d²
2
and
જ
9 го
(1000)
3
= 2.822.
From the Tables the values of X, Y, and T are found for
y = 2·822
8
8
X2.82210575; °Y₂ -'006788; °T ='12156,
2.822
2'822
whence x=2783.2 ft.; y=-178.6 ft.;t=3"•476.
But
0.
3559.7 ft.; 5y+181.9 ft.;t=3"077;
therefore x = 6342.9 ft.; y =+ 3·3 ft.; t =6"·553 ;
and
v=u,' sec8° = 715 f-s.
When the shot is 3.3 feet above the level of the muzzle of
the gun in the descending branch, its course is inclined at 8°
to the horizon. Hence the range of the shot on the hori-
zontal plane = (6342.9 +3.3 cot 8°) feet.
But suppose that we wished to calculate the exact range,
time of flight, striking velocity and angle of incidence. In
the first place we remark, that as the shot has risen to the
height of 1819 feet, it must fall through an equal height..
Let be the angle of incidence, then in the descending.
branch
u oy $;
Уф = 181.9 feet
therefore
But
and
0
Y
2.822
2.822
2
$
2.822
='006910.
9
0
• Y
9
Y
2.822
8
=·006910·006788 —·000122.
_°Y
•Y
2.822
2.822
··008329 — ·006788 =
•006788 =·001541,
-
Hence, by proportional parts, we find = 8°.079 the angle of
incidence
X
8.079
2.822
∙10657; and 'T,
8.079
2.822
=12264,
60
EXAMPLES OF THE METHODS OF FINDING
which give 3.07928049;
But
therefore
X
= 3559.7;
0*8.079 = 3"-507.
sto=3"077.
6"-584:
X =6364.6, Y= 0, T
The striking velocity v'.079 is found from the equations
and
1000\3
8.079
v'
8'079
8
1000 ³ K d2
U
* + X × P8-0797
g W
=u' sec 8°.079.
8.079
71. We will now proceed to explain how the tables may
be used in calculating the trajectory of an ogival-headed shot
when successive arcs are supposed to be described under the
action of a retarding force varying as the cube of the velocity.
The coefficient of resistance for any arc must be taken ap-
proximately, so as to give the mean value of the coefficient
for that arc of the trajectory. It will be convenient to change
the coefficient of resistance at points where the tangents to
the trajectory are inclined to the horizon at angles expressed
exactly in some number of degrees, because the values of X, Y,
and T are given in all those cases in the Tables. Afterwards,
for comparison of the results, the range, &c. will be calculated
by the use of only one coefficient for the ascending and one
for the descending branch.
An ogival-headed shot 6:34 inches in diameter, weigh-
ing 70-6 lbs. is projected at an angle of 5° with an initial
velocity of 1335 feet per second; find the path described by
the shot, its velocity at various points, its range, and angle of
incidence. Here
V=1335f-s, u=1335 cos5°
d2
=
=
1329.9 f-s, w=70-6.Ibs.,
1329·9f-s,
d²
9:75538.
พ
d = 6.34 inches, •56935, log
W
The coefficient of resistance for the first arc of the trajectory
is that due to a velocity of 1300 f-s; therefore K=107.9
K
and log =5253,
g
(1000) * = (2000)*
ისი
3
+
K d²
P
9 го
92736, or u=1025·5 f-s.
THE TRAJECTORIES OF PROJECTILES.
61
We must now find the value of u
or
(1000) * = (1000)
・
•52635,
3
K d²
P.
9 го
u=1238.5 f-s, and v 1238.5 sec4°.
The value of
u
Ο
9
2
5X
K d²
3
y= 1000) = 2.058,
2.058
9 го
897.7 feet,. sy₁ =
5t4
5 4
Uo 5 T
05T
g
2.058
4
2
u 5y += 70·8 feet,
g
= 0".700.
2.058
For the arc 4° -3° we suppose the angle of projection to
be 4, the velocity of projection v, and the coefficient of
resistance to be that due to a velocity of 1200 f-s; therefore
K=108.9 and log
or
And
or
(1000) *
K
9
=
•5293,
(1000) *
3
+
47
K d²
PA
•93107,
9 го
U₁ = = 1024.1 f-s.
Ио
0
(1000) * = (1000) *
The value of
2
4°C 3
10 4X
3
2.069
9
3 K d²
·g w
u₂ = 1167·8f-s.
Из
K ď²
3
=
P¸ = '62797,
y = 1000) = 2·069,
9 го
=786.2 feet, y3=
U
0 4Y
9
3
=48 2 feet,
2.069
4 3
04 T
2.069
9
3
=0"·654.
U
For the arc 3° -2° the coefficient of resistance is that
which is due to a velocity of 1135f-s; therefore K=107·7
and log
K
g
5244. The new value of u to be used for this
arc is found, as before,
y=2·053, which give
=
0
1025-3 f-s and u₂ = 1111·6f-s,
x=704.8 feet, 3,309 feet, t=0"619.
2
3 2
62
EXAMPLES OF THE METHODS OF FINDING
For the arc 2° -1° the coefficient of resistance is that which
is due to a velocity of 1090f-s; therefore K=105·1 and
log
K
g
5139, which gives the new value of u=1027·1 ƒ-s,
u₁ = 1066·0 ƒ-s, and y=2·014, from which it is found that
1
=
x=642.6 feet, 27, 16.9 feet, t, 0"-590.
2Y1
1
=
21
For the arc 1° – 0° the coefficient of resistance is that due
K
to a velocity of 1047 f-s; therefore K=91·4 and log *4532,
g
which gives u=10319f-s for the calculated velocity with
which the projectile is moving at its highest point, and
y= 1.776, from which it is found that
x=5960 feet, y=5.3 feet, t=0"-568.
0
0
For the descending branch we shall make use of intervals
of 2°.
For the arc 0° to - 2° the coefficient of resistance is that due
K
g
to a velocity of 1010f-s; therefore K = 77.5 and log
•3817,
and the shot is supposed to be projected horizontally with
a velocity u=1031-9f-s. It must be remarked that for the
descending branch of the curve P_4= − Pp, for
P_¿ = 3 tan (− 6) + tan³ (− 4) = − (3 tan 4 + tan³ 4) = − P$,
(1000) * = (1000) *
3 K d²
+
9 го
P
•91020+ ·14370=1·05390,
or u'=982.7 feet and y = 1.506; from which we find
0
。=1099·1 feet, 32-18.9 feet, t=1"·092.
2° to - 4° the coefficient of resistance is that
K
For the arc
due to a velocity of 970 f-s; therefore K=69.6, log •3349
9
2 4
-
=
4
2 4.
which gives u=1026'4 f-s, u', 9454 f-s, and y = 1331, from
which we
we find x = 1009.6 feet; Y 52.7 feet; t = 1".048.
For the arc 4° to 6° the coefficient of resistance is
that due to a velocity of 935f-s; therefore K=66.2 and
log
=31345, which gives the new value of u
9
u'=914·3 f-s; and y = 1.250, from which we find
K
10218 f-s,
+x=944.5 fcet, 1824 feet, t=1".016.
THE TRAJECTORIES OF PROJECTILES.
63
It is now necessary for us to find the height of the shot;
since
5Y4
70·8 feet, and 。Y¿
y
18.9 feet,
2
52.7
Y3
48.2
2Y4
82.4
3Y 2
30.9
Yo
— 154.0
Y 1
= 16.9
+ 172.1
190
5.3
Height of vertex = 172.1
+18.1
When, therefore, the shot has fallen through an additional
vertical height of 18.1 feet it will be in the horizontal plane
passing through the point of projection. Suppose y = 1·250,
we find
6X1.25
7
=
6
01393; Y.-001587; T25
7
=⚫01570.
Supposing that & denotes the inclination of the trajectory
when it meets the horizontal plane passing through the point
of projection, then to determine we have
น
1 2
0
or
1.25 7
6Yp=
18.1 feet
=
6 Y $
9
6 Y $
1.25
u
g
1 2
0
× 18.1
•000558.
Hence it is found by proportional parts that = 6°352.
BUC
6.352
1.25
6.352 =159.1 feet,
# 2
u
0
6 6.352
g
6 T 6,352 = 0″·175.
1.25
! 2
И
• X
6
g
897.7 feet
4
Xz
x
4X3
3X2=
786.2
704.8
X
11
642'6
X
2
SOC A
1099.1 feet
= 1009.6
4006 = 944·5
159.1
2 1
6 6.352
1x0
x = 596.0
3627.3
+ 3212·3
+3627.3
therefore x = 6839·6 feet
st=0"-700
4
4t3 =
3 2
at
0.654
0.619
t 0.590
21
t =
to
10
to
= 1"·092
0°2
et₁
= 1.048
24
sto
= 1∙016
4 6
0.175
6 6.352
0.568
3.331
3.131
3.131
therefore T = 6·462 seconds
Y = 0
64
EXAMPLES OF THE METHODS OF FINDING
72. We will now proceed to calculate the last example
by two steps, the division being at the vertex of the trajectory.
For the ascending branch take the coefficient due to a
K
velocity of 1180 f-s, therefore K=108.9 and log •5290,
therefore
3
(1000)² = (1000)
which gives u=1023:9 f-s, y=
9 го
9
3 K ď²
+
K d2
9
го
(100) = 2·066,
and 5=3617·5; 1719;
¿Yo
and t=3"127.
50
For the descending branch we shall use the coefficients
due to a velocity of 950 f-s; therefore K 674 and
K
g
log 3210. This gives y=1·280, and the angle of in-
cidence on the horizontal plane passing through the muzzle
of the gun = 6°349.
Y6-349
oy6.349
=-1719 ft.; and of.949=3"328;
Yo = +171.9 ft.; and t₁ =3"127;
х
0 6.349
but 500
=3203.5 ft.;
=3617.5 ft.;
X
= 6821·0 ft.;
Y
The former values were
Y
50
0 ft.; and T6"455.
0; and T =6"·462.
X = 6839.6.;
The coefficients given in Table I. for ogival-headed elongated
shot, and other kindred forms, and those in Table II. for
spherical shot, have been found for the highest velocities
likely to be obtained in practice. But the coefficients for
velocities below about 1000 f-s require to be determined by
a special set of experiments. These are, however, of far
less importance than those for higher velocities.
Also, if any different form of projectile should be required
for use, then it would only be necessary to find a new table
of values of K for that particular form of head. All the
remaining tables would then be applicable.
If the tables should be required for some place on the
earth's surface, where the force of gravity g' was sensibly
different from g, the force of gravity at Greenwich, then all
the values of log would be required to be increased by log 4.
K
9
9
73. The preceding investigations of the path described
by a projectile moving through the air are based upon the
supposition that the resistance of the air acts through the
THE TRAJECTORIES OF PROJECTILES.
65
centre of gravity of the shot, and in the direction of a
tangent to its path. These would apply strictly to the case
of a spherical shot which did not rotate and whose centre
of gravity coincided with its centre of figure, and its motion
would lie entirely in the vertical plane of projection. But
we should only obtain by this means an approximation to
the path described by an elongated shot fired from a rifled
gun. Practically, a shot of this kind is found to deviate to
the right, when the rifling communicates a right-handed
rotation as seen from the gun. If there were no resistance
of the air, then the centre of gravity of the shot would
describe a parabola, and the axis of the shot would continue
parallel to its original direction, as is represented in fig. 8°,
just as the earth moves in her orbit and maintains the
parallelism of her axis of rotation. The resistance of the
air alters all this. It causes elongated shot to move approxi-
mately point foremost. Thus when an elongated ogival-
headed shot escapes from a rifled gun, it is moving in the
direction of its axis, or very nearly so. The action of gravity
very soon causes the axis of the shot GB, (fig. 9"), to make
a small angle with CD, the direction in which G, the
centre of gravity of the shot, is moving. The resultant
resistance of the air then tends to turn the shot about an
axis through G, perpendicular to the plane BGD, so as
to increase the angle BGD. If the shot had no rotation,
the shot would fly end over end. But if the shot seen from
C had a right-handed rotation about its axis, this motion would
be combined with that which the resistance of the air tended
to impart to the shot, and, in consequence, the point A as
seen from C would be slightly deflected to the right. Suppose
this continued, then the consequence would be that the axis
GA would describe a cone about CD, rotating in the same
direction as the shot rotates about its own axis. We have
been considering the motion about the centre of gravity G,
supposed fixed. If we now consider the motion of translation
of the shot, we shall easily see that the resistance of the air,
acting normally to every part of the surface of the shot,
will have a resolved part which will be effective in a slight
degree in pushing the shot away from CD, the direction
of its motion. Thus the shot will have a sinuous motion.
But as the first deflection of an ogival-headed shot spinning
with right-handed rotation is to the right, and afterwards,
as its point is directed more to the right than to the left,
the shot will have a deviation on the whole greater to
the right than to the left.
F
66 EXAMPLES OF THE TRAJECTORIES OF PROJECTILES.
74. If the form of the head of the shot, or the position
of the centre of gravity of the shot, was such that the
resistance of the air tended to diminish the angle BGD,
then, the shot having a right-handed rotation as seen from C,
the axis GA would have a left-handed rotation about the
tangent CD, and the resulting deviation of the shot from
its vertical plane of projection would be to the left. This
is the case with a flat-headed shot.
Professor Magnus* has shewn how to illustrate the mo-
tion of rifled shot by means of the gyroscope, and General
Mayevskit has published a mathematical investigation on
the subject. For practical illustrations see a paper on the
Derivation of Elongated Projectiles, by Major Owen, R.A.‡
* Ueber die Abweichung der Geschosse, 1860; and first edition translated
in Taylor's Scientific Memoirs, 1853.
+ De l'influence du Mouvement de Rotation sur la Trajectoire des Projectiles
oblongs, dans l' Air. Revue de Technologie Militaire, t. v., p. 13, 1866.
+ Proceedings of the R.A. Institution, Vol. IV., p. 185; and Treatise on
Modern Artillery, p. 224.
( 67 )
CHAPTER V.
DESCRIPTION AND USE OF THE GENERAL TABLES.
75. Ir will be found sufficient for many practical purposes
to neglect the effect of gravity and treat the motion of a shot
as if its path were a straight line. This will suffice for ex-
perimental purposes when it is desired to find the loss
of velocity, or the time of flight for a limited space, the
initial velocity being high. The less the shot is affected
by the resistance of the air the more accurate will be the
results. Therefore this method will apply better to pointed
elongated shot than to spherical shot, and better to solid
shot than to shell of the same external form.
76. The equation of motion for the cubic law of re-
sistance is
and therefore
d's
dv
dt2
2bv³,
dt
1 1
=
C+ 2bt.
2 v2
Suppose that v=V when t=0, s= 0, then CV, and
or
1
1
d² 1
4bt = 2tK
22
V2
w (1000)³
if K=262 (1000)³,
d²
500
t
י
K
{(1000) *
(1000) *)
.(1),
which connects t and v.
Again,
dv
ช
2bv³,
ds
1
and integrating,
= C' + 2bs,
v
F2
68
or
DESCRIPTION AND USE OF
ར
1
1
d2
1
2bs = sK
V
V
w (1000)³,
or
d2
พ
S=
(1000)³ {(1000) - (1000))
K
which connects s and v.
If in the equation
we have
……….. (2),
1
1
dt
1
บ
T
2bs, we substitute
for
ds
V
dt
1
ds
V
+2bs,
and integrating, we have
which connects t and s.
If we divide
by
we obtain
S
t=
= = = + bs ²
2
...(3),
1
1
4ht
V2
1.
1
æ '
V V
= 2bs,
1
1
2t
+
ย V S
..(4),
which connects v, t, and s independently of 26 the coefficient
of resistance.
77. In determining the velocity of a shot it is usual to
measure the time in which a given short range is described,
and then, dividing the space in feet by the time in seconds,
the result is adopted as the approximate velocity at the
middle point. If the cubic law of the resistance of the air
be supposed sufficiently near the truth, this may easily be
shewn to be strictly correct for any range, so long as the
path of the shot may be considered to be a straight line.
We have seen that when V is the initial velocity and v
the velocity at the distance S, then
1 1
V V
+2bs,
or if v' be the velocity at the distance
S
then
2'
1
1
+ bs.
v'
THE GENERAL TABLES.
69
Also
space in feet
S
S
time in seconds
t
by equation (3),
S
+bs
1
ຈ່,
+bs
the true velocity at the middle point of the range s.
78. Inasmuch as the resistance of the air does not vary
strictly as the cube of velocity, when formulæ (1), (2), and (3)
are used for considerable differences of V and v, it is necessary
to use several numerical values of K. But as this would
be a troublesome operation to perform in each case, general
for spherical
and ogival-headed shot, Tables (IX.) and (XI.), and also of
the values of (1000)* (1000
500
K
2
Tables of the values of {(1000) * - (1000)"}
K
V
1000)
Ꮴ
for spherical and elon-
gated shot, Tables (VIII.) and (x.) have been calculated.
It is manifest that these quantites depend upon v and V,
which are quite independent of the nature of the shot, while
K is a coefficient dependent on the form of the projectile but
not at all upon its weight and diameter.
2
3
79. Suppose the initial velocity to be V, and that the
velocity falls from V to v, in space s,, and in time t₁; from
v, to v, in spaces, and in time t; from v, to v, in space s,
and in time t; and from v
in space s, and in
time t. Let K, K, K,...K, be the particular values of K
due to the mean of velocities Vand v₁, v,
and v v¸ and
and v. Then we have from equation (1)
n
3
d²
t₁
500
K₁
W
d²
/13
d²
го
&c.
d"
го
t₂
$3
28-1
2
to vn?
{(1000) * - (1000)"}
'
2
500 ((1000)-(1000)}
K
500
K
500
១
2
{(1000) * - (1000)"}
&c.,
2
500 (1000) - (1000))}.
K
υ
21 2
Vg•••Vn−1
70
EXAMPLES OF THE USE OF
And summing each side of the equation we have
2
d²
Στ = 500Σ
n
พ
1
K
{(1000)* - (1000) *}
(I).
22-1
Proceeding in the same way with equation (2) we have
d² (1000)* (1000 1000
S
1
W
K, V
d²
1
V
(1000)² (1000 1000)
9
S
ω
K
"
V₂
&c.
&c.;
d²
therefore
22
Σε = (1000) Σ
1 (1000
1000)
W
n
K V₁₂
(II).
v
N
7~1
=
3
1
3
In calculating the numerical values of the right-hand
members of the above equations, V was taken (for elongated
shot) = 1700 f-s; v₁ = 1690; v=1680; v = 1670, &c., and
K, the coefficient corresponding to the velocity 1695 f-s, K to
d2
1685, K to 1675, &c. Tables of the values of t and S
were thus formed corresponding to a loss of every ten feet in
d2
d2
the velocity. By interpolation the values of t and of S
which have been given in the Tables, were found for every
foot lost in velocity.
พ
W
พ
го
It unfortunately happens that equation (3) does not admit
of the formation of a general Table to connect directly space
and time. But the use of the two general Tables connect-
ing velocity and space, and velocity and time, enable us to
calculate the time occupied in describing a given space, or the
space described in a given time, the initial velocity being
known.
EXAMPLES OF THE USE OF THE GENERAL TABLES.
80. (1) Suppose it was asked in what range an 11:52 inch
ogival-headed shot weighing 600 lbs. would have its velocity
reduced from 1400 to 1300 f-s. Here d'÷w= '2212.
s denote the required space, then
d²
W
s = '2212s = 1865 – 1348 = 517,
Let
THE GENERAL TABLES.
71
the difference of the ranges opposite 1400 and 1300 f-s in
Table VIII. Hence
s=517 ÷ •2212 = 2337 feet.
(2) Suppose now it was asked in what time (t) the velocity
of the same shot would be reduced from 1400 to 1300 f-s.
d²
Here
พ
t='2212t=1"-258-0"-8750"-383, the difference of
the times opposite 1400 and 1300 f-s in Table IX. Hence
t=0"·383÷ ·2212 = 1″·732.
(3) If, on the other hand, the initial velocity being given
1350 f-s, it was required to find what would be the loss of
velocity in 500 yards = 1500 feet, we should have given
d2
พ
s=•2212 × 1500=331.8, the reduced range. Now opposite
the initial velocity 1350 f-s in Table VIII. we find 1599, to
which must be added the reduced range 331·8, making 1930'8;
and corresponding to this we find the velocity 1288.2f-s, by
the same Table. Hence the velocity of an 11.52 inch 600 lb.
elongated shot would fall from 1350 to 1288-2f-s in 500 yards.
(4) In like manner, if it was required to find how much
the velocity of the same shot would be reduced in half a
second, its initial velocity being 1334f-s, we must find the
d²
reduced time = t=•2212 × 0″•5=0"1106. Adding this to
1":120, the number opposite the velocity 1334 f-s in Table IX.
we obtain 1"-2306. And opposite 1"-2306 we obtain by pro-
portional parts 1306.6 f-s, which is the velocity the shot will
retain at the end of half a second.
го
(5) Suppose a Rodman shot weighing 452 lbs. to be fired
with an initial velocity of 1400 f-s at a target 500 yards
off, to find the striking velocity. Here d=14.88 inches
d²
го
d²
W
=
··4898. Then s='4898 × 1500 7347, the reduced
range. Opposite the velocity 1400 in Table X. we find 1501,
and adding 7347 to this, we have 2235.7, opposite which, in
the same Table, we find the velocity 1215-8f-s, which is the
required striking velocity.
81. The general Table may be used to test the accuracy
of some of the early experiments made with the chronograph
represented in the frontispiece. From experiment alone the
72
EXAMPLES OF THE USE OF
average velocity of eleven shot at every ten feet from the
gun was calculated and published in 1866, for distances
between 130 and 1000 feet. Here d=3 inches, w = 11·7 lbs.,
d²
and ='7692. Suppose the velocity to be 11413f-s at
W
distance 200 feet, to find the velocities at 600 and 1000 feet.
Here V=11413, opposite which in Table VIII. we find:
2850, to which must be added s =400 ×·7.692 = 307·7.
d2
W
giving 3157.7. Opposite this number we find 1100'0f-s.
Also 800 × 7692 + 2850 = 3465·4, opposite which number we
find 1063-0 f-s.
Distances.
Velocities deduced from actual)
experiment in 1866.........
Velocities calculated from the
general Table VIII.
Differences
200 feet. 600 feet.
1000 feet.
1141·3f-s 1098·7f-s 1061·7f-s
1141.3
1100.0 1063.0
0.
+1.3
+1.3
Also Tables of velocities of variously formed heads of shot
at distances varying from 0 to 4500 feet, were appended to
the second Report dated 1866.† The initial velocity was
supposed to be 1500 f-s in all cases. Taking the Table for
an ogival head (1 diam.) where w=39.56 lbs. and an ogival
head (2 diam.) where w=38.50 lbs. and d=4·7 inches, we find
Distances.
Ogival (1 diam.) (2
•5584
Calculated by the general
Table VIII.
Differences
'd²
2
Ogival (2 diam.)
-='5738
W
0
2000 feet.
4000 feet.
| 1500 f-s 1267-9 f-s 1098′1f-s
1,500
1277.2
1106.2
0
+9.3
+8:1
Calculated by the general
Table VIII..
Differences
1500 f-s 12717f-s 1103-7 f-s
1500
1271.7
1098.2
0.
0
– 5.5
The general Table VIII. was deduced from experiments
made with ogival-headed shot struck with a radius of one
diameter and a-half. The consistency of these results is very
striking.
* Description of a Chronograph, p. 25.
† Reports, &c. p. 15..
THE GENERAL TABLES.
73
82. For high initial velocities (V), and low angles of
projection (a), the general Tables may be used to find
approximately the time of flight and trajectory of the shot.
Thus, suppose V the initial velocity, and v the velocity when
the shot would have described the space Op (fig. 8) in time t
if gravity had not acted. Then, by the general Tables
VIII. to XI., it is possible to find Op and t. If x, y be the
coordinates of p at time t, then
X = Op cosa
y = Op sina - 1gt"
become known because Op and t are known approximately.
Table XII. will be useful to find the values of 1gt.
On the principle of these General Tables, special Tables
of the velocity, time of flight and energy of each kind of
rifled service shot have been calculated for intervals of one
hundred feet, and recently published under the title:—
Tables of Remaining Velocity, Time of Flight, and Ener
of various projectiles calculated from the results of experiments
made with the Bashforth Chronograph, 1865–1870. London:
E. and F. N. Spon, 1871.
( 74 )
CHAPTER VI.
ON THE LAW OF PENETRATION OF PROJECTILES.
83. A KNOWLEDGE of the destructive effects of projectiles
is of so much importance that it does not seem desirable to
omit all consideration of the subject, although I have no
new results to communicate.
A Commission, appointed by the French Minister of
War, carried on experiments at Metz in 1834 and 1835
with a view to determine the law of penetration of spherical
shot into various kinds of wood, masonry, and earth. The
conclusions arrived at were, first, that the resistance of the
same substance to spherical shot of different diameters, varied
as the square of the diameter of the shot; and, secondly,
that the resistance of different substances to the same shot.
varied as a +B × (velocity)², where a and B were constant
for each substance. If, therefore, d be the diameter of the
shot in inches, w its weight in pounds, and v its velocity
in feet per second, then the resistance to the shot will be
expressed by πd² (a + ßv²) = d² (λ+ µv²), and the retarding
force by d* (λ+ µv²).
9
พ
λ
---
н
The following are the values of λ, µ, and calculated
from the values a and B, as given by Didion,* and adapted
to English measures.
* Didion, Traité, pp. 301, 302, and 304.
ON THE LAW OF PENETRATION OF PROJECTILES. 75
Substances.
λ
μ
u =
√ (1)
Oak, Beech, and Ash
2329.4 004328
734
Elm
1787.5 003322
734
Fir and Birch
12960 002408
734
Poplar
1217·7 002263
734
Sand, mixed with Gravel
486·0
009031
232
Earth, mixed with Sand
and Gravel
Clayey soil
Earth from an old Parapet
670.3 012456
232
1167.5
003799
554
782.0 ·004360
424
Damp Clay
297.2
002209
367
Moistened Clay
102.4 000762
367
Masonry of good quality
6166.9 008595
847
Masonry of medium quality 4915-7 006851
847
Brickwork
3530.4 004920
847
84. Suppose that is the striking velocity of a spherical
shot, and that when it has penetrated a distance s, in time t,
its velocity is v. Let s and T denote the values of s and
respectively when the shot comes to rest, that is, when v0.
dv
gd2
We have
V
ds
(λ + μv²).
W
Integrating,
C-
2d gus
= logε (λ + µv²),.
พ
and since s=0, when v V, C=logɛ(λ+µV²),
or
S
2d gμ
พ
logε
дн
λ + μv²
+μ2
+ μν
2
(x + μ V²
2
(1),
1.1513 w
d&gμ
дн
And total penetration = $=
log 10
λ+ μυ
1.1513w
d'gμ
1.1513w
log₁0 (1 + ½ V³)
010
V
{1+ (~)}⋅
log101+
76
ON THE LAW OF PENETRATION OF PROJECTILES.
85. Again, to find the time t in which the velocity is
reduced from V to v, we have
dv
dt
Integrating, C'-g.d²μt
But when v=
W
t=
gd2
(λ + µv²).
W
√ (~)
tan-¹v
√(~).
V, t=0, therefore
C' =
-1
√(5) tan√(E).
Hence
W 1
t=
gd² √(λµ)
{
tan™¹ V
√(~)-
tan¹v
(2).
And the total time of penetration is found by making v=
when we obtain
09
W
1
W
1
V
T=
tan¹ V
2
gi virus
tan-¹
-1
gd² √(λµ) W
86. Between equations (1) and (2) we might elimate v,
and thus express s in terms of t, or t in terms of 8, and known
ds
quantities; or, writing for v in equations (1) and (2) we
dt
might integrate again. But it will be found more convenient
to give certain values to v in equations (1) and (2), and
thence calculate the corresponding values of s and t.
87. When a spherical shot was fired into a substance
which retained its form, the diameter of the cross section
of the opening was found to decrease gradually, till it be-
came at last equal to that of the shot. And the section made
by a plane along the axis of the opening had its convexity
turned towards the axis. From the Metz experiments, it
was also concluded that when a given shot was fired into a
given substance, but with different velocities, that the volume
of the opening made varied as the vis viva of the shot on strik-
ing. Let y be the radius of a section of the opening at depth
s when the velocity of the shot is v. Then we have
w
-
v²)
2 g
1 ½ (V² — o") = k "J" y'de.
ON THE LAW OF PENETRATION OF PROJECTILES. 77
W
V
g
Differentiating,
dv
ds
-
= ky" = d² (λ + µv²), by Art. (84),
d²
or
y²
k
(λ + μv²).
And when
v=0, 2y=d,
d2
d2
therefore
λ, or k=4λ.
4
k
Hence
(3).
y²=jd" (1 + 2 v)
(1+1)
And the radius of the mouth of the opening
d
= 1/2 √/ (1 + 2 v²).
By equation (1), we have
W
Ꮴ
* = 2dgu {log. (1+ * V*) – log. (1+))
W
дн
2d gμ
дн
λ
μ
{log. (1 + 2 V³) — 2 log.
2y)
-
,
d
which gives the equation to the section of the opening made
by a plane along its axis.
88. Additional experiments were made at Gâvre,* in
1835, 1843, and 1844, to determine the penetrations of sphe-
rical shot. The penetrations obtained at Metz were a little
deeper than those obtained at Gâvre. But it is plain that no
great precision can be arrived at in such experiments.
Substances of the same name may
have very
different
degrees of consolidation. As the ballistic pendulum was in
use when these experiments were made, it was not possible
to measure the velocity of the shot which actually made the
openings in the substances fired at.
89. M. Didion assumest that the formulæ used in calcu-
lating the penetration of a spherical shot into any substance,
may be employed to calculate the penetration of an elongated
shot into the same substance, provided the coefficients X and
* Helié, Traité, p. 192, 196.
† Didion, Traité, p. 296.
78 ON THE LAW OF PENETRATION OF PROJECTILES.
μ are replaced by λ and μ. The resistance to an elon-
gated shot, d inches in diameter, will then be expressed by
}ď² (λ+µv²). But it is manifest that much would depend
upon the form of the head of the elongated shot. The results
of a few experiments with 7-inch, 70-pr., 40-pr., 20-pr., and
12-pr. elongated shot, when fired into compact loam, are
given in the Handbook for Field Service, 1867, p. 110.
90. Although a vast number of random shots have been
fired against iron plates, no systematic series of experiments.
have been made to determine, in a satisfactory manner, the
law of perforation of iron plates by elongated shot. Before
anything of the kind could be attempted with success, it
would be necessary (1) to perfect the rolling of thick plates
of iron so that they should not be caused to separate into
thin layers when struck by the shot, and (2) to improve
the manufacture of the elongated shot so that they should
not break up or sensibly change their form in penetrating
iron plates. In the next place it would be necessary (3) to
determine the best form of head for penetration, by firing,
with equal velocities, a number of elongated shot of equal
weights and diameters, provided with variously formed heads,
against given plates. After these preliminaries were settled
there would still remain the following four quantities subject
to variation: (4) the weight of the shot; (5) the diameter
of the shot; (6) the velocity of the shot; and (7) the thick-
ness of the plate of iron. In experimenting it would be
necessary to keep two of these four quantities constant, to
give to a third variations following a law, and then by
experiment to determine the corresponding values of the
remaining quantity, which must of necessity follow some law.
Thus, in the first place, the weight and diameter of the shot
might be constant, the striking velocity of the shot might
be made in succession 900, 1000, 1100, 1200, and 1300 f-s, and
the corresponding thicknesses of plate, which the shot would
just perforate, might be found by experiment. In cases
where the projectile passed completely through a plate,
its remaining velocity would be best measured by taking
the depth of its penetration into sand, or some more suitable
substance.
If such a series of experiments were to be undertaken,
I should particularly advocate the use of guns with bores
of to 3 inches, in connection with suitable thicknesses of
iron plates. I believe the law of penetration would, in this
way, be more accurately and far more cheaply determined
2
ON THE LAW OF PENETRATION OF PROJECTILES.
79
than by the use of the large guns now in the service. In
confirmation of this opinion, I refer to the comparatively
good results obtained by Robins and Hutton in experiments
made with small calibres, by the aid of such a rude instru-
ment as the ballistic pendulum.
The late Ordnance Select Committee were accustomed
to calculate the perforating power of a shot on the suppo-
sition that the vis viva of the shot must vary as the diameter
of the shot, and as the square of the thickness of the plate.*
But on the Continent it is thought necessary to make the
vis viva of the shot vary as the square of its diameter. The
approximate law, however, must remain in doubt till a
systematic course of experiments has been made with small
calibres, and with shot which will perforate without breaking
up, or sensibly changing their form.
* British Association Reports for 1866, p. 411.
( 80 )
APPENDIX.
I.
ADAPTATION OF THE CHRONOGRAPH SHEWN IN THE
FRONTISPIECE TO MEASURE A SUCCESSION. OF
EXTREMELY SHORT INTERVALS OF TIME.
WHEN I brought forward my original scheme for a Chronometric
Instrument adapted for Ballistic Experiments, I lithographed a rough
sketch to explain my plans to the late Ordnance Select Committee.
The following explanatory description, which was appended, will be
sufficiently intelligible without the diagrams.
"In all experiments relating to the velocities of projectiles, to the
"resistances of the air to their motion, or to the effects of projectiles
upon iron plates, it is necessary,
66
(
'(1) that the time should be registered by a clock of known rate of
“going, and
66
(2) That the time of flight over several successive equal spaces
"should be measured by the same instrument under the same con-
"ditions.
"The adjoining plan has been compiled with a view to secure these
"two results. The revolving cylinder is in use at Greenwich; the
spark from a Ruhmkorff's Coil is used by Martin de Brettes; and
"Prof. A. Vignotti, of Metz, teaches how to prepare the paper so that
"the hole burnt by the spark may have a black rim and thus become
easily visible.
66
Fig. 1. abed is a cylinder having a metallic surface always kept in
"connection with the wires by, cz, by contact springs at b and c.
66.
e, f are points from which the clock and screen sparks respectively
"proceed, whenever the currents vuy, wxz, are interrupted for an instant.
"g is a fly wheel mounted on an axis h which drives the cylinder
abcd.
"i, are toothed wheels which turn the screw o, and thus give a uni-
"form motion to the carriage k along the bar Im. Thus the holes made
by the sparks from the points e, f, will be arranged in two spirals.
י
"As each experiment will be completed in a few seconds, it seems
"undesirable to complicate the instrument with any more machinery
"for driving the cylinder. If the fly wheel be set in motion (as a
"gyroscope) the rotation of the cylinder will diminish very slowly,
"and the diminishing distances of the seconds marks will point out the
"nature and amount of the necessary corrections. The cylinder abcd
"is covered with prepared paper.
"Fig. 2. w, r, are two galvanic batteries, and y, z, two Ruhmkorff's
"Coils.
The current from v is broken for every beat of the Clock
“Pendulum, and at the same instant a spark from e makes a hole in the
APPENDIX.
81
!
((
"
paper covering of the cylinder. When a ball from a cannon passes
through the screens S1, S2, S3..S-1, §,, it will interrupt the current in
wxz, and thus give a spark at ƒ.
"current.
(C
Fig. 3 shews an arrangement for making and breaking the
GH is a strong but light chain fixed at G and attached to
"the lever ED at H. The lever is kept in contact with CB at C by a
strong spring K. The cross wires of the screen are fixed at one end
"to the frame, whilst the other end, after being bent downward over a
pulley, is fixed to the chain GH. When a ball passes through the
screen it bends the wires, pulls the chain, raises the lever ED about
66 1
2% of an inch from C, and causes it to rest against C'. The wire
"breaks and the spring K brings D in contact with C again and so
(C renews the current.
20
"April 19, 1864.
""*
(Signed)
F. BASHFORTH."
Thus it is plain (1) that the rotating cylinder and (2) that the spark
of a secondary coil were employed in my original plan of 1864, which
I candidly described as a "compilation." The real secrets, however, of
my success were (1) the spinning of the cylinder "as a gyroscope" with
a view to secure perfect smoothness of motion, (2) the contrivance of
means for measuring the varying motion of the cylinder, and (3) the
invention of a proper form of screen. Unfortunately the Ordnance Select
Committee in 1864 entertained a strong objection to any rival of their
pet Navez, and, in particular, to all chronographs with "rotating
cylinders.' But when they had in their hands Reports I. and II. of
my experiments, dated respectively December, 1865, and October, 1866,†
containing abundant evidence of the practical success of the rotating
cylinder employed in my instrument, the Committee thought it fit and
becoming shortly afterwards to bring forward the Noble Chronograph,
which was, in fact, a very poor imitation of my instrument, only it must
be borne in mind that I used neither toothed wheels to drive the cylinder
nor yet a stop-clock to measure its angular velocity. Under these cir-
cumstances I could have had no hesitation in entering on a competitive
trial with my instrument as originally constructed. But in order to
adapt the principle of my chronograph to measure, in the most satis-
factory manner possible, extremely short intervals of time, I had the
machine constructed which is represented in fig. 16. The whole change in
the original instrument was about equivalent to the addition of a seconds
dial to a clock which previously indicated only hours and minutes.
The short cylinder F about 33 inches in diameter is mounted on
a vertical axis C, which is driven in the manner shewn in fig. 16. The
endless band passes round the pulley E shewn in plan and section
in fig. 17. F is a fixed centre supporting one end of the axis C of
the cylinder F. D is a pall moving freely about its pivot. When
the pulley E is spun the heavy end of D has a tendency to fly outwards,
so that the other end catches in a notch in the axis C and drives the
cylinder F. When this cylinder has attained its required velocity the
driving power is stopped, and the axis C, now moving faster than the
pulley E, pushes the pall D out of the way, and thus the cylinder Fis
left spinning "as a gyroscope" under the retarding forces of the friction
at the pivots, and the resistance of the air. Its motion will then be free
from vibration and perfectly steady, so that the loss of its angular
* Proceedings of the Ordnance Select Committee, 1862, p. 15.
† Reports, &c., p. 1—17.
G
82
APPENDIX.
velocity must follow some simple law. If it be afterwards found desirable
to increase the angular velocity of F", by turning the handle in the
right direction, the pall D will be caused to catch the notch in C,
and so drive the cylinder till the required velocity is attained. Any
other driving arrangement might be used, provided it did not impart
a vibratory motion to the frame and was capable of being completely
detached from C before the experiment was made.
20
The records are received on paper stretched upon the cylinder F'.
They are made by sparks discharged at the point P from one or more
secondary coils. In cases where the records are required to succeed
each other at intervals less than ‚¹th of a second it will be necessary to
use several independent primary coils, generally as many coils as there
are records to be made. This is in itself a disadvantage, but there does
not appear to be any remedy. If now, with a view to obtain all
the discharges from one point, one end of each secondary coil were
attached at A, and all the remaining ends at B, it is very probable that
no record whatever would be obtained at P. For
For suppose two coils a, b
to be so connected as shewn in fig. 18. When the primary wire of a was
broken, it is quite probable that the secondary discharge would not take
place directly from A to B, but through b to B, which course affords a
continuous metallic circuit. So also when the primary current of b was
ruptured, the secondary discharge would take place from A to B through
a, and not directly from A to B. If, however, we make breaks in the
secondary circuits as at c, d (fig. 19) we may render the resistance
of the direct course from A to B less than that of any other. This
arrangement was tried with two coils, and proved quite satisfactory.
There is no doubt it would answer equally well for any number of coils.
It is plain that the angular velocity of the cylinder F' could not vary
sensibly during the short time an experiment was in progress. The
distances between the records would therefore shew the relative intervals
at which the primary currents were actually cut, abstraction being made
of the errors to which the records made with the spark are known
to be liable.
In order to find the absolute intervals of time between the records
it would be necessary for the exact velocity of the fly wheel to be
known at the instant when the experiment was made. This is
satisfactorily provided for as follows. The chronograph shewn in the
frontispiece is used to measure the time of every revolution of the
cylinder F'. For this purpose a short arm attached to the axis C
interrupts, once for every complete revolution, the galvanic current con-
nected with the electro-magnet E. In order to shew during what
particular revolution the experiment was made, one or more of the
secondary circuits is arranged to give a record near the markers m, m'.
This record serves to point out the particular revolution required, and
knowing the exact times of several successive revolutions before and
after, we can calculate the exact motion of the cylinder F' at the instant
when the records were made. If necessary, it would be possible to
calculate the varying motion of the cylinder during the progress of the
experiment.
As I have already remarked (Art. 11), the great drawbacks to the
use of such an instrument in ballistic experiments are (1) the uncertain
action of the spark as a recording agent, and (2) the impossibility
of securing accuracy in space. When the successive intervals of time to
be measured are subject to rapid variation, as in determining the motion
of a shot in the bore of a gun, such an instrument is perfectly powerless,
APPENDIX.
83
and at present I cannot think of any practical case in which it could be
used advantageously. As, however, I found that Captain Andrew Noble,
in conjunction with the Committee on Explosives, was employing the
principles of my chronograph not only to an improper purpose, but also
in a most rude and unscientific manner, I felt it to be my duty, in con-
sequence of my official position, to explain clearly how to obtain the best
possible result. In order to shew how little novelty there was in the
Noble Chronograph, I offered (March 20, 1869) to place my original
chronograph in the same circuit with it, using the fly wheel as
my rotating cylinder to obtain a sufficient velocity of rotation. My
chronograph was reported ready for the competitive trial offered me
May 28, 1870, in the following terms: "The chronograph has been
"removed from Shoeburyness, and is now ready for the inspection of the
"gentlemen who are to examine into the comparative merits of my chrono-
graph and that subsequently brought forward by Captain Andrew Noble
"when applied under like circumstances to any experiment whatever.”*
It will be seen that I was careful not to offer to determine the motion
of a shot in the bore of a gun by means of a chronograph, which I believe
to be an impossibility, and therefore I consider it an absurdity to attempt
it. But by putting both instruments into the same circuits the results
would have been free from any question about the precise time when the
primaries were cut, that is, from any question about accuracy of space.
The real trial would thus have been limited to the comparative merits
of the two chronographs as measurers of very short intervals of time.
To my regret the competitive trial did not take place, for, if the records
of the two instruments had been read off and reduced by competent
mathematicians, the true value of such instruments as the Committee
on Explosives have been using would have been authoritatively settled.
But if this question had been decided in favour of one or both the
chronographs, before the results of the experiments of the Committee
on Explosives could be accepted as conclusive, there would remain
the question whether the shearing instrument employed by them, and
represented in fig. 4, could be expected to secure sufficient accuracy
in the intervals of space to allow the rapidly varying pressure of fired
gunpowder to be accurately calculated.
The following is a statement of the times in which a few successive
revolutions of the cylinder F' were made in a trial experiment:
No. of
Revolutions.
Time
seconds.
23
13.352
seconds.
A₁
24
+ 0.550
13.902
25
14.453
+ 551
Δη
+ •001
+ .003
26
+ ·554
15.007
+ '003
27
+ ·557
15.564
28
+ 561
+ ·004
16.125
29
16.688
+ ·563
+ •002
30
+ ·566
+ •003
17.254
+ .003
+ ·569
31
17.823
+ '003
32
+ ·572
18.395
33
+ ·575
+ ·003
18.970
+ .003
34
+ •578
19.548
+ •002
+ ·580
35
20.128
36
20.712
+ ·584
+ •004
+ .004
37
21.300
+ ·588
&c.
&c.
* Reports, &c., p. 162.
84
APPENDIX.
In order to determine the value of the spark from the secondary.
coil as a recording agent the following experiments were made. The
discharging point P (fig. 16) was caused to move accurately in a
line parallel to the axis of the cylinder F". The primary current of
the coil was arranged so that it was momentarily interrupted at every
complete revolution of the cylinder F". Consequently the secondary
sparks discharged from P ought to have given records arranged in
a straight line. When the cylinder was stationary, a tracer, attached
to the same frame as P, was used to mark a line on the paper.
The distances of all the records were measured carefully from this
line as a base, and then their mean was found. Afterwards the dif
ference between this mean and each reading was taken as the error
of that reading. The sum of the numerical values of these errors
was divided by the number of records to obtain the mean error.
The
velocity of the surface of the cylinder on which the records were
made was about 300 inches per second and under.
Experiment I. The successive errors were +010, -·003, +·013, 000,
- *004, - •018, + ·006, &c., giving the mean error of 34 records = 0.0107 inch..
Experiment II. The successive errors were +001, − ·020, − ·048, - •012,
+ •020, +·006, &c., giving the mean error of 32 records = 0·0112 inch.
Experiment III. The successive errors were † ·006, +·012, ·000, +·020,
+·008, -·006, &c., giving the mean error of 26 records = 0.0118 inch.
Numerous other trials of the same kind were made, but the general
results were manifestly the same as those given above, no matter which
pole was attached to the discharging point.
In the next place, the marker m, in the instrument shewn in the
frontispiece, was replaced by a discharging point properly connected
with a secondary coil. The fly-wheel was then spun as a gyroscope,
and, the primary current being rapidly interrupted in the usual manner,
the records on the cylinder ought to have ranged in a spiral line.
In this case the lateral deviations of 100 successive records were
measured, and the mean was found to be 0.0075 inch. In this
case the error may have been in some slight degree due to the
motion of the paper, which was moving at the rate of not more
than 10 inches per second.
Lastly, the cylinder F" (fig. 16) was kept perfectly stationary.
While the primary current was being rapidly interrupted in the
usual manner, the discharging point P was caused to move parallel
to the axis of the cylinder. The mean error of 46 successive records.
so obtained was found to be 0·0069 inch.
II.
DESCRIPTION OF A GRAVITY CHRONOGRAPH.
AFTER having made preliminary experiments, I expressed an opinion*
that the ballistic instruments of Navez, Leurs, Boulengé, &c., might
be replaced for ordinary purposes by a simpler, cheaper, and more
reliable instrument. Shortly afterwards I received several communica-
tions on the subject from the War Office. The carrying out of my
plans proved more troublesome than I at first expected, and, as my
time was then fully occupied by experiments on the resistance of
* Philosophical Transactions, 1868, p. 419.
"
85
APPENDIX.
the air to the motion of projectiles, I was not disposed to allow my
attention to be diverted from that important work merely to perfect
an instrument to which no scientific interest was attached. When,
however, the reports on those experiments had been sent in, I enquired
whether a simple form of my chronograph was still required. Being
encouraged to proceed with my invention, and seeing no reason why
the simple form of instrument should not succeed, I felt obliged to
persevere.
Although the original instrument, shewn in the frontispiece, would
have given results of extreme accuracy, yet the exact reduction of
each round would have been attended with a great waste of labour.
After much trouble the original scheme for a simple form of chrono-
graph was made perfectly successful, which was dependent on a time
scale given by a body falling freely through a given height. The
firing of the gun was so timed that the records of the two end
screens were nearly equidistant from the time records. Thus A, B are
the records of the beginning and end of the time, generally half-
a-second; and a, b, c, d are the records given by four equidistant screens.
Au must be made nearly equal to dB.
Å
ä
b
B
The records are received on a vertical cylinder K, the motion of
which is regulated by a fly-wheel A, as shewn in the frontispiece. As
the stage S is now stationary for each experiment, there is no necessity
for the lowering machinery MCD; and the slide L may be made
of wood. There is only one electro-magnet E, and one marker m,
which is used to give both the time and screen records. The cylinder K
is about 9 inches in diameter, and the speed must be such that the
distance AB representing half-a-second may be from 15 to 20 inches.
The arrangement for dropping the ball W, which gives the scale
of time, is shewn in fig. 11. The wires C₁, C₂ must be connected with
the wires + and - of the electro-magnet E, shewn in the frontispiece. A
wire must be carried from B, to the galvanic battery, to the arrangement
for dropping the weight by which the gun is fired, through three or
more screens constructed as shewn in fig. 12, when arranged at equal
distances as shewn in fig. 13, and finally connected with B. The
galvanic circuit is completed as follows, when arrangements have been
made for firing a round: through the screens, &c., to B,, h, i, g, f, d, e,
c, b, a, v, u, coil of the electro-magnet D, r, o, B₁.
The firing of the gun is caused by the fall of a weight. When the
weight is dropped a momentary interruption of the above galvanic
current is caused. The height through which the body falls before
it fires the gun requires to be adjusted so as to throw the screen records
a, b, c, d in the proper position relatively to the time records A, B.
The instrument is represented in fig. 11, ready prepared for the
firing of a round, the electro-magnet E of the chronograph being
excluded from the galvanic circuit. When the weight intended to
fire the gun is liberated, the momentary interruption of the galvanic
current destroys the magnetism in D, and the lever OA, having nothing
to support it, begins to fall by turning about its pivot O. The switch
descends from the position of to ot', and so turns the galvanic current
into the course B₁, o, s, C₁, through the chronograph, C, w, a, b, c, e,
d, f, g, &c. When the lever OA comes into the vertical position OA',
it strikes a projection on the bolt eA', and thus suddenly withdraws,
86
APPENDIX.
in a horizontal direction, the support of the ball W. Since the galvanic
current passed through c, e, d, we are quite sure that the interruption of
the galvanic current and the dropping of the ball would be two perfectly
simultaneous events. The ball W is connected with ce by a flexible
chain. The distance between the lower side of the ball A and the
horizontal disc F is accurately adjusted according to the time required,
for which purpose Table VII. will be found convenient. When the ball
strikes the stage F the galvanic current is instantaneously broken at g.
It is plain that the first interruption of the galvanic current, namely,
when the firing weight was liberated, would not be recorded, because
the chronograph was then excluded from the circuit. There would
be two time records A, B corresponding to the interruptions of the
current at e and These would be separated by a known interval
of time, generally half-a-second. But in the meantime there would be
brief interruptions of the galvanic current when the shot passed three
or more equidistant screens. If the record a was too near A, this
distance would be. increased by slightly lengthening the fall of the
weight used for firing the gun, and vice versâ. When the distance Aa
was nearly equal to the distance dB, the slight loss of velocity of the
fly-wheel during half-a-second would be of no importance, and the
calculation of the velocity might be proceeded with just as if the
angular velocity of the cylinder had been perfectly uniform. The use of
one or two intermediate screens b, c is to check the accuracy of the
instrument. If the screens were equidistant, then the spaces ab, bc, cd, &c.,
would gradually increase in a satisfactory experiment. In calculating
the velocity of the shot, however, the time in ad should be used. Now
time in ad time in AB:: ad: AB, or time in
seconds; and
if S denote the distance in feet between the
records a and d, then velocity
=
S
time in ad
=
ad
=
1 ad
2 AB
screens which gave the
AB
25 ×
ad
feet per second.
In proving gunpowder and making ordinary experiments 28 may
be made a constant quantity by placing the extreme screens always
AB
ad
at the same distance. The value of 2S may be read off with
sufficient accuracy by inspection from a slide rule, or a table of four-
figure logarithms may be used when greater accuracy is required.
To prepare for another round the weight which fires the gun is
replaced, the ball W is suspended in its proper position, and the
lever OA being raised up by hand and pressed upwards, the galvanic
current is turned from the chronograph to the electro-magnet Ď, which
holds up the lever OA until the current is broken on liberating the
weight which fires the gun.
With a view to exclude the disturbing effects of remaining magnetism,
care is taken not to allow the electro-magnet E to come in contact
with its keeper which actuates the marker m. This distance can be
adjusted according to the strength of the galvanic current. It will
be remarked also that the galvanic current is only turned through
the chronograph just soon enough to give the first time record, and
that during the half-second the experiment lasts the galvanic current
is being rapidly interrupted. Under these circumstances we are entitled
to assume that the loss of time in making each record is practically
constant, and therefore that the resulting velocity will be unaffected by
remaining magnetism.
APPENDIX.
87
The marker used in connection with the gravity chronograph is
made to strike the paper as the clock hammer strikes its bell, the
needle point being free to open slightly so as to avoid tearing the
paper. As soon as the marker is raised, the needle is restored to
its proper position and maintained there by the action of a spring.
Care must be taken to check the vibration of the marker preparatory
to making a record. For this and other reasons it is not desirable
for the screens to be placed at intervals of less than 75 or 100 feet.
It should be borne in mind that this instrument was contrived merely
to measure initial velocities, &c., while the instrument shewn in the
frontispiece was designed for determining the resistance of the air
to the motion of projectiles, for which purpose it alone should be
used. The gravity chronograph was used to measure the velocities
of small arm bullets-it was officially inspected-it was confessed that
it was easily manipulated and furnished results of a high degree of
accuracy.
66
(6
The late Ordnance Select Committee were at one time in raptures
with the performance of Navez' electro-ballistic pendulum, simply
because they did not attempt to deduce any laws from their experi-
ments. The course of events led them quietly to adopt Leurs' improved
pendulum as soon as it was brought out. In the description of this
instrument, printed under their auspices in 1868,* it was confessed that
its advantages were simplicity, as the instrument does not consist of
so many parts, the conjunctor, and consequent use of mercury has
"been abolished, the large magnet suppressed, and a mechanical
arrangement substituted; cheapness, as the instrument costs less than
"the original Navez; greater accuracy and ease of manipulation," &c.
Two years later a description of Boulengé's chronoscopet was issued.
This is undoubtedly far simpler to use and superior to both the Navez'
and Leurs' instruments, because in every experiment the time scale
is given by direct comparison with the time occupied by a body in
falling freely. In the Navez' and Leurs' instruments the body falling
freely is replaced by a pendulum rotating about a pivot, and its
indications are reduced by reference to a scale of time previously
deduced by trial. But all three instruments depend upon the dropping,
in consequence of the rupture of the galvanic current, of two weights
directly supported by two electro-magnets. Unless the two independent
currents required for each instrument be regulated with the greatest
nicety there will be a variable loss of time in dropping the two weights.
Now the galvanic currents are apt to vary greatly in strength in the
course of a day, and especially when the battery is newly charged.
Every round fired also ruptures the circuits, and the repairing of these
circuits may affect their resistances unequally.
So unsatisfactory were these instruments, owing partly to the dis-
turbing effects of remaining magnetism, and partly to their inability
to afford any test of their own accuracy, that it has been found necessary
to employ two Boulengé instruments to measure the velocities of shot
in the proof of pebble powder where charges of 35 lbs are used. If
the same pair of galvanic currents be used to work both instruments,
no satisfactory test of accuracy will be obtained; and if the two
instruments be used to measure independently the velocity of the same
shot, then four galvanic batteries must be used, the currents of which
* 84, v., 188,
† 84, L., 459.
88
APPENDIX.
would require regulating with the greatest nicety. If any one of the
four became unduly strong, one velocity would be vitiated, and then
how would it be possible to decide which of the two results was the
correct one? As the number of such instruments as those of Navez,
Leurs, Boulengé, &c., is increased, the liability to discordant results
is also increased.
III.
ON INTERPOLATION AND QUADRATURES.
WE have already seen (Art. 54) that
n (n − 1) (n − 2)
1.2.3
-
▲³ux + &c. ... I.
N
Ux+ni
= U x +
ī
Дих +
n (n - 1)
1.2
Also
Ux-1= Ux - AUx-19
-
-
- A
Ux-21= Ux-1 - AUx-21= Ux — AUx-1 — ▲ (Ux-¿ — AUx-21),
= U
Ux − 2Aux-ı + ³Ux-219
Ux-31 = Ux − 3AUx_1 + 3A³Ux-21 − ▲³Ux-319
&c.
&c.
n
Ux-ni = U
Их
www
Aux_l +
n (n-1)
1.2
▲²Ux-¿ì
▲³u¸µ - n (n − 1) (n − 2)
1.2.3
x-21
▲³ux_x + &c...II.
x 1
Uz+} = U₂ +
AU₂ −
Putting n =
in I. and II. we have
5
128
1²u₂ +
1
1
16
A³ux - Aux + A³½ - &c....(a),
T
256
and
thu
=
Au HAU TA SỐ TSAU 4
1▲Ux-1 – 14³Ux-21
A³ux-31
1
16
5
128
2 66
Aus- &c...(b).
Ux-l
x-41
Thus, in Art. 55, we find by interpolation the proper place of beat
3.5 seconds, for
by (a)
U3.5 = U3 + AU3 – §A²Ug +
1
164
1
16
73.637 + 1 × 23·794 + 1 × ∙054 - 1 × ·003 = 85·541.
8
U4
16
Ճա
or by (b) U3.5 = U₁ - Az - ▲²₂ – 1²Ã³µ₁
= 97·431 − × 23·794 + 1 × ·051 + 1 × ·003 = 85·541.
=
1
16
Again, if after having completed the interpolations for half-seconds,
we desire to find the readings corresponding to the one-tenth of a
second, or to the one-fifth of half-a-second,
n = }, U x+] = U x + & Aux - 24² + £54³u
2
?
x
12
-
u.
843
U x
n = 3, Uzzi = Uz + & Au₂ - 2ª² + 1½¿ª³u
Also
&c.
x +
&c.
25
126
Mở Al-A - I AU -
-
21
6 2 5
2¹¸
625
26
625
A
A¹u₂+ &c....(c₁),
A¹u₂+ &c....(C2),
▲³ux-17
- &....(d),
Uz - Z1 = U₂ ZA Ux_1234²ul-A³ux-st- & Aux-11 - &c....(d₂),
U.
Ux-
= U
Aux-
U2-1
스리
​Δ
8
и
Δ
626
26
&c.
&c.
APPENDIX.
89
Supposing the divisions corresponding to the half-seconds to be
known in Art. 55, let us find the divisions corresponding to 4.2
and 4.3 seconds.
3
By formula (c2) U₁.2 = 97.431 + × 11·877 + 2 × 014 = 102-184,
- 2
3
25
and by formula (d₂) u₁., = 109.308 x 11.877 +2% × 014 = 104·559.
It will be observed that the quantities u, Au, A³ux, A³ur, &c., used
in I., are arranged in a diagonal line pointing downwards from ux
in the scheme at page 39, and also that the quantities u Au
▲³Ux-219 A³µx-31, &c., used in II., are arranged in a diagonal line pointing
upwards. We can replace these formulæ by two others depending
upon the terms
Ux
A'u
+ All x-² + A²Ux-² + A³ U x-l
+ 4³U --21
A³u.
+ A¹Ux-??
A³Ux-l
+ Aux
+ A³Ux-çl
+
+ A¹Ux_1?
+ AºUx-31
+ A³ux-42•
+ A¹Ux-31
'x-21
Since
Uz = U x 1 + Aux-79
Au = A®up_z + Aut
x
c-l 'x-21 + A³µx_29
5
▲¹ux = ▲¹Ux_21 + 2A³ux-¿ì + Au₂ + A¹Ux-3¹9
'x-sl
▲³Ûx = A³Ux_21 + 2AºUx-31 + 3A¹Ux-31 + A³Ux-11 + A³Ux-419
A³ux = A®ux-32 + 3A¹µx-31 + &c.,
доих
&c.
&c.
Substituting now these values of A³u, A³u₂, &c., in I., we get
n
Ux+m² = U x +
Ux
Aux +
n (n - 1)
1.2
A²Ux-1 +
(n + 1) n (n − 1)
A³ux_1
1.2.3
(n + 1) n (n − 1) (n − 2)
+
▲¹Ux.21 +
(n + 2) (n + 1) n (n - 1) (n-2)
1.2.3.4
1.2.3.4.5
(n + 2) (n + 1) n (n − 1) (n − 2) (n − 3)
+
▲°Ux_31 + &c. ……..
1.2.3.4.5.6
Ux-2
III.
n (n − 1)
1.2
A²ux-i
+
▲⭑Ux-2l
Or, expressing the values of A²ux-219 A³ux-31, &c., in terms of
A³Ux-19 A³ux-219 &c., and substituting in II., we shall find
Ux-nl = Ux Aux-¿ +
n
1
(n + 1) n (n − 1) (n − 2)
(n + 2) (n + 1) n (n − 1) (n-2)
-
K
(n + 1) n (n − 1)
1.2.3
A³ux-2
A³ux-31
1.2.3.4
1.2.3.4.5
(n + 2) (n + 1) n (n − 1) (n − 2) (n − 3)
+
A°ux3 ~ &c.
IV.
1.2.3.4.5.6
It thus appears that having obtained values of ...Ux-19 Ux, Ux÷19...&c.,
by calculation or observation, we are able to express the value of
Ux+nl from ux- to wx+, or, more correctly, from ux- to ux which is
all that is really needed.
By the help of the preceding formulæ of interpolation we are enabled
to obtain the numerical values of integrals of functions which cannot
H
90
APPENDIX.
be integrated in the ordinary way. Suppose ya to be the ordinate of
a curve corresponding to the abscissa x. Then the area bounded by
the curve, the axis of x, and the ordinates at the distances a and a + 1,
will be equal to a fa+¹y,dx = °f'Ya+ndnl = °f¹lya+nidn = °f'Ua+nıdn, suppose,
n (n
1.2
n
1
- "J' (v + A + " (51)
Alla
A&ua +
n (n-1) (n-2)
1.2.3
A
4³ua + &c.) dn
19
= Ua + ½▲ua - ³½³ua + 2²44³µa - 7¹² A¹ua
+185a-8634°ua + &c.
α 60
α
..V.
In the same manner we may find the area of the curve enclosed be-
tween the ordinates ya+t and Ya+2l; between Ya+21 and ya+zi; and so on.
And taking the sums of these partial areas we find the area of the curve
enclosed by ordinates at the distances a and a + rl where r is any integer.
But this method, applied in the first instance to find the areas of curves,
may be used to find the values of the integrals of any continuous
functions between limits.
+
The above formula (V.) was employed in the calculation of the
tabular values of X, Y, and T. Each of these quantities is a function
of y and . A numerical value was given to y, and then was made
Ø.
The
in succession 0, +1°, +2°, +3°, &c., and 0, -1°, - 2°, - 3°...- 45°.
value of l or do was the circular measure of one degree. But, when the
values of up became irregular, l = do was made equal to the circular
measure of th of a degree. In this way the values of Au, ▲³u, &c.,
were made to decrease rapidly, which allowed the formula (V.) to
be used in spite of the slow convergence of the coefficients.
Another formula may be deduced from III.
a-41 μa+³¹y xdx = − + f + ³ly a+nıɑn = −4f+³ua+midn
-
{u
Ua + Aula +
n (n − 1)
1.2
A²ua-i +
(n + 1) n (n − 1)
1.2.3
A³ua-1+ &c.} dn
n
1
9 367-0▲³ua-31 - &c.
= U« + 2¹4▲²Ua-1 − 577*Ua-21 + 567880
ERRATA.
Page 40, formula (2), for "32" read "1"
""
49
90
9
14, Tables, for 43 Y.02° "-676277,” read “·677277.”
'08
I. COEFFICIENTS FOR THE CUBIC LAW OF
RESISTANCE.
ELONGATED PROJECTILES WITH OGIVAL HEADS.
(Reports &c. p. 152.)


Log
Kv
80
Kv
K
V
K
Log
V
go
K
V
К.
V
g
مع
f-s
f-s
900
64.4
2.001
.3012
1300
107.9
3.352 .5253
910
64.8
2.014
.3041
920
65.3
2.029
1310 107.7
.3073 1320 107.4
3.345 .5244
3.337
.5234
930
65.9
2.047
.3III
1330 107.I
3.328
.5222
940
66.6
2.069
.3158
1340 106.8
3.317
.5207
950
67.4
2.094
.3210
1350 106.4
3.305
.5192
960
68.4
2.125
.3274
1360 || 106.0
3.293
.5176
970
69.6
2.162
•3349
1370 105.6
3.280
.5159
980
71.0 2.206
.3436
1380 105.I
3.265
.5139
990
72.8 2.262
.3545
1390
104.6
3.249
.5118
1000
75.0
2.330
.3674
1400 104.0
3.231
.5093
ΙΟΙΟ 77.5
2.408
.3817
1410 103.4
3.212
.5068
1020 80.4
2.498
.3976 1420 102.8
3.193 .5042
1030 83.9
2.606
.4160
1430 I02.2
3.175
.5017
1040 88.2 2.740
1050 92.8 2.883
.4378
1440 101.6
3.155
.4990
.4598
1450 100.9
3.134
.4961
тобо
97.2
1070 100.8
3.019 .4799
3.131 .4957
1080
103.4 | 3.212 .5068 1480 98.7
1090 105.I
3.265
1460 100.2
3.112
.4930
1470 99.4
3.089
.4898
3.065
.4864
.5139 1490 97.9
3.041
.4830
ΙΙΟΟ 106.0
3.293
.5176 1500
97.2
3.018
.4797
IIIO 106.6 3.312 .5201 1510 96.4
II20 107.I 3.327 .522I 1520
I130 107.5 3.339 .5236
1140 107.9 3.35I
1150 108.2 3.361
1160 108.5 3.37I
1170
108.7 3.377 .5285
1180 108.9 3.381 .5290 1580
1190 108.9 3.383 .5293 1590
2.994
.4763
95.5
2.968
.4725
1530 94.7
2.942
.4686
.5252
1540 93.8
2.915
.4646
.5265 1550 93.0
2.889
.4607
.5278
1560 92.2
2.864
.4570
1570 91.4
2.839
.4532
90.6
2.814
.4493
89.8
2.790
.4456
1200
108.9 3.383
.5293 1600 89.0
2.765 .4417
1210 108.9 3.383
.5293
1610
1220 108.9 3.382 .5292
1230 108.8 3.381 .5290
88.2 2.740 •4378
1620
1630
86.7
87.4 2.715 •4338
2.693 .4302
1240 108.8 3.380
.5289
1640
86.0
1250 108.7 3.378 .5287
1650
2.672 .4268
85.4 2.654 .4239
1260 108.6 3.375
.5283
1660
85.0
2.640 .4216

1270 108.5 3.370
1280 108.3 3.364 .5269
1290 108.1 3.358 .5261
.5276
1670
1680
1690
84.I
84.6 2.628 .4196
84.3 2.619 .4181
2.613 .4171
1300 107.9 3.352
.5253 1700
83.9 2.606 .4160
1
1
II. COEFFICIENTS FOR THE CUBIC LAW
OF RESISTANCE.
SPHERICAL PROJECTILES.
Reports &c. p. 114.

K.
Kv
▼
Kv
Log
V
V Ky
K
K,
Log
g
80
go
go
f-s
f-s
850
138.4
4.299
.6334
1250 151.I
4.694
.6715
860
138.3
4.296
.6331
1260 150.5
4.674
.6697
870 138.3
4.294
.6329
1270
149.8
4.654
.6678
880 138.2
4.293
890 138.2 4.293
.6328 1280 149.1
.6328
4.632
.6659
1290
148.4
4.611
.6638
900 138.2
4.294
.6329 1300 147.8
4.591
.6619
910
138.3
4.296
.6331 1310 147.2
4.572
.6601
920 138.4
4.299
.6334
1320
146.5
4.552
.6582
930 138.5
4.302
.6337
1330
145.9
4.533
.6564
940
138.6
4.306
.634I
1340
145.3
4.514
.6546
950 138.8 4.312
.6347
1350
144.7
4.495
.6527
960 139.1 4.322
.6357
1360
144.I
4.475
.6508
970 139.5
4.334
.6369 1370
143.4
4.455
.6489
980 139.9
4.346
.6381
1380
142.7
4.433
.6467
990 140.4
4.362
.6397
1390 142.0
4.410 .6444
1000
141.I
4.383
.6418
1400 141.3
4.388
.6423
ΙΟΙΟ 141.9
4.408
.6442
1410 140.6
4.366
.6401
I020 142.8
1030 143.8 4.467
4.436
.6470 1420 139.8
4.343
.6378
.6500 1430 139.1
4.320
.6355
1070
1040 144.9 4.501
1050 146.1 4.539
1060 147.3
4.576
148.5 4.613
.6533
1440 138.4
4.299
.6334
.6570 1450
137.7
4.277
.6311
.6605
1460
137.0
4.254
.6288
.6640 1470
136.2
4.231 .6264
1080
149.6 4.647
.6672
1480 135.5
4.209
.6242
1090
150.6 4.677
.6700 1490 134.8
4.188
.6220
IIOO 151.4 4.703
.6724 1500 134.I
4.166
.6197
4.775
IIIO 152.I 4.725 .6744 1510 133.5
1120 152.7
4.744
1130 153.I 4.757
1140 153.4 4.766
1150
153.6 4.772
1160 153.7 4.774
1170
153.7
1180 153.7 4.774
1190 153.6 4.77I
1200 I53.4 4.765
1210 153.I
4.756
1220 152.7 4.744
1230 152.2 4.728
4.146
.6176
.6762 1520 132.8
4.125
.6154
.6773 1530 132.I
4.105
.6133
.6782 1540
131.5
4.085
.6112
.6787 1550
.6789 1560 130.1
130.8
4 064
.6090
4.043
.6067
.6790
.6789
1570 129.5
1580 128.8 4.003 .6024
.6786 1590 128.2 3.983 .6002
.6781
1600 127.5
3.961 .5978
.6772 1610 126.8 3.940 .5955
.6762 1620 126.2
3.920 .5933
4.023
.6046
.6747
1630 | 125.5
3.899 .5910
1240 151.7 4.712
1250 151.I 4.694
.6732
.6715 1650
1640 124.8
124.1
3.877
.5885
3.856
.5861
2

K,
K₁
K
Ky
V
K
V
Log
T
▼
K.
مع
Log
go
g
مع
f-s
f-s
1650
1660
I24.I
3.856
.5861
1900
108.7
3.377 .5285
123.5 3.836
.5839
1910 | 108.2
3.361
.5265
1670
122.8
3.815
.5815
1920 107.7
3.346
.5245
1680
122.I
3.793
.5790
1930
107.3
3.332
.5227
1690 121.4
3.772
.5766
1940 106.8
3.317
.5208
1700
120.8
3.752
.5743 1950 106.3
3.302
.5188
1710
120.I
3.73I
-5718
1960
105.8
3.287
.5168
1720
119.4
3.710
.5694
1970 105.3
3.272
.5148
1730 118.8
3.689
.5669
1980 104.9
3.258
.5130
1740
118.1
3.669
.5646
1990
104.4
3.243
.5110
1750 II7.4 3.648
.5621
2000 103.9
3.228
.5089
1760 116.8
3.628
.5597
2010
103.4
3.212
.5068
1770 116.1 3.608
.5573
2020
102.9
3.197
.5047
1780 115.5
3.588
.5549
2030
102.5
3.183
.5028
1790 114.9 3.568
.5524
2040
102.0
3.169
.5009
1800 I14.2
3.548
.5500
2050 101.5 3.154
.4989
1810 113.6
2060
IOI.I
3.529
.5477
3.140
.4969
1820 113.0 3.5II
.5454
2070
100.6
3.125
.4949
1830 112.5 3.494
.5433
2080 100.I
3.110
.4928
1840 III.9 3.476
1850 III.3 3.459
1860
II0.8
3.442
2090 99.7
.54II
3.096
•4908
-5390 2100 99.2
3.081
.4887
•5368
2110 98.7
3.066
.4866
1900
1870 I10.3 3.425
1880 109.7 3.408
1890 109.2
108.7
.5347 2120 98.3
3.053
.4847
.5325
2130 97.8
3.039 .4827
3.392
3.377 .5285
.5305
2140 97.4
3.025 .4807
2150 96.9
3.010 .4786
Log Po
I
C
1 2 3
0
9.94636
9.97579 32
0.00392 33
456
78 9
9.62777
23 0.13030
III. Log Pp = Log (3 tan & + tan ³4).


Log Po
9-
16
8.71909
9.02038 17
9.19691
78
18
9.32247 19 0.03093
9.42018 20 0.05695
9.50034 21 0.08212
333
9-
456
9.56844 22 0.10654 37
ww
78
9-
Log Po
Log Po
3I 0.30525 46
0.62501
0.32605 47
0.64839
0.34676 48
0.67226
34
0.36743 49
0.69666
35
0.38809 50
0.72164
36
0.40877 51
0.74725
0.42952 52
0.77355
38
0.45037 53
0.80059
9.68045 24 0.15349 39 0.47135 54
0.82844
ΙΟ 9.72792 25
0.17618 40 0.49250
+
13
14
II 9.77121 26
12 9.81109 27
9.84813 28
9.88280 29
15
9.91545
0.19844 4I 0.51385
0.22033 42
0.24191 43 0.55732 58 0.94937
0.26322 44
30 0.28432 45
0.53545
5555
55
0.85717
56
0.88685
57
0.91755
0.5795I 59
0.60206 60
0.98239
1.01671
3
1 *
IV. VALUES OF X, Y & T FOR INTERVALS OF 0°.2.

Y = 0.00
9-
y =
= 0,01
X
Y
T
9-
X
Y
T
0
1.73205 | 60.0 1.77949
1.71818 59.8 1.76457
1.70446 59.6 1.74984
1.69091 59.4 1.73530
1.67752 59.2 1.72096
1.66428 59.0 1.70679
1.55885
1.75552
1.53311
1.74112
1.50791
1.72691
60.0 1.73205 1.50000
59.8 1.71818 1.47606
59.6 1.70446 1.45259
59.4 1.69091 I.42959
59.2 1.67752 1.40703
59.0 1.66428 1.38492
58.8 1.65120 1.36322
58.6 1.63826 1.34195
58.4 1.62548 1.32109
58.2 1.61283 1.30062
58.0 1.60033 1.28054
57.8
1.58797 1.26083
57.6 1.57575 1.24149
57.4 1.56366 1.22251
57.2 1.55170 1.20388
57.0 1.53987 1.18559
56.8 1.52816 | 1.16764
56.6 1.51658 1.15001
56.4 1.50512 1.13270
56.2 1.49378 1.11569
56.0 1.48256 1.09899
55.8 1.47146 1.08259
55.6 1.46046 | 1.06648
55.4 1.44958 | 1.05065
55.2 1.43881 1.03509
55.0 1.42815 1.01980
54.8 1.41759 1.00478
54.6 1.40714 .99002
54.4 1.39679 .97550
54.2 1.38653 .96124
54.0 1.37638 .94722
53.8 1.36633 .93343
53.6 1.35637 .91987
53.4 1.34650
1.65120 58.8 1.69281
1.63826 58.6 1.67899
1.62548 58.4 1.66535
1 61283 58.2 1.65188
1.60033 58.0 1.63857
1.58797 57.8 1.62543
1.57575 57.6 1.61244
1.56366 57.4 1.59960
1.55170 57.2 1.58691
1.53987 57.9 1.57437
1.52816 56.8 1.56198
1.51658 56.6 1.54973
1.48323 1.71288
1.45906 | 1.69902
1.43539 1.68533
1.41221
1.67180
1.38949 1.65843
1.36723 | 1.64523
I.34542 1.63218
1.32404 1.61928
1.30308 1.60653
1.28253 1.59393
1.26238 1.58147
1.24262 1.56915
1.22324 1.55697
1.20423
I.I1440 1.48665
1.09742
I.54493
1.18558
1.53302
1.50512 56.4 1.53762 | 1.16728
1.52124
1.49378 56.2 1.52564
1.14932
1.50958
1.48256 56.0 1.51380
1.47146 55.8 1.50209
1.46046 55.6 1.49051
1.44958 55.4 1.47905
1.13170 1.49806
1.47537
1.08075
1.46420
1.43881 55.2 1.46772
1.06438
1.45315
1.42815 55.0 1.45651
1.41759 54.8|| 1.44541
1.04831
I.44222
1.03253
1.43140
1.40714 54.6 1.43444
1.39679 54.4 1.42357
1.38653 54.2 1.41282
1.01703
1.42068
1.00180
1.41008
.98684
1.39958
53.2 1.33673
53.0 1.32704
52.8 1.31745
52.6 1.30795
.85536
1.37638 54 1.40219
1.36633 53.8 1.39165
1.35637 53.6 1.38123
.90653 1.34650 53.4 1.37091
.89342 1.33673 53.2 I.
1.36069
.88052
1.32704 53.0 1.35058
.86783
1.34056
1.30795 52.6 1.33064
1.38919
.92956 1.35862
.91585 1.34863
.90238
1.33873
1.31745 52 8
.88914 1.32891
.87612
1.31921
52.4 1.29853
.84309 1.29853 52.4 1.32081
.86332
1.30959
52.2 1.28919
.83102 1.28919 52.2 1.31108
52.0 1.27994
.81913 1.27994 52.0 1.30144
51.8 1.27077
.80743
1.27077 51.8 1.29189
51.6 1.26169 .79592 1.26169 51.6 1.28243
51.4 1.25268 .78460
51.2 1.24375 .77346
.85073
1.30007
.83834 | 1.29063
.82616 1.28127
.81418
1.27200
1.25268 51.4 1.27306
.80240
1.26282
1.24375 51.2 1.26378
.79081
1.25371
.97214
•95770 1.37890
.9435I 1.36871
(
4

y = 0.02
Y = 0.03
❤
X
Y
T
9-
X
Y
T
60.0 1.83254| 1.62567
59.8 1.81636
59.6 1.80041
59.4 1.78468
59.2 1.76919| 1.51766
59.0 1.75391 1.49213
58.8 1.73885 1.46716
58.6 1.72399 1.44273
58.4 1.70934
1.41882
58.2 1.69488 I.39541
58.0 1.68062
1.37250
57.8 1.66655 1.35007
1.32810
57.6 1.65266
57.4 1.63896| 1.30658
57.2 1.62542 1.28550
57.0 1.61206 1.26485
56.8 1.59887 1.24461
56.6 1.58584 1.22478
1.59774
I.57045
1.54376
1.11367
54.8 1.47542 1.06276
56.4 1.57298 I.20534
56.2 1.56027 1.18628
56.0 1.54771
1.16759
55.8 1.53530 1.14927
55.6 1.52304 1.13130
55.4 1.51093
55.2 1.49895 1.09638 | 1.46841 55.2 1.53291
55.0 1.48712 1.07941 1.45718 55.0 1.52035
1.06276
1.78117 60.0 1.89266
1.76618 59.8 1.87490
1.75139 59.6| 1.85743
1.73680 59.4 1.84025
1.72239 59.2 1.82334
1.70817 59.0 1.80671
1.69413 58.8 1.79033
1.68027 58.6|| 1.77421
1.66658 58.4 1.75833
1.65306 58.2 1.74270
1.63971 58.0 1.72729
1.62652 57.8 1.71211
1.61349 57.6 1.69716
1.60062 57.4 1.68241
1.58790 57.2 1.66788
1.57533 57.0 1.65355
1.56290 56.8 1.63941
1.55062 56.6| 1.62547
1.53847 56.4 1.61172
1.52647 56.2 1.59815
1.70264 1.80950
1.67200
1.79380
1.64211
1.77832
1.61294 | 1.76306
1.58447 1.74802
1.55667 1.73318
1.71854
1.52952
1.50301 1.70410
1.47710 | 1.68985
1.45178 1.67579
1.42703 1.66191
1.40283 | 1.64822
1.37917
1.35603
1.63469
1.62134
1.33339 1.60815
1.31123 1.59513
1.28955 1.58227
1.26833
1.56956
I.24755
1.55701
1.22721
1.54461
1.51460 56.0 1.58476
1.20728
1.53235
1.50286 55.8 1.57154
1.18776
1.52023
1.49125 55.6 1.55850
1.16864
1.50826
1.47977 55.4| 1.54562
I.14990
I.49642
1.13154 1.48472
I.I1354
1.47315
1.44607 54.8 1.50795
1.09589
1.46171
54.6 1.46385| 1.04642
1.43507 54.6 1.49570 1.07859
1.45039
54.4 1.45241 1.03038
1.42419 54.4 1.48360 | 1.06163
I.43920
54.2|| 1.44109
.44109| 1.01464
1.41342 54.2 1.47164
I.04499
1.42813
54.0 1.42991
53.8 1.41884 .98401
53.6 1.40789 .96911
53.4 1.39706 .95447
53.2 1.38635 .94010
.92598
53.0 1.37575
52.8 1.36526 .91211
.99918
1.40276 54.0 1.45983
1.39221 53.8 1.44815
1.38177 53.6 1.43661
1.02867
1.41718
1,01266 1.40634
.99695 1.39562
1.37143
1.36120 53.2 1.41393
53.4 1.42521
.98153
1.38501
.96640
I.37452
1.35107 53.0 1.40278
.95155
1.36413
52.6 1.35488
52.4 1.34460
1.34104 52.8|| 1.39175
.89848 1.33111 52.6 1.38085
.88509
1.32127 52.4 1.37007
.93697
1.35385
.92266 1.34367
.90860
I.33359
52.2 1.33442
.87192 1.31152 52.2 1.35940
.89480
1.32361
52.0 1.32435
.85898
51.8 1.31438 .84627
51.6 1.30451 .83378 1.28284 51.6 1.32808
.82149 1.27346 51.4 1.31786
.80941 1.26416 51.2 1.30774
1.30187 52.0 1.34885
88125
1.31372
1.29231 51.8 1.33841
.86794
1.30393
.85486
I.29424
51.4 1.29474
51.2 1.28506
51.0 1.27548 .79753 1.25494 51.0 1.29773
50.8 1.26598 .78585 1.24581 50.8 1.28782
50.6 1.25658 .77436 1.23676 50.6 1.27801
.79278 1.24718
.84201
1.28465
.82938 1.27515
.81697 1.26574
.80477 1.25642
5

Y = 0.04
Y = 0.05
9-
X
Y
T
Ф X
Y
T
60.0 1.96194 1.79302
59.8 1.94216 1.75889
59.6 1.92275 1.72568
59.4 1.90372 1.69336
59.2 1.88503 1.66189
59.0 1.86669 1.63124
58.8 1.84867 1.60138
58.6 1.83097 1.57227
58.4 1.81358 1.54388
58.2 1.79648 1.51620
58.0 1.77967 1.48920
57.8 1.76314 1.46284
57.6 1.74688 1.43712
57.4 1.73088 1.41201
57.2 1.71513 1.38748
57.0 1.69963| 1.36351
56.8 1.68437 1.34010
56.6 1.66933 1.31721
56.4 1.65453 1.29484
56.2 1.63994 1.27297
56.0 1.62556 1.25157
55.8 1.61139 1.23064
55.6 1.59742 1.21017
55.4 1.58365 1.19013
55.2 1.57007 1.17051
55.0 1.55667 1.15130
54.8 1.54346 1.13250
54.6 1.53042
I.I1408
54.4 1.51755 1.09604
54.2 1.50485 | 1.07837
1.07837
54.0 1.49231
1.06105
53.8 1.47994 1.04408
53.6 1.46772 I.02744
53.4 1.45565 1.01113
53.2 1.44373 .99514
53.0 1.43195 .97945
52.8 1.42032 .96406
52.6 1.40882 .94897
52.4 1.39746 .93417
52.2 1.38624 .91965
52.0 1.37514 .90540
51.8 1.36417 .89141
51.6 1.35332 .87768
51.4 1.34260 .86420
51.2 1.33199 .85095
1.84116 60.0 2.04361
1.82459 59.8 2.02112
1.80828 59.6 1.99913
1.79222 | 59.4 1.97764
1.77640 59.2 1.95662
1.76082 59.0 1.93605
1.74546 58.8 1.91591
1.73033 58.6 1.89618
58.6 1.89618
1.71542 58.4 1.87685
1.70072 58.2 1.85790
1.68623 58.0 1.83932
1.67193 57.8 1.82108
1.65783 57.6 1.80319
1.64392 57.4 1.78563
1.63020 57.2 1.76838
1.61666 57.0 1.75143
1.60329 56.8 1.73478
1.59010 56.6 1.71842
1.57707 56.4 1.70233
1.56421 56.2 1.68650
1.55151 56.0 1.67093
1.90178
1.90178 1.87718
1.86297 1.85951
1.82535 1.84215
1.78886
1.78886
1.82509
1.75346
1.80831
1.71908 1.79181
1.77557
1.68569
1.65325 1.75960
1.62170 1.74388
|
1.59102 1.72840
1.56116 1.71316
1.53210
1.69814
1.50379 1.68336
1.47622
1.47622 | 1.66878
1.44935 1.65442
1.42316 1.64026
1.39762
1.39762 | 1.62630
1.37270 1.61253
1.34839 1.59895
1.32466
1.32466
1.58556
1.30150
1.57234
1.53897 55.8 1.65562
1.52657 55.6 1.64054
1.51433 55.4 1.62570
1.27887
1.55930
1.25677
1.54643
1.23517
1.53372
1.50224 55.2 1.61108
1.49029 55.0 1.59669
55.0 1.59669
1.47847 54.8 1.58251
1.46680 54.6 1.56854
1.45526 54.4 1.55477
1.44385 | 54.2 1.54120
1.21407
1.52117
1.19344
1.50878
1.17326
1.49655
1.15353 1.48446
1.43256 54.0 1.52782
1.42141 53.8 1.51463
1.41038 53.6 1.50162
1.39946 53.4 1.48878
1.38867 53.2 1.47612
1.37799 53.0|| 1.46362
1.36743 52.8 1.45129
1.35698 52.6 1.43911
1.34664 52.4 1.42709
1.33640 52.2 1.41522
1.32626 52.0 1.40351
1.31623 51.8 1.39194
1.30630 51.6 1.38051
1.29647 51.4 1.36921
1.28674
I.13423 I.47253
1.11534 1.46073
1.09686
1.09686 | 1.44908
1.07877 1.43756
1.06105
1.42617
1.41492
1.40380
1.04370
1.02671
1.01007 1.39280
.99376 1.38192
.97778 1.37116
.96212 1.36052
.94676 I.34999
.93171 1.33958
.91695 1.32928
.90247 1.31909
.88827 1.30901
1.28674 51.2 1.35804
.87434 1.29903
1.27711 51.0 1.34701
.86067 1.28914
1.26757 50.8 1.33611
.84726 1.27935
51.0 1.32150
.83795
50.8 1.31112 .82518
50.6 1.30085 .81264 1.25812 50.6 1.32533
.83410 1.26966
6

Y = 0.06
Y = 0.07
9-
X
Y
T
9-
X
Y
T
60.0 2.14308 2.03739
59.8 2.11669 1.99187
59.6 2.09106 1.94799
59.4 2.06613 1.90568
59.2 2.04187 1.86482
59.0 2.01825
1.82534
58.8 1.99522
1.78717
58.6 1.97276 1.75023
58.4 1.95084 1.71445
58.2 1.92943
1.67979
58.0 1.90851 1.64619
1.91913 60.0 2.27039
1.89999 59.8 2.23783
1.88124 59.6 2.20652
1.86286 59.4 2.17634
1.84484 59.2 2.14722
1.82716 59.0 2.11908
1.80980 58.8 2.09184
1.8098058.8
1.79276 58.6 2.06545
1.77601 58.4 2.03986
2.21597
1.96977
2.15980
1.94851
2.10621
1.92779
2.05498 1.90757
2.00594
1.95891
1.88782
1.86852
1.91376 1.84964
1.87036 1.83117
|
1.82859 1.81308
1.75956 58.2 2.01501
1.78835
1.79535
1.74339 58.0 1.99085
1.74955
1.77798
57.8 1.88806
57.6 1.86806
57.4 1.84848
57.2 1.82931 1.52135
57.0 1.81053 1.49233
56.8 1.79213
1.46410
56.6 1.77409
1.43663
56.4 1.75639 I.40990
56.2 1.73903
1.38386
56.0 1.72199 1.35850
55.8 1.70526 I.33379
55.6 1.68883 1.30970
55.4 1.67268 1.28621
1.61359
1.72749 57.8 1.96736
1.71209
1.76093
1.58194
1.71186 57.6 1.94448 | 1.67591
1.74421
1.55121
1.69647 57.4 1.92220 1.64092
1.72780
1.68133 57.2 1.90046
1.60707
1.71167
1.66642 57.0 1.87926
1.57430
1.69584
1.65175 56.8 1.85856
1.54254
1.68027
1.63729 56.6 1.83833
1.51175 1.66497
1.62306 56.4 1.81857
1.48188 1.64991
1.60902 56.2 1.79923
1.45289 1.63511
1.59520 56.0 1.78031
1.42473
1.62054
1.58156 55.8 1.76179 | 1.39738
1.56812 55.6 1.74365 1.37078
1.55487 55.4 1.72587
1.60620
1.59207
I.34491
1.57816
55.2 1.65682 1.26330 | 1.54180 | 55.2 1.70844
|
1.31974 1.56446
54.8 1.62588
55.0 1.64122 I.24094
1.21911
1.52890 55.0 1.69135
1.29524 1.55096
1.51617 54.8 1.67458
1.27138
1.53766
54.6 1.61079 1.19780
54.4 1.59595 1.17699
54.2 1.58134 1.15666
1.5036154.6 1.65812 | 1.24813
1.52454
1.49122 54.4 1.64196 1.22548
1.51161
1.47898 54.2 1.62609
1.20339
1.49885
54.0 1.56696
1.13679
1.46690 54.0 1.61049
1.18185 1.48627
53.8 1.55280
1.11738
1.11738
1.45496 53.8 1.59517
1.16083
1.47386
53.6 |
1.53885 1.09839
1.44318
53.6 1.58010
1.14032 1.46161
53.4 1.52512 1.07983
53.2 1.51158 | 1.06167
1.43154
1.4315453.4 I.
53.4 1.56529
1.12030 I.44952
53.0 1.49824 1.04390
52.8 1.48509 1.02652
52.6 1.47213 1.00950
52.4 1.45935
.99284
52.2 1.44674 .97653
52.0 1.43431 .96056
51.8 1.42204 .94491
51.6 1.40993 .92958
51.4 1.39798 .91455
51.2 1.38619 .89983
51.0 1.37454
50.8 1.36304
50.6 1.35168
1.42004 53.2 1.55071 1.10075 1.43758
1.40867 53.0 1.53638 1.08165
|
1.42580
1.39744 52.8 1.52227 1.06299 1.41417
1.38634 52.6 1.50838 1.04475
1.40268
1.37537 52.4 1.49470
1.36452 52.2 1.48122
1.35379 52.0 1.46794
1.3431851.8 1.45486
1.33269 51.6 1.44197
1.32232 51.4 1.42926
1.31206 51.2 1.41674
1.02691 1.39133
1.00948 1.38011
.99244 1.36902
-97577 1.35807
.95945 1.34725
.94348 1.33655
.92784
1.32597
.88540 1.30191 51.0 1.40439
.87125 1.29186 50.8 1.39220
.85737 | 1.28192 50.6 1.38018
.91252
1.31552
.89752
1.30519
.88284 1.29497
7

y = 0.08
y = 0.09
X
Y
T
9.
X
Y
T
60.0 2.44854 2.47491
59.8 2.40421 2.39843
59.6 2.36247 2.32699
59.4 2.32298 2.25995
59.2 2.28550 2.19682
59.0 2.24980 2.13716
58.8 2.21570
2.21570 2.08063
2.08063
58.6 2.18304 2.02693
|
58.4 2.15171 1.97579
2.03488 59.0 2.43774
2.01008 58.8 2.38974
1.98615 58.6 2.34500
1.96302 58.4 2.30304
1.94061 58.2 2.26349
2.40308
2.40308
2.32351 1.9597I
2.24992 1.93565
1.98477
2.18144
1.91249
2.11740
1.89013
1.91887 58.0 2.22605
2.05726
1.86850
1.89775 57.8 2.19049
2.00056
1.84753
1.87720 57.6 2.156591.94694
1.82718
1.85719 57.4 2.12420
1.89609
1.80738
1.92700
1.83767 57.2 2.09316
1.84775
1.78812
1.88037
1.81862 57.0 2.06336
1.80168
1.76934
1.83573
1.75770 1.75102
58.2 2.12158
58.02.09255
57.8 2.06455
57.6 2.03749 1.79292
57.4 2.011ZI 1.75182
57.2 1.98594 1.71231
57.0 1.96134 1.67428
56.8 1.93745 1.63764
56.6 1.91424 1.60230
56.4 1.89166 1.56818
56.2 1.86968 1.53522
56.0 1.84826 1.50334
1.80002 56.8 2.03470
1.78183 56.6 2.00707
1.76404 56.4 1.98040
1.74662 56.2 1.95462
1.73314
1.71564
1.67535 1.71566
1.63670 | 1.69856
1.68182
1.72956 56.0 1.92967 | 1.59956
1.7295656.0 |
1.71284 55.8 1.90548 1.56384 | 1.66544
1.69644 55.6 1.88202 I.52944 1.64938
1.68036 55.4 1.85923
1.66457 55.2 1.83707
1.64906 55.0 1.81551
1.49628 1.63363
1.46428 1.61818
1.43338 1.60302
1.63383 54.8 1.79451 1.40350
1.61887 54.6| 1.77405 1.37460
1.60415 54.4 1.75409 | 1.34661
1.58968 54.2 1.73461 1.31950
1.57545 54.0 1.71558 1.29321
1.56144 53.8 1.69698 1.26771
1.54765 53.6 1.67880
1.53407 53.4 1.66101
1.52069 53.2 1.64359
1.50752 53.0 1.62653
1.49454 52.8 1.60982
1.48174 52.6 1.59344
||
1.46912 52.4 1.57737
55.8 1.82737 I.47249
55.6 1.80699 | 1.44261
55.4 1.78709 1.41366
55.2 1.76765 1.38559
55.0 1.74864 1.35834
54.8 1.73006 1.33189
54.6 1.71187 1.30620
54.4 1.69406 1.28123
54.2 1.67661 | 1.25695
54.0 1.65951 1.23333
53.8 1.64275 1.21034
53.6 1.62630 1.18796
53.4 1.61017 1.16615
53.2 1.59433 I.14491
53.0 1.57878 1.12419
52.8 1.56351 1.10399 1.43230 51.8 1.53095
52.6 1.54850 1.08429 1.42036 51.6 1.51603
52.4 1.53375 1.06507 1.40858 51.4 1.50137
52.2 1.51924 1.04630
1.04630
52.0 1.50497 1.02797
51.8 1.49094 1.01007 1.37410 50.8 1.45887
51.6 1.47713 .99258
51.4 1.46354 .97549
51.2 1.45015 .95878
51.0 1.43696 .94244
50.8 1.42397 .92646
50.6 1.41118 .91083
1.58813
1.57350
1.55913
1.54500
1.53110
|
1.51742
I.24295
1.50397
1.21891
1.49072
1.19554 1.47767
1.17283 1.46482
1.15073
1.45216
1.12922
1.43968
1.10828
1.42738
1.45668 52.2 1.56161
1.08789
1.41525
1.44441 52.0 1.54614
1.06802
1.40328
1.04865
1.39147
1.02976 | 1.37982
1.01133
1.36834
1.39694 | 51.2|| 1.48696
.99335
1.35700
1.38545 51.0 1.47280
.97579
1.34581
.95865
1.33476
.94191
1.32385
1.36290 50.6 1.44517
1.35183 50.4 1.43169
1.34090 50.2 1.41842
1.33010 50.0 1.40535
1.31943 49.8 1.39248
1.30888 49.6 1.37980
.92556 1.31307
.90957 I.30241
.89393 1.29186
.87865 1.28144
.86370 1.27115

8

Y = 0.10
=
Y = 0.II
9-
X
Y
T
58.0 2.42326 2.32865
57.8 2.37144 2.24604
57.6 2.32368 2.17049
57.4 2.27931 2.10084
57.2 2.23780 2.03618
57.0 2.19876 1.97583
56.8 2.16188
1.91925
56.6 2.12689| 1.86598
56.4 2.09358 1.81566
56.2 2.06179 1.76799
56.0 2.03136 1.72270
55.8 2.00217 1.67958
55.6 1.97411 1.63844
55.4 1.94708 I.59911
55.2 1.92100 1.56146
55.0 1.89581 1.52534
54.8 1.87143 | 1.49066
54.6 1.84782 1.45730
54.4 1.82491 1.42519
54.2 1.80268
1.39424
54.0 1.78106 1.36438
53.8 1.76003 I.33555
53.6 1.73956 1.30768
53.4 1.71961 1.28071
53.2 1.70015 1.25461
53.0 1.68116
1.22932
52.8 1.66262 1.20480
52.6 1.64450 1.18101
52.4 1.62678 I.15792
52.2 1.60945 1.13550
52.0 1.59249 I.I1371
51.8 1.57587 1.09252
51.6 1.55959 1.07191
51.4 1.54363 1.05185
51.2 1.52798 1.03232
51.0 1.51262 1.01330
50.8 1.49755 .99476
50.6 1.48276 .97669
50.4 1.46823 .95906
50.2 1.45396 •94186
50.0 1.43993 .92508
49.8 1.42613 .90870
49.6 1.41256 .89270
49.4 1.39921 .87707
49.2 1.38607 .86180
49.0 1.37315 .84687
X
Y
T
2.25049 1.88501
2.16518
1.85950
2.08810
1.83528
1.81218
1.93480 57.0 2.40402
1.9348057.0
1.90959 56.8 2.34841
1.88543 56.6| 2.29779
1.86226 56.4 2.25119
1.83998 56.2 2.20795
1.81849 56.0 2.16753
1.79772 55.8 2.12955
1.77759 55.6 2.09367
1.75805 55.4 2.05966
1.73906 55.2 2.02730
1.72059 55.0 1.99642
55.0 1.99642
1.70258 54.8 1.96687
2.01771
1.95286 I.79004
1.89271 1.76874
|
1.83660 1.74820
1.78402 1.72835
1.73453 1.70911
1.68779
1.69044
1.64352
1.67229
1.60147 1.65463
1.68502 54.6 1.93852 | 1.56144 | 1.63742
1.66787 54.4 1.91128
1.65111 54.2 1.88504
1.63472 54.0 1.85974
1.61868 53.8 1.83529
1.60297 53.6 1.81163
1.58757 53.4 1.78872
1.52324 1.62062
1.48673 1.60422
1.45177
1.41824 1.57252
1.58819
1.38603
1.55717
1.35506
1.54213
1.57247 53.2 1.76649
1.32525
1.52739
1.55766 53.0 1.74491
1.29650
1.51294
1.54312 52.8 1.72394
1.52884 52.6 1.70353
1.51481 52.4 1.68367 1.21609
1.26877 1.49875
1.24199
1.48482
|
1.47114
1.50102 52.2 1.66431
1.48746 52.0 1.64542
1.47412 51.8 1.62699
1.46100 51.6 1.60900
1.44808 51.4| 1.59141
1.43536 51.2 1.57422
1.42283 51.0 1.55739
1.41049 50.8 1.54092
1.39832 50.6 1.52479 1.01610
1.19104
1.16679 | 1.44448
1.14328
1.45770
1.43149
1.41870
1.03580
1.36949
1.35765
1.38633 50.4 1.50898
I.37451 50.2 1.49349
1.36286 50.0 1.47829
.99692
1.34597
.97826
I.33445
.96008
1.32310
1.35136 49.8 1.46338
94238
1.31190
.92512
1.30085
.90829
1.28994
.89188 1.27917
1.34001 49.6 1.44875
1.32881 49.4 1.43438
1.31776 49.2| 1.42026
1.30685 49.0 1.40639
1.29608 48.8 1.39275
1.28544 48.6 1.37934
1.27493 48.4 1.36616
1.26454 48.2 1.35319
1.25428 48.0 1.34042
1.12050
1.09839 1.40611
1.07692 | 1.39372
1.05607 1.38151
•
.87587 | 1.26855
.86024 1.25806
.84498 1.24770
.83007 1.23746
.81551 1.22734
.80128 1.21734
.78737 1.20746
48.8 1.36042 .83227 1.24414 47.8 1.32785
48.6 1.34788 .81800 1.23412 47.6 1.31547 -77377
1.19769
9
Y = 0.12
Y = 0.13
9-
X
Y
T
56.0 2.37888 2.16734
55.8 2.31979 2.08005
55.6 2.26665 2.00215
55.4 2.21820
1.93166
55.2 2.17357 1.86719
55.0 2.13210 1.80775
54.8 2.09331 1.75256
54.6 2.05683 1.70103
54.4 2.02237 1.65271
54.2 1.98967 1.60720
54.0 1.95854 | 1.56419
53.8 1.92882 1.52344
53.6 1.90037 1.48471
53.4 1.87307 1.44781
53.2 1.84682
1.41260
53.0 1.82153 1.37892
52.8 1.79714 1.34666
52.6 1.77356 1.31570
52.4 1.75074 1.28596
52.2 1.72862 1.25735
52.0 1.70717 1.22979
51.8 1.68634 1.20322
51.6 1.66608 I.17757
51.4 1.64637 1.15279
51.2 1.62717 1.12883
51.0 1.60846 | 1.10564
50.8 1.59021 1.08318
50.6 1.57239 1.06142
50.4 1.55499 1.04030
50.2 1.53798 1.01981
50.0 1.52134 .99992
49.8 1.50506 .98058
49.6 1.48912 .96179
0
X
1.83487 55.0 2.34664
1.80925 54.8 2.28474
1.78508 54.6 2.22971
Y
T
2.07817 1.78424
I.99009
1.75868
1.91236
1.73470
1.76213 54.4 2.17996
1.84260
1.71201
1.74020 54.2|| 2.13441
1.71917 54.0|| 2.09231
1.69894 53.8 2.05309
1.67941 53.6 2.01633
1.66052 53.4 1.98169
I.77922
1.69040
1.72105 1.66973
1.66727 1.64987
1.61722
1.63074
1.57041
1.61225
1.64221 53.2 1.94891
1.52643
I.59435
1.62444 53.0 1.91777
1.48495
1.57699
1.60715 52.8 1.88809
1.44571
1.56011
1.59032 52.6 1.85972 | 1.40846
1.54369
1.57390 52.4 1.83253 1.37303
1.52769
1.55789 52.2 1.80642
1.33925
1.51208
1.54224 52.0 1.78130
1.30698
1.30698
1.49683
1.52694 51.8 1.75708
1.27609 1.48193
1.51197 51.6 1.73370
1.24648
1.46735
I.4973I 51.4 1.71108
1.21805
1.45308
1.48294 51.2 1.68918
1.19071
I.43909
1.46886 51.0 1.66795
1.16439
1.42538
1.45504 50.8 1.64734
1.13903
1.41194
1.44147 50.6 1.62731
I.I1457
1.39874
1.42814 50.4 1.60783
1.41505 50.2 1.58887
1.40218 50.0 1.57040
1.38953 49.8 1.55238
1.09094
1.38577
1.06810
I.37304
1.04600 1.36052
1.02460
1.34821
1.37707 49.6 1.53480 | 1.00387
1.00387
1.33609
1.36482 49.4 1.51763
1.35276 49.2 1.50085
.98377
1.32417
.96426 1.31244
1.34088 49.0 1.48445
1.32917 48.8 1.46840
1.31764 48.6 1.45269
1.30627 48.4 1.43731
1.29507 48.2 1.42223
1.28401 48.0 1.40745
.94533 1.30088
.92693 1.28950
.90905 1.27828
49.4 1.4735I .9435I
49.2 1.45820 .9257I
49.0 1.44320 .90839
48.8 1.42848 .89152
48.6 1.41403 .87508 1.26234 47.6 1.37873
48.4 1.39985 .85905 1.25172 47.4 1.36477
48.2 1.38593 .84342 1.24124 47.2 1.35106
48.0 1.37225 .82817
1.23090 47.0 1.33759
47.8 1.35880 .81329
47.6 1.34558 .79876
1.21060 46.6 1.31134
47.4 1.33258 .78457 1.20063 46.4 1.29855
47.2 1.31979 .77071 1.19077 46.2 1.28597
47.0 1.30720 .75716
1.18103 46.0 1.27358
46.8 1.29481 .74392 1.17140 45.8 1.26139
46.6 1.28261
46.6
•73097
1.16188 45.6 1.24939
.89166
1.26722
.87473
1.25631
.85826
1.24556
1.27310 47.8 1.39295
.84222
I.23495
.82659 1.22448
.81136
.79650 1.20395
1.21415
.78200
1.22069 46.8 1.32435 -76785
.75404 I.17409
•74056 1.16438
.72739
1.15478
1.19387
1.18392

.71452
1.14530
.70194 I.13593
.68964 1.12666
10
Y = 0.14
y = 0.15

X
Y
T
54.0 2.30632 | 1.98239
53.8 2.24268 1.89511
53.6 2.18662
1.81878
53.4 2.13627 1.75073
53.2 2.09040 1.68919
53.0 2.04816
9
0
X
1.73297 53.0 2.25742
1.70768 52.8 2.19343
1.6840552.6 2.13738
1.66177 52.4 2.09724
1.64059 52.2 2.04169
1.62037 52.0 1.99983
Y
T
1.68106
1.88017
1.79555 1.65629
1.72197 1.63321
1.65662 | 1.61148
1.59768 | 1.59086
I.54391 1.57118
1.63293
52.8 2.00893 | 1.58106 | 1.60097 | 51.8 1.96103
52.6 1.97224 1.53290 1.58230 51.6 1.92480
52.4 1.93775 1.48795
I.49442
1.55232
1.44854
1.53417
1.56427 51.4 1.89076 1.40575
1.51667
52.2 1.90516| 1.44578
52.0 1.87425 1.40607
1.54683| 51.2 1.85864 | 1.36565
I.49973
1.52992 51.0 1.82819 1.32791
1.48331
51.8 1.84482 | 1.36853
1.51350 50.8 1.79922
1.29226
1.46737
I.12545
51.6 1.81672 1.33295
51.4 1.78982 1.29913
51.2 1.76400 1.26691
51.0 1.73918 1.23615
50.8 1.71527 1.20672
50.6 1.69219 1.17852
50.4 1.66988 1.15146
50.2 1.64829
50.0 1.62737 1.10043
49.8 1.60706 1.07632
1.45195 50.0 1.69536
50.0 1.69536
1.43746 49.8 1.67187
1.42329 49.6 1.64921
1.49752 50.6 1.77157
1.25849
1.45186
1.48195 50.4 1.74512
1.46677 50.2 1.71975
1.22640
1.43675
I.19584
1.42202
1.16667
1.40764
1.13877
1.39358
1.11205
1.37983
1.40941 49.4 1.62731
1.08641
1.36636
1.39582 49.2 1.60612
1.06177
1.35317
1.38250 49.0 1.58558
1.03806
I.34024
1.36943 48.8 1.56566
1.01522
1.32756
49.6 1.58734 | 1.05306
1.35660 48.6 1.54631
.99320 1.31511
49.4 1.56816 1.03060
1.34400 48.4 1.52750
.97194
1.30288
49.0 1.53132
49.2 1.54950 | 1.00890 | 1.3316348.2 1.50919
.98792 1.31946 48.0 1.49136
.95139
1.29086
.93151
1.27905
48.8 1.51359
.96760
1.30749 47.8 1.47397
.91228
1.26743
48.6 1.49630 .9479I
48.4 1.47941 .92883
48.2 1.46292 .91031
1.29572 47.6 1.45702
1.28414 47.4 1.44046
|
.89364 1.25599
.87557
1.24474
1.27273 47.2 1.42428
.85804
1.23366
48.0 1.44679 .89233
47.8 1.43101 .87487 1.25042 46.8 1.39300
1.26149 47.0 1.40847
.84102
I.22275
.82449
1.21199
47.6 1.41557
.85790 1.23952 46.6 1.37786
.80843
1.20139
47.4 1.40045 .84140 1.22876 46.4 1.36303
47.2 1.38563 .82534
1.21816 46.2 1.34851
47.0 1.371II
.80971 1.20770 46.0 1.33427
46.8 1.35688 .79449 1.19738 45.8 1.32030
46.6 1.34292 .77966 1.18719 45.6 1.30660
46.4 1.32920 .76520
1.17713 45.4 1.29315
46.2 1.31572 .75110
46.0 1.30247 •73736
45.8 1.28945 .72394
45.6 1.27666 .71084
45.4 1.26409 .69805
45.2 1.25173 .68556
1.16720 45.2 1.27995
1.15741 45.0 1.26697
1.14774 44.8 1.25421
1.13818 44.6 1.24167
1.12873 44.4 1.22935
I.I1939 44.2 1.21723
.79280
1.19094
.77760 1.18064
.76281 I.17047
.74839 1.16044
•73435
.72067
1.15053
1.14076
.70732 1.13110
.69430 1.12157
.68159
1.11216
.66918 1.10286
.65707 1.09366
.64524 1.08457
45.0 1.23959
.67336 I.I1015 44.0 1.20531
.63368
1.07558
44.8 1.22765
44.6 1.21588
.66143 I.IO102 43.8 1.19358
.64976 1.09200 43.6 1.18203
.62238
1.06669
.61133
1.05790
11
Y = 0.16
Y = 0.17
9-
X
Y
T
9-
X
Y
T

52.0 2.20019
1.77260
51.8 2.13736 | 1.69247
1.69247
51.6 2.08241 1.62287
1.62287
51.4 2.03328 1.56111
51.2 1.98869 1.50545
51.0 1.94773 I.45469
1.62859 51.0 2.13564
1.60460 50.8 2.07535
1.58225 50.6 2.02246
1.56122 50.4 1.97508
1.54127 50.2 1.9320I
1.52223 50.0 1.89240
50.8 1.90978 1.40798
50.6 1.87434| 1.36468
50.4 1.84107 1.32431
50.2 1.80966 1.28648
50.0 1.77989
49.8 1.75158 1.21725
1.50398 49.8 1.85566
1.48642 49.6 1.82133
1.46947 49.4 1.78907
1.66158 1.57577
1.58739 1.55277
1.52276 1.53132
1.46529 1.5IIII
1.41340 1.49191
1.36603| 1.47358
1.32239 1.45600
1.28191 I.43908
1.24414 I.42274
1.45308 49.2 1.75860
1.20872 1.40692
1.25088
1.43719 49.0 1.72971I
1.17536
1.39159
1.42176 48.8 1.70222
1.14385
1.37668
49.6 1.72456
1.18539
1.18539
1.40674 48.6 1.67597
I.11397
1.36218
49.4 1.69871
1.15512
1.39211 48.4 1.65085 1.08557
1.34805
49.2 1.67391
1.12629
1.12629 | 1.3778548.2 1.62674 | 1.05852
1.33426
49.0 1.65007
48.8 1.62712 1.07246
48.6 1.60497 1.04725
48.4 1.58356 1.02305
48.2 1.56285 .9998 I
48.0 1.54278
47.8 1.52331 .95588
47.6 1.50439 .93510
47.4 1.48600 .91503
47.2 1.46810
.89563
47.0 1.45067 .87687
46.8 1.43367
1.09878
1.09878
1.36392 48.0 1.60356 | 1.03268
1.32079
1.35030 47.8 1.58123
1.00797
1.30762
1.33698 47.6 1.55968
.98428 1.29473
.97743
1.32393 47.4 1.53885
1.31115 47.2 1.51868
1.29862 47.0 1.49914
46.6 1.41709
•
46.4 1.40090
1.28632 46.8 1.48017
1.27424 46.6 1.46175
1.26238 46.4 1.44383
1.25073 46.2 1.42639
1.23927 46.0 1.40939
.85871 1.22800 45.8 1.39282
.84111 1.21690 | 45.6
.82405
.96155 1.28211
.93970 1.26974
.91866 1.25760
.89840 I.24570
.87884
.85996
I.23400
1.22251
.84171 1.21122
.82405
.80695 1.18918
1.2001I
1.37665
.79038 1.17843
46.2 1.38508
.80749
1.20598 45.4 1.36086
1.19523 45.2|| 1.34543
.7743I 1.16783
.75872 1.15740
46.0 1.36962
.79142
45.8 1.35449 .77581
1.18463 45.0 1.33034
1.17419 44.8 1.31558
45.6 1.33968 .76064 | 1.16389 44.6 1.30113
45.4 1.32518 .74588 I.15374 44.4 1.28698
45.2 1.31097 .73152
45.0 1.29703 .71754
44.8 1.28335 .70392
44.6 1.26993 .69065
44.4 1.25677 .67772
44.2 1.24386 .665II
44.0 1.23118 .65280
43.8 1.21872
I.14373 44.2 1.2731I
1.13385 44.0 1.25950
1.12410 43.8| 1.24616
I.I1447 43.6 1.23307
1.10497 43.4 1.22021
1.09559 43.2 1.20758
1.08632 43.0 1.19518
.64079 | 1.07716||42.8 1.18299
43.6 1.20646
.62907 1.06811 42.6 1.17100
43.4 1.19439 .61763 1.05916 42.4 1.15920
43.2 1.18251 .60644 1.05031 42.2 1.14760
43.0 1.17082 .5955I 1.04157 42.0 1.13618
42.8 1.15931 .58485 1.03292 41.8 1.12494
42.6 1.14798 .57439 1.02437 41.6| 1.11387
.74358 1.14712
.72887 1.13698
.71457 1.12699
.70066 1.11714
.68712 I.10741
.67394 1.09782
.66110
1.08835
.64858
1.07900
.63638
1.06976
.62449 1.06064
.61288 1.05163
.60155
1.04272

.59048 I.03392
-57968 1.02522
.56912
1.01661
.55880 1.00811
.54872
.99969
.53885
.99136
12

Y = 0.18
Y = 0.19
9-
X
Y
T
❤
X
Y
T
50.0 2.06533| 1.54956
49.8 2.00860 1.48218
49.6 1.95852
1.42312
49.4 1.91346 1.37036
49.2 1.87235 1.32256
49.0 1.83445 1.27881
1.52292 49.0 1.99111
1.50107 48.8 1.93855
1.48063 48.6 1.89175
1.46132 | 48.4 1.84937
1.44295 48.2 1.81053
1.42538 48.0 1.77459
1.43884
1.47035
1.37858
I.44975
1.32530
1.43038
1.27740 1.41203
1.23380 1.39452
1.19374
1.37775
1.23841
1.20087
1.20087
48.8 1.79921
48.6 1.76624
1.76624
48.4 1.73520 1.16579 1.37655 47.4 1.67997
1.40851 47.8 1.74107
1.15665 1.36162
1.39226 47.6 1.70963
I.12209 1.34605
1.08972
1.33099
48.2 1.70586 1.13285 | 1.36134 47.2 1.65187
1.05928
1.31639
48.0 1.67800 1.10181
1.34657 47.0 1.62516
1.03053
1.30220
47.8 1.65146 I.07244
1.33221 46.8 1.59969
1.00330
1.28840
47.6 1.62611
47.4 1.60182
47.2 1.57850
1.04457
1.01806
.99279 1.29130 46.2 1.52948
1.31824 46.6 1.57531
.97744 I.27495
1.30461 46.4 1.55194
.95281 | 1.26183
.9293I I.24902
47.0 1.55606 .96864 | 1.27830 | 46.0|| 1.50785
.90683
1.23648
46.8 1.53443
.94553
46.6 1.51355 .92337
1.26559 45.8 1.48698
.88529
I.22422
1.25314 45.6 1.46682
.86463
1.21221
46.4 1.49335
46.2 1.47379
46.0 1.45483 .86191
45.8 1.43642
.90209
1.24094 45.4 1.44730
.84477
I.20043
.88162
1.22898 45.2 1.42839
.82566
1.18888
1.21724 45.0 1.41005
.80725
1.17754
.84291
1.20572 44.8 1.39223
.78949
1.16641
45.6 1.41852
45.4 1.40112 .80686
.82457
1.19441 44.6| 1.37490
.77235
I.15547
1.18329 44.4| 1.35804
.75577
1.14471
45.2 1.38417
.78973
1.17235 44.2 1.34161
.73974
1.13413
45.0 1.36765
.77315
44.8 1.35153 .75710
44.6 1.33581 .74153
44.4 1.32044 .72644
44.2 1.30543 .71178
44.0 1.29074
43.8 1.27637
43.6 1.26230
1.16160 44.0 1.32559
.72422
I.12372
1.15101 43.8 1.30996
1.14058 43.6 1.29470
1.13032 43.4 1.27978
.70918
I.I1347
.69459 1.10337
.68044 1.09342
I.12020 43.2 1.26520
.66670 | 1.08362
.69755 1.11023 43.0 1.25093
.68372
1.10040 42.8 1.23697
.67027 1.09071 42.6 1.22328
.65335 1.07396
.64037
1.06442
.62774
1.05502
43.4 1.24851
.65718
43.2 1.23499
.64445
43.0 1.22173
1.0811442.4 1.20988
1.07170 42.2 1.19673
.63204 1.06239 42.0 1.18384
.61546 1.04574
.60350
1.03658
.59185
1.02754
42.8 1.20872
42.6 1.19595
.61994 1.05320 41.8 1.17118
.58049
1.01861
.60815 1.04412 41.6 1.15875
.56942
1.00979
42.4 1.18342 | .59666
1.03515 41.4 1.14655
.55862
1.00107
42.2 1.17III .58547
42.0 1.15901 .57455
41.8 1.14711 .56388
41.6 1.13541 .55346
41.4 1.12390 •54328
41.2 1.11258 .53333
41.0 1.10143 .52360
40.8 1.09045 .51409
40.6 1.07964 .50479
1.00029 40.6 1.09978 .51796
.99183 40.4 1.08856
.98346 40.2 1.07752
.97518 40.0 1.06665
.96699 39.8 1.05595
.95889 39.6 1.04541
.48091
.93482
.47215 .92693
1.02628 41.2 1.13456
.54808
.99246
1.01751 41.0 1.12277 .53780
1.00885 40.8 1.III18
.98395
52776
.97554
.96722
.50838 .95899
.49901
.95085
.48986 .94279
13

y = 0.20
Y = 0.3
9-
X
Y
T
48.0 1.91475 1.33144
47.8 1.86658 1.27812
47.6 1.82326 1.23051
47.4 1.78375 1.18740
47.2 1.74734 I.14794
47.0 1.71350 1.III5I
46.8 1.68182 1.07766
46.6 1.65202 1.04603
46.4 1.62383 1.01633
9
X
Y
T
1.41840 40.0 1.47905.776117 | 1.08479
1.39905 39.8 1.44479
1.38078 39.6| 1.41331
1.36340 39.4 1.38411
1.34678 39.2 1.35684
1.33081 39.0 1.33120
.747469 1.07054
.721330
1.05691
1.04383
.697262
.697262
.674936 1.03122
.654103 1.01903
1.31542 38.8 1.30699
1.30055 38.6 1.28402
.634565
1.00722
.616164
•99574
1.28614 38.4 1.26216.598772
.98458
46.2 1.59708
.988332 1.27215 38.2 1.24127
.582280
•97370
46.0 1.57159
.961847 1.25855 38.0 1.22128
.566599
.96308
45.8 1.54724
.936718 1.24530 37.8 1.20208
.551653
.9527I
45.6 1.52391
.912812 1.23238 37.6 1.18361
|
.537377
.94256
45.4 1.50151
.890016 1.21977 37.4 1.16580
.523714
.93262
45.2 1.47995
.868234 1.20743 37.2 1.14861
.510615
.92289
45.0 1.45917
.847381 1.19537 37.0 1.13198.498037
.91333
44.8 1.43911
.827384 1.18355 36.8 1.11587
.485942
.90396
44.6 1.41970
.808178 1.17197 36.6 1.10025
.474297
.89475
44.4 1.40090
.789706 1.16062 36.4 1.08507
.463071
.88569
44.2 1.38267
.771917 1.14947 36.2 1.07033
.452236
.87679
44.0 1.36498
.754766 1.13853 36.0 1.05597
.441769
.86803
43.8 1.34777
.738212 1.12777 35.8 1.04199
.431647
.85940
43.6 1.33104
.722218 1.11720 35.6 1.02836
.421851
.85091
43.4 1.31474
.706751 1.10680 35.4 1.01506
.412363
.84254
43.2 1.29885
.691780 1.09657 35.2 1.00207
.403166
.83429
43.0 1.28336
.676277 1.08650 35.0
.98937
.394244
.82615
42.8 1.26823
.663218 1.07658 34.8
.97696
.385584
.81813
42.6 1.25345
.649579 1.06680 34.6
.96481
.377172
.81021
42.4 1.23900
.636339 1.05717 34.4
.95291
.368996
.80239
42.2 1.22486
.623479 1.04767 34.2
.94126
.361046
.79467
42.0 1.21103
.610979 1.03830 34.0
.92984
.353312
.78704
41.8 1.19748
.598823 1.02906 33.8
.91863
•
345784 .77951
41.6 1.18420
.586996 1.01995 33.6
.90764
.338453
.77206
41.4 1.17119
.575483 1.01095 33-4
.89685
.331311
.76470
41.2 1.15843 .564270 1.00206 33.2
.88625
.324350
.75743
41.0 1.14590
40.8 1.13361 .542692 .98462 32.8
40.6 1.12153 .532304 .97605 32.6
40.4 1.10966 .522170 .96759 32.4
.553344 .99328 33.0
.87584
.317563
.75023
.86561
.310942
.743II
.85555
.304481
.73607
.84565
.298174
.72910
39.8 1.07526 .493196
39.4 1.05324 .474982
39.2 1.04249 .466182
39.0 1.03190 .457578 .91093 31.0
38.8 1.02147 .449162 .90318 30.8
38.6
38.6 1.01119 .440928 .89551 30.6
40.2 1.09800 .512280 .95922 32.2 .83591
40.0 1.08653 .502625 .95095 32.0 .82632 .286000
.94278 31.8 .81688
39.6 1.06416 .483984 .93469 31.6 .80758
92668 31.4
|
.79842
.292015
.72220
.71537
.280124
.70861
.274381
.70191
.268767
.69528
.91877 31.2
.78939 .263277
.68871
.78049.257908
.68220
.77171 .252656
.67575
.76305 | .247516
.66936
14

Y = 0.4
Y = 0.5
9-
X
Y
T
❤
X
Y
T
0
0
34.2 1.24777
34.0 1.21572 .514384
33.8 1.18649
33.6 1.15954 .476774
33.4 1.13449 .460188
33.2 I.III03 .444779
33.0 1.08894 .430379
32.8 1.06805.416859
32.6 1.04819 .404114
.536087
.89410 29.6 1.06591
.384666
.75459
.88133 29.4 1.03607 .367783
.74287
.494743
.86916 29.2 1.00900
1.00900352587
.73172
.85750 29.0 .98413
.338743
.72106
.84629 28.8 .96107
.326013
.71081
.83546 28.6
.93953
.314219
.70093
.8249828.4
.91928
.303226
.69136
.81481 28.2
.90015
.292927
.68208
.80492 28.0
.88201
.283238
.67306
32.4 1.02927
32.2 1.01117
.392057
.79528 27.8 .86473 .274088
.66427
.380618
.78588 27.6
27.6
.84822 | .265421
.65570
32.0.99383
.369734
.77669 27.4
.83240 257188
.64732
31.8 .97715
31.6 .96110
.359357
.76770 27.2
.81722.249348
.63913
.349440
.75891 27.0
.80260 .241866
.63110
31.4
.94561 .339947
.75028 26.8
.78850 .234713
.62323
31.2
.93063.330844
.74182 26.6
.77488.227862
.61552
31.0 .91614 .322101
.73351 26.4 .76170 | .221289
.60794
30.8
.90209 | .313693
.72535 26.2.74892 .214976
.60049
30.6
.88846.305596
.71733 26.0
26.0 73653
.73653 .208903
.59316
30.4
.87521
.297791
.70944 25.8
.72448 .203055
.58595
30.2 .86232
.290258
.70167 25.6
.71277
.197418 .57886
30.0.84976
.282981
.69402 25.4
.70136
.191977 .57187
29.8 .83753 .275945
.68648 25.2
25.2
.69025
.69025 .186722
.56498
29.6 .82559 .269136
.67905 25.0
.67940 .181643
.55818
29.4
.81393 .262542
.6717224.8
.66882
.176728
.55148
29.2
.80255 .256152
.66449 24.6
.65847
.171970
•54487
29.0 .79141 .249954
.65735 24.4
.64836
.167361
.53834
28.8
78052
.243939
.65031 24.2
.63846 | .162892
.53189
28.6
.76985
.76985
.238098
.64335 24.0
.62877 .158557
.52552
28.4 .75940 .232424
.63648 23.8
.61928
.154350
.51922
28.2 .74916
.226908
.62969 23.6
.60997
.150265
.51300
28.0
27.8
.73911 .221543
.72925
.216323
.62298 23.4 .60084
.146297
.50684
.61634 23.2 .59189
.142440
.50075
27.6
27.4
.71007
.71957 .211242
.206294
.60977 23.0 .58309
.138689
.49473
.60328 22.8 .57446 .135040
.48877
27.2
.70073 201473
•
.59686 22.6 .56597 .131490
.48287
27.0
.69155.196775
.59050 22.4
.55763.128034
.47703
26.8
.68252 .192195
.58420 22.2
.54942 .124668 .47124
26.6
.67364
.187728
.57797 22.0
.54135
.121390
.46551
26.4 .66489
.66489 .183370 .57180 21.8
.53341
.118196
.45983
26.2 .65629 .179117
.56568 21.6 .52559
.115083
.45420
26.0 .64781 .174966
.55963 21.4
.51788
.I12048
.44862
25.8
.63946 .170913
.55363 21.2 .51028
109088
.44310
25.6
.63124.166954
.54768 21.0 .50280 .106200
.43763
25.4
.62313.163086
.54178 20.8
·
25.0 .60726 .155615
24.8 .59948
25.2 .61514 159307
.53594 20.6
.48815
20.6
•53015 20.4
.152005
.52440 20.2
.49542 .103384 .43220
.48815.100636 .42682
.48097 .097954 .42148
.47389.095336 .41618
15

Y = 0.6
Y = 0.7
9-
X
Y
T
9-
X
Y
T
0
26.0
.93344 .290751
.65330 23.0
.81319 .220114
.56922
25.8
.90438 | .276640
.64211 22.8
.78636.208780
-55871
25.6
.87825.264063
.63151 22.6
.76226 .198698
.54877
25.4
.85442 .252696
.62141 22.4
.74029.189597
.53929
25.2.83245
.242310
.61172 22.2
.72004 .181290
.53020
25.0 .81202
.232737
.60239 22.0
.70121 .173643
.52145
24.8
.79288
24.6 .77486 .215563
24.4 .75780
.223854
.59338 21.8
.68358
.166554
.51300
.58465 21.6
.66697
.159945
.50480
.207789
.57617 21.4
.65126 .153754
.49684
22.2
22.0
.60003
24.2 74159
24.0 .72613 .193554
23.8
.71134 .187002
23.6 .69716.180776
23.4 .68352 .174848
23.2 .67039 .169191
23.0 .65771.163783
22.8 .64545 .158605
22.6 .63358 .153639
22.4 .62207.148871
.61089 .144286
.139873
.200470
.56792 21.2
.63632.147930
.48909
.55987 21.0
.62208 .142434
.48153
.55201 20.8
.60845 .137231
.47415
.54433 20.6
.53681 20.4
.52943 20.2
.59539 .132293
.46693
.58282
.127596
.45986
.57072 .123118
.45293
.52220 20.0
.55903.118842
.44613
.51510 19.8
.54774 .II4752
.43945
.50812 19.6
.53679 .110835
.43289
.50126 19.4
.52618
.107077
.42643
.4945I 19.2
.51588 .103468
.42008
.48786| 19.0
.50586.099998
.41382
21.8
.58945
135622
.48131 18.8
.49611 .096659
.40765
21.6
.57915 .131522
.47486 18.6
.48660 .093443
.40157
21.4
.56910.127564
.46849 18.4
.47733 .090343
.39557
20.8
21.2 .55929 .123741
21.0 .54972
.54035
.46221 18.2
.46829.087352
.38966
.I20045
.45602 18.0
.45945 .084464
.38381
.116470
.44990 17.8
.45081 | .081674
.37804
20.4
20.2
20.6 .53119 113009
.52223 .109657
.51345.106409
•
.44385 17.6
.44236 .078976
.37234
.43788 17.4 .43409
.076368
.36671
.43198 17.2
.42598
.073843
.36113
17.8
17.6
16.8
16.6
20.0
.50484.103259
19.8 .49640 .100204
19.6 .48812 .097238
19.4 .47999.094360
19.2 .47201 .091563
19.0 .46416 .088847
18.8 .45645| .086206
18.6 .44887.083639
18.4 .44141 .081143
18.2 .43406 .078715
18.0 .42683 .076352
.41971 .074053
.41270 .071815
17.4 .40579 .069636
17.2 .39897 .067513
17.0 .39225 .065444
.38562.063430
.37908.061468
.42614 17.0
.41804
.071399
.35562
.42037 16.8
.41466 16.6
.41025 .069031
.35017
.40260
.066738
.34478
.40901 16.4
.40342 16.2
.39788 16.0
.39509 .064514
.33944
.38772.062358
.33416
.38047 .060267
.32892
.39239 15.8
·
37335 .058237
.32374
.38696 15.6 .36634 .056268
.31860
.38158 15.4
.35945
.054356
.31351
.37625 15.2 .35266
.37096 15.0 .34598
.36572 14.8
.052500
.30847
.050697
.30346
.33940
.048946
.29850
.36053 14.6
.33292
.047245 29358
.35538 14.4
.32653
.045592
.28870
.35027 14.2
.32023
.043985
.28386
.34520 14.0
.31401
.042424
.27906
.34017 13.8 .30788
.33518 13.6 .30183
.040907 .27429
.039432
.26956
16

Y = 0.8
y =
=0.9
Ф
0
X
Y
T
9-
X
Y
T
20.4
20.2
20.6 .72328 .173439
.69753 .163809
.67451 .155295
.50514 18.6 .64747
.138865
.45242
.49502 18.4
.62289
.130637
.44265
.48546 18.2
.60098
.I23395
.43344
20.0
.65361 .147646
.47637 18.0
.58115 116911
.42469
19.8
.63440.140692
.46766 17.8
.56295.111033
.41631
19.6 .61658 .134310
.45928 17.6
|
.54609.105653
.40826
19.4
•59992
.128411
.45119 17.4
•53035.100690
.40049
19.2
.58426.122924
.44335 17.2
.51556 .096084
.39297
19.0 .56945 117797
•
.43575 17.0
.50159 091787
.38566
18.6
18.8 .55539 .112984
.54200.108450
18.4
.52920
.104167
.42834 16.8
.421 [2
.41407
.48834.087761
-37855
16.6
.47573 .083976
.37162
16.4
.46367 .080405
.46367.080405
.36486
18.2 .51693
.100109
.40718 16.2
.45212 .077027
.35824
18.0
.50514 .096255
.40043 16.0
.44103
.073825
.35177
17.8
.49378.092587
.39382 15.8
.43035
.070782
.34542
17.6 .48283 .089091
.38733 15.6
.42005 .067886
.33919
17.4 .47224 .085752
.38095 15.4
.41009.065125
•33307
17.2
.46199 .082559
.37468 15.2
.40045.062488
.32706
17.0
.45205
.07950I
36852 15.0
.39111.059967
.32114
16.8
.44240
.076569
.36246 14.8
.38204.057554
•31532
16.6
.43302 .073754
.35648 14.6
.37323 .055242
.30958
15.4
15.2
13.8
13.6
13.2
13.0
16.4 .42389 .071050
16.2 .41499 .068450
16.0 .40632.065947
15.8 .39786.063537
15.6 .38960 .061213
.38152 .058972
.37361.056810
15.0 .36587
.36587.054722
14.8 .35829.052705
14.6.35087 .050756
14.4 .34358 .048872
14.2 .33643 .047050
14.0 .32942 .045287
.32252 .043581
.31575 .041929
13.4 .30909 .040330
.30254.038782
.29609.037282
.35060 14.4
.36465.053024
.30393
.34479 14.2
.35630.050894
.29836
.33906 14.0
.34815.048848
.29286
.33341 13.8
.34020 .046881
.28743
.32783 13.6
.33244
.044989
.28208
.32232 13.4
.32485 .043167
.27678
.31688 13.2
.31743
.041412
.27155
1149 13.0
.31016 .039721
.26638
.30617 12.8
.30305.038091
.26127
.30091 12.6
.29607 | .036518
.25621
.29570 12.4
.29054 12.2
.28923 .035001
.25120
.28251
.033537
.24625
.28544 12.0
.27592
.032124
.24134
.28039 11.8
.26945 .030760
.23648
.27538 11.6
.26308.029442
.23167
.27042 II.4
.25683 .028170
.22690
.26551 II.2
.25068 | .026940
.22218
.26064 11.0
.24462.025753
.21749
12.8
12.6
11.8
.28974 .035829
.28349 .034421
12.4 .27734 .033057
12.2 .27128 .031735
12.0 .26531 .030454
.25942 .029213
.25581 10.8
|
.23866 .024606
.21285
.25102 10.6
.23279 .023497
.20825
.24627 10.4
.22701
.022426
.20368
.24156 10.2
.22132
.021391
.19914
.23690 10.0
.21571
.020391
.19464
.23227
9.8
.21018 .019425
.19018
11.6
II.4
II.2 .24222
.24788 .026844 .22310 9.4
.025713 .21856 9.2
.25361.028010 .22767 9.6
.20472.018492
.18575
.19934 .017591
.18135
.19403
.016721
.17699
17
2

Y = 1.0
Y = 1.I
9-
X
Y
T
9-
X
Y
T

17.0
-59414 .116140
.41265 154
154
.51899 .090241
.36756
16.8 .56934 .108606
16.6 .54752
16.4 .52792
.52792.096251
.102059
.40293 15.2
.39382 15.0
15.0
.49780.084445
.35864
1
.47885 .07933I
.35022
.38519 14.8
14.8
.46163.074748
.34219
16.2 .51004
.091024
.37696 14.6
.44578 .070591
•3345I
16.0 .49357
.086268
.36907 144
.43108.066788
.32711
15.8 .47824
.47824.081903
.36147 14.2
.41732 .063282
.31996
15.6 .46389 .077869
.35412 14.0
.40438.060032
.31302
15.4
15.2
15.0
.42541
.45037
.43758 .070618
.067336
.074119
.34699 13.8
.39215
.057004
.30629
.34006 13.6
13.6
.38053
.054171
.29974
.3333113.4
13.4
.36945
.051512
.29334
14.8
.41381 .064249
.32672 13.2
13.2
.35886
.049008
.28709
14.6
.4027I .061337
.32029 13.0
13.0
.34871.046645
.28098
14.4
.39206.058582
.31399 12.8
•33895 .044409
.27499
14.2
.38182.055972
.30782 12.6
.32954 .042291
.26912
14.0
.37195
.053493
.30177 12.4
.32047
.040279
.26336
13.8
.36242
.051134
.29583 12.2
.31169
.038366
.25769
13.6
.35320 .048887
.28999 12.0
.30320
.036544
.25212
13.4
.34427
.046744
.28425 11.8
.29496.034807
.24664
13.2
.33561.044696
.27860 11.6
.28695.033150
.24124
13.0
.32720
.042738
.27303
11.4
.27917
.031567
.23592
12.8
.31902.040864
.26755 11.2
.27160
.030053
.23068
12.6
.31105.039069
.26214 II O
.26422.028605
.2255I
12.4
.30329 .037348
.25681
10.8
.25702
.027219
.22040
12.2
12.0
11.8
.2957I
.28832
.034III
.28109 .032588
.035696
.25155
10.6
.24999
.025891
.21536
.24635 10.4
.24313 .024618
.21039
.24122 10.2
.23641.023398
.20547
11.6
.27403
.031125
.23615 10.0
.22985
.022228
.20060
II.4 .26711 .029718
.23113 9.8
.22341
.021105
.19579
II.2
.26034 .028365
.22618
9.6
.21711
.020028
.19103
!!
II.O
.25371 .027063
.22127
9.4
21093
.018994
.18632
10.8
.24720
.025811
.21642
9.2.20487
.018001
.18166
10.6 .24082.024605
.21162
9.0 .19892
.017049
.17704
10.4
.23456 .023444
.20686
8.8.19308
.016134
.17247
10.2
10.0
.22841
.022326
.20215
8.6 18734
.015255
.16794
.22236
9.8 .21642 .020212
9.6 .21058 .019213
9.4 .20483 .018252
.021249
.19749 8.4
.18170
.014412
.16346
.19286 8.2 .17615 .013603
|
.15901
.18828 8.0 .17069
.18374
.012826
.15460
7.8 .16532
.012080
.15023
oi oioo
9.2 .19918 .017325
.17924
7.6.16003
.16003
.011365
.14589
9.0
.19361 .016433
17477
7.4 .15482
.010680
.14159
8.8
∞∞∞
7.6
8.2
.18812 .015574 .17034
8.6 .18271 .014747 .16594
8.4 .17738 .013951 .16158
.17213 .013185 .15726 6.6
8.0.16696 .012447 .15297 6.4
.16185 .011738 .14871 6.2
.14448
.15681 .011056
7.8
7.2.14969
.010022
.13733
7.0 14463
.009392
.13309
6.8
.13965 .008788
.12888
.13473 .008210
.12471
.12988.007657
.12057

6.0 .12037
.12509 .007129
.006625
.I1646
.11238
18

= 1,2
Y = 1.3
9-
X
Y
T
9-
X
Y
T
0
14.6 .52776 .089263
.35707 13.4
.46665 .071208
.32266
14.4
.49953
.081960
.34683 13.2 .44277
.065561
.31329
14.2
.47586
.075927
.33745 13.0
.42223
.060781
•30459
14.0
.45528
13.8 .43694 .066215
.070755
.32871 12.8
.40407
.056621
.29643
.32047 12.6
.38770
.052932
.28868
13.6
.4203I .062161
.31263 12.4
.37274 .049614
.28128
13.4 .40505
13.2 .39091
13.0 -37770
.058497
.30512 12.2
.35892.046600
.27417
.055153
.29790 12.0
.34604 .043839
.26731
.052079
.29093 11.8
•33396 .041294
.26068
12.8
.36528
.049236
.28418 11.6
.32258
.038936
.25424
12.6
.35355 .046592
.27762 11.4
.31178
.036740
.24797
12.4
.34242 .044125
.27123 II.2
.30152
.034688
.24187
12.2
12.0
11.8
.33182 .041813
.26501 11.0
.29172 .032765
.23591
.32169 .039641
.25892 10.8
.28233 .030958
.23008
.31198 .037595
.25297 10.6 .27333 .029256
.22438
II.4
11.6 .30265 .035663
.29367.033835
.24715 10.4
.26466
.027650
.21878
.24143 IO.2
.25630 .026131
.21329
II.2
.28500 .032103
.23582 10.0 24823
.024693
.20790
II.O
.27662.030459
.23031 9.8 .2404I
.023329
.20260
10.8
.26851
.028897
.22489 9.6||
.23284
.022034
.19738
10.6 .26065
.0274II
.21956 9.4 .22549
.020804
.19224
10.4
.25301
.025995
.21431 9.2
21835 .019635
.18718
oi oico ∞ ∞ ∞
.20481
10.2
10.0
9.8
9.6
.23836.023359
.23132 .022130
.22446 .020957
.019836
9.4.21776
9.2
9.0
.21121
.24559 .024646
.20913 9.0
.2114I .018522
.18220
.20403 8.8
.20464
.20464
.017462
.17728
•
19900 8.6
.19804 .016453
.17242
.19404 8.4 .19160 .015491
.16763
.18913 8.2 .18532 .014573
.16290
.018764
.20481 .017738 .17950 7.8.17316 .012865
8.8.19855 .016758
|
.18429 8.0 .17917 .013699
.15822
.15359
.17477 7.6 .16728
.012069
.14902
8.6
.19242 .015819
.17009
7.4 .16151 .011310
.I4449
8.4
.18641 .014922
.16546
7.2.15586
.010586
.14002
8.2
.18052 .014062
.16088
7.0
.15032 .009896
.13558
8.0
.17474 .013240
.15635
6.8
.14489.009238
.13120
.15186 6.6
7.8 .16908 .012454
7.6 .16351 .011701 .14741 6.4
7.4 .15804 .010981 .14300
6.2 .12915 .007444 .11828
7.2 .15267.010293 .13864 6.0 .12408 .006903
7.0 .14738 .009634 .13431 5.8 .I1910 .006387
.11405
.I3955
.008611
12685
•
13430
.008013
.12254
.10985
6.0
5.8
5.6 .11261
.11261.005788
6.2.12706 .007282 .I1735
.12218 .006760 .11320
.11736.006262 .10908
4.8
4.6
6.8
.14218 .009005 .13002 5.6
6.6 13706 .008404
13706.008404 .12576 5.4 .10937
6.4 .13203 .007830 .12154 5.2.10461
.11419
.005898
.10570
.005433
.10157
.004992
.09748
5.4
5.2
5.0.09993
.09532 .004178 .08939
.09077 .003804 .08539
.10499 4.4 08628 .003451 .08142
.10793 .005337 .10093 4.2 .08185 .003118 .07748
.10331 .004908 .09689 4.0.07748 .002805 .07357
.004574
.09342
19

Y = 1.4
Y = 1.5
9-
X
Y
T
9-
X
Y
T
0
12.2
12.0
11.8
.40049 .054244
.38169 .050213
.36495 .046685
.28670 11.4
.27842 II.2
.37183 .046879
.26693
.35357 .043229
.25879
.27061 II.O
.33734 .040046
.25112
11.6
.34979
.043544
.26318 10.8
10.8
.32267
.32267.037220
.24384
II.4
•33588
.040714
.25607 10.6
.30923 .034679
.23687
II.2
.32299
.038138
.24923 10.4
.29678 .032373
.23016
II.O
.31095
.035777
.24262 | 10.2
.28517.030263
.22369
10.8
.29964.033599
.23623 10.0
.27427 .028320
.21743
10.6
.28896 .031580
.23001 9.8
.26398 .026524
.21134
10.4
.27882 .029701
.22396 9.6
.2542I
.024855
.20542
IO.2
10.0
.26917.027946
.21806 9.4
.24492
.023299
.19964
.25994 .026303
.21230 9.2
.23604
.021845
.19400
9.8
.25110.024760
.20666 9.0
.22753 .020482
.18848
9.6
.24260.023307
.20113 8.8
9.4
.23442.021938
.19572 8.6
∞6
.21936.019202
18308
.21149 .017998
.17777
9.2
.22653.020646
.19040 8.4 .20390.016864
.17257
9.0
.21890 .019424
.18517 8.2 .19657 .OI5794
.16746
8.8
.21152
.018267
.18003
8.0.18946 .014783
.16243
∞∞∞
8.6
.20435 .017171
.17497 7.8 .18258
.013827
.15748
8.4
.19740 .016131
.16999 7.6
.17590
.012924
.15260
8.2
.19064 .015145
.16508 7.4 .16940
.012068
.14780
7.0
8.0 .18405 .014209
7.8 .17764 .013319
7.6
.17139 .012474
7.4 .16529 .011670
|
7.2 15933 .010906
.15350 .010180
.16024
7.2.16308
.011258
•
14306
.15546 7.0 .15692
.010491
.13839
.15075 6.8 .15091
.009764
.13378
.14610 6.6 .14505
.14150
.13695 6.2
.009075
.12922
6.4
.13932
.008422
.12472
.13372.007803
.12028
6.8
6.6
.14779.009489 13246
.14221 .008833 .12801 5.8
6.4.13673 .008209 .12361 5.6
6.0
.12824
.12824 .007218
.11588
.12288 .006663
.II153
11762
.006139
.10722
6.2
.13137.007617 .11926 5.4 .11247
.005642
.10296
5.8
.12094
6.0 .12610 .007054 .I1495 5.2
.006520
.10741
.005173
.09874
.11068 5.0 10245 .004731
.09457
5.6
.11586
.006014
.10645
4.8
5.4 .I1088
.005534
.10226 4.6
86
.09758
.004313
.09043
.09279 .003919
.08633
5.2.10598
.005080
.09810 4.4
.08809 .003549
.08226
5.0
.10116
.004650
.09399 4.2
.08347.003201 .07823
4.8
.09642
.004244
.08990 4.0
4.6
.09176
003860
4.4
.08716 .003499
.08184
士
​4.2
.08264.003158
4.0
.07818 .002839
.07390 3.2 .06139
.08585 3.8
3.6
.07785 3.4 .06568 .002018
.07891 .002875
.07423
.07443
.002569
.07027
.07002
.002284
.06634
.06244
.001771
.05856
on on on
3.8
3.6
.07379 .002539
.06997 3.0 .05717
.001543
.05472
.06946 .002259
.06607 2.8 .05301 .001332
.05090
3.4 .06519 .001998
.06220 2.6
.04891 .001138
.047II
33 2
O∞
3.2
.06098 .001755 .05836 2.4
.04487.000961
.04335
3.0
2.8
05681
.001530 .05455 2.2
.05270 .001322 .05076 2.0
.04088 .000801
.03962
.03693.000657
.03591
20

Y = 1.6
Y = 1.7
9-
X
Y
T
9
о
X
Y
T
10.8
•35727.042866
IO.2
10.6
.33811
.039244
.254II
.24577 10.0 .31860
·33774 .038251
.23987
.034841
.23157
10.4
.32135
.036137
.23799
9.8 .30198
.031938
.22384
10.2
10.0
.30635
.033411
.23064 9.6 .28718 .029407
.21655
9.8
.22363
.29272.030982
.28018 .028793 .21691
9.4
.27377 .027162
.20961
9.2
.26146.025147
.20297
9.6
.26853 .026803
.21045 9.0
.25006 .023321
.19658
9.4
.25764 .024980
.20419 8.8
.23942
.021653
.19041
9.2
.24740 .023302
.19813
8.6
.22941
.020122
.18444
9.9
.23770 .021749
.19224
8.8
.22850 .020307
.18650
8.6
.21972 .018964
.18091
∞∞∞
8.4
.21996 .018710
17863
8.2
.21099 .017401
.17297
8.0
.20245 .016185
.16746
∞∞∞
8.4
.21133 .017709
.17543 7.8
.19429
.015052 .16207
8.2
.20328 .016535
.17008 7.6
.18646
.OI3994 .15679
8.0
.19554 .015433
.16483 7.4
.17894
.013004
.15163
7.8
.18808 .014398
.15968 7.2 .17170
.012077
.14656
7.6
.18088 .013425
.15462 7.0 .16472 .011206
.14158
7.4
.17392
.012509
.14965 6.8
.15796
.010389
.13669
7.2
.16718 | .011645
.14475 6.6
.15142
.00962I
.13188
7.0
.16064 .010830
.13994
6.4.14508
.008898
.12715
6.8
.15428 .010061
.13519
6.2.13892
13892
.008218
.12248
6.6
.14811.009336
.13052 6.0 .13294
.007578
.11788
6.4
.14209
.008650
.12590 5.8 .12711
.006976
.II335
6.2
.13623
.008003
.12135 5.6
.12143 .006409
.10887
6.0 .13051
.007392
.11686
5.4
.11590.005876
.10446
5.8
.I2493
.006815
.I1242 5.2
.II049
.005374
.IOOIO
5.6
.I1947
.006270
.10803 5.0
.10521
.004903
.09579
5.4 .11413
.005756
.10370 4.8
.10004
.004460
.09152
5.2 .10891 .005271
.09941 4.6 .09499
.004044
.08731
4.8
4.6
4.4
5.0.10380.004815 .09517 4.4
.09878 .004385 .09097 4.2
.09387.003981 .08681
.08904 | .003601
.09004 .003655
.08314
.08270
WAA
4.0
3.8
20∞
.08519
.003290
.07901
.08043 .002949
.07493
.07577 .002631
.07088
4.2
4.0
.08431
.003245
.07862
3.8
.07966 .002912 .07458
.07509 .002600 .07058
333
3.6
.07119 .002335
.06687
3.4
.06669 .002060 .06290
3.2 .06227 .001805
.05897
3.6
.07060 .002310 .06661 3.0 .05793 .001570
.05507
ભેંસઁ
2.6
3.4 .06618
.002039 .06267
3.2
.06183 .001789
3.0 .05755 .001557
2.8 .05333 .001343
.04918 .001147
d
2.8
.05366 .001354
.05120
.05877
2.6
.04945 .001156
.04737
2.4
.04509.000969
.05490 2.4 .04532
.05106 2.2
.04725 2.0 .03723
.04346 1.8
.000975
.04357
.04124
.0008 II
.03979
.000664
.03605
2.2
.04106.000807
.03971
1.6
2.0
.03708.000661
.03598
1.4 .02553 .000316
.000417
.03327 .000533 .03233
.02937
.02864
.02497
1.8 .03316 .000530 .03227 1.2
1.6 .02929 .000415
1.4 .02547 .000315
.02174 .000230
.02859 I.O ,01800 .000159 .01772
0.8 .01431
.02494
ΙΟΙΟΟΟ
.01413
.02133
21

Y = 1,8
Y = 1.9
X
Y
T
X
Y
T
9.6
.31398 .033273
.22453 9.09.28737
.028267
20836
9.4
oi ooo oo
9.2
.29579 .030230
.27994 .027634
.21645
.20891
8.8
.27075
8.6 .25612.023425
.025664
.20065
.19342
9.0
8.8
8.6
986
.20180
.26580.025369
8.4
.24299 .021461
.18658
.25298.023361
.24120
.021558
.19503
8.2
.18854 8.0
.23101
.21997
.019714
.18004
.018142
.17377
8.4
.23028
.019926
8.2
.22008
.018437
8.0 .21048
7.8
.20141
7.6 .19281 .014649
.017071
.015813
.18230 7.8 .20969
.17627 7.6
.20007 .015415
.17042 7.4 .19100 .014221
.016716 .16773
.16188
.15620
.16474 7.2
.18242
.013121
.15068
.15921 7.0 .17426
.012104
.14531
7.4
.18460 .013569
.15382 6.8 .16647
.011162
.14005
6.2
7.2 .17676 | .012564
7.0 .16925 .011628
6.8 .16202
.010754
6.6 15506 .009936
6.4 .14834 .009171
14185 .008453
·
.14854 6.6 .15902
.010287
.13492
.14338 6.4 .15187
.009472
.12989
.13832 6.2
.13336 6.0
.12848
.14499
.008712
.12496
.13836 .008003
.12012
5.8
.13195 .007341
.I1537
.12369 5.6
.12575
.006722
.11069
6.0
5.8
.13556.007781
.12946 .007151
.11897 5.4 .I1975
.006144
.10609
.II434 5.2
.I1392
.005603
.10157
5.6
ம் ம்
.12354 .006559
.10977 5.0
.10826
.005098
.09710
5.4.11778.006005
5.2
.11217 .005484
5.0.10670.004997
4.8
.10526 4.8
.10275
.004625
.09270
.10082 4.6
.09739
.004184
.08836
.09643 4.4
.09216 .003773
.08408
3.6
3.4
.10137 .004540
4.6 .09617 .004112
4.4 .09108
4.2 .08611 .003338
4.0
.08125 .002989
3.8 .07648
.07181
.06723
3.2 .06274
.09210 4.2 .08706
.08783 4.0.08207
.003389
.07984
.003032
.07566
.003712
.08360 3.8
.07720
.002700
.07153
.07942 3.6 .07244 .002391
.06744
.07529 3.4.06777.002106
.06340
.002664
.07120 3.2 .06320 .001842
.05940
.002362
.06716 3.0 .05872 .001600
.05544
.002082
.06315
.001823 .05918
तं तं
2.8
2.6
86
.05433
.05002
.001377
.05152
.001174
.04763
2.6
3.0.05833
.05833 .001585
2.8 .05400 .001365
.04974 .001164
.05525 2.4
.04578.000989
.04379
.05136 2.2
.04162 .000822
.03997
.04750 2.0
.03753.000672
.03619
2.4
.04555 .000982
.04368 1.8
.03352 .000538
.03245
2.2
.04144 .000816
.03989 1.6 .02956 .000421
.02873
2.0
.03739 .000668
.03612 1.4 .02567 .000319 .02504
1.8
1.6
1.4
.03340.000536
.02947 .000419
.02560.000318
.03239 I.2 .02184 .000232
.02869 I.O .01807 .000160
.02139
.01776
.02501 0.8 .01435 .000102
.01416
I.2
1.0
.02179
.01803
.00023 1
.000159
0.81
.01433
101000*
.02136 0.6 .01069
.01774 0.4 .00708
.01415 +.2 .00352 .000006 .00350
.000057
.01058
.000025
.00703
0.4
0.2
0.6 .01068 .000056 .01058 0.0
.00708
.000024 .00703
.00352 .000006
.00350
.00000 .000000
.00000
2
.00347 .000006 .00348
0.4 .00689 .000024 .00694
22

Y = 2.0
y = 2.1
9-
X
Y
T
9-
X
Y
T
C
8.6 27661 .026059
.19960 8.2
·
26354 .023652
.19018
8.4 .25962
.023518
.19182
80
.24666
.021250
.18243
8.2 .24482
.021358
.18455
7.8
.23203
.019219 .17522
8.0.23162
.019478
.17769
7.6
.21901
.017459
.16842
7.8 .21964
.017816
.17117
7.4 .20724
.015908
.16195
7.6 .20864
.20864 .016328
.16491
7.2.19644
19644
.014525
.15576
7.4.19843
.014985
.15889
7.0 18644
.013279
.14981
7.2.18890
.013763
.15308 6.8 .17711
.012149
.14406
7.0
.17994
.012646
.14744 6.6
.16834
.OIII19
.13849
6.8 .17146
.17146
.011621
.14196 6.4
.16005 .010175
.13308
6.6.16342
.010676
.13663 6.21
.15220
.009308
.12781
6.4 .15575
.009802
.13142 6.0
.1447I .008508
.12267
6.2.14843
.008993
.12633 5.8
·
13756.007769
13756 .007769
.11764
6.0 .14140
.008242
.12135 5.6
.13071 007085
.11273
5.8
.13465 .007544
.11647
5.4 .12412 .006451
.10791
5.6 .12815 .006895
.II168
5.2.11778.005862
.10319
5.4
.12187.006291
.10698
5.0.11167 .005316
.09855
5.2
.11580 .005727
.10236
4.8
.10575 .004809
.09399
5.0
4.8
.10992
.10422
.005203
.09781 46
.10002 .004338
.08950
.004714
.09333 4.4
.09447 .003901
.08508
4.6
.09868
.004259
.08892 4.2
.08908
.08908.003496
.08073
4.4 .09329
.003835
.08457
4.0
.08383 .003120
.07645
4.2 .08805
.003441
.08028
3.8
.07873
.002772
4.0.08294
.003074
.07605 3.6
.07222
.07376
.002450 .06804
3.8 .07796
.07796.002734
.07187 3.4
.06891
.002153 .06392
3.6
.07309
.002420
.06774 3.2
.06418
.001880 05985
3.4 .06834
.002129
.06366 3.0
.05955
.05955
.001630
.05582
3.2
.06369
.001861
.05962
2.8
.05503 .001400
.05184
3.0
.05914
.001615
.05563
2.6
.05060
.001192
.04791
2.8
.05468 .001389
.05168 2.4
.04627
.001003
.04402
2.6
.05031 .001183
.04777 2.2
.04202 .000832
.04016
2.4
.04603.000996
.04390 2.0
.03785.000679
.03635
2.2
.04182 .000827
.04007
1.8
.03377 .000544
.03257
2.0
.03770.000675
.03627
1.6 .02976 .000425
.02883
1.8
.03364 .000541
.03251
1.4 .02582
.000322
.02512
1.6 .02966 .000423
.02878
I.2 .02194 .000234
.02144
1.4
1.2
1.0.01811
0.8
+.2
0.0
.02575 .000320
.02508
.01814
.02189 .000233 .02141 0.8 .01440
.000160 .01778 0.6
.01071
.01438 .000102 .01417 0.4 .00709
0.6 .01071 .000057 .01059 +.2
.00709 .000025 .00704
0.4
.00352
.000006
.00000 .000000
1,0
191000*
.01779
.000102 .01418
.000057
.01059
.000025
.00703
.00352
.000006
.00350
0.0
.00000
000000
.00000
.00351 .2
.00000
2
.00345 .000006
.000024
0.4
.00348 0.6
.00347
.00689
.01026
.000006 .00347
.00692
.000053
.01035
0.4
.00686 .000024
.00694 0.8 .01358
.000094
.01376
0.6 .01024 .000053
.01037 1.0 .01685
.000145
.01715
0.8 .01358 .000094
.01378 I.2 .02008
.000207
.02051
23

Y = 2.2
Y = 2.3
9-
X
Y
T
9-
X
Y
T
8.0 .26801
.023841
.18851 7.6 .25022
.020996
.17785
7.8
.24848
.021130
.18019 7.4 .23197
.018592
.16980
7.6
.23217.018925
7.4
.21803 .017062
7.0
6.8
7.2 .20544 .OI5449
.19404 .014029
.18359 .012764
6.6 .17390 .011626
.17259 7.2 .21660
.16550 7.0 .20318
.15882 6.8 .19120
.15246 6.6 .18032
.016622
.16241
.014950 .15552
.013500 .14900
.012221 .14280
.14638
6.4 .17031
.011081
.13685
.14053
6.2.16104
.010057 .13113
6.4
.16486
.010596
.13487
6.0.15236
.009129 .I2559
6.2 .15637 .009658
.12940
5.8 .14420
.008286
.12023
ம்ம் ம்
6.0
.14834 .008800
.12407 5.6 .13648
.007516
.11501
5.8
.14073 .008013
.11889 5.4 .12916
.006810
.10993
5.6
13348.007289
.11383 5.2 .12217
.006162
.10497
5.4
.12654 .006622
5.2
.11990.006005 .10405 4.8
.10889 5.0 .II550
.005566
.10012
.10909 .005017
.09538
5.0
4.8
4.6
980
.I1352
.005436
.10737 .004909
.09932 4.6
.09467 4.4 .09701 .004044
.I0294
.0045II
.09073
.08616
.IO144 .004421
.09010 4.2 .09128
.003613
.08168
4.4
.09571 .003970 .08561
4.0.08575
.003216
.07727
4.2
.09016 .003552 .08120 3.8
.08038
.002850
.07294
4.0
.08477 .003166 .07685
3.8
.07954
.002810
.07257
حب حب
3.6
.07518
.002514
.06866
3.4
.07013
.002205 .06446
3.6
.07446 .002481
.06835
3.2
.06521
.001921 .06031
3.4
.06951.002178
.06419
3.2
.06469 .001900
.06008
लं तं
3.0
.06043 .001662
.05622
2.8
.05576.001426
.05218
2.8
2.6
IOZIOO'
3.0 .05998 .001645 .05602 2.6
.05539 .001412 .05201 2.4
.05090
.05121 .001212 .04819
.04677
.001018
.04424
2.2
2.0
2.4 .04651 .001009
.04222 .000837
.03802 .000683
1.8 .03389 .000546
1.6 .02985.000426
.03642 1.6
.03263 1.4 .02596
.02887 1.2 .02205
.04805
.04413 2.0
.04026 1.8
2.2
.04243 .000843
.04035
.03818.000687
.03649
.03402 .000550
.03268
.02995 .000429
.02891
I.2 .02200
Ι.Ο
1.4 .02589 .000322
.000234
.01817
191000*
0.8 .01442 .000102
0.6
.01072 .000057
0.4 .00709 .000025
+.2 .00352 .000006
.000324 .02518
.000235 .02149
.02515 I.O .01821 .000162 .01782
.02146 0.8 .O1444 .000102
.01781 0.6
.01074 .000057 .01060
.01419 0.4 .00710 .000025
.01060 +.2
отобо
.00352
.000006
.00704
0.0 .00000 .000000
.01420
.00704
.00351
.00000
0.0 .00000 .000000
.00351 .2 .00346 .000006
.00000 0.4 .00687
.00348
.000024
.00693
2 .00347 .000006
.00348 0.6 .01023 .000053
.01035
0.4
0.6
.00689 .000024
.01026 .000053
0.8 .01357 .000094 .01377 I.2
.00694 0.8
.01354 .000094
.01375
.01037 Ι.Ο .01680 .000145
.01712
.02002. .000206
.02047
I.4
1.0 .01683 .000145 .01714
1.2 .02005 .000207 .02049
.02323 .000280
.02382
HHH
I.4 .02319 .000277
.02380
1.6 .02632 .000359
.02711
1.8
.02940 .000451
.03039
24

Y = 2.4
y = 2.5
9-
X
Y
T
9-
X
Y
T
0
7.4
.25182 .020836
.17534 7.0 .23004
.017756
.16361
7.2 .23130 .018206
.16682 6.8
.21205
.015577
.15563
7.0
.21464 .016130
.15914 6.6
.19704
.013813
.14834
6.8
6.6
86
.20040
.014406
.15204 6.4
.18402
.012329
.14155
.18786 .012933
.14538 6.2
.17243
.OI1049
.13515
6.4
.17660 .011649
•
13906 6.0
.16193
.009927
.12907
6.2
.16632
.010514
.13304 5.8
6.0
.15685
.009501
·
5.8
.14803 .008590
.12726 || 5.6
.12168 5.4
∞64
.15231
.008933 .12324
.14340
.008043
.11764
.13508
.007241
.I1222
5.6
5.4
5.2
642
.13977 .007765
.11628 5.2
.13198 .007016
.12461 .006332
.II104 5.0
.10595 4.8
20∞
.12726
.006516
.10697
.11987 .005856 .10188
.11286 .005255
.09691
5.0
.11760 .005706
.10098 4.6
.10618 .004706
.09207
4.8
Ι
4.6
.11091
.10451
.005132
.09613 4.4
.09980.004204
.08733
.004606
.09139 4.2
.09369.003744
.08270
4.4
.09836 .004122
.08674 4.0
.08781 .003322
.07816
4.2
.09245
.003677
4.0
3.8
.08675 .003269
.08125 .002893
.08219 3.8 .08215 .002937
.07370
.07772 3.6
.07669 .002583
.06933
.07332 3.4
.07141 .002260
.06503
3.6
.07592 .002549
.06900 3.2
.06630 .001965
.06079
3.4
.07076
.002232
.06475 3.0
.06134 .001696
.05663
3.2
.06574
.001943
.06056 2.8
3.0 .06087 .001680
.05643 2.6
.05652.001452
.05252
.05184 .001232
.04848
2.8
.05614 .001439
.05236 2.4
.04728
.001032
.04448
2.6
.05152
.001222
.04834 2.2
.04284
.000854
.04054
2.4
.04702 .001025
.04437 2.0
.03851
.000695
.03665
2.2
.04263 .000849
.04045 1.8
.03428 .000555
.03281
2.0
.03834.000692
.03658 1.6
.03015
.000432
.0290I
HH
1.8 .03415 .000553
.03275 I.4
.0261I
.000326
.02525
1.6
.03005 .000431
.02897 I.2
.02215
.000237
.02154
I.4
.02603 .000325
.02522 I.O
.01828 .000162
.01786
1.2
.02210 .000236
I.O .01824 .000162
.02152 0.8
.01448 .000102
.01422
.01076
0.8
.01446 .000102
0.6
.01075 .000057
0.4 .00710 .000025
+.2 .00352 .000006
.00351 -.2
0.0 .00000 .000000
-.2
.00346 .000006
.00000 0.4
.00348 0.6
.01021
.01785 0.6
.01421 0.4 .007II
.01061 +.2 .00352
.00704 0.0
.000025
.000006 .0035I
.00000 .000000
.00000
.00348
.00346 .000006
.00686 .000024 .00692
.000053 .01034
.000057
.01062
.00704
0.4
0.6
468
.00687
.000024
.01022
.000053
.00693 0.8
.01035 1.0
.000093
.01350
.01675 .000144 .01710
.01373
0.8 .01352 .000094
.01374 I.2
.01995 .000205
.02044
I.2
1.0 .01677 .000145 .01711 I.4
.01998 .000206 .02045 1.6
.02310.000276
.02376
.02620 .000357
.02705

1.4
.02314 .000277
.02377 1.8
.02925
.000448
.03031
1.6
.02626.000358
.02707 2.0
.03225 .000548
.03355
1.8
2.0
.02933 .000449
.03235 .000551
.03035 2.2
.03521.000657
.03677
.03360 2.4
.03814 .000775
.03997
25
3

Y = 2.6
Y = 2.7
X
Y
T
9-
X
Y
T
6.6
.20876 .014962
.15189 6.4
.20465
.014281
.14792
6.4
.19307 .013173
.I4445 6.2
.18842
.012488
.14035
6.2
.17964 .011690
.13756 6.0
.17470 .OIIO21
.13339
6.0
.16780 | .010425
.13110 5.8
.16271.009782
.12689
5.8
.15716 .009324
.12497 5.6
.15199 .008711
.12074
5.6
.14744 .008355
.I1912 5.4
.14224
.007773
.I1488
5.4
•
13848
.007491
.11350 5.2
.13328
.006941
.10927
5.2
.13014 .006717
.10808 5.0
.12496
.006198
.10386
5.0
.12232.006019
.10283 4.8
.11718
.00553I
.09862
4.8
.11494 .005387
.09774 4.6
.10985
.004929
.09355
4.6
.10796.004813
.09279 4.4
.10293 .004383
.08862
4.4
.10132 .004290
.08796 4.2
.09635.003889
.08381
4.2
.09499
.003814
.08324 4.0
.09007 .003439
.07912
4.0
3.8
.08892 .003379
.08309 .002982
.07863 3.8
.074II
mmm mŇ
3.6 .07749
.06968 3.4
www
.08407
.003030
.07453
3.6
.07832
.002657
.07004
3.4 .07209
3.2 .06687 .001988
.06105 3.0
.002620
.002289 .06533 3.2 .06745 .002011 .06132
.05707
.07278 .002319
.06564
.06230 .001732
2.8
2.6
3.0 .06182 .001714
.05692.001466
.05217 .001242
.05685 2.8
.05271 2.6
86
.05732 .001480 .05290
.05250 .001253
.04879
.04863 2.4
.04782 .001048
.04474
2.4
2.2
2.0
1.6
HHH
864
1.8
1.4
.04756 .001040
.04306.000860
.03869 .000699
.03442 .000558
.03026 .000434
.02619 .000328
.04461 2.2
.04327 .000866
.04075
.04065 2.0
.03885 .000704
.03682
.03674 1.8
.03455 .000561
.03294
.03288 1.6 .03035 .000436
.029II
.02906 1.4
.02626 .000329
.02533
.02529 I.2
.02226
.000238
.02159
I.2
1.0
.02221 .000237
.01832 .000163
.02156 1.0
.01835 .000163
.01790
.01788 0.8
.01453 .000103
.01424
0.8
.01451 .000103
.01423 0.6
.01078
.000057
.01063
0.6
.01077
.000057
.01062 0.4
.00712 .000025
.00705
0.4 .007II
.000025
.00705.2
.00352 .000006
.00351
+.2
.00352
.000006
0.0
-.2
.00346
.00000 .000000
.000006
.00351 0.0
.00000
.00000 .000000
.00000
.2
0.4
.00686 .000024
.00348 0.4
.00692 0.6
.00346 .000006
.00686 .000024 .00692
.01019 .000053 .01033
.00348
0.6
.01020 .000053
.01034 0.8
.01347
.000093
.01371
0.8
.01349 .000093
1.0 .01672 .000144 .01708 1.2
.01372 1.0
.01669 .000144
.01707
.01986 .000204
.02040
I.2
I.4
240 ∞
1.6
HH
I.4 .02298 .000274
1.6 .02606 .000354 .02698
.02909 .000444 .03023
.01990 .000205 .02042
.02303 .000276 .02373
.02612 .000357 .02701 1.8
.000447
.02370
1.8
2.0
2.2
.02916
.03216 .000547
.03512.000656
.03027 2.0
.03350 2.2
.03671 2.4
.03207 .000544
.03345
.03501 .000652
.03665
.03791.000769
.03983
2.4
2.6
.03804 .000773
.04092 .000898
2.8 .04376 .001031
.03990
2.6
.04077 .000894
.04299
.04307 2.8
.04622 3.0
.04359
.001027
.04614
.04636
.001167
.04927
26

Y = 2.8
y = 2.9
9-
X
Y
T
9-
X
Y
T
0
6.2
.19970 .013534
.14368
6.0
.19388
.012730
.13921
6.0
.18312
.011761
.13603
5.8
.17717
.OIIO02
.13154
5.8
.16925 .010328
.12903
5.6
.16330
.009618
.I2455
555
5.6
5.4
5.2
642
.15721 .009125
.12252
5.4 .15131
.008463
.11805
.14649 .008092
.11637 5.2 .14067
.007475
.II193
.13678 .007191
.I1053 5.0
.13105
.006616
.10611
5.0
.12786.006395
.10493 4.8
.12223
.005861
.10054
4.8
.11961 .005687
4.6
.II190
.005053
.09433 4.4 .10647
64
.09954 4.6 .11408 .005190
.09519
.004591
.09002
4.4
.10465
4.2
004483
.09780.003968
.08929
4.2
.09933
.004054
.08501
.08439 4.0
.09258
.003570
.08015
4.0
.09130 .003502
.07961
3.8
.08618
.003134
.07541
3.8
.08511
.003080
.07495 3.6
.08009
.002739
.07079
3.6
3.4
.07919 .002697
.07352
.07040 3.4
.07427
.002383
.06628
.002350
.06594 3.2
.06870.002062
.06186
3.2
.06807 .002036
.06157 3.0
.06334 .001772
.05753
3.0
.06282 .001751
.05728 2.8
.05818
.001510
.05328
2.8
.05775 .001495
.05307 2.6
.05320
.001275
.04910
2.6
.05285
.001264
.04893 2.4
.04839 .001065
.04500
2.4
2.2
.04811 .001056
.04350.000872
.04486 2.2
.04373 .000878
.04096
.04085 2.0
.0392I .000712
.03699
2.0
1.8
986
.03903
.03469 .000564
.000708
.03689 1.8
.03483.000566
.03307
.03300 1.6
.03056 .000440
.02921
1.6 .03046 .000438
.02915 I.4
.02641 .000331
.02540
I.4
.02634.000330
.02536 I.2
.02237
.000239
.02165
1.2
.02232 .000238
.02161 1.0 .01842 .000164
.01793
1.0
.01839 .000163
.01791 0.8 .01457 .000103
.01427
0.4
0.8 .01455 .000103
0.6 .01080 .000057
.00712 .000025
.01425 0.6 .01081
.000057
.01064
.01063 0.4
.00713
.000025
.00706
.00705 +.2
.00353
.000006
.00351
+.2
.00353 .000006
0.0
.2
.00000 ,000000
.00346.000006
.0035I
.00000
0.0
,00000 .000000
,00000
—
.2
.00346.000006
.00347
.00347 0.4
.00685 .000024
.00691
0.4
0.8
1,0
.00685 .000024
0.6 .OI018 .000053 .01033 0.8
.01345 .000093 .01371 1.0
.01667 .000143 .01706 I.2
I.2 .01983 .000203 .02038 I.4
I.4 .02294 .000273 .02368 1.6
1.6 .02600 .000353
1.8 .02901
2.0 .03197
.00692 0.6
.01017 .000052
.01032
.01343
.01664
.000092
.01369
.000143
.01704
.01979 .000203 .02036
.02289.000273
.02364
.02594 .000353
.02691
2.2
2.4
.02695 1.8
.000443 .03019 2.0
.000542 .0334I 2.2
.03489 .000649 .03661
.03777.000765
.02893 .000442
.03014
.03188 .000540
.03335
.03479.000646
.03654
2.4
.03978 2.6
2.6 .04061 .000889 .04293 2.8 .04326
468
.03766.000760
.03970
.04048.000883
.04284
.001015
.04596
233
2.8
3.0
3.2
.04342 .001021 .04606 3.0 .04600 .001155
.04618 .001161 .04917 3.2 .04871 .001303
.04890.001308 .05226 3.4 .05138
.04906
.05214
.001458 .05520
27

y = 3.0
Y = 3.I
X
Y
T
4
X
Y
T
5.6
.17062 .010218
.12684 5.6
·
17983 .010990
.12957
5.4 .15690 .008897
.11989 5.4
5.2
5.0
.13457
4.8
.12510
4.6
.I1644
4.4
.10842
.14506 .007798
.006862 .10736 5.0
.006050 .IO159 4.8
.005337 .09608 4.6
.004706
.II344 5.2
42
.16358 .009425
.12201
.15014 .008177
.11513
.13855 .007143
.10875
.12829
.006262
.10274
.I1901 .005499
.09703
.09077 4.4
.I1053 .004831
.09157
4.2
.10094 .004144
.08565 4.2
.10268 .004241
.08632
4.0
.09393 .003641
.08068 4.0
.09536 .003716
.08126
www
3.8
3.6
86
.08731 .003189
.07586 3.8
.08849
.003248
.07635
.08103 .002783 .07117 3.6
.08201
.002829
.07158
3.4
.07505 .002417
.06660 3.4
.07587
.002453
.06694
33
3.2
.06934.002088
.06212 3.2
.07001
.002115
.06241
3.0
.06387
.001791
.05775 3.0
.06442
.001812
.05799
2.8
.05861
.001525
.05346 2.8
.05907
.001541
.05366
હેં હૈં હૂઁ
2.6
.05355
.001287
.04925 2.6 .05393
.001298
.04942
2.4
.04867 .001074
.045T2 2.4 .04897
.001082
.04526
2.2
.04396.000884 .04106 2.2
.04420
.000890
.04117
2.0
.03939 .000717
.03707 2.0
.03958
.000721
.03715
1.8
.03497 .000570
.03313 1.8
.035II .000573
.03320
1.6
.03067
.000442
.02926 1.6
.03078 .000444
.0293I
1.4
.02649
.000332
.02543 1.4
.02657 .000334
.02548
I.2
1.0
.02242 .000240
.01846 .000164
.02167 1.2
.02248 .000241
.02170
.01795 Ι.Ο
.01850 .000165
.01797
0.8
.01459 .000103
.01427 0.8
.01462
.000104
.01429
0.6
.01082
.000057
.01064 0.6
.01083
.000057 .01065
0.4
.00713
.000025
.00706 0.4
.00714
.000025 .00706
+.2
.00353
.000006
.00351 +.2
.00353
.000006
.00351
0.0
.00000 .000000
.00000 0.0
.00000
.000000
.00000
.2
.00346 .000006
.00347 -.2 .00345 .000006
.00347
0.4 .00684 .000024
0.6 .01016
0.8
.01342
.00691 0.4
.00684 .000024
.00691
.000053
.01032 0.6
.000092
.01369 0.8
.01015
.01340 .000092
.000052
.01031
.01368
Ι.Ο
.01662
.000142
.01703 1.0
.01659 .000142
.01701
I.2
.01976
.000203
.02034 1.2
.01972 .000202
.02032
I.4
.02284 .000273
.02363 1.4
.02279
.000272
.02360
HH
1.6
1.8
.02587
.02886
.000352
.000441
.02688 1.6
.03011 1.8 .02878
.02581
.00035 I
.02684
.000439
.03006
2.0
.03179 .000538
.0333I 2.0 .03171
.000536
.03326
2.2
.03468.000646
2.4
.03753 .000761
.03649 2.2
.03964
.03458 .000641
.03643
2.4
2.6
.04034
.000883
.04277 2.6
46
.03741
.000755
.03957
.04019 .000876
.04270

तं लंलं
2.8 .04310 .001012
.04588 2.8
.04294 .001006
.04579
3.0
3.2
.04582 .001149
.04850 .001295 .05204 3.2
.04897 3.0
.04564
.001143
.04887
.04830
.001288
*05193
www
3.4
3.6
3.8
468
.05115
.05377
.05635.001775
.001448
.001608
.05509 3.4 .05093
.001440
.05497
.05812 3.6 .05353
.001599
.05799
.06113
3.8
.05609
.001764
.06098
28

Y = 3.2
Y = 3.3
9-
X
Y
T
9-
X
Y
T
0
5.2
5.0
4.8
.15612 .008631
.14309 .007467
.13183 .006502
.I1704 5.2
.16339 .009192 .I1926
.I1027 5.0
.14836.007850
.II199
.10398 4.8
.13583 .006775
.10535
4.6
.12184 .005680
.09805 | 4.6
.12496
.005880
.09917
4.4
.I1281 .004969
.09242 4.4
.I1529
.005119
.09334
4.2
.I0454 .004347
.08703 4.2
.10653 .004460
.08780
4.0
3.8
3.6
986
.09689 .003798
.08975 .003311
.08304 .002878
.08185 4.0
.09850.003885
.08249
.07685 3.8
.09107 .003377
.07738
.07200 3.6
.08412 .002928
.07245
33
3.4
.07671 .002490
.06729 3.4
.07760.002529
.06766
3.2
.07071 .002144
.06271 3.2
.07143 .002174
.06301
3.0
.06400 .001835
.05824 3.0
.06559 .001857
.05849
2.8
.05954 .001558
.05387 2.8
2.6
.05431
.001311
.04959 2.6
8 6
.06002
.001575 .05407
2.4
.04928 .001092
.04539 2.4
ΙΟΙΙΟΟ
.05469 .001323 .04976
.04959
.04553
2.2
.04444
.000897
2.0
.03977 .000726
.04128 2.2
.03724 2.0
.04468 .000904 .04139
.03996 .000730
.03733
1.8
.03526.000576
.03327 1.8 .03540
.000579
.03334
1.6
.03089.000446
.02936
1.4
.02665 .000335
.02551
H H
1.6
1.4
64
.03100
.000448
.02941
.02673 .000336 .02555
I.2
.02253
.000242
.02172 I.2
.02259 .000243
.02175
1.0
.01853 .000165
.01799 1.0 .01857
.000165
.01800
0.8
.01464
.000104
0.6
.01085 .000058
.01430 0.8 .01466
.01066 0.6
.000104
.01431
.01086
.000057
.01066
0.4
.00714 .000025
.00706 0.4
.00715
.000025 .00706
+.2
.00353 .000006
.00351 +.2
.00353
.000006
.00351
0.0
.00000 .000000
.00000 0.0
.00000
.000000
.00000
.2
0.4
.00345
.00683
.000006
.00347
.000024
0.6
ΟΙΟΙ4
.000053
0.8
.01338
.000092
1.0 .01656
.000142
1.2
.01968
.000202
HHH
1.4
.02275
.000272
1.6
.02575 .000350
.02357 I.4
.02681 1.6
1.8
.02871.000438
.03003 1.8
468
.01367 0.8 .01337
.01700 I.O .01654
.02030 1.2 .01965
.02270 .000271 .02354
.02569 .000349 .02678
.00345 .000006 .00347
.00691 0.4 .00683 .000024 .00690
.01030 0.6 .01013 .000052 .01030
.000092 .01366
.000142 .01698
.02028
.000201
.02864 .000436
.02999
2.0
.03162.000535
.03321 2.0 .03153 .000532 .03317
2.2
.03447 .000639
.000752
तं तं लं
2.8
A A w www
3.4
3.6
3.8
4.0
4.2
∞ ON
2.4 .03729
2,6 .04006 .000873
.04278
3.0 .04547 .001137
3.2 .04812 .001280 .05183
.05073 .001430 .05486
.05331 .001587 .05786 3.6
.03637 2.2
.03951 2.4
.04262 2.6
.03437
.000636 .03632
.03717
.000749 .03945
.03992 .000869
.04255
ΙΟΟΙΟΟ
.04571
2.8
.04263 .000997
.04563
.04878
3.2
246
33
3.4
.05585
.001752 .06084
3.8
.05835
.05560
.06082
.001132 ..04869
.04793 .001274
05052 .001423 .05474
.05308 .001579 .05773
.001742
.001924 .06380 4.0 .05808 .001912
.002102 .06675 4.2 .06053
3.0 .04530
.05173
.06070
.06365
.002088 .06658
29

Y = 3.4
Y = 3.5
9-
X
Y
T
4
X
Y
T
5.0
.15467 .008315
4.8
4.6
4.4
.14042 .007093
.12845 .006108
.11801 .005286
.I1392 4.8
.10685 4.6
.10036 4.4 .12102
.14583
.007473
.10854
.13241
.006369
.10167
.005473 .09535
4.2
.10869
.004585
.09430 4.2
.08859 4.0
.III04
.004722
.08943
.10210
.004080
.08383
4.0
.10023 .003979
.08314 3.8
.09395
.003525
.07849
3.8
.09246
.003449
.07792
3.6
.08646
.003040 .07336
3.6
3.4
64
33
.08525
.002983
.07289 3.4 .07949
.002614
.06842
.07852.002570
.06803 3.2
.07297
.002238
.06364
लंलं तं
3.2 .07218
.002205
.06332
3.0
.06683
.001905
.05901
3.0 .06619 .001881
.05875 2.8
.06102
.001611
.05450
2.8
.06051 .001592
.05428 2.6
.05550 .001350
.05010
2.6
.05508
.001337
.04993 2.4
.05023 .001120
.04581
2.4
.04990
ΟΙΙΙΟΟ
.04567 2.2
.04519.000918
.04161
2.2
.04493
.000910 .04150 2.0
.04036 .000740
.03750
2.0
1.8
1.6
.04015 .000735
.03555 .000582
.03III .000451
.03742 1.8
.03571 .000586
.03347
.03340 1.6
.03122 .000453
.02952
.02946 1.4
.02689.000340
.02563
1.4
1.2
1.0
.02681 .000338
.02265.000244
.01861 .000166
.02559 I.2
.02270 .000244
.02180
.02178 Ι.Ο
.01865
.000166
.01804
.01802
0.8
.01471
.000105 .01433
0.8
.01468 .000104
.01432 0.6
.01088
.000058 .01067
0.6
.01087 .000058
.01067 0.4
.00716
.000025 .00707
0.4
.00715 .000025
.00707+.2
.00354
.000006 .00351
+.2
.00353 .000006
.00351 0.0
.00000
.000000
.00000
0.0
-.2
.00000
.000000
.00000
2
.00345 .000006
.00347 0.4
.000024
.00345 .000006 .00347
.00682
.00690
0.4
.00682 .000024
.00690 0.6
ΟΙΟΙΙ
.000052
.01029
0.6
.01012
.000052
.01029 0.8
.01333
.000092 .01365
0.8
.01335 .000092
.01365 Ι.Ο
.01649
.000141
.01696
1.0
.01651
.000141
.01697 1.2
.01958 .000201
.02025
1.2
.01961
.000201
.02026 I.4
.02260.000269
.02350
I.4
.02265.000271
.02352 1.6
.02558 .000347
.02672
H H
1.6
.02563
.000349
.02675 1.8
.02849
.000434
.02991
1.8
.02856.000436
.02995 2.0
.03135
.000529
.03307
2.0
.03144 .000531
.03312 2.2
.03417
.000632
.03621
2.2
.03427
.000635 .03626 2.4
.03693.000743
.03932
2.4
.03705
2.6
तं लंलं
∞ 0 2
3 3
468
.05031 .001416
.05285 .001571
.05535 .001733
.000747 .03938 2.6
.04247 2.8
.03978.000866
2.8 .04248
3.0
3.2
3.4
3.6
3.8
.000993 .04554
.04513 .001127 .04859
4.0 .05781 .001901
4.2 .06024 .002075
.06264.002255
4.4
.03965.000861
.04240
.04233.000988
.04546
.04774 .001268 .05161
333
3.0
3.2
024
.04496
IZIIOO'
.04850
.04756 .001261
.05151
3.4 .05012 .001408
.05450
.05461 3.6 .05264 .001562
.05747
.05759 3.8 .055II .001723
.06042
.06055 4.0 .05755
.001890
.06335
.06349 4.2 .05996
.002063
.06626
.06641 4.4
.06234 .002242
.06915
.06931 4.6
.06469 .002427
.07202
30

y = 3.6
Y = 3.7
9-
4.6
4.4
4.2
X
.13697 | .006674
.I2439 .005683
.11361.004872
Y
T
9-
X
Y
T
.10314 4.6
.14235 .007040 .10480
.09650 || 4.4
.09035 4.2
.12820
.005925 .09775
.11645 .005041 .09133
ww3
4.0
.10410 .004190
.08457 4.0
3.8
.09554.003606
.07909
3.8
3.6
.08772 .003101
.07386 3.6
986
.10627
.0043II
.08536
.09723
.003695
.07973
.08906
.003166
.07437
3.4
.08050.002659
.06883 3.4
.08157
.002708
.06925
3.2
.07378 .002272
.06398 3.2
.07463
.002308
.06432
3.0
.06748
.001931
.05928 3.0
.06816
.001957
.05956
2.8
2.6
86
.06154 .001630
.05472 2.8
.05592 .001364
.05029 2.6
86
.06208 .001649
.05495
05635
.001378
.05047
2.4
.05056 .001130
.04596 2.4
.05090 .001140
.04611
2.2
.04545 .000925
.04173 2.2
.04572
.000932
.04185
2.0
.04056.000745
.03760 2.0
.04076 .000750
.03769
1.8
.03586.000590
.03354 1.8
.03602.000593
.03362
HHH
1.6 .03134 .000455
.02957 1.6
.03145 .000457
.02962
1.4
.02698 .000341
.02567 I.4
.02706 .000342
.02571
1.2
.02276.000245
.02183 I.2
.02282
.000246
.02186
1.0
.01868 .000167
.01806 I.O
.01872
.000167
.01808
0.8
.01473 .000105
.01434 0.8
.01476
.000105
.01435
0.6
.01089.000058
.01068 0.6
.01091
.000058
.01069
0.4
.00716 .000025
.00707 0.4
.00717
.000025
.00708
+.2
0.0
.00354
.000006
.00000 .000000
.00351 +.2 .00354
.000006 .00351
.00000 0.0
.00000
.000000
.00000
.2
0.4
0.6
46
.00345 .000006
.00681
.00347 .2
ΟΙΟΙΟ
0.8
1.2
.000024
.000052
.01332 .000092
1.0 .01646 .000141
.01954 .000200
HHH
46∞
I.4
1.6
1.8
.02256 .000269
.02552 .000346
.02842 .000432
468
.02347 I.4
.02669 1.6
.02987 1.8
.01951 .000199 .02021
.02252 .000268 .02346
.02546 .000345 .02666
.02835 .000431 .02984
.00690 0.4
.01028 0.6
.01363 0.8 .01330
.01695 1.0 .01644
.02023 I.2
.00345
.00681 .000024
.01009
.000006
.00347
.00690
.000052 .01028
.000091 .01363
.000140 .01694
2.0
2.2
.03127 .000527
.03407 .000629
2.4 .03682 .000740
.03303 2.0
.03119 .000525
.03299
.03615 2.2
.03397 .000627
.03611
.03925 2.4
.03670.000737
.03920
~ ~ m m m m
2.8
3.0
.04479
911100*
2.6 .03952 .000858 .04233 2.6
.04218 .000983 .04538 2.8
.04840 3.0
.03939 .000854 .04226
.04203 .000978
.04530
.04463
ΟΙΙΙΟΟ
.04832
3.2
3.4
3.6
3.8
246
.04737 .001255 .05140 3.2
.04719 .001249
.05131
.04991 .001401 .05438 3.4 .04971 .001394
.05428
.05241 .001554
.05734 3.6 .05219 .001546
.05723
4.0
∞ 0 2
.05487 .001713
.05729 .001878
.06028
3.8 .05463 .001704
.06016
4.2 .05968 .002049
4.4
4.6
4.8
468
.06204 .002226
.06898 4.4
.06437 .002409
.06667 .002598 .07468 4.8
.07184 4.6
468
.06175
.06320 4.0 .05704 .001868
.06610 4.2 .05941 .002038 .06594
.002214 .06881
.06406.002396
.06634 .002583 .07449
.06306
.07166
31

Y = 3.8
y = 3.9
9-
X
Y
T
X
Y T
0
4.4
.13261 .006209
4.2
11962.005231
4.0
.10864 .004444
3.8
3.6
3.4
6+
.09905 .003790
.09048
.08268
.003235
.002758
.09915 4.4 .13783
.09240 4.2 .12321
.08619 4.0
.III26
.08039 3.8 .IOI02
.07491
3.6
.06969 3.4
.006551
I
.10072
.005449 .09357
.004592
.08709
.003894
.08110
.09199
.003310
.07548
.08387 .002813
.07014
3.2
.07552 .002345
.06468 3.2
.07645 .002385
.06504
3.0
.06886 .001984
.05985 3.0
.06960 .002013
.06015
2.8
.06264 .001669
.05519 2.8
.06322.001690
.05542
2.6
.05679 .001393
.05066 2.6
.05724 .001408
.05085
2.4 .05125 .001151
2.2 .04599 .000939
.04626 2.4
.05160 .001162
.04641
.04197 2.2
.04626.000947
.04209
2.0
.04097 .000755
.03778 2.0
1,8
.03617.000596
.03369 1.8
1.6
.03157 .000460
.02968 1.6
986
.04118 .000761
.03788
.03633 .000600
.03376
.03169.000462
.02973
1.4
1.0
0.8
0.6
.02714 .000344
1.2 .02288
.000247
.01876 .000168
.01478 .000105
.01092
.02574 1.4
.02723 .000345
.02579
.02189 1.2 .02294 .000248 .02191
.01809 1.0 .01880
.01436 0.8
891000*
.01811
.01480 .000105 .01438
0.4
.00718
.000058
.000025
.01069 || 0.6
.00708 0.4
.01093
.000058
.01070
.00718
.000025
.00708
+.2
Ι.Ο
I.2
.00354
0.0 .00000 .000000
-.2 .00345 .000006
0.4
.00681
.000024
0.6 .01008 .000052
0.8
.01329 .000092 .01362 0.8
.01641 .000141 .01693 1.0 .01639
.01948 .000199 .02020 1.2 .01944
.000006 .00351 +.2
.00354
.00000 0.0 .00000
.00347 .2 .00344
.000006 .00352
.000000
.00000
.000006
.00347
.00689 0.4
.00680 .000024
.00689
.01028 0.6
.01007 .000052
.01027
.01327 .000091 .01361
I.4
.02247 .000267
1.6
.02541 .000344
1.8
.02828 .000430
2.0 .03III .000523
.000140
.000198 .02017
.02343 1.4 .02242
.000266 .02340
.02663 1.6
.02535 .000342
.02660
.02980 1.8 .02821 .000427 .02976
.03294 2.0 .03102 .000521 .03289
.01691
2.2
.03388.000625
.03605 2.2
2.4
.03660.000734
2.6
.03927 .000850
.03913 2.4
.04219
2.6
246
.03377 .000622
.03600
.03648 .000730
.03907
.03913
.000846
.04212
तं लंलं
2.8
.04189 .000974
.04522 2.8
3.0
.04448 .001105
.04823
3.0
∞ O
.04175 .000969
.04514
.04431 .001099
.04814
3.2
www
3.4
.05417
3.6
.05198 .001537 .057II
3.6
3.8
.05440.001694
.06002
3.8
46∞
333
3.4
.04702 .001242 .05121 3.2
.04952.001386
.04931 .001378 .05406
.05176 .001528 .05699
.001684 05989
.04683 .001235
.05III
.05417
4.0
.05678 .001857 .06291 4.0
.05654
.001846
.06277
4.2
.05913 .002026
.06578 4.2 .05888 .002014 .06563
4.4
.06145 .002200 .06864 4.4
.06118
.002187
.06847
4.6
.06374.002380
.07148 4.6
.06345
.002366
.07130
4.8
.06600
.002566
.07430
4.8
.06569 .002550
.07411
5.0
.06824
.002757 .07710
5.0
.06790
.002739
.07690
32

Y = 4.0
Y = 4.1
9-
X
Y
T
9
X
Y
T
4.2
4.0
.I2735 .005704
.11417 .004759
.09487 4.2
.08807 4.0
3.8
.10317 .004008
.08186
3.8
20∞
.13224 .006010
.09634
.11746 .004949
.08914
.10552
.004135
.08268
3.6
.09362.003390
.07608
3.6
.09537 .003478
.07671
3.4
.08512 .002870
.07062 3.4
.08645 .002932
.07112
3.2
.07743 .002426
.06543 3.2
.07845
.002470 .06583
3.0
.07036
.002044
.06046 3.0
2.8
.06382
.001712
.05567 2.8
2.6
.05771
.001424
.05105 2.6
986
.07116
.002075
.06078
.06444 .001735
.05593
.05819
.001440 .05125
2.4
.05197
.001173
.04657 2.4
.05235 .001185
.04673
21.21
.04655 .000955
2.0
.04140.000766
.04221 2.2
.03797 2.0
.04684 .000963
.04234
.04162.000772
.03807
1.8
.03650.000604
.03383
1.6
.03181
.000464
.02979
I.4
.02732 .000347
.02582
HHH
1.8
.03666.000607
.03391
1.6
.03193
.000467
.02984
I.4
.02740.000348
.02587
1.2 .02300 .000249
Ι.Ο .01884 .000169
0.8 .01483 .000106
0.6 .01095 .000058
.02194 I.2
.02306.000250
.02197
.01813 1,0
.01888 .000169
.01815
.01439 0.8
.01485 .000106
.01440
.01071 0.6 .01096
.000058
.01071
0.4
+.2
.00719 .000025
.00354
.000006
.00708 0.4 .00719
.000025
.00709.
.00352 +.2 .00354
.000006
.00352
0.0
.00000
-.2
.00344
.000000
.000006
.00000 0.0 .00000
.00347 -.2
.000000
.00000
.00344
.000006
.00347
0.4
.00680 .000024
.00689 0.4
.00679 .000024
.00689
0.6 .01006 .000052
.01027 0.6
.01006
.000052
.01026
I.4
1.6
46
0.8
.01325 .000091
1,0 .01636 .000140
I.2 .01941 .000198
.02238 .000266
.02529 .000342
.01360 0.8
.01690 1.0 .01634 .000139
.02016
.01324 .000091
.01360
.01689
.02338 1.4
1.2 .01937
.02234 .000265
.000198
.02014
.02336
.02657 1.6
.02523 .000341
.02654
1.8
2.0
∞ O
.000427
2.2
2.4
2.6
2.8
3.0
3.2
3.4
.02814
.03094 .000519
.03368 .000620
.03637.000728 .03901 2.4
.03901 .000843 .04205
.04160.000965 .04506
.04415 .001094 .04805
.04666 .001229 .05IOI
.04913 .001371 .05395
.02972 1.8
.02807
.000425
.02969
.03284 2.0
.03085 .000517
.03281
.03594 2.2
.03358.000617
.03590
तं तं
2.6
2.8
468
.03626
.000725
.03896
.03888 .000839
.04199
.04146.000961 .04499
333
3.0
3.2
3.4
924
.04399
.001089
.04797
.04649 .001224
.05092
.04893 .001365
.05385
3.6
3.8
4.0
.05156 .001520
.05395 .001675 .05976
.05629 .001836 .06263
.05687 3.6 .05134 .001512
.05676
3.8
.05372 .001666
.05964
4.0
.05605 .001825
.06250
4.2 .05860
4.4
.06088
.002002 .06548 4.2
.002174 .06831
4.4
4.6 | .06313
.002351 .07112
4.6
246
.05835 .001990 .06534
.06062
.002160 .06816
.06286.002336 .07096
∞ 0 2
4.8
5.0
.06756
5.2 .06973
.06536.002534 .07392
.002722 .07670 5.0
.002915 .07946

4.8
.06506
.002517
.07374
.06723 .002704
.07651
5.2
.06938 .002896
.07926
33
4

r = 4.2
Y = 4.3
9-
X
Y
T
9-
X
Y
T
0
4.0
3.8
.12122 .005169
.10813 .004276
.09032 4.0
.12561
.005431
.09165
.08355 3.8
3.6
.09726 .003573
.07738 3.6
.11103 .004435
.08450
.09932.003677
.07810
3.4
3.2
mmm
420
.08787
.07953 .002517
.002998
.07165 3.4
.08938.003069
.07220
.06624 3.2
.08067 .002566
.06667
3.0
.07199 .002109
.06110 3.0
.07286
.002143
.06144
2.8
2.6
2.4
∞64
.06509 .001759
.05619 2.8
.06575
.001783
.05646
.05870 .001457
.05146 2.6
.0592,I
.001474 .05167
.05273 .001197
.04689 2.4
.05312
.001209
.04706
2.2
.04713
.000972 .04247 21.2
.04743
.000980
.04260
2.0
.04185.000778
.03817 2.0 .04207 .000783
.03827
1.8
.03683
119000*
.03398 1.8 .03700
.000615
.03406
1.6
.03205
.000469
.02990 1.6
.03218
.000472
.02996
1.4
.02749 .000350 .02591 I.4 .02758
.000351
.02595
1.2
.02312 .00025I
.02200 1.2
.02318
.00025 I
.02203
1.0
.01892 .000170
.01817 1.0
.01896
.000170
.01819
0.8
0.6
.01487 .000106
.01097
.000059
.OI441
0.8
.01072 0.6
.01490
.000106
.01442
.01098 .000059
.01072
+.2
0.4 .00720
.000026
.00354 .000006
.00709 0.4
.00720 .000025
.00709
.00352+.2
.00354 .000006 .00352
0,0
.00000
.000000
.00000 0.0
.00000
.000000
.00000
.2
0.4
0.6
.00344
.00679 .000024
.01004 .000052
.000006
.00347 .2
.00689 0.4
.01026
.00344
.000006
.00347
.00678
.000023 .00688
0.6
.01004 .000052
.01025
0.8
.01322
.000091
1.0
.01631
I.2
.01934
.000139
.000197
.01359 0.8
.01687 1.0
.01629
.02012 1.2 .01930
.01320 .oooogo
.01358
.000139
.01686
.000197
.02010
I.4
.02229 .000264 .02333 1.4
.02225
.000263
.0233I
1.6
.02518 .000340
.02651 1.6
.02512 .000339
.02648
1.8
.02800 .000424
.02965 1.8
.02794
.000422
.02961
2.0
.03077 .000516
.03276 2.0
.03070
.000514
.03271
2.2
.03348.000615
.03584 2.2
.03339 .000612
.03579
2.4
.03614 .000722
.03889 2.4
.03604 .000719
.03883
2.6
.03875.000836
.04192 2.6
.03863.000832
.04184
2.8
.04132
.000957
.04491 2.8
3.0
.04384
333
3.2
.04631
3.4 .04874
.001219
.001359
3.6
.05114
.001505
3.8
.05349
.001658
4.0
4.2
4.4
4.6
4.8
.001085 .04788 3.0
.05082 3.2
.05374 3.4 .04856
.05664 3.6 .05094
.0595I 3.8
.05581 .001816 .06236 4.0 .05558 .001805
.05809.001979 .06519 4.2
.05784 .001967
.06034 .002148 .06800 4.4 .06007 .002135 .06784
.06256 .002322 .07079 4.6
.06227.002308 .07062
.06475 .002502 .07356 4.8
.06444 .002486
.06504
∞ o
.04118
.000952
.04483
.04368
.001079
.04779
.04614
.001212 .05072
.001351
.05363
.001497
.05652
.05328
.001648
.05938
.06222

.07338
5.0
5.2
.06691 .002687
.06904 .002877
5.4 .07114
.07632 5.0
.07906 5.2 .06869
.08178
.003072
5.4
.07077
.06658
.002670
.07612
.002859
.07885
.003052
.08156
34

Y = 4.4
y = 4.5
9-
X
Y
T
X
Y
T
0
0
www
3.8
.11434
.004621
3.6
3.4
www
3.2
3.0
2.8
2000
.IO159 .003794
.09102 .003147
.08188 .002620
.07378
.002180
.06645 .001809
.07278 3.4
.08554
.11816 .004833
3.8
.08671
.07886 3.6 .10409 .003923 .07969
.09278 .003230 .07340
.06712 3.2
.06180
.05673 2.8
.08317 .002676
.06760
3.0
.07474 .002219
.06216
.06718.001836
.05702
2.6
.05974 .001493
.05189
2.6
.06030
.001511
.05212
2.4
.05354 .001222
.04723 2.4
.05396 .001234
.04741
2.2
.04775 .000989
.04273 2.2
.04807.000998
.04287
2.0
1.8
.04231 .000789
.03837 2.0
.04255 .000795
.03848
.03718.000619
.03414 1,8
.03736.000623
.03422
1,6
.03231 .000474
.03001 1.6
.03244 .000477
.03007
I.4
.02767.000353
.02599 I.4
.02777
.000354
.02603
1.2
.02324 .000252
.02206 1.2
.0233I
.000253
.02209
I.O .01900
.000171
.01821 1.0
.01904
.000171 .01823
0.8
.01492
.000107
.01443 0.8
.01495
.000107 .01445
0.6
.OIIOO
.000059
.01073 0.6
ΟΙΙΟΙ
.000059 .01074
0.4
.00721
.000025
.00709 0.4
.00721
.000026
.00710
+.2
.00355
.000006
.00352 +.2
.00355
.000006
.50352
0.0
-.2
.00000 .000000
.00344
.000006
.00000 0.0
.00347
.00000 .000000
.00000
.00344 .000006
.00346
0.4 .00678 .000024
.00688 0.4
.00678.000023
.00688
.01002
.000052
.01025 0.6
.000090
.01357 0.8
1.0
I.2
1.4
0.6
0.8 .01318
.01626 .000139
.01685 1.0
.01927 .000197 .02009 1.2
.000196 .02007
.02220 .000263 .02329 1.4 .02216 .000262 .02326
.01002
.000052
.01024
.01317 .000090
.01356
.01624 .000138 .01683
.01924
1.6
1 2
1.8
2.0
.02507 .000338 .02645
.02787 .000421 .02958
.03061 .000512 .03268
H H
1.6
.02501 .000337 .02642
1.8
.02781 .000420
.02954
2.0 .03054 .000510
.03263
2.2
.03330 .000611
.03574
2.4 .03593 .000717 .03878 2.4
2.6 .03851 .000830 .04178 2.6
2.2
.03321
.000608
.03569
.03583 .000713
.03872
.03839.000825
.04171
23 m
2.8
3.0
3.2
∞02
.04105 .000949 .04476 2.8
.04091 .000944
.04468
.04598
www
3.4
3.6
3.8
.05306
.04353 .001075 .04771
.001207
.04838 .001346 .05353
.05074 .001490
.001640
.05063
ww
3.0
.04338 .001069
.04762
3.2
.04581
.001201
.05054
.05641
.05926 | 3.8
www
3.4
3.6
468
.04820.001338
.05343
.05054
.001482
.05630
.05285.001631
.05914
4.0
.05535 .001796
.06209
4.0
.05512.001785
.06196
4.2
.05760 .001957
.06490 4.2 .05735 .001945
.06476
יד
4.4
.05982 .002123
.06769 || 4.4
.05955 .002110
.06754
4.6
4.8
6∞0
555
5.2
5.4
5.6
246
5.0
.06200 .002295
.06415 .002472
.06627 .002654
.06836 .002840
.07043 .003031
.07247 .003227
.07594
.07865 5.2
.07046 | 4.6
.06172 | .002281
.07030
.07321
4.8
.06386
.002457
.07304
5.0
.06596 .002638
.07576
.06804.002824 .07846
.08134 5.4
.08402 5.6
.07009
.07211 .003208
.003014
.08114
.08381
35

y = 4.6
y = 4.7
X
Y
T
9-
X
Y
T
www
3.8
3.6
.12268 .005092
.10690.004070
.08803 3.6
.I1009 .004240
.08158
.08060 3.4
.09679.003425
.07476
3.4 .09470 .003323
.07406 3.2
.08601 .002803
.06862
3.2
.08454
3.0
.07575
2.8
.06794
.002737
.002260 .06255 2.8
.001864 .05732 2.6
.06810 3.0
.07682
.002304
.06295
.06873
.001894
.05763
.06146 .001551
.05258
2.6
.06087 .001531
2.4 .05439 .001248
.05235 2.4
.05483.001262
.04777
.04759 2.2
.04872 .001017
.04315
2.2
.04839 .001007
.04301 2.0
.04304 .000808
.03869
2.0
.04280.000802
.03858 1.8
.03772.000631
.03438
H H
1.8
1.6
.03754.000627
.03257.000480
.03430
1.6
.03013 1.4
.03270 .000482 .03019
.02795
.000358
.02612
1.4 .02786 .000356
.02607 I.2
.02343
.000255 .02215
1.2
1.0
.02337 .000254
.02212 1.0
.01912
.000172 .01827
.01908
.000172
.01825 0.8
.01499
.000108
.01447
0.8
.01497
.000107
.01446 0.6
.01103
.000059
.01075
0.6 .01102
.000059
.01075 0.4
.00722
.000026 .00710
0.4
.0072,2
.000026
.00710 +.2
.00355
.000006
.00352
+.2
.00355
.000006
0.0
.00000 ,000000
.00352 0.0
.00000
.00000 .000000
.00000
.2
.2
.00344
.000006
.00346 0.4
.00344 .000006
.00677
.00346
.000023 .00687
0.4
.00677.000023
.00688 0.6
00010'
.000052
.01023
0.6
ΙΟΟΙΟ
.000052
.01024 0.8 .01314 .oooogo
.01354
0.8
.01315
.oooogo
.01355 Ι.Ο .01619
.000138
.01681
1.0
I.4
I.2 .01920
.02212
.01622 .000138
.000196
.01682 I.2 .01917
.000195
.02004
.02005 I.4 .02207
.000261
.02322
.000262
.02324 1.6
.02490 .000335
.02636
1.6
.02496
.000336
.02639 1.8
.02767
.000417
.02947
1.8
.02774 .000419
.02951 2.0
.03038
.000507
.03255
2.0
.03046.000509
.03259 2.2
.03303 .000604
.03559
2.2
.03312 .000606
.03564 2.4
.03562.000708
.03860
2.4
2.6
.03572 .0007II
.03828
2.8 .04078 .000941
.03866 2.6
.03816 .000819
.04158
.000823
.04164 2.8
.04065 .000936
.04453
.04460 3.0
.04309
.001060
.04745
ww
3.0
.04324
.001065
.04754 3.2
3.2
.04565
.001196
.05044 3.4
.04549
.04784 .001325
.001189 .05035
.05322
www
3.4
.04802
.001332 .05332
3.6
.05035
3.8
.05264
.001475 .05618
.001623
33
3.6
3.8
6∞
.05016 .001467
.05607
.05243 .001614
.05890
.05901 4.0 .05467 .001766
.06170
4.0
.05490 .001777
.06182 4.2
4.2
.05712 .001936
.06461 4.4
4.4
.05931 .002100
.06738 4.6
246
.05687 .001923 .06448
.05904 .002086 .06724
.06117 .002254 .06998
4.6
.06146.002269
4.8
5.0
.06357 .002443
.06565 .002621
.07013
.07286
4.8
5.0
.07557 5.2
∞0 2
.06327
.002427
.07270
.06534 .002606
.07539
.06738 .002789
.07807
555
5.2
5.4
5.6
246
.06771 .002804
.06974 .002992
.07174 .003185
.07826 5.4
.08093 5.6
.07138 .003167
.08359 5.8 .07334
.06939 .002976
.08073
.08338
.003362
.08601
36

Y = 4.8
Y = 4.9
9-
X
Y
T
X
Y
T
3.2
333
642
3.6
3.4
.09911
.11380.004438
.08760.002874
.08269 | 3.6
.11823
.004679
.08395
.003539
.07552 3.4
.06917
.10171
.003667
.07635
3.2
.08932
.002951
.06977
3.0
.07795 .00235I
.06336 3.0
.07915 .002400
.06380
2.8
.06955 .001925
.05794 2.8
.07043 .001958
.05828
2.6
.06207
.001572
.05282 2.6
.06272 .001595
.05308
2.4
.05529
.001276
.04796 2.4
.05577
.001291
.04816
2.2
2.0
.04907 .001026
.04330 .000814
.04329 2.2
.04942 .001036
.04344
.03880 2.0
.04356 .000821
.03892
1.8
.03791 .000635
.03446 1.8
.03810.000640
.03455
1.6
.03284.000485
.03025 1.6
.03297 .000487
.03032
1.4
.02805 .000359
.02616 I.4
.02814 .000361
.02621
I.2
1.0
.02350
.000256
.02217 I.2
.02356
.000257
.02221
.01916 .000173
.01828 1.0
.01921 .000173
I
.01831
0.8
.01502 .000108
.01448 0.8
.01505
.000108
.01449
0.6 .01105 .000059
.01076 0.6
.01106
.000059
.01076
0.4
.00723
.000026
.00710 0.4
.00724
.000026
.00711
+.2
.00355 .000006
.00352 +.2
.00355
.000006 .00352
0.0
.00000
.000000
.00000 0.0
.00000 .000000
.00000
.2
0.4
.00343 .000006
.00676 .000023
.00346 -.2
.00687 0.4
.00343 .000006 .00346
.00676.000023
.00687
0.6
.00999 .000051
.01023 0.6
.00998 .000051
.01022
0.8
.01312 .oooogo
.01353 0.8
.0131I
.oooogo
.01353
1.0
.01617 .000138
.01679 1.0
.01615
.000137
.01679
H H
I.2 .01913 .000195
.02001 I.2
.01910
.000194
.02000
1.4
.02203 .000260
.02319 1.4
.02199
.000260 .02317
1.6
.02485 .000334
.02633 1.6
.02480
.000333
.02631
1.8 .02761 .000416
.02943 1.8 .02754
.000415
.02940
2.0
.03030 .000506
.03250 2.0
.03022 .000504
.03247
2.2
.03293 .000602
.03554 2.2
.03285.000600
.03549
2.4
.03551.000706
03854 2.4
.0354I .000703
.03849
2.6
.03804.000816
2.8
.04052 .000933
.04151
2.6
.04446 2.8 .04039
.03792
.000813
.04145
.000929
.04439
333
3.0
.04294 .001056
3.2
3.4
3.6 .04997 .001460
.04737 3.0
.04533 .001185 .05026 3.2
.04767 .001320 .05312 3.4
.05596|3.6
.04280
.001051
.04730
.04517
.001179 *05018
.04749 .001313
.05303
.04978
.001453
.05586
3.8
.05223
4.0
.05445
.001607 .05877 3.8
.001758
.05202
.001598
.06157 4.0 .05423
.05867
.001748
.06145
4.2
.05663
.001914
.06434 4.2
4.4
.05878
.002075
.06709 4.4
4.6
.06982 4.6
5.0
5.2
5.4
5.6
5.8
∞ 0 2
.06090.002242
4.8 .06299 .002414 .07253 4.8
.06504 .002590 .07521 5.0
.06707 .002771 .07788 5.2
.06907 .002957
.07104 .003147
.07298 .003341
.06874 .002939 .08034
.08296
246
.05640 .001903
.06421
.05854 .002063 .06695
.06064
.002228
.06967
.06271.002399
.07237
.06475
.002575
.07504
.06676
.002755
.07770
.08053 5.4
.08316 | 5.6
.08578 5.8
.07070 .003127
.07263 .003319 .08557
37

Y = 5.0
Y = 5.0
9.
X
Y
T
9-
X
Y
T
3.4
.10465.003815
.07725 I.4
3.2
.09119
.003038
.07039 1.6
3.0
.08044
46∞
.02194 .000259
.02315
.02474 .000332
.02628
.002455
.06425 1.8 .02748 .000414 .02937
2.8
.07134
.001993
.05861 2.0
.03015 .000502
.03242
2.6
.06338 .001618
.05334 2.2
.03276.000598
.03544
2.4
.05626 | .001306
.04834 2.4
.0353I .000700
.03843
2.2
.04978 .001046
.04359 2.6
.03781.000809
.04139
2.0
.04383 .000828
.03902 2.8
.04026 .000925
.0443I
1.8
H H
1.2
.03829 .000644
1.6 .033II .000490
I.4 .02824 .000362
.02363.000258
.03463 3.0
.04266 .001046
.04721
.03037 3.2
3.6
.02625 3.4 .04732
.02224
246
.04501
.001174
.05008
.001307
.05293
.04959 .001446
.05575
1.0 .01925
.000174
.01832 3.8
0.8
.01507
801000*
0.6
.01108
.01451 4.0
.000059 .01077 4.2
∞0 2
.05182
.001590
.05854
.05401 .001740
.06131
.05616 | .001895
.06406
0.4 .00724
.000026 .007II 4.4
+.2
.00355 .000006
.00352 4.6
0.0
.00000
.000000
.00000 4.8
468
.05828.002054
.06679
.06037 .002218
.06950
.06243 .002387
.07219
.2
0.4
0.6
.00343
.00675 .000023
.00997 .000051
.000006
.00346 5.0
.06446 .002560
.07486
.00687 5.2
.06646
.002738
.0775I
.01022 5.4
.06843 .002920
.08014
0.8
1.0
I.2
.01309 .000089
.01612 .000137
.01907 .000194
.01352 5.6
.07037 .003107
.08275
.01677 5.8
.07228.003298
.08535
.01998 6.0
.07416 .003493
.08793
V. Po 3 tan tan ³4.

.0
.2
.4
.6
.8
000 000
.OIO 472
.020 945
.031 418
.041 893
1 2 3
456
789
I
.052 371
.104 805
.157 367
.062850
.073 333
.083 819
.094 310
.115 305 .125 811
.136 323
.146 842
.167 901
.178 442
.188 993
.199 553
.210 122
.220 702
.231 293
.241 895
.252 509
.263 136
.273 775
.284 428
.295 095
.305 777
.316 474
•327 186
.337 915
•348 661
.359 424
.370 205 .381 004
.391 823
.402 661
.413 519
.424 398
.435 299
.446 222
.457 167
.468 135
9
.479 126
.490 143
.501 184
.512 251
.523 344
ΙΟ
.534 463
•545 610
-556 785
.567 989
.579 222
II
•590 485
601 779
.613 104
.624 461
.635 850
I2
.647 273
.658730
.670 221
.681 748
.693 310
13
.704 910
.716 547
.728 222
•739 936
•751 690
38
Pç = 3 tan 4 + tan ¾q.

9-
.0
.2
.4
.6
8
456
HH
14
15
.763 483
.775 319
.787 196
.799 115
.811 078
.823 086
.835 138
.847 236
.859 381
.871 573
16
.883813
.896 103
.908 442
.920 832
.933 275
17
18
78
•945 769
.958 317
.970 919
.983 577
.996 291
1.009 062
1.021 891
1.034 779
1.047 727
1.060 736
19
1.073 807
1.086 941
1.100 139
I.I13 402
1.126 731
20
1.140 127
I.153 592
1.167 126
1.180 730
I.194 407
21
1.208 155
1.221 978
1.235 875
1.249 849
1.263 901
22
1.278 031
1.292 241
1.306 532
1.320 906
1.335 363
23
1.349 906
1.364 535
1.379 252
1.394 058
1.408 955
222
24 1.423 943
1.439 025
1.454 202
1.469 476
1.484 847
25
26
56
1.500 318
1.515 890
1.531 565
I.547 344
1.563 229
1.579 221
1.595 323
1.611 535
1.627 861
1.644 301
27
1.660 857
1.677 532
1.694 327
1.7II 244
1.728 284
28
1.745 450
1.762 745
1.780 169
1.797 726
1.815 417
29
1.833 244
1.851 209
1.869 315
1.887 564
I.905 959
30
1.924 501
1.943 193
1.962 038
1.981 038
2.000 195
33 3
31
2.019 513
2.038 993
2.058 639
2.078 452
2.098 437
32
2.118 596
2.138 931
2.159 446
2.180 143
2.201 026
33 2.222 098
2.243 361
2.264 820
2.286 477
2.308 337
333
34
2.330 401
2.352674
2.375 160
2.397 862
2.420 783
35
2.443 928
2.467 300
2.490 904
2.514 743
2.538 821
36
2.563 143
2.587 714
2.612 536
2.637 615
2.662 957
www
37
38
7∞
2.688 563
2.714 441
2.740 594
2.767 029
2.793 748
2.820 759
2.848 066
2.875 675
2.903 591
2.931 820
39
2.960 368
2.989 240
3.018 444
3.047 983
3.077 866
40
3.108 099
3.138 688
3.169 640
3.200 962
3.232 661
4I
3.264 745
3.297 220
3.330 095
3.363 377
3.397 074
42
3.431 194
3.465 746
3.500 739
3.536 180
3.572 079
43
3.608 446
3.645 289
3.682 618
3.720 444
3.758 776
44
3.797 624
3.837 000
3.876 914
3.917 378
3.958 402
45
4.000 000
4.042 183
4.084 962
4.128 352
4.172 365
46
4.217 014
4.262 314
4.308 277
4.354 920
4.402 257
47
4.450 304
4.499 074
4.548 585
4.598 854
4.649 898
48
4.701 734
4.754 380
4.807 854
4.862 177
4.917 366
49
4.973 442
5.030 427
5.088 340
5.147 206
5.207 045
50
5.267 881
5.329 737
5.392 639
5.456 611
5.521 681
51
5.587874
5.655 218
5.723 742
5.793 475
5.864 448
52
5.936 691
6.010 238
6.085 117
6.161 368
6.239 026
53
6.318 124
6.398 702
6.480 797
6.564 452
6.649 706
5505
54
6.736 602
6.825 183
6.915 496
7.007 588
7.101 509
55
7.197 304
7.295 031
7.394 739
7.496 488
7.600 333


56
7.706 333
7.814 550
7.925 047
8.037 894
8.153 153
57
58
59
78 9
8.270 898
8.391 203
8.514 142
8.639 795
8.768 244
8.899 574
9.602 605
9.033 869
9.171 225
9.311 731
9.455 492
9.753 175
9.907 314
10.065 136
10.226 756
39
VI. VALUES OF X, Y & T FOR INTERVALS OF 1º.

Y = 0.00
9-
X
Y
T
Y = 0.00
X
Y
T
60
1.73205 I.50000
1.73205
18
55
59
1.66428 1.38492
1.66428
17
58 1.60033 1.28054
1.60033
16
10∞ NO
HH
.32492 .052786 .32492
.046736 .30573
.28675 .04III2 .28675
.30573
២២ ទី២៨
57 1.53986 1.18559
56 1.48256 1.09899 1.48256 14
55 1.42815 1.01980 1.42815
|
13
1.53986
15
.26795 .035898
.26795
.24933 .031082
.24933
.23087.026650
.23087
54
1.37638
53
1.32704
52
1.27994
.947215 1.37638 12
.880524 1.32704 II
.819125 1.27994 ΙΟ
.21256 .022590
.21256
.19438 .018892
.19438
.17633 .015546
.17633
51
I.23490
.762486 1.23490
50
1.19175
.710138 1.19175
49
I.15037
.661674 1.15037
987
.15838 .012543 .15838
.14054 .009876
.14054
.12279 .007538
.12279
48
I.1 1061
.616730 1.11061
www www www ✡ ✡ * ☆ ☆ ☆ ☆±
47
1.07237
46
1.03553
.574987 1.07237
.536162 1.03553
654
.10510 .005524
10510
.08749.003827
.08749
.06993 .002445
.06993
45 1.00000
44 .96569
43 .93252
.500000 1.00000
.466278 .96569
42
.90040
4I .86929
.434792 .93252 +
.405363 .90040 O
.377830 .86929
321
.05241 .001373 .05241
.03492
.000610 .03492
.01746
.000152
.01746
Ooooo
Oooooo
00000
40
.83910
352044 83910
39
.80978
.327875 .80978
38
.78129
.305204 .78129
Y = 0.01

X
Y
T
37
-75355
.283922
.72654
75355
.263932 72654
0
.70021
.6745I .227481
.245145 .70021
.67451
5050
.64941
.62487
.60086
.210865 .64941
.195231
.62487
.180517 .60086
555
57
56
76
55
60 1.77949 1.55885 1.75552
59 1.70679 I.43539 1.68533
58 1.63857 1.32404 1.61928
1.57437 1.22324 1.55697
1.51380 1.13170 1.49806
1.45651 1.04831
1.44222
32 2
2 2 2
•
30 .57735 166667 .57735 54
29
.5543I
28
.53171
1.40219
153629 .5543I 53 1.35058
.141358
.97214
1.38919
.90238 1.33873
.53171 52 1.30144
.83834 | 1.29063
27
.50953 .129808 .50953
26
.48773 .I18942 .48773
577 57
5I
1.25458
.77941 1.24468
50
1.20980
.72508
I.20073
25
.4663I
.108721 .46631 49
1.16694
.67488 1.15861
24
.44523
23
22
.42448
.40403
.099114 .44523 48 1.12585
.ogoogo .42448 47 1.08640
.081619 .40403 46 1.04847
.62843
1.11820
.58536 1.07935
.54537
1.04196
21 .38386
.073676 .38386 45
1.01192
.50818 1.00593
20
.36397
19
.34433
.066237 .36397 44 .97669
.059281 .34433 53 .94267
.47355
.97116
.44127.
.93757
40

Y = 0.01
Y = 0.01
9-
X
Y
T
X
Y
T
42
.90978 .41114
.90507
41
.87795 .38297
.87361
40
.84710 .35662
.84309
456
~ ~ ~ ~ ~ ~ ~ ww www www www
96
.81718 .33195
.81347
.78812 .30883
.78469
.75986 .28715
.75670
789
.06988 .002443 .06990
.08741 .003823 .08745
.6 .10499
.005516 .10505
.12264 .007525 .12271
.14034 .009857 .14044
.15813 .012516
.15826
.73237
.2668I
.70559.24769
.67947.22974
.65398
.62908
.21287
.19701
.67698
.65169 13
12
.21211
.62697 14
.60473.18209
.58091 .168051
.55758.154847
.60279 15
.57913 16
.55594
.72944 ΙΟ .17602
.015509 .17618
.70290 II .19400 .018843 .19420
.022526 .21234
.23034 .026568 .23060
.24870 .030978 .24902
.26723 .035769
.26759
.53470
.142427
.53320 18
HHH
17
629
.28592 .040952
.28633
.30479
.046543 .30526
.32385
.052554
.32439
.51226
.130745
.51089 19
.34313
.059005
.34373
26
.49023
.119760
.48898 20
.36262
.065909
.36330
.46859.109433
.46744 21
38236
073290
.38311
.44729
.099732
.44626 22
.40236
.081168
.40319
23
.42635 .090623
.42541 23
.42263 .089565
.42355
.40572.082077
.40487 24
.44320.098508
.44421
21
.38539 .074067
.38462 25
.46407 .108022
.46519
20
.36534 .066569
.36465 26
48528.118139
.48650
19
.34555.059561
.34494 27
.50684 .128889
.50818
18
.32601.053021
.32546 28
.52878 .140309
.53024
16
15
II
NO HER HO a∞nont
17 .30669 .046931
.30621 29
.55III .152437
.55270
.28758 .041272
.28717 30
.57387.165314
.57560
14
.26868.036029 .26831
.24996 .031187 .24964
13 .23141
12
.026734 .23114 33
.21302 .022655 .21279 34
.19476 .018941 .19457 35
IO .17664 .015583
.17649 36
.15863 .012570 .15850 37
www.3
.59707.178985
.59896
.62075
.193499
.62281
.64494.208911
.64717
456
.66967.225279
.67208
.69497 .242668
.69758
.72087.261147
.72370
.14074 .009894 .14064
7
.12294 .007550 .12286
333
38
39
7∞ a
.74742
.280795
.75048
77465 .301694
-77796
.80262 .323939
.80619
6 .10521 .00553I .10516 40
.83135 .347631
.83521
5
4
.08757 .003832 .08752 4I
.06998 .002447
.86092 .372884
.86509
.06995 42
.89136 .399820
.89586
2
3 .05244 .001374
.03493 .000610
+ I .01746 .000152
.05242 43
.92274.428576
.92761
.03493 44
.01746 45
+
+
+
O
00000
000000
+
I
2 3
0
46
OOOOO 47
48
.01745 .000152 .01745 49
.03490
.05238
.000610 .03491 50
.001373 .05239 51
.95512 .459307 .96038
.98856.492182 .99426
1.02315 -527388 1.02931
1.05896.565132 | 1.06564
1.09608 .605648 | 1.10331
1.13461.649198
I.14245
1.17464.696076
1.17464 .696076 1.18315
1.21629 .746612 1.22554
41
5

γ
Y = 0.02
Y = 0.01
9.
X
Y
|
T
X
Y
T
1.25968.801177 | 1.26976
52
53
54
55
56
1.30496 .860189 1.31593 26
1.35226.924126 | 1.36424 25
1.40176.993530 1.41486 24
1.45363 1.069017 | 1.46799 23
57 1.50807 1.151292 1.52384 22
556
58 1.56531 1.241166 | 1.58267
1.62559 1.339565 | 1.64476
1.68919 1.447563 1.71042 19
59
60
Y = 0.02
20
18
17
16
.44939
1.26976 27
.51505
.131702 .51227
.49277
.I20594 .49024
.47090 .I10159 .46859
.100361 .44730
.42825 .091165
.42635
.40743 .082542
.40572
21
.38693.074464.
.38539
.36672.066906
.36534
.34678059845
.34555
HHH
∞ 76
.32710 .053259
.32600
.30765.047128
.30669
.28843 .041434
28759
9-
X
Y
T
H H
15
.26942.036161
.26868
14
.25060 .031293
.24996
13
.23196.026818
.23141
60
59
นา นา
96
58
1.83254 1.62567
1.75391 1.49213
1.68062 1.37250
57 1.61206 1.26485
པ་ག་
56 || 1.54771 1.16759 1.51460
1.48712 1.07941 1.45718
1.78117 12
1.70817 II
1.63971 ΙΟ
.21348 .022720
.21302
.19515 .018991
.19476
.17696 .015620 .17665
1.57533
55
или
54
1.42990 | .99918
1.40276
9∞ 76
53
1.37575 .92598
1.35107
5
.15889 .012597
.15863
.14094 .009913 .14074
.12309 .007563 .12294
.10532 .005539 .10521
.08765 .003836
.08756
52
1.32435
.85898
1.30187
4
.07003 .002450
.06998
55
51
1.27548 .79753
I.25494
32
.05247 .001376 .05244
119000°
50 1.22890 .74101 1.21012
49 1.18443 .68892
48 1.14188 .64082
47 1.IOIII .59632
46|| 1.06198|| .55508
.03494
.03493
1.16721 +1
.01746 .000153
.01746
+
+
1.12609
1.08660
1.04863
+
00000
Oooooo
00000
+
44
45 1.02436 .51679
.98814 .48119
1.01207
.97682
43
.95322 .44805 .94278
42
.91950 .41716 .90988
4I
.88692.38832
.87804
40
.85537.36138
.84718
1 2 3
456
I
.01745
.000152
.01745
.03489.000610
.0349I
.05235
.001372
.05238
.06983
.002440
.06988
.08734 .003818
.08741
.10488 .005508
.10499
mmm
39
.82481 .33618
.81725
38
.79516 .31259
.78818
37
.76636.29049
.75992
78 9
.12249
.007513
.12264
.14015
.009839
.14034
15788 | .012490
.15813
36
73836 .26977
.73241 ΙΟ
.17571
.015473
.17602
35
34
5.4
.7IIII .25032
.70563 II .19362
.018794
.19401
.68455 .23207
.67950 12 .21166 .022462
.21211
33
.65866 .21493
.65401
13
32
.63339.19883
.62911 14
31
.60869 | .18370
.60476
15
30
.58454.169469
.58093
29
.56091 .156094
.55759
28
.53775 .143521
.5347I 18
HHH!
16
17
34 in 6 7∞
.22981 | .026486
.23034
.24808.030875
.24871
.26651 | .035641
.26723
.28510
.040795
.28592
.30385.046352
.30479
.32279
.052324
.32386
42

Y = 0.02
Y = 0.03
9.
X
Y
T
X
Y
T
19
.34194 .058732
•34314 60
1.89266 1.70264
1.80950
20
21
.36129.065586
.38088 .072911
.36263 59
1.80671 1.55667
1.73318
.38237 58
1.72729 1.42703
1.66191
~ ~ ~ ~ ~ ~
22
23
.40072.080724
.40237 57
1.65355 1.31123
1.59513
.42082.089050
.42264 56 1.58476 1.20728
I.53235
24 .44119 .097912
.44320 55
1.52035 1.11354
1.47315
25
.46187 .107336
.46408 54
1.45983 1.02867
1.41718
26
27
.48287.117351
.48529 53
1.40278 .95155
1.36413
.50420.127989
.50685 52 1.34885.88125
1.31372
28
.52590.139284
.52879 51
H
1.29773 .81697
1.26574
2 3
29 .54797 .151271
.55113 50 1.24916 .75803
1.21996
30
.57045.163992
.57389 49 1.20291
70387
1.17621
333
31
32
33
1 2 3
.59336.177489
.59710 48
1.15877 .65397
I.13432
.61673 .191811
.62078 47
1.11657 .60792
1.09414
.64058
207008
.64497 46
1.07614 | .56532
1.05555
www www
34
.66495 .223138
.66971 45
1.03735 52585
1.01843
35
.68986.240262
.69501 44
1.00008
.48921
.98267
36
.71535 .258446
.72092 43
.96419 .45514
.94817
.74146.277765
.76822.298300
.74747 42
.92960.42344
.91484
.77471 4I
.89621 .39389
.88260
.79567 .320137
.80268 40
.86393 .36632
.85139
40
.82385 .343375
.83143
43
44
* * * * * * no
4I
42
.85282 .368121 .86100
.88262 .394495 .89145 37
༢ ༢གུུུུ
45
46
47
48
.91331
.94494 .452645
.97758.484721
1.01129 -519028
1.04614.555761
1.08221 .595136 | 1.09628 31
|
49 1.11960 .637396 | 1.13483
50 1.15838 .682811
1.17489
51 1.19866 731683 1.21658 28
| |
52 1.24054 .784351
.422624 .92284 36
.95523
.98870 34
333
987
64
333
39
.83269 .34057
.82113
38
.80242 .31648
.79175
.77305 .29395
.76321
.74452
.27283
.73545
35
.71678.25304
.70842
.68977
.23447
.68208
1.02330 33
1.05914 32
321
.66346 .21705
.65638
.63780 .20070
.63128
.61274
.18535
.60676
N NW
30
29
.58825.170923
.58277
96
.56430.157371
.55927
.54085.144639
.53625
53
54
1.26002 27
1.28415 .841196 I.30534 26
1.32961.902648 1.35270 25
.51788 | .132679
.51367
.49535
.121445
.49152
.47324
.110898
.46976
55
1.37707 .969193 1.40226 24
56 1.42667 1.041380 1.4542I 23
57 1.47859 1.119831 1.50875 22
.45152
.IO1000
.44836
.43017 .091716
.4273I
.40917.083014
.40659
58
59
60
1.53299 1.205257 1.56611
1.59010 1.298469 | 1.62654 20 .36811 .067249
1.65012 1.400388 1.69032 19 .34802
21
.38849 .074867
.38617
.36603
.060134
.34617
HHH
9 26
18
.32820.053501
.32655
17
.30863.047329
.30717
16
.28928
.041599
.28801
15
14
13
i43
.27016 | .036295
.26905
.25124 .031401 .25028
.23251.026903
.23168
43

Y = 0.03
= 0.03
γ
9-
X
Y
T
X
Y
T
12
.21394
.022786
.21325
II
.19553 .019041
.19496
IO
.17727 .015657
.17681
333
34
35
36
456
.66035
.68490
.221054
.66738
.237922 .69249
.70999 | .255823
.71820
98 7
.15914 .012624
.14114 .009932
.12324 .007575
.15876 37
.73567 .274828
•74454
.14084 38
.76197 .295012
.77154
.12301 39
.78893 .316461
.79926
6
.10543 .005547
.10527 40
.81659 | .339267
.82774
5
4
.08772
.07008 .002452
.003841
.08760 41
.84500 .363530 .85702
.07000 42
.87419 .389362
.88716
32
.05249 .001376
.03496.000611
.05245
.90423
43
.03494 44
.416885
.91822
.93515 .446235
.95024
-I .01747 .000152
.01746 45
.96702
.477561
.98331
о
+
+
+
46
.99991
.511027
1.01749
00000
000000
00000
47
1.03386 .546814
1.05286
+
1 2 3
I
.01745
.000152 .01745 49
.03488 .000609 .03490 50
.05232 .001371 .05237
업​영웅 ​fo
48
1.06897.585125
1.08950
1.10529 .626186
1.12750
I.14291
.670247
1.16696
51
1.18193
.717586
1.20799
456
78 9
.06978 .002438 .06985 52
.08726 .003814 .08738 53
.10477 .005501 .10494 54
.12233 .007501
.13995.009820
.15763 .012463
I.22243
.768515
1.25070
1.26452 .82338 2
1.29523
1.30832.882577
1.34171
.12256 55
.14025 56
567
I.35393 .94654!
1.39031
1.40150 1.015767 1.44118
.15801 57 1.45116 1.090812
1.49453
ΙΟ .17540 .015437
II
12
.21121
.19325 .018746
.022399
.17587 58 1.50307 1.172306
.19382 59 1.55738 1.260959 1.60949
.21189 60 1.61429 1.357583 1.67159
1.55055
кни
13
.22928 .026405
.23007
14
.24747 .030773
.24840
15
.26580035514
.26687
Y = 0.04
HHH
16
.28428.040640
.28551
17
.30292 .046163
.30432
X
Y
T
18
.32174.052098
.32333
19
.34076.058462
.34255
60 1.96194 1.79302
1.84116
20 .35998.065267
.36197 59 1.86669 1.63124
1.76082
21
.37942.072536
.38163 58
1.77967 1.48920
1.68623
22
.39910.080287
.40155 57
1.699631.36351
1.61666
23
.41903 .088543
.42174 56
1.62556 1.25157
1.55151
24
.43923
.097325
.4422I 55
1.55667 1.15130
I.49029
25
.4597I .106662
.46299 54
1.49231 1.06105
1.43256
26
.48050.116579
.48410 53
1.43195 .97945
I.37799
27
.50162.127107 .50555 52
1.37514 .90540
1.32626
28
.52308
29
.54490
30
.138279
.150130 •54958 50
.56712 .162699 .57220 49
.52737 51
1.32150 .83795
1.277II
1.27071 .77632
1.23030
1.22249 .71985
1.18563
31
.58974.176029 .59527 48
1.17661.66798
1.14292
32
.61281 .190164 .61880 47
1.13285.62022
33
.63633 205154 .64282 46
1.09102 | .57614
1.10201
1.06275
44

Y = 0.04
Y = 0.04
9-
X
Y
T
9-
X
Y
T
42
4I
38
N N N N NW www www www ☆ ☆ ☆ ££*
45
1.05097 -53538
44 1.01256 .49762
43
.97564 .46258
.940II
1.02503
.98873
.95373
.43002
.91996
.90587.39971
.88731
40
.87281 .37147
.85572
.84085 .34513
.82511
.80992 .32052
.79542
37
.77995 .29752
.76659
123
456
78 9
I
.01745 .000152
.01745
.03487.000609
.03490
.05230 .001370
.05235
.06973
.002435
.06983
.08719
.003809
.08734
.10467 .005493
.10488
.12219
.007489
.12249
.13976
.009802
.14015
.15739 .012437
.15788
.75086
.27599
.73856 IO
.17510 .015401
.17571
.72261.25584
.71128 II
.19288 .018698
.19363
.69513
.23695
.68471 12
.21076 .022336
.21166
.66838
.21924
.65880 13
.22875 .026325
.22981
.64231.20263
.63350 14
.24686.030671
.24810
.61688
.18704
.60879 15
.26509 .035388
.26652
30
26
.59204 .172413
.56777.158678
.54402 .145783
.52077 .133677
.49798 .122314
.58463 16
.28347.040486
.28510
.56098
17
.30200 .045976
.30386
.53782 18
.32071.051874
.32281
.51510 19
.33960 .058195
.34196
.49282 20
.35868.064953
.36131
25
.47563.111652
.47094 21
.37798 .072167
.38091
24
.45369 .101651
.44943 22
•39750 .079857
.40075
23
.43213.092276
.42828 23
.41726 .088044
.42085
2.2.
.41093 .083494
.40746 24
.43728.096749
.44123
21
.39007.075276
.38695 25
.45758 .105999
.46191
20
.36953 .067596
.36673 26
.47817 .115820
.48292
19
.34928.060427
.34679 27
.49907.126241
.50426
18
.32932 .053746
.327II 28
.52030 .137294
.52596
17
.30961 .047532
.30766 29
.54188 .149013
.54805
16
.29015.041766
.28844 30
.56383.161436
.57054
15
.27091 .036430
.26942 31
.58618 .174602
.59346
14
.25189 .031509
.25060 32
.60895.188556
.61684
13
.23306 .026988
.23195 33
.63216.203346
.64071
12
.21441 .022853
.21348 34
II
.19592 .019092
.19515 35
IO
.17759 .015695
456
.65585 .219024
.66509
.68003 .235647
.69002
.17697 36 .70474 .253275
.71553
+I
98 7 654
32 1
.15940 .012651 .15889 37
.14134 .009951 .14094
.007588 .12309 39
.12340
.10555
.005555
.08780.003845
www
38
78 9
.73001
.271977
.74166
.75588.291827
.76844
.78237
.312903
.79592
.07012
.002454
.05252 .001377
.03497
.10532 40 .80953 .335295 .82414
.08764 4I .83740
.07003 42 .86602
.05247 43
.000611 .03494 44
.01747 .000152 .01746 45
.359099
.384420 .88298
.85314
+
+
+
O
00000
000000
46
00000 47 1.02205 .538280 1.04679
.94539
.95683.470706
.97809
.98894 .503379
1.01186
48
1.05624 .575598 1.08295
.89543 .411374 .91371
.92568 .440090
45

Y = 0.04
Y = 0.05
Ф
X
Y
T
9-
X
Y
T
49
1.09158 .615542
50 1.12813 .658346
I.12043 30
1.15932 29
.59593
.173941
.58653
.57131.160017
.56272
51
1.16598.704267
1.19973 28
.54726.146954
.5394I
52
1.20521 -753593
1.24176 27
.52372 .134698
.51655
53
1.24591 .806644
1.28555 26
.50067 .123201
.49414
54 1.28817.863779
1.28817 .863779
1.33122 25
.47806
.I12420
.47214
55
56
57
58
5678
1.33212 .925397
1.37891 24
.45589.102314
.45052
1.37785 .991947
1.42879 23
.43412.092847
.42926
1.42548 1.063932
1.47515 1.141917 1.53584 21
59 1.52700 1.226536 1.59342 20
60 1.58116 1.318507 1.65401 19
1.48104
22
.41272.083983
.40834
.39168 .075692
.38774
.37096 .067948
.36744
.35056.060724
·34742
18
.33045 .053993
.32767
17
.31061 .047737
.30815
γ
y = 0.05
16
.29103 .041934
.28887
15
.27167.036566
.26980
9-
X
Y
T
14
.25254.031618
.25093
13
.23362 .027075
.23223
0
60 2.04361 1.90178 1.87718
12
.21488.022920
.21372
55
58
1.83932
59 1.93605 1.71908 1.79181 II
1.56116
1.56116 1.71316 ΙΟ
.19631 .019143
.19535
.17791
.015733
.17713
57 1.75143
1.42316 1.64026
56 1.67093 1.30150 1.57234
86
.15966 .012678
.15902
.14154 .009970
.14104
55 1.59669
1.19344 1.50878
7
.12355 .007600
.12316
53
54 1.52782
1.46362
52 1.40351
1.01007
.93171 1.33958
1.09686❘ 1.44908 6
1.39280
.10566.005563
.10538
54
.08788.003850
.08768
.07017
.002457
.07005
51 I.3470I
50
1.29373
49
1.24333
.86067 1.28914 3
.79602 1.24119 2
.73699 1.19552 +1
.05255 .001378
.05248
.03498 .000612
.03495
.01747
.000152
.01747
48
1.19551
.68293 1.15192
+
+
+
45
www www www £ £ £ ££☆ ☆±·
1.15004
46 1.10669
.63329 I.I1022
.58760 1.07025
O
000000
Ooooo
00000
+
1.06527
44 1.02563
.54544 1.03189 I
.50647
.01745 .000152
.01745
.99501 2
.03486.000609
.03489
.98759
.47039
.95949 3
.05227.001369
.05234
.95106 .43690
.91591 .40579 .89217
.88202 .37684 .86018
.92524
4I
40
38
.84931
.81768
37
36
35
34
78707 .30122
.75740 .27926
.72861 .25873
.70064
.23950
.34987 .82921
.32471 .79919
.77006
.15714 .012411
.15776
.74175 ΙΟ .17479 015365
.17556
.71421 II .19251
.68740 12
.018651
.19345
.21032 .022273
.21144
333
33
32
31
.67343 .22149
.64694 .20461
.62112 .18878
.66127 13
.22823.026245
.22955
.63577 14
.61087 15
.24625.030571
.24779
.26439 .035263
.26616
456
.06968
.002433
.06980
.08711 .003805
.08730
.10456 .005485
.10483
78 9
.12204 .007477
.12241
.13956 .009784 .14005
46

Y = 0.05
Y = 0.06
23
.41552 .087553
24
.43537.096183
25
.45548 .105349
26
.47587.115076
X
Y
T
16
17
.30110
18
.28267.040334
.31969 .051653
19 .33845 .057932
.35740.064643
20
21 .37655 .071804
22 .39592
~ ~ ~ ~ ~ ~N~
.079434
9.
X
Y
T
.28470 60 2.14308 2.03739 1.91913
.045792
.30340
.32229
55
59
58
86
2.01825
1.90851
1.82534 | 1.82716
1.64619| 1.74339
50555
.34138 57 1.81053
.36067 56
I.49233 1.66642
1.72199
1.35850 1.59520
.38018 55 1.64122
1.24094 1.52890
.39995 54
1.56696
1.13679 1.46690
|
.41996 53
.44026 52 1.43431
1.49824
1.04390 1.40867
.96056| 1.35379
.46085 5I
I.37454
.88540 1.30191
.48175 50
1.31844
.81732 1.25271
27
.49656 .125393
.50299 49
1.26558
.75542 1.20594
28
.51757.136330
.52457 48
1.21561
.69894 1.16137
29
.53891 .147920
30
.56061.160200
.54654 47 1.16824
.56890 46 1.12322
.64723 1.11881
•59977
1.07808
333
123
31
32
.60518
33 .62809
.58269.173208
.59168 45 1.08031
.55610 1.03903
.186988
.61492 44 1.03933
.51582
1.00153
.201584
.63863 43 1.00010
.47860
.96546
33 3
34 .65145
.217048
35
.67529 .233433
.66284 42 .96249
.68759 41
.44412 .93071
.92636
.41215
.89718
36
.69963.250799
.71291 40
.89158 .38245
.86478
www
37
.72451.269211
73884 39
.85807
.3548x
.83343
38
.74996
.288738
.76540 38
.82571
.32907
.80307
39
.77600
.309458
.79264 37
.79443
.30506
-77361
40
.80268
.331455
.82061 36
.76415
.28265
.74501
4I
.83004 .354821
.84935 35
-73479
.26171
.71721
42
.85811 .379656
.87890 34
.70631
.24213
.69015
43
.88693.406069 .90932 33
.67862
.22380
.66379
44
.91655 434183
.94067
45
.94702 .464131 .97300
ww
32
.65169
.20665 .63808
3I
.62547
.19057 .61299
46
.97839.496058
1.00639 | 30
.59990
.175509 .58847
47
48
49
52
53
+ Dom Nm
1.01071 .530125
1.04089 29
.57494
.161389 .56449
1.04405
.566510 1.07660 28
.55057
.148151 .54102
1.07846.605410 1.11358 27
.52673
.135741 .51803
I.I1401 .647042 1.15194 26
.50340
.124107 .49548
5I 1.15077 | .691646 | 1.19176|| 25
.48054
.I13204 .47335
1.18882 -739489
1.23316| 24
.45813
.102991
.45162
1.22824 .790867
1.27625 23 .43614
.093428 .43025
54
1.26911.846111
1.32116 22 .41454
.084480 .40923
55
1.31153.905589 1.36802 21 .3933I
.076116 .38854
56
1.35559.969709 | 1.41698
20
.37242
57
1.40139 1.038928 | 1.46821 19
.35186
少
​.068306 .36816
.061024 .34806
58
59
1.44905 1.113758 1.52190 18 .33160
1.49869 1.194772
60 1.55041 1.282610
.054244 .32823
1.57823 17 .31162
.047945 .30865
1.63744
16 .29191
.042104 .28931
15 .27244
.036704 .27018
13
14 .25320
.23418
.031728 .25125
.027162 .23251
47

y = 0.06
Y = 0.06
9-
X
Y
T
9.
φ
X
Y
T
12
.21535 .022988
II
ΙΟ
.19670 .019194
.17824 .015771
.21395 34
.19554
.17729 36
www
35
456
.64715 .215120
.66063
.67065.231277
.6852I
.69464 .248390
.71035
7
098 .15992 .012706
.14175 .009989
.12371 .007613
.15915 37
.14114 38
.12324 39
333
.71915
.266522
.73607
.74419.285740
.76242
.76981 .306119
.78944
6
.10577 .005571
.10544 40
.79603.327738
.81717
+I
in 4
32 H
5
.08795 .003854
.08772 4I
.82289.350684
.84564
.07022 .002459
.07007 42
.85044 .375055
.87492
.05257 .001379
.03499 .000612
.01748 .000152
.05249 43
.87870.400955
.90504
.03496 44
.90772 .428499
.93607
.01747 45
.93754
.457812
.96806
+
00000
+
+
46
.96822
.489034 1.00108
oooooo
00000
47
.99980
.522316| 1.03518
48
+
.000152 .01745
2 3
.001368
.03488 50
.05233 51
456
78 9
.06978 52
.08726 53
.10478 54
.12234 55
I
.01744
.03484.000608
.05224
.06963 .00243I
.08703.003801
.10445 .005478
.12189 .007465
.13937 .009766 I3995 56
.15690 .012385
•
.15764
57 1.37871 1.015585 1.45601
.17541 58 1.42456 1.087557 1.50866
.19326 59 1.47220 1.165322 1.56386
1.52175 1.249463 1.62181
Y = 0.07
1.17320
1.21144
49|| 1.06588
1.03233 .557825 1.07046
1.06588.595747 1.10697
1.10049 .636285 1.14482
1.13625 .679664 | 1.18410
.726132 I.22490
.775964 1.26734
1.25102 .829467 | 1.31153
56
1.29204 | .886981 | 1.35761
1.33457 .948882 1.40572
IO
II
12
.17449 .015330
.19214 .018603
.20988
.022212
.21122 60
13
.22772 .026165
.22929
14
.24565 .030471
.24749
15 .26370 .035140
.26581
16
.28188 .040183
.28430
17
.30020 .045610 .30295
X
Y
T
18
.31867 .051435
.32178
19
33732.057673
.34080
60 2.27039
2.21597 1.96977
20
.35614 .064337
.36002 59 2.11908
1.95891 1.86852
21
.37515.071446
.37947 58 1.99085
1.74955 1.77798
22
.39437 .079016
.39916 57 1.87926
1.57430 1.69584
23
魚丸
​24
.41381 .087070
.43348 .095626
.41909 56 | 1.78031 1.42473 1.62054
.43930 55
1.69135
1.29524
1.55096
25
.4534I .104709
.45980 54
1.61049 1.18185 1.48627
26 .47361 .II4345
.48060 53
1.53638
1.53638
1.08165
1.42580
27 .49410 .I24559
.50173 52
1.46794
.99244 1.36902
28
.51489 .135383
.52320 51
I.40439
.91252 1.31552
29
32
33
123
31 .57928
.60149
.53600 .146849
30 .55746 .158990 .56729 49
.171845 .58994 48
1.23705
.185455 .61303 47 1.18757
.62410.199864 .63658 46
.54505 50
1.34506
.84053 1.26492
1.28942 .77537 1.21694
.71616 1.17131
.66215 1.12781
1.14070
.61274 1.08626
48

Y = 0.07
Y = 0.07
X
Y
T
५
9-
X
Y
T
45
1.09615 .56741
1.04648
44
1.05372 .5257I
1.00832
43
1.01320 .48725
.97165
42
.97443 .45171
93637
4I
.93724 .41882
.90236
40
.90153
.38830
.86952
38
86
33
39
.86716.35996
.83777
37
.83402
.80203
·
33360
.80705 8
.30905
•77726
1 2 3
456
7∞ a
I
.01744 .000152
.01745
.03483 .000608
.03488
.05222
.001367
.05231
.06959.002428
.06975
.08696 .003796
.08722
.10434
.005470
.10472
.12174
.007453
.12226
.13918 .009748
.13986
9
.15666 | .012360
.15752
ww
ww
36
.77110
.28616
.74836 ΙΟ
35
.74116 .26480
34
.71215 .24485
.72028
.69296 12
II
.17419
.19178 .018556
.015294
.17526
.19308
.20945
.022151 .21100
33
33 .68395 .22619
.66637 13
32
.65656.20874
.64044 14
31
.62992 .19241
.61515
15
3 2
30
29
28
0 000
.60396
.177118
.59044
16
.57864.162794
.55394
.149377
.56629 17
•54266
18
m4 in o 70
.22721
.026088 .22903
.24506 .030373 .24719
.26301 .035018 .26546
.28110 .040034
.28390
.29931 .045430 .30250
.31767 .051219
.32127
27
.52980.136807
.51952 19
.33620 .057416
.34023
26
.50618 .125032
.49684 20
.35488.064035
.35939
25
.48306
.114005
.47458 21
.37376 .071092
.37877
24
.46041 .103681
.45273 22
.39283.078605
.39837
23
.43820
.094020
.43125 23
.4I2II .086593
.41823
22
.41639 .084986
.41014 24
.43162.095078
.43835
21
.39496 | .076546
.38935 25
.45138.104080
.45876
20 .37390 .068670
.36889 26
.47139 .113626
.47946
19
.35317 .061330
.34871 27
.49168
.123742
.50049
HHH
96
18
17
16
.33276.054498
.32880 28
.51226
.134455
.52186
.31264 .048154
.29280.042276
.30915 29 .53315 .145799
.28975 30 •55436
.54359
.157805
.56570
15
.27321 .036843
.27056 31
14
.25386 .031840
.25158
32
13
.23475 .027250
.23279 33
1 2 3
.57593 .170511 .58822
.59787
.183957 .61117
.62020 .198184
.63457
I2
.21583.023056
.21419 34
II
IO
.17856
.19710 .019246
.015809
.19574 35
.17745 36
456
.64295
.213240 .65847
.66613 .229175 .68288
.68978 .246045
.70783
987 654
321
.16018
.012734
.15928
.14195 .010008
.14124
38
.12386.007626
.12332 39
3 3 3
78 9
37
.71391 .263907
.73337
.73857
.282829
.75951
.76377.302880
.78631
.10588
.08802
.005579 .10550 40
.78956 .324136
.81381
.003859 .08776 41
.81595 346683
.84203
.07027
.002461 .07010 42
:
.84300
.370611
.87104
.05260.001380
.03500
+ I
.01748
+
.05251 43
.000612 .03496 44
.000153 .01747 45
.87072
.396021 .90088
.89917 .423022 .93160
.92839.451733 .96326
+
+
46
.95841 | .482286 .99592
O
00000
Oooooo
Ooooo
47
.98928
.514825
.514825
1.02965
48
1.02106
.549509
1.06451
49
6

Y = 0.07
Y = 0.08
9-
X
Y
T
.
X
Y
T
49
50
5I
55
Он
1.05379
1.08754
.586512 1.10058
.626026 I.13795
1.12234
.668261 1.17670
3 2 2
30
.60811
.178769
.59245
29
.58243 .164235
.56813
28
.55738.150632
.54433
55 55 5
2 34
52
53
54
1.15828 .713449 1.21693 27 .53292 .137898
1.19541 .761847
1.25876
1.23381 .813741 1.30228 25
222
765
.52104
26
.50901
.48562 .114822
.125978 .49821
•47583
58
60
567 ∞ ∞ o
5503
55 1.27353 .869445 1.34763 24
56 1.31467 .929307 I.39494 23
57 1.35729 .993714 1.44436 22
.46272.104384
.45386
.44028 .094623
.43227
.41826 .085501
.41105
59
1.40148 1.063094 | 1.49605 21
1.44733 1.137924 1.55019 20
1.49492 1.218733
.39664 | .076983
.39017
.37539 .069039
.36962
1.60698 19
.35449 .061639
.34936
18
.33393 .054756
.32938
17
.31367.048366
.30966
Y = 0.08
16
.29369.042449
.29019
9-
X
Y
T
13
H H
543
15
.27399 .036984
.27094
14
.25453
.031952
.25191
.23532
.027338 .23307
60 2.44854 2.47491
2.03488
58
57
56
55
No 5
.21631 .023125
59 2.24980 2.13716 1.91887 I I .19750 .019299
2.09255 1.88037 1.81862 IO .17889 .015848
1.96134 1.67428 1.72956
|
12
.21442
.19593
.17761
1.84826
1.74864
1.50334 1.64906
1.35834 1.57545
54 1.65951 1.23333 1.50752
53
52
1.57878
I 50497
I.12419 1.44441
I.02797 1.38545
5I
50
но
1.43696
.94244 1.33010
I.27794
1.37389 .86591
49 1.31509 .79704
98 7 6 5 4
.16044 .012761
.15941
.14216 .010028 .14135
32
.12402
.007639
.12339
.10600
.005587
.10555
.08811 .003864
.08780
.07032
.002463 .07012
.05263
.001381
.05252
.03502
.000612
.03497
.01748 .000153
.01747
+
+
+
Ooooo
Oooooo
00000
1.22860+I
48
1.25999
73476 1.18180
47
1.20817
.67819 1.13728
46
I.15924
.62662 1.09483
40
45 I.I1290 .57945 1.05425
44 1.06888 .53619 1.01539
43 1.02695 .49639
42 .98692
4I .94861 .42581
.91188
.97809
123
I
.45970 .94224
.39443
.90771
.87442
+36
5
.08688
.10424
38
333
39
286
.87660 ·36534 .84225
.84264 33832 .81114
.80990 .31320 .78101
7∞ a
.01745
+
.01744 .000152
.03482 .000608 .03487
.05219 .001366 .05230
.06954 .002426 .06973
.003792 .08719
.005462
.10467
.12160 .007441 .12219
.13899 .009730 .13976
37
9 .15641 .012334
.15739
36
35
.77829 .28980 .75179 ΙΟ .17389 .015259
.74773 .26800
.72342 II
34 .71813 .24765
.17510
.19142
.018509
.19290
.69584 12
.20902 .022090
.21079
333
321
33
32
31
.68943 .22865 .66900
.66156 .21090 .64285
.63447 .19430 .61735 15
50
H H
13
14
345
.22670.026010 .22877
.24447
.030275
.24689
.26233
.034897
.26512

Y = 0.08
Y = 0.09
9.
X
Y
T
9-
X
Y
T
16
.28032 .039887
.28351
+
+
0
+
17
18
.29843 .04525I
.31668 .051006
.30205 59
2.43774 2.40308
1.98477
.32077 58
2.22605 2.05726
1.86850
19
.33509.057164
.33967 57
2.06336 1.80168
1.76934
20
·35365 .063737
•35876 56
1.92967 1.59956
1.68182
21
.37238
.070744
.37807 55
1.81551 1.43338
1.60302
22
.39131 .078200
.39760
23
.41044 .086125
.41738
24
.42979
.094538
.43742
ir ir i
54
53
432
1.71558 1.29321
1.53110
1.62653 1.17283
1.46482
52 1.54614 1.06802
1.40328
22
25
.44937
26 .46920 .I12920
.103462
.45773 5I
1.47280
.97579
1.34581
.47834 50
1.40535 .89393
1.29186
27
.48929
28
.50967
.122939 .49927 49
.133545
1.34287.82076
1.2410I
.52053 48
1.28467 .75496
I.19291
29
30
·53034 .144770
.55132.156645
.54214 47
1.23020 .69549
1.14727
.56413 46
1.17897 .64151
1.10383
333
1 2 3
31
32
33
.57265.169207
.58652 45
1.13065
.5923I
1.06239
•59432 .182492
.60933 44
1.08488
.54733
1.02276
.61638 | .196544
.63260 43
1.04141
.50607
.98479
mmm
456
34
333
7∞
35
36
37
38
.63883
.68502
.70881.261365
.73309
.211406
.65634 42
1.00002 .46813
.94833
.66170 .227128
.68058 4I
.96050.43316
.91326
.243762
.70536 40
.92268.40086
.87947
.73071 39
.88642.37096
.84687
.280001
.75666 38
.85159 .34324
.81536
39
.75790
.299737
.78325 37
.81805
.31751
.78487
40
.78326.320646
.81052 36
.78572
.29358
.75532
4I
.80921 .342809
.83850 35 .75450
.27131
.72664
42
.83577
.366314
.86725 34
.72430 .25055
.69878
43
.86299
.391257
.89682 33
44
.89090 .417741
.92724
45
.91953 .44588I
.95859 3I
www
32
32 H
.69506 23119
.67169
.66669 .21313
.64531
.63914
.19624
.61959
46
.94893 .475801
.99091 30
.61237.180466
.59450
47
.97914 .507638
I.02427 29
.58630 .165714
.56999
48
1.01020 .541543
1.05873 28
.56089
.151918
.54603
49
1.04217
.577681
1.09438 27
ли
5I
50 1.07509
1.10901
.616232
1.13129 26
.657393
1.16955 | 25
765
.53610
.139014
.52258
.51190
.126944
.49961
.48823.115655
.47710
10 10 10
55 5
पूज
52
53
I
1.18011
54
I.14400 .701385 1.20925 24
.748448 1.25049 23
1.21740 .798848 1.29338 22
.46508 .105100
.45500
55
56
57
I.25593 .852879 1.33804
1.29577 .910864 1.38461 20
I.33700 .973161 1.43321 19
21
.44239 .095236 .43330
.42015
.086024 .41198
.39833 .077427 .39100
.37690.069413 ·37036
.35583 .061952
.35002
556
58
1.37968 1.040167 1.48401
18
.335II
.055016
.32996
59
60
1.42389 1.112321 1.53718 17
1.46970 1.190111 1.59290
16
76
.31470
.048581
.31017
.29460 | .042625
.29063
15
.27478.037126
.27133
14
.25521
.032066
.25224
13
.23589
.027428
.23336
51

y
y = 0.09
= 0.09
987 654
.16070
9-
X
Y
T
12
.21679
.023194
.21466
II
ΙΟ
.19790 .019351 .19613
.17922 .015887 .17777
.012790
9-
X
Y
T
0
mm m
34
35
36
456
.63480 .209616
.65425
.65738.225131
.67833
.68037 .241538
.70294
.15954 37
.14236 .010047
.12417.007652
.14145 38
78
.70382
.258891
.72811
.72775 .277251
.75387
.12347 39
.75218.296684
.78025
.10611
+I
3 & H
2
.005595
.08819 .003868
.07037 .002466
.05266 .001382
.03503
.000612
.01749 .000153
.10561 40
.77713 .317260
.80730
.08784 41
.80265.339056
.83506
.07015 42
.82876 .362156
.86356
.05254 43
.85549 .386652
.89286
.03497 44
.88287.412643
.92300
.01747 45
.91095 .440239
.95404
+
+
+
46
.93976 .469558
.98603
O
Ooooo Oooooo
00000 47
.96934 .500731
1.01904
1 2 3
456
I
.01743
+
.000152 .01745
.08681
.10413 .005455
.003787 .08715 53
.10461 54
.03481.000607
.05217
.001365
.06949 .002424 .06970 52 1.13031
.03487
.05228 5I
영웅 ​fo
48
.99972 .533901
1.05313
49
1.03097
.569223
1.08837
50
1.06311
.606867 1.12485
1.09621
.647021 1.16263
.689892
1.20182
1.16546
.735708 I.24251
1.20172
.784717 1.28481
78 9
.12145 .007429
.13880.009713
.15617
.122II 55
13967
9
IO
II
12
.012309
.17359 .015224
.19106 .018462
.20859 .022029
HHH
13
.22619.025932
14
.24388 .030179
.24654
15
.26166 ,034779
.26478
HHH
16
.27955 .039741 .28312
ллл
དང་
56
.893440 1.37468
.15727 57 1.31773 .953792 1.42252
.17495 58 1.35902 1.018617| 1.47249
.19271 59 1.40173 1.088323 1.52475
.21057 60 1.44592 1.163360 | 1.57947
.22852
1.23914 .837193 1.32882
1.27779
17
.29756 .045075 .30161
18 .31570 .050795 .32027
19
20
.33399 .056914 .339II
.35242 .063443
21 .37103 .070400
22
.38981 .077800
23
.40879 .085663
24
.42798 .094007
25
.44740.102854
26
.46704 .I12227
27
.48695
.122151
28
.50712 .132653
29
.52758.143761
30
.54834
.155509
1 2 3
31
32
33
www
.56942
.59085
.167930
.181061
.61263.194942
.35814
Y = 0.10

X
Y
T
.37737 58 2.42326 2.32865 1.93480
.39684 57 2.19876
.41654 56 2.03136
.43649 55 1.89581
1.97583 1.81849
1.72270 1.72059
1.52534 1.63472
1.36438 | 1.55766
1.22932 1.48746
1.11371
1.42283
1.01330 1.36286
.925075 1.30685
.846868 1.25428
.45672 54 1.78106
.47724 53 1.68116
.49807 52 1.59248
.51922 51 1.51262
.54072 50 I.43993
.56259 49
1.37315
.58486 48 1.31137 .777020 1.20472
.60754 47 1.25387 .714251 1.15783
.63066 46 1.20007 .657546 1.11331
52
Y = 0.10
Y = 0.10

9-
X
Y
T
9-
X
Y
T
45
1.14951
.606087
1.07093 I
44
43
1.10182 .559206
1.05666 | .516347
1.03047 2
.99177
3
OHN M
.01743 .000152
.01744
.03480.000607
.03486
.05214 .001364
.05227
42
1.01378 .477046
.95466
4I
.97295 .440911
.91902
40
.93396
.407608
.88471
456
.06944 .002422
.06968
.08673 .003783
.08711
.10402 .005447 .10456
333
39
38
86
.89665.376850
.85164
.86088
.348387
.81971 8
7∞
.12130 .007417
.12204
·
13861.009695
.13957
37
.82650
.322004
.78884 9
.15593 .012284
.15715
www.
36
.79341
.297513
.75894 IO
.17329 .015190
.17480
35
.76149
.274747
.72995 II
.19070 .018416
.19253
34
-73067
.253559
.70180 12
.20816 .021969
.21035
333
32 1
33
32
31
.70085
.233820
.67445 13
.22569.025856
.22826
.67196
.215416
.64782 14
.24330 .030083
.24629
.64394.198242
.62188 15
.26099.034660
.26444
322
ه الحماد
30
.61673.182208
.59659 16
.27879 .039596
.28273
29
.59026.167230
.57189 17
.29670 .04490I
.30117
28
.56448 .153234
•54776 18
.31473 .050587
.31977
222
27
•53936 .140154
.52415 19 .33290 .056667
.33855
26
.51484 .127928
.50103 20 .35121 .063153
.35752
24
.46747
25 .49089 .116503
.105828
.47838 21
.36969 .070061
-37669
.45616 22
.38834 .077406
.39608
23
.44454 .095858
.43435 23-
.40717 .085208
.41570
22
.42208.086553
.41292 24
.42620.093484
•43558
21
.40005 .077875 .39184 25
20
.37843.069790
.37110
19 .35719 .062268
222
.44545 .102256
.45572
26
.46492 .III545
.47614
.35068 27 .48464 .121376
.49687
H H
876
H
18 .33630 .055279 •33054
17 .31575
16
.048799 .31069 29
.2955I .042803 .29108 30
28
.50461 .131776
.51793
.52487 .142772
.53932
.54541.154396
.56108
15
14
13
543
.27557
.037270 .27172 31
.25589.032181
.032181..25258
32
.23647 .027518
.23364 33
333
1 2 3
.56626
.166680
.58322
.58744.179660
.60577
.60897 .193376
.62875
9876 in t
.010067
.10623.005603
.003873
4
.002468
I2 .21728 .023264 .21490
ΙΟ
II .19831 .019404 .19633
.17955 .015926 .17793 36
16097 .012818 .15967
.14257
.12433.007665
5 .08827
.07042
333
34 .63086 .207868
.65219
35
ал
.65314 .223183
.67612
.67583
.239370
.70057
.14155
38
ww
78
37
.69896 .256482 .72556
.72254
.274577
.75113
.12355 39
.74660
293718
.77732
.10567 40 .77117
.313973
.80415
.08788 4I .79627
.335417
.83168
.07017 42
.82194
.358129
.85994
+I
32 H
.05269.001383
.05255 43
.84820 .382198
.88898
.03504
.000613
.03498 44
.87510.407719
.91885
.01749
.000153
.01747 45
.90265 .434796
.94960
+
+
+
46
.93089 .463544
.98128
00000
Oooooo
OOOOO 47
48
.00
.95987 .494086 1.01395
.98962 .526557
.526557
1.04768
53

Y = 0.10
= 0.11
γ
Y =
X
Y
T
9-
X
Y
T
49
50
5I
1.02018 .561106 1.08254 27
1.05160 .597895
1.08391
.54268
.141327
.52575
1.11860 26
.637102 1.15593 25
.51784 .128940
.50248
.49360.117373
.47969
53
54
52 1.11717 .678923 | 1.19464 24
1.15143 .72357I 1.23481 23
1.18672.771281 | 1.27654
.46991 | .106575
.45734
.44673.096497
.4354I
22
.42404 .087096
.41387
55
56
57
567
1.29941
1.22312 .822311 1.31994
1.26066
.876946 | 1.36514 20
.935498 1.41226 19
21
.40180 .078334
·39269
·
37998 .070177 .37186
.35856 .062591
.35135
58
1.33942 .998312
1.46145
59 1.38075 1.065767 1.51286
60
1.42345 1.138282 1.56665
|
928
H H
18
.3375I .055547
•33114
17
16
.31682
.049020
.31121
.29644 .042983
.29154
Y = 0.II
γ
HHH
15
27637 .037416
.27211
14
.25658 .032297 .25292
13
.23706 .027609
.23393

12
.21777
.023335 .21514
X
Y
T
II
.19872 .019457
.19653
ΙΟ
.17989 .015965
.17809
765
57
56
0
2.40402
2.25049
2.16753| 1.89271
55 1.99642
1.88501
1.76874
1.64352 | 1.67229
98 7
.16124 .012845
.15980
.14278 .010086
.14165
.12449 .007678
.12363
лыы
53
52
5I
54 1.85974 1.45177 1.58819 6
1.74491 1.29650 1.51294 5
1.64542 1.16679 1.44448 4
1.55739 1.05607 1.38151
.10635 .005611
.10573
.08835.003878
.08792
50 1.47829
.96008
1.32310
49|| 1.40639
.87587
1.26855 +1
32 H
.07047 .002470 .07020
.05271 .001384 .05256
.03505
.000613 .03499
.01749 .000153 .01748
48
I.34042
.80128
1.21734
47
I.27944
.7347I
1.16905
46 1.22271
.67491
I.12334
45 1.16965
.62091 1.07992
44
1.11981 .57192 1.03857
43
1.07279 .52730 .99907
1 2 3
I
42 1.02827
.4865I .96127
40
41 .98601
.94576
.44910
.92501
.41471 .89015
456
.01743 .000152
03478.000607
.05211 .001363
.06939 .002418
.08666 .003779
.10391 .005440 .10451
+
+
+
Ooooo
oooooo
OOOOO
+
.01744
.03485
.05225
.06965
.08707
86
333
39
38
.90733
.38304
37 .83528
.87055 .35377
.32669
76
.85658
.12116
.007405
.82420 8 .13842 .009677
.79293 9 .15569 .012259
.12197
.13948
.15703
33 3
35
34
36
.80137
.76873
.73724
.30160
.76267 ΙΟ .17300
.015156
.17465
•
27832
.25668
.73335 II
.19034 .018370
.19235
.70490 12
.20774 .021910
.21014
www
www
33
32
31
-70682 .23654
.67739
.64887
.67727 13
.22519 .025780
.22801
322
30 .62120
.21779
.20030
.184005
.65039 14
.24272 .029988
.24600

.62422 15
.26033 .034543
.26411
.59872|| 16
29
28
.5943I
.56815
.168790 .57383
.154589 .54952
67∞
H H
.27804
.039453 .28234
17
.29585 .044728
.30073
18
.31377 .050381
.31928
54

Y = 0.II
Y = 0.12
9.
X
Y
T
9-
X
Y
T
19
.33182 .056423
·33800
56 2.37888 2.16734
1.83487
20
.35002 .062866
•35691 55 2.13210 1.80775
1.71917
21
.36836 .069726
.37601
23
222
234
22 .38688 | .077018
.39533
.40556.084760
24
.42444 .092970
.41488
.43467
+32
in in in
54
1.95854 1.56419
1.62444
53
1.82153 1.37892
1.54224
52
1.70717 1.22979
1.46886
51
222
25
.44353 .101668
.45473 50
но
1.60846 1.10564
1.40218
1.52134 .99992
1.34088
26
.46283 .110875
.47506 49
1.44320.90839
1.28401
27
.48237
.120616
.49570
48
1.37225 .82817
1.23090
223
28
.50215
.130916
.51665 47
1.30720 .75716
1.18103
29
.52220 .141803
.53793 46
1.24710 .69382
1.13398
30 .54253
.153305
.55958
45
1.19122 .63693
1.08942
♡ ♡ m
3I
.56315.165456
.58160 44
1.13897 .58557
1.04708
32
.58410 .178290
.60402 43
1.08989 .53899
1.00672
33 .60537
.191845
.62687
42
1.04359 .49655
.96817
www
34
35
36
456
.62700
.206161
.65017 4I
.99976 .45777
.93124
.64900
.221282
.67394 40
.95814 .42221
.89579
.67139 | .237256 .69823
W w
37
38
7∞
.69420
.254135 .72305
.71744
.271974 .74844
ww w
39
.91850 | .38953
.86170
38
.88064
.3594I
.82885
37
.84440
.33160
.79715
39 .74115 .290835 .77443
36
.80963
.30587
.76651
40
.76535 .310781
.80106 35
.77622 .28203
-73684
4I
.79006 •331886
.82837 34
.74403 .25991
.70808
42
.81530.354226
.85640 33
43
.84112 .377886
.88520
32
44
.86754.402958
.91480 3I
32H
.71298 .23935
.68016

.68298 .22023
.65303
.65394
.20243
.62662
45
.89459 .429541
•94526
30
.62579 .185852
.60090
46
.92230
.457744 .97664
29
.59847
.170393
.57581
47
.95071
.487685 | 1.00899
28
.57192 .155976
•55131
48
.97985
.519494 1.04237
27
49
5. In in in
5I
1.00977
50 I.04049 .589292 1.11252 25 .49636
1.07207 .627606 I.14943
.5533II 1.07686 26
76
.54608
.142525
.52737
.52091 .129973 .50395
.118261
.48102
24
.47239 .107335
.45854
52
53
I.13796
1.10455 .668438 1.18768 23
.711989 1.22735 22
.44896
.097145
.43648
.42603.087647
.41483
54 1.17236.758483
1.26854 21
.40358.078800
.39355
56
555
55
56
57
I.20779 .808164
I.24430 .861298
1.28194 .918179
1.31137 20
I.35594 19
1.40239 18
.38156 .070569
.37263
.35996
.062919
.35203
•
33875 .055820 .33174
58
1.32076| .979130
1.45084 17
59
1.36082 1.044507 1.50145
16
96
.31790 .049244
.31173
60 1.40216 1.114701 1.55438 15
14
13
int m
.29738 .043166
.27718.037563 .2725I
.29200
.25727
.23765
.032414
.027701
.25326
.23422
12 .21827
.023406 .21539
II
.19917 .019511
.19673
IO
.18022 .016005
.17826
55

Y = 0.12
Y = 0.13
9-
X
Y
T
9-
X
Y
T
0
987 654
.16151 .012874
.15993
.14299
.010106
14176
.12465
.007691
.12371
m m m
37
.68955 .251848
.72059
38
.71248 .269440
.74580
39
.73584 .288029
.77161
.10647
.005619
.10579 40
.75968 .307679
.79804
.08843
.00388 2
.08796
4I
.78400 .328458
.82514
.07052 .002473
.07022 42
.80885 .350442
.85294
+I
32 1
123
.05274
.001385
.05258 43
.83424 .373710 .88150
.03507 .000613
.03499 44
.86020
.398351
.91084
.01750 .000153
.01748 45
.88677
.424461
.94104
+
+
+
46
.91397
.452144
.97212
OOOOO
Oooooo
୦୦୦୦୦ 47
.94184
.481513
1.00416
48
.97040 .512692
1.03722
+
.01742 .000152 .01744 49
.99970
.545816
1.07135
.03477
.000606 .03485
.05208 .001361
.05224
ли
50
1.02978
.581032
1.10663
51
1.06066
.618502
1.14313
456
789
.06935
.08658
.002417 .06963 52 1.09239
.003774 .08703 53
.10380.005432 .10445 54
.658401
1.18094
.I2IOI
.007394
.13823 .009660
.15546
.12190
.012234
13938
.15691
56
In in i
567
55
57
1.12501
1.15857 .746275 1.26082
I.19310 .794691 1.30310
1.22864 .846423 1.34708
1.26526 .901747
.700922 1.22014
1.39288
ΙΟ
.17271 .015121
.17450 58
II
.18999 .018325
.19217 59
6∞
1.30298 .960968
1.44064
1.34185 1.024421
I.49050
12
.20732 .021851
.20993 60 1.38193 1.092474
1.54262
13
.22470 .025705
14 .24215 .029894
.22777
.24571
15
.25967 .034427 .26377
16
.27729 .03931I
.28196
H H
17 .29500
.044558
.30030
18
.31282 .051178 .31879
Y = 0.13

19
.33076 .056182 -33745
❤
X
Y
T
20
.34884 .062583 35630
21
.36706 .069396
22
.38543 .076635 .39459
23
.40398
.084318
.41406
24
.42271.092463
.43377
5 in
.37534 55 2.34664 2.07817 1.78424
54 2.09231 1.72105 1.66973
53 1.91777 1.48495 1.57699
52 1.78130 1.30698 1.49683
56
2 2
27
25
.44164
26 .46077
.48013.119869
.101089
.45375
.110216
.47400
55
51
50
1.66795 | 1.16439| 1.42538
1.57040 1.04600 1.36052
.49454 49
1.48445
.94533 1.30088
28
.49973 .130072
29
.51958 .140851
30
.53970.152236
3I
.56011 .164257
www ww.
32
.58082
.176949
33
60185.190347
.51539 48 1.40745
.53657 47 I.33759
.55810 46 1.27358
.58000 45 1.21445
.60230 44 I.15948
.85826 1.24556
.78200 1.19387
.71452 I.14530
.65434 1.09946
.60030 1.05604
.62502 43 1.10809
.55152 1.01474
456
.62322.204492
.64817 42 1.05981
.50727 .97537
.64494
.219426
.66705
.235194
.67180 41 1.01426 .46697 .93773
.69593 40 0.97114
.43013
.90165
56

Y = 0.13
Y = 0.13
X
Y
T
Ф
X
Y
T
www
98 76
333
39
38
37
36
35
.93019.39636
.86700
.89117
.85391
.36533
.83365
78
.12087 .007382
.12182
.13804 .009642
.13928
.33673
.80150
9
.15522 .012209
.15679
.81821
.31032
.77046 ΙΟ
.17242 .015088
.17436
.78398.28589
.74043 II
.18964 .018280 .19199
34
.75106.26326
WWW
33 .71934
.24226
32
.68874 .22276
31
.65915
.20462
.71134 12
.68312 13
.65572 14 .24158
.62907 15
.20690
.021792
.2097I
.22421
.025630
.2275I
.029801
.24542
.25902
.034312
.26344
NO N W
30
.63050.187754
.60312 16
.27655
.039171
.28158
29
28
.60273 .172040
.57577 .157400
.57783 17
.29417 .044389
.29987
.55314 18
.31188 .049977
.31831
NNN
27
26
25
.54956.143753
.52405 131030
.49918.119169
.52902
19
.3297I .055944
·33691
•
.50544 20
.34767.062304
•35570
.48236 21
.36576 .069070
.37467
24
.47492 .1081II
.45975 22
.38401 .076258
.39385
23
.45123 .097806
.43757 23 .40242 .083883
.41325
22
.42806 .088208
.41580 24 .42100 .091964
.43289
21
.40538
.079274 .39443 25
20
.38316 .070968
.37340
19
826
HH
18 .34000
.31899
17
16
.36138.063252 .35272 27
.056096 .33234 28
.049471 .31226 29
.29833 .043350 .29246 30
56
222
.43977 .100518
.45278
26
.45874.109568
.47294
.47793 .I19134
.49339
.49734
.129242
.51414
.51700 .139917
.53522
.53692.151186
.55664
H
15
14
.25797 .032532
.27800 .037712 .27291 31
.25361
32
13
333
123
.557II
.163082
.57843
.57759 .175635
.60061
.23824 .027794 .2345I 33 .59838 .188882 .62319
12 .21877 .023478 .21563
II .19955
.019566 .19694
ΙΟ
.18056 .016045 .17842
W W W
456
34 .61950
.202861
.64622
35
.64096
.217613
.66970
36
.66279.233182
.69368
.16178
.14320
.001386
.03508.000614
96
7
654
321
.010126 .14186
.12481 .007704 .12378 39
.10658 .005627 .10584 40
.08851.003887 .08800 4I
.07058
.05277
+I .01750
+
.012903
.16007 37
www
38
7∞ a
.68500
249618
.71817
.70761 .266971
.74321
73065
.285299
.76883
.75414.304663
.79507
.77810 .325129
.82197
.002475
.07025 42
.80256 .346769
.84956
.05259 43
.82754.369661
.87788
.000153
.03500 44
.01748 45
•
85307.393890
.90698
.87917 .419547
.93691
+
+
46
.90589.446733
.96772
00000
Ooooo
47
48
I .01742
2
oooooo
+
.03476.000606
.93324 .475557 .99946
.96125 .506137 1.03219
.000152 .01744 49 .98997 .538603 1.06598
.03484 50 1.01943 •573094 1.10089
.05223 51| 1.04965 | .609765 1.13701
4 .06930 .002415
.06961 52 1.08068 .648783 I.17439
.08651 .003770 .08700 53 1.11256 .690331
.10370 .005425 .10440 54 1.14532 .734609 1.25334
3 .05206 .001361
+36
1.21314
57
7

γ
Y = 0.14
= 0.13
9-
X
Y
T
५
9-
X
Y
T
0
55
56
57
567
1.17900 .781837
1.21364.832254
1.24929 .886123
1.29510
.832254 1.33852 17
.886123
18
.34126.056376
.33295
.32009
.049700
.31280
1.38371
16
.29929
.043537
.29293
58
1.28598 .943732
1.43082 15
.27883 .037862
.27332
59 1.32376 1.005397
1.47997
14
25868.032652
.25395
60 || 1.36267|1.071464
1.53132 13
.23884 .027888
.23480
I2
.21927 .023550
.21588
II
.19996 .019621
.19714
γ = 0.14
ΙΟ
.18090 .016086
.17859
9.
X
Y
T
987
.16205
.012932 .16020
.14341
.010146
.14197
.12497
.007717
.12386
0
555
54 2.30632 1.98239
1.73297 6
.10670 .005635
.10590
53
52
2.04816 1.63293
1.87425 1.40607
1.62037 5
.08859.003892
.08804
1.52992 4
.07062 .002478
.07027
51
A+ A AU
50
но
1.73918 1.23615
1.62737 1.10043
49
1.53132 .98792
1.45195
1.38250
1.31946 +1
321
.05280 .001387
.05261
.03509 .000614
.03501
.01750 .000153
.01748
+
+
+
48
1.44679
.89233
1.26149
47
1.37III
.80971
I
1.20770
O
00000
oooooo
Ooooo
46
1.30247
-73736
1.15741
+
43
+++
45
1.23959 .67336
44
1.18152
.61627
I IIOIK
1.06551
1.12752 .56502
1.02319
42
1.07703 .51874
.98292
4I
1.02959
.47676
.94450
40
.98483.43853
.90775
1 2 3
456
I
.01742 .000152
.03475 .000606
.01744
.03484
.05203 .001360
.05222
.06925 .002413
.06958
.08643 .003766
.08696
.10358 .005417
.10434
www
39
38
86
.94245
.40358
.87250
.90218 .37155
.83862
78
.12072
.007370
.12175
.13787
.009625
.13919
37
.80600
.86381 .34210
9
.15499 .012185
.15667
www
36
.82714 .31496
.77453
ΙΟ
.17213
.015054
.17421
35
.79203 .28991
.74412
II
.18929
.018235
.19181
34
.75833 .26674
.71469 12
.20649
.021734
.20950
333
.72591
33
.24528
.68616 13
.22372
.025556
.22726
32
31
2 1
.69467 .22538
.65848 14
.24102 .029708
.24513
.66451 .20689 .63157 15
.25838.034198
.26311
32 2
30
.63535 .189715
.60538 16
.27581 .039032
.28120
29
.607II .173734
.57988 17
28
.57972 .158861
.55500 18
78
.29334 .044222 .29944
.31095
.049778 .31783
27 .55312 .145012 .53071 19
26
222
.52725 .132113
25 .50206 .I20097 .48373
24
23
22
.47750 .108904
.45354 .09848I
2I .40722
20
19
21
.079757 .3953I 25
.38479 .071373 .37418 26
.36282.063590 .3534I 27
.50696 20
.32867 055709
.34651 .062028
.36449.068749
•
.33637
.35510
.3740I
.46098 22
.38260.075886
.39312
.43868 23
.40087.083454
.41245
.43012.088780 .41679 24
.41931 .091472
.43201
.43793 .099958
.45182
.45674.108931
.47190
.47576.118413
.49225
58

Y = 0.14
Y = 0.15
9-
X
Y
T
Ф
X
Y
T
223
28
.49499
.128427
29
.51446 .139000
.51291 45
•53389 44 1.20531
1.26697 .69430
1.12157
.63368
1.07558
333
1 2 3
30
31
32
33
.53418.150158
.55520 43
1.14836 .57963
1.03212
.55416.161930
.57688 42
1.09539 .53107
.99087
.57443
.174349
.59893 4I
1.04585.48723
.95160
.59499
.187449
.62139 40
.99928.44746
.91412
333
456
34
35
.61586.201266
.64428 39
.95534 .41122
.87822
36
37
38
.63707
.65862
.215841
.66763 38
.91371 .37810
.84378
.231218
.69145 37
.87414
.34774
.81066
78
.68055
.247442
.71579 36
.83643
.31982
.77874
.70286.264565
.74066 35
39
.72558
.282641
.76611 34
54
.80038
.294II
.74793
.76585 .27037
.71814
40
.74873 .301729
.79216
4I
.77234.321893
.81886
42
.79643 .343203 .84623 31
333
33
32
3 2
.73269.24842
.68929
.70078 .22809
.66131
.6700320923
.63413
43
.82101 365734 .87433 30
.64032 .191738
.60770
44
.84612.389568
.90319
45
.87179 .414793
.93287
2 2
29
.61160
28
86
.175479
.58198
.58376.160365
.55690
46
.89803 .441504
.96341 27
55675.146306
.53242
47
.92489 .469807
.99486 26
.53051 .133222
.50851
48
.95238 .499816
1.02729 25
.50499 .121046
.48512
49
.98055 .531655
1.06075 24
.48013
.109714 .46223
50 1.00941 .565458
1.09531 23
.45588
.099169
.43980
53
54
5I 1.03901 .601371
52 1.06938 .639555
1.10055
.680185
1.13256 .723452
1.13105
22
.43221 .089363
.41779
1.16803 2I
.40908.080249
.39620
1.20635
20
.38644 .071785
.37497
1.24609 19
.36427.063933
.354II
ллы
55
56
57
567
1.16545 .769562 1.28735
I.19924 .818746
18
.34254 .056659
•33357
1.23399
1.33023 17
.871253 1.37485
.32121 .049933
.31334
16
.30026 .043726
.29340
58
1.26973 .927355 1.42134 15
59
1.30648
1.30648
.987352 | 1.46982
14
60
1.34430 1.051571 1.52045
13
W Ai
.27967.038014
.27372
.25940
.032773
.25430
.23945
.027983
.23510
12
II
H
ΙΟ
210
.21978 .023623
.21613
.20038 .019676 .19735
.18124 .016126
.16232 .012961 .16034
.14362 .010166 .14207
.17875
Y = 0.15
9-
X
Y
T
ir ir
53 2.25742 1.88017
I.54391 1.57118
52 1.99983
1.68106
9∞ 76n+
8
.12513
.007730
.I2394
6
.10681 .005643
.10596
5
.08867 | .003896
.08808
4
.07067 .002480 .07030
49
5I 1.82819 1.32791 1.48331
50 1.69536 1.16667 | 1.40764
1.58558 1.03806 1.34024 +I
321
.05283 .001388 .05262
.035II
.01751 .000153
.000614
.03502
.01749
+
+
+
48 1.49136 .9315I I.27905
47
1.40847 .84102
46 I.33427 .76281
1.22275 O
I.17047
Ooooo
Oooooo
Ooooo
59

Y = 0.15
Y = 0.15
9-
X
Y
T
4
X
Y
T
0
.000152
123
.01741
.03474.000605
.05200 .001359
.06920 .002412
.08636 .003762
.10349 .005410
.12059
.007358
.13767.009608
.15475 .012160
.01744 49
.03483 50
.05220 51
.06956 52
.08692 53
.I0429 54
.12168 55
.13910 56
56
.97143 .524947 1.05566
.99973 .558094 1.08989
1.02874 .593287 1.12526
1.05848.630680 1.16186
1.08898 .670440 I.19977
1.12028 .712750 1.23907
I.15242 .757808 1.27985
1.18542 .805832 1.32223
.15655 57 1.21932 .857060 1.36630
456
78 9
IO
II
12
.17184 .015021
.18895 .018191
.20608 .021676
.17406 58
1.25415 .911751
1.41220
.19164 59
.20929 60
1.28996 .970188 1.46005
1.32676 1.032683 | 1.50999
13
.22324 .025483
.22701
HH
14
.24046 .029616
.24484
H H
7∞
15
16
17
18
.25774.034085
.26278
.27509.038895 .28083
γ
Y = 0.16
.29252
.044057
.29902
.31003
.049581 .31735

19
.32764.055476
X
Y
-33584
T
20
.34537 .061755
.3545I
21
.36322.068431 .37336 52
2.20019
1.77260
1.77260 1.62859
22
.38121 .075518
.39240 51
1.94773
1.45469 1.52223
23
.39935
.083032
24
.41765.090988
.41166 50
.43115 49
1.77989 1.25088 1.43719
1.65007
1.09878
1.36392
222
567
25
26
27
.43612.099405
.45088 48 1.54278
.97743
1.29862
.45477 .108303
.47087 47 1.45067
.87687
1.23927
.47362
.117703
.49113 46
1.36962
.79142 1.18463
223
28
.49268 .127627
.51170 45
1.29703
.71754
1.13385
29
.51197 .138100
.53258 44
1.23118
.65280
1.08632
30
.53149 .149148 .55379 43
1.17082
.5955I 1.04157
333
31
.55128
.160801
.57535 42
1.11504
•54438 .99925
32
33
2 3
•57133
.173088
.59729 4I
1.06314
.49845
.95906
.59166.186045
.61963 40
1.01458
.45697
.92078
333
34
.61230.199705
.64239 39
.96893
.41933
.88419
35
36
.63325
.65455
.214109
.66560 38
.92582
.38503
.84914

.229298
.68928 37
.88496
.35367 .81548
W W W)
37
.67619 .245318
.71346 36
.84612
.32492 .78309
38
39
.69822 .262218 .73817
.72063 .280050
.76345
54
33
35
.80907
.29850 .75186
34
.77366
.27415 -72169
40
•
74346 .298871
.78932
4I
.76673 .318745 .81582 32
42
.79045 .339737
.84299
www
33
.7397I
.25168 .69249
.70710
.23090 .66421
3 I
.67571
.21166 .63676
43
45
.81466 .361920 .87087
44 .83937 385373 .89950
.86461 410182 .92893
32 2
30
.64544
.193826 .61007
29
.61620
.177278 .58412
28
.58790
.161911 .55883
46 .89041
.436439
47
48
78
.91680.464244
.94379
.493707
.9592I 27
26
.99038
1.02251 25 .50798
.56047
.147632 .53416
.53385
.134358 .51008
.122017 .48653
60

Y = 0.16
Y = 0.16
9-
X
Y
T
4
X
Y
T
1
0
24
23
22
.45827.099872
.43434 .089958
21
.41097
.080750
20
.38811.072204
19
.36574.064281
56
222
.48281 .II0541
.46350 2,2 .37984 .075156 .39169
.44093 23 .39784 .082615 .41087
.41881 24 .41600 .090511 .43029
26
.39710 25
.37578
.3548I 27
.43433 .098862
.44994
.45283.107686
.46985
.47151 .117005
.49003
HHH
96
17
18 .34383 .056947
.32234 .050169
.33419 28
.4904I .126840
.51050
.31388 29
.5095I .137215
.53128
16
.30124
.043918
.29387 30
.52885.148157
.55239
15
.28051 .038168
.27413
3I
H
.54844 .159693
-57385
14
.26012 .032895
.25465
13
.24006 .028078
.23539
33
32
.56828 .171853
.59567
33
.58839 184670
.61789
I 2
.22029 .023696
.21638 34
II
.019731
987 6 in 4
.14384
.20081
ΙΟ .18159 .016167
.16260
.012990 .16047 37
.010186 .14218
.19756 35
.17892 36
456
.60880 .198178
.64052
.62952 .212416
.66359
.65055
.227423
.68713
.12529 .007744 .12402 39
www
38
7∞
.67193 .243245
.71116
.69367 .259928
.73572
.71579
.277525
.76082
.10693 .005652 .10602 40
.73831
.296089
.78652
5
.08875
.003901
.08812 4I
76125
•
315682
.81283
.07073 .002482
.07033 42
.78463
.336368 .83980
3
.05286 .001389
.05263 43
.80847
358217
.86747
2
+I
.03512 .000614
.01751 .000153
.03502 44
.83280 .381306
.89588
.01749 45
.85763.405716
.92507
+
Ooooo
+
+
46
.88301 .431537
.95510
Oooooo
+
Ooooo 47
48
.90894 .458866
.98600
.93545 .487808
1.01785
- I
.03473
.01741 .000152
.000605
.01743 49
.03482 50
.05198 .001358 .05219 51
.96259 .518477 1.05069
.99036 .550998 1.08459
1.01880.585506 | 1.11962
1 2 3
456
7∞
.06916
.002409
.06954 52
I.04794 .622148 1.15585
.08629
.003757 .08688 53
1.07782
.661084 1.19336
.10338
.005403 .10424 54
1.10845
.702489
.702489
1.23224
•
12044 .007347
.12161
55
1.13987
1.13987
.746553
.746553 1.27257
8
.13749 .009591 .13901 56
1.17212
.793483 1.31446
9
.15452 .012136
.15644 57
1.20523
.843506 1.35801
ΙΟ
II
.17156 .014987
.18860 .018146
I2 .20567 .021619
.17392 58 1.23922
.896870
I.40335
.19146 59
1.27412 .953845
1.45060
.20908 60
1.30998 1.014728
1.49989
13
HHH
345
.22276 .025410 .22677
14
15
.23991 .029525 .24456
.25710 .033973 .26246
16 .27436 .038759
HH Hol
6780
17
.29170.043894
18 .30911 .049386
.28046
.29860
.31688
19 .32662 .055246 .3353I
20 .34424 .061486 .35392
.36197 .068118 .37271
21
61

Y = 0.17
Y = 0.17
9-
X
Y
T
.
X
Y
T
5I
2.13564 1.66158
1.57577
50 1.89240 1.36603
1.47358
49
1.72971 1.17536
1.39159+
32 H
.05288
.001390
.05265
.03513
.000615
.03503
.01751 .000153
.01749
48
1.60356 1.03268
+
+
+
1.32079
47
1.49914 .91866
1.25760
O
OOOOO
Oooooo
Ooooo
46
1.40939.82405
1,200II
+
45
1.33034 -74358
I.14712
44 1.25950
.67394
1.09782
43
1.19518 .61288
1.05163
42
1.13618
.55880
1.00811
4I
1.08162 .51053
.96690
40 1.03083.46714
.92775
123
456
.01741
.03472
.000152
.01743
.000605
.03482
.05195
.001357 .05218
.06911 .002406 .06952
.08621 .003752
.08685
.10328
.005395 .10419
W W W
38
9∞ 765
333
39
.98329 .42795
.89042
.93856 .39236
.85471
.12031
78
.007335
.12154
.13731
.009574
.13891
37
.89630 .35992
.82048 9
.15429 .012112
.15632
32
31
32H
36
35
34
33
.85624 33028
.81813 .30309
.78177 27809
.74698.25506
.71363 | .23381
.78759 IO
.17127 .014954
.17377
•75591 II
.18826 .018102
.19129
.72534 I 2
.20526 .021562
.20887
.69579 13
.66718 14
.68157 .21417
.63944 15
345
.22229
.025338
.22652
.23936
.029435
.24428
.25647
03386 2
.26213
322
30
29
28
00∞
.65071.195981
.61250 16
.27364
.038624
.28009
.62093.179130
.58631 17
.59215.163500
.56080 18
7∞
.29089 043732
.29818
.30821 .049193
.31641
26
AOM
27
.56428
.148994
25
.53727 .135523
.51104. .123011
•53594
51167 20
19
.32561
.055019
-33479
.34312 .061220
.35334
.48797 21
.36074.067808
.37206
2.3
22222222
24
.48554.111386
.46479 2,2,
.37848.074798
.39098
.46072.100590
.44208 23
.39636.082204
.41010
.43651.090564
.41984 24
.41438 .090041
.42944
2I
.41289
.081259
.39801 25
.43256.098326
.44901
20
.38981 .072629
.37659 26
.45091 .107078
.46884
19
.36724 | .064634
.35553 27
.46944 .116318
.48893
18
.34514.057238
.33482
28
.48817 .126066
.50931
17
16
96
.32348
.30224
.050407
.044112
.31443
29
.50710.136347
.53000
.29435 30
.52625 .147184
.55IOI
15
H H
14
13
543
.28137 .038324
.27454
.26085 .033019
.25500
.24067 .028175 .23569 33
www
31
32
123
.54564.158606
.57236
.56528.170642
.59407
.58518
.183323
.61617

12 .22081 .023770
II .20123 .019788
.21663 34
.19777 35 .62584
.60537 .196683
.63868
.210759
.66162
IO
.18193 .016208 .17909 36
.64663 225591
.68502
987
.010206
.16287 .013019 .16061
.I4405
.14228
.12545
.007757 .12410
78
3 3 3
37
.66776 .241220
.70890
38
.68922 257694
•73330

39
.71106 .275062
.75825
6
.10704
.005660 .10608 40 .73327 .293378
.78377
54
.08883 .003906 .08816 4I
.75589 .312700
.80990
.07078.002485 .07035 42 .77894
333091
.83668
62

Y = 0.17
Y = 0.18
9-
X
Y
T
9-
X
Y
T
43
.80243
.354618
.86414 27
.56819
.150393
.53775
44
.82639
.377356
.89233 26
.54077
.136718
.51330
45
.85083 .401384
.92130 25
.51417
.124028 .48943
46
.87579 .426787
.95108 24
.48833 .112251
.46609
47
.90129 .453660
.98172 23
.46320
.101322
.44325
48
.92735
.482104
1.01329 22
.43872.091182
.42088
49
.95400 .512229
ཕུཔ པ་་ ་
HO
50
51
.98127.544154
1.00917 .578010
1.04584
1.07943
1.11413 19
21
.41485 .081777
.39894
20
.39154 .073061
.37741
.36876.064992
.35626
56
1.15931 .781655
57
1.03774 .613937
1.06702 .652090 1.18714
1.09701 | .692636 | 1.22561
18
17
16
55 1.12777 .735758 1.26551 15
1.19166.830542
I.15001
.34647.057533
·33546
76
H
.32464
.050649
.31499
.30324
.044308
.29483
1.30693 14
I.34999 13
543
.28223.038482
.27496
.26158 .033144
.25536
.24129 .028272
.23599
58
1.22485.882656
1.39479 12 .22133
.023845
.21688
59
1.25892
.938257
1.44146 II .20166 .019844
.19798
60
I 29388
.997630
I 49014 ΙΟ
.18228 .016250
.17926
γ
0.18
❤
X
Y
T
987
6534
.16315 .013049
.16074
.14427 .010227
.14239
.12561 .007771 .12419
.10716 .005669
.10613
.08891 .003911
.08820
.07083 .002487
.07038
0
49
50 2.06533 I.54956
1.83445
1.52292
3 2
.05291 .001391
.05266
.03514 .000615
.03504
1.27881
1.42538 +1
.01751 .000153
.01749
48
1.67800 1.10181 1.34657
+
+
+
47
1.55606
.96864 | 1.27830
О
.00000 .000000
.00000
46
1.45483
.86191 1.21724
+
45
1.36765
.77315
1.16160
44
1.29074
43
1.22173 .63204
.69755 I. 11023
1.06239
2 3
42 1.15901
•57455 1.01751
4I I.10143 .52360
.97518
40
1.04814 .47807 •93508
333
900 7 to in
39
38
37
36
www)
35
456
789
.99850 .43714 .89693
.95199 .40014 .86052
.01740 .000152 .01743
.03470
.000604 .03481
.05192 .001356 .05216
.06906 .002405 .06949
.08614 .003749 .08681
.10317 .005388 .10413
.12016 .007324 .12146
.13712 .009557 .13882
.012088 .15620
.90821 .36653 .82568 9 .15406
.86683 .33591 .79224 IO .17099 .014921 .17363
-30789
I
.82757
34 .79020 .28221
I 2
33 .75452 .25859
32
.72038
31
.68763
.76009 II 18792 .018058 .19111
.72910
.20486 .021506
.69918 13
.23684 .67023 14
.21676
.20866
.22182 .025266
.22628
.23881 | .029345
.24400
322
30
.65614.198209
.61498 16
.64219 15 .25584 .033752
.27293
.26181
29
28
I
.59651 | .165136
.62580 181040
.58854 17
.56281 18
70
.038491
.29009 .043572 .29777
.27972
.3073I
.049003
.31595
63

Y = 0.18
y = 0.19
X
9-
Y
T
X
Y
T
19
.32461
.054794
.33428
49
1.99111 1.43884
1.47035
20
.3420I
21
.060957
.35952.067503
.35277 48
1.77459 1.19374
1.37775
.37143
47
1.62516 1.03053
1.30220
23
2222222222
.37714 .074445
.39028 46
1.50785.90683
1.23648
.39489 .081798
.40933
24
.41278 .089577
.42860
45
1.41005 .80725
I.17754
44
I.32559 .72422
1.12372
56
222
2 3
333
30
3I
32
33
123
25
26
43082 097798
.44902.106480
27 .46740 .115643
28 .48596 | .125306
29 .50472 .135494
.52369.146230
.54289 157541
.61448
.58203 .182004
.44810 43
1.25093
.65335
1.07396
.46784
42
.48785 4I
1.18384 .59185
1.02754
1.12277 .53780
.98395
.50814 40
1.06665 .48986
.94279
.52874
•54964
www
39
1.01466
.44698
.90376
38
.96619
.40841
.86659
-57089 37 .92074
.37353
.83109
.56233 .169455 .59249
333
456
333
78 9
.70642 .272661
34
35
36 .64280
37
38
39
.223799
.68294
.66367|| .239242 .70668
.68487
.255514 .73093
.60200 .195220 .63687 34
.62225 .209139 .65968
333
36
.87792 .34184
.79707
35
.83742.31293
.76442
.79897 .28650
-73298
W W W
33
.76235 | .26227
.70267
32
.72737 .23999
.67337
31
.69388 .21945
.64501
.75572
40
.72835 .290737
.78107
32 2
30
.66173
.200513
.61752
29
.63080
.183011
.59082
28
.60098
.166821
.56487
41
.75066.309797
.80702
42
-77339 .329903
.83361
43
.79654 .351120
.86088 25
222
27
.57219 .151832
·53960
26
.54434
.137944 .51496
.51736
.125070
.49092
44 .82014 .373521 .88886
45
.84421 .397179 .91760
46
.86878 .422181
.94714 22
222
24
.49117
•
113135
.46742
23
.46572
.102069 .44444
.44096
091811 .42194
47
.89386 .448617
48
.91949
.476583
.97754 21
1.00884
.41683
.082304
.39988
20
•39329 .073501
.37824
49
50
556
vice were of
888
.94567
.506186
1.04III 19 .37029 .065357
.35699
.97245 .537542 I.07440
.99984.570776 | 1.10878
18
.34781 .057833
.33610
17
567
1.02788 .606023 1.14431 16
1.05658 .643431 1.18108
54 1.08597 .683162 1.21916
1.11609
1.11609 .725390 1.25864 13
1.14695 .770306 1.29962 12
1.17859
.818118
96
.32581 .050895
.31555
.30425 .044507
.29532
15
14
55
56
57
58
59
I.2443I
60
.22185 .023921 .21713
1.34221 II .20209 .019901 .19819
1.21104 .869053 1.38650 IO .18263 .016292 .17943
.923360 1.43262
.16343
1.27844 .981309 1.48072
98 7 6 5 4
.013079
.16088
.14448 .010247
.14249
.12577 .007784
.12427
.10728 .005676
.10619
.08899.003915
.07088
.08824
.002490
.07040
in 43
.033270
.28310 .038641
.26233
.24192 .028371 .23630
.27538
.25572
64

Y = 0.19
Y = 0.19
9-
X
Y
T
9-
X
Y
T
+I
32H
.05294
.001392
.05268 43
.79078
.347714
.85768
.03516.000615
.03504 44
.81404 .369788
.88546
.01752 .000153
.01749 45
.83776.393092
.91398
+
+
+
46
.86194.417708
.94329
O
.00000 .000000
.00000 47
.88663 .443723
.97345
+
48
.91183
.471231 1.00449
123
456
.01740 .000152
.01743 49
.93758
.500335 1.03649
0.3470
.000604
.03481 50
.96389
.531146 | 1.06949
.05190 .001356
.05215 5I
.99079.563785 I.10355
.06902 .002402
.06947 52
.08607 .003745 .08678
.005381 .10408
.10307
.12002 .007312 .12139
234
53
505050
1.01831.598384 | 1.13876
1.04646 .635083 | 1.17518
1.07528
54
.674038 1.21289
78
55 I.10479
.715418 1.25197
.13694 .009539
13872 56
I.13502
.759406
1.29253
9
.15383 .012064
.15608 57
1.16599
.806203
1.33466
ΙΟ
.17071 .014889
.17348 58
1.19773
.856028
1.37846
ΙΙ
.18758 .018016
.19094 59
1.23025 .909118
I.42407
12
.20446 .021450
.20846 60
1.26359.965730
1.47160
13
.22135 .025195
.22604
14
.23827.029257
.24372
15
.25522.033643
.26149
Y = 0.20

16
.27223.038359
.27936
HH
17
18
78
.28930
.30642
.043413
.29736
X
Y
T
.048814
.31549
19
20
.32362.054572
.33377 48
1.91475 1.33144
1.41840
.34092 .060697
.35831.067201
.35220 47
56
2 2
.44715 .105891
28
29
25
26
27 .46538.114977
.48378.124558
.50237 .134654 *52749
21
22
23
.37581.074097
.39344 .081398
24 .41120 .089120
.42910 .097278
.44719 42, 1.21103
.46686 4I 1.14590
.37080 46
.961847
.38958 45
1.71350 I.III5I
1.57159
1.45917 .847381 1.19537
1.33081
1.25855
.40857 44
.42776 43
1.36498.754766 | 1.13853
1.28336 .677277 1.08650
.610979
1.03830
.553344
.99328
.48679 40 1.08653 .502625
.95095
.50699
30
.52117
.145291
.54830
33 3
39
1.03190 .457578
.91093
38
.98123 .417263
.87294
37
.93394 .380962
.83673
31
.54019 .156493
.56945 36
.88956 .348113
.80210
32
33
.55943 .168290 .59094 35
.57893.180709
.84771.318258
.76891
.61281 34
.80810 .291027
.73700
333
34
.59868 .193785 .63508
35
.61871
.20755I
.65777
36
.63903.222045 .68090 31
www
33
-77047 .266113
.70627
32
.73460.243260
.67660
.70032 .222250
.64791
333
78 9
37
38
39
.65965 .237308 .70450
.68060 .253382
.72860
.70181 .270315 .75323
32 2
30
.66748.202899
.62012
29
.63593.185048
.59316
28
.60557
.168559
.56697
40
.72353 .288158
.77842 27
.57628
.153312
.54148
42
41 •74555 .306965
.76796 | .326796
.80420 26
.54799
•
.83061 25
.52061
139203
.126139
.51665
.49243
65
8

y = 0.2
y = 0.2
9-
X
Y
T
4
X
Y
T
0
24
.49406
.I14040
23
.46829
.10283 2
.46877
.44564 23
22
.37450 .073753
.38890
.39201
.081003
.40782
22
.44323
.092451
.4230I
24
.40963 .088669
.42694
21
.41884 .082840
.40083 25
20
.39506.073948
.37908
19
37185 .065728
.35773 27
222
.42740.096766
.44629
26
.4453I .105311
.46589
.46339
.114323
.48573
18
.34917
.058139
33675 28
.48163
.123822
·50585
17
.32698
.051144
.31612 29
.50006 .133830
.52626
16
.30526.044708
.29581 30
.51869.144370
.54698
15
.28397 .038802
.27580 31
.53753 .155466
.56802
14
.26307 .033397
.25608 32
.55659 | .167147
.58941
13
.24255
.028470
.23661 33
.57589 .179441
.61117
I2
.22238
.023997
.21739 34
.59543
.192380
.63332
II
.20253 .019958
.19840 35
.61524
.205997
.65589
ΙΟ
.18298 .016334
.17961 36
.63534 .220330
.67889
9∞ 7 on t
.16371 .013109
.14470 .010268
.12594 .007798 .I2435 39
.16102 37
.65572.235417
.70236
.14260 38
-77642 .251300
.72631
.69745 .268026
.75079
6
5
.10740 005685
.08907 .003920
.10625 40
.71882
285644
.77582
.08828 4I
.74055 .304206
.80143
4
.07093 .002492
.07043 42
.76266
.323771
.82766
+I
32 H
24
.05297 .001393
.03517
.01752 .000153
.05269 43
.78517 .344399
.85455
.000615
.03505 44
.80810
.366159
.88213
.01749 45
.83146
.389121
.91044
+
+
+
.00000
.000800
.00000
忐​忐
​46
.85528
.813365
.9395 3
47
.87959
.438974
.96945
48
I
1 2 3
456
7∞
.01743 49
.03480 50
Մ Մ
53
54
234
+
.01740 .000152
.03468
.000604
.05187 .001355 .05214 5I
.06897 .002400 .06945 52
.08600
.003741 .08674
.10297 .005374 .10403
.11988 .007301 .12132 55
.13676 .009523 .13863 56
.15360 .012040
1.00903 .591001
1.03667 .627024
1.06494 .665241 | 1.20678
1.09388.705815 I.24548
.90439 .46604I
1.00025
.92971.494666
.494666
1.03198
•95557 .524955
1.06469
.98201 | .557024
1.09846
I.13335
1.16943
9
.15597 57
ΙΟ .17042 .014856
.17334 58
II
I 2
.18724 .017972
.20406 .021394
.19076 59
.20825 60
.843521 1.37066
1.21671.895452
1.2493I .950797 | 1.46276
1.12350 .748922 | 1.28563
1.15383 .794754 1.32732
1.18489
1.41576
13
14
15
лаш
.22088
.025124
.22580
.23773 .029169
.24344
=
Y = 0.3
.25461 .033535
.26117

HHH
16
17
.27153 .038228 .27900
.28851 .043256
18 .30554 .048628
9-
X
Y
T
.29695
19
20
.32264 .054352
.33983.060440
.31503
40 1.47905 .776117 1.08479
•33325 39 1.33120 .654103 1.01903
.35163 38 1.22128 .566599 .96308
21
.35712 .066903
.37017 37 1.13198
.498037
.91333
66

Y = 0.3
Y = 0.3
9-
X
Y
T
9-
X
Y
T
0
www
36
1.05597
.441769
.86803
ΙΟ
.16770
.014542
.17194
35
.98937
.82615
.394244
II
.18398
.017557
.18908
34
.92984 353312
.78704 12
.20020
.020858
.20826
N N N N N N
No Nw www
.87584 .317563
.75023 13
.21639
.024448
.22347
32
.82632 .286000
-71537 14
.23256
.028329
.24075
.78049 257908
.68220 15
.24871
.032506
.25809
30
.73777 .232740
.65051 16
29
.69772 .210072
.62013
28
.65996
.189569
.59092
18
679
.26485
.036984
.27551
17 .28100
.041768
.29302
.29717 0.46866
.31064
247
.62422.170959
.56276 19
.31336.052284
.32836
.59026 .154020
.53556 20
.32958 .058029
.34622
.55787.138568
.50922 21
.34585
.064112
.36421
24
.52689.124448
.48366
2,2
.36218
.070542
.38235
23
.49718
22
•
.46862 .099695 .43466
.III527 45883
2 2
23
.37856 .077330
.40065
24
•39502 .084486
.41914
21
.44IIO .088852
.4IIII
20
19
.078915
.41453
.38882 .069810
.38811
.36564 27
222
25
.41156
0.92025
.43781
26
.42819
.099959
.45669
.44493 .108304
.47579
18 .36390 .061472 .34365 28
.46178 .117075
.49512
HH
17
-33972 .053845
16 .31620 .046879
.322II 29
.47875.126290
.51470
.30098 30
.49585 | .135967
.53455
H
15
.29331 .040529
.28024 31
.51310.146127
.55469
14
.27099
.034755
.25985 32
13
.24920
.029523
.23980
33
23
·53050 .156791
.57513
.54807.167983
.59589
I 2
.22790 .024800
.22005 34
II
.20706
.020559
.20058
IO
.18663 .016773
.18138 36
www
35
456
.56581
.179727
.61699
.58374
.192050
.63846
.60187 .204982
.66031
986
.16661
.013421
.16243 37
.62020
.218553
.68257
.14694
.010482
.14370 38
.63876 .232796
.70525
7
.12762
.007937
.12517 39
.65756.247748
.72840
6
.10861 .005771
.10684 40
.67660 .263447
.75203
in t
5
.08989 .003968
.08868 4I
.69590 .279935
.77616
4
.07145
.002516
.07068 42
.71548
.297255
.80085
+ I
321
.05326
.001403
.05283 43
.73534 .315457
.02610
.03529
.000618
.03511 44
.75550
.334591
.85196
+
0.1755 .000153
+
.01750 45
.77597 .354713
.87847
+
46
.79678 .375883
.90565
.00000
.000000
+
,00000 47
81792.398164
•93356
48
.83942 .421628
.96224
}
1 2 3
456
78 9
I
.01736 .000151
.01741 49
.86129 .446352
.99172
.03456
.000601
.05161
.001345
.002379
.06921 52
.08638 53
.10195
.11851 .007191
.12062 55
.13498
·
13772 56
.009359
.15138 .011809 .15483 57
10 10 1n
567
1.00113 .626124 1.18876
1.02601
.662329 1.22556
1.05137
.700659 1.26368
.06851
.08529.003699
.005304 .10351 54 .97672 .591904
1.15322
234
.03474 50 .88355 .472413
.05200 5 I .90620 .499900
.92928 .528908 1.08557
.95278 -559539 1.11885
1.02207
1.05334
67

Y = 0.3
Y = 0.4
γ
X
Y
T
58
9-
X
Y
T
.06806
.002358
.06898
59
60
1.07724 .741270 1.30323
1.10362.784329
I.34430
1.13053 .830018 1.38701
456
.08459 .003660 .08602
.10097
.005236 .10301
Y = 0.4
78 9
.11719
.007085
.I1995
.13329 .009203
.13685
.14926 .011590
.15373
9-
X
Y
T
IO
.16512 .014245
.17061
II
.18089 .017168
.18748
12
34
1.21572 .514384
.88133
.19658 .020359
.20436
32
31
.91614
33 1.08894 .430379
.99383 .369734
.322101
.73351 15
345
.82498 13 .21219
.77669 14
.023820
.22127
.22774
.027553
.23821
.24323
.031560 .25520
322
30
.84976 .282981
29
.79141
.249954
28
.73911 .221543
18
6790
.69402 16 .25869 .035846
.65735 17 .274II
.62298
.27224
.040415
28935
.28951.045270
.30654
27
.69155.196775
.59050 19
.30490 .050419
.32382
26
.64781 .174966
.55963 20
.32028.055867
.3412I
25
.60726 .155615
.53015 21
-33567 .061621
•35870
24
23
22
.56937.138345
.53377 .122862
.50015 108932
.50189 22
.35108.067690
.87633
.47471 23
.36651 .074082
.39409
.44849 24
.38197 .080806
.41201
21
2 H
20
19
.46826.096367
.43789 .085011
.40888 .074735
.42313 25
.39748 .087874
.43009
.39855 26
.41304 .095296
.44834
.37468 27
.42866.103084
.46679
HHH
17
16
926
18 .38108 .065432
•35146
28
.44436 .II1252
.48545
.35437 .057009
.32882 29
.46013 .119815
.50433
.32865 .049387
.30672 30 .47599 .128787
.52344
HH
in 43
15
14
13
.30382 .042499
.27980.036286
.25653.030697
.28511 31
.26396
.24324 33
32
333
1 2 3
.49194
.50800
.138186
.54281
.148030
.56245
.52418.158337
.58237
12
II
.23393 | .025686
.22289 34
.54048
.169128
.60260
IO
.21196 .021215
.19056 .017248
.20291 35
.55692 .180427
.62316
.18326 36
-57351.192256
.64405
987
.16969 .013755
.16391
.1493I
.010709 .14484
33
37
.59024 .204642
.66532
38
.60715 .217613
.68697
.12938 .008085 .12602
39
.62422 .231196
.20903
6 .10987 .005861
.10745
40
.64148.245426
.73152
5
4
.09075 .004019
.07198 .002541
.08910 41
.65894 .260335
.75448
.07094 42
.67660 | .275961
.77792
321
.05355 .001413
.03542 .000621
+ I .01758
+
.000154
+
.05297 43
.03517 44
.01752 45
.69447 .292342
.80188
.71258
.309522 .82638
.73091
.327545 .85147
+
46 .74950
.346460 .87716
.00000 .000000
.00000 47
.76835
.366321 .9035I
+
I
48
.78746
.387184
.93055
|
1 2 3
.01733 .000151
.01739 49
.80686 .409110
.95832
.03444 .000599 .03468 50
.05135
.82654 | .432165
.98686
.001337 .05187 51
.84654 .456419
1.01623
68

y = 0.4
Y = 0.5
9-
X
Y
T
9-
X
Y
T
0
1.01866
1.04185
52
53
54
.86684 .481950 1.04647
.88747.508840 | 1.07765
.90844 .537178 1.10981
.92975.567061 I.14302
.95142 .598594 | 1.17736
57 .97346 .631891 1.21290
55
56
ли
58
∞ ao
59
60
.99587.667075 1.24971
.704281 1.28789
743656| 1.32753
123
.01731 .000151
.01738
.03433
.000596 .03462
.05109
.001328 .05174
456
78 9
.06761
.002338
.06876
.08392 .003621
.08568
.10001 .005170
.10252
.11592
.006982
.11929

.13166
.009054
.13600
.14724 .011382
.15268
ΙΟ
.16267 | .013965
.16932
II
.17798 .016801
.18594
Y =
=
= 0.5
12
.19316
.019891
.20255
13
14
9-
X
Y
T
15
34in
.20824
.023234
.21917
.22323 .026832
.23580

.23813.030687
.25246
29
28
0.98413
.88201
.338743 .72106
.67306
H
16
17
6 7∞
.25297 .034800
.26916
.26774
.039175
.28590
.28245 .043816 .30271
27
~ N N N N &
.283238
18
.80260
.241866
.63110 19
.29713 .048726 .31959
73635
.208903
.59316 20
.31178 .053912
.33655
.67940
.181643
.55818 21
.32640.059379
.35360
24
.62877.158557
.52552 22
.34101 .065134
.37077
.58309
.138689
.49473 23
.35561.071183
.38805
22
.54135 .121390
.46551 24
.37022 .077535
.40546
2I
.50280.106200
.43763 25
20
.46691 .092779 .41092
19
.43328 .080866
.38521
222
.38484
.084199
.42301
26
.39949
.091184
•44073
27
·
41416.098500
.45861
876
HH
13
543
H H
.31579 .044778
.28971
.038031
18 .40158 .070258
.37157 .060791
17
16
15
14
.34303 .052335 .31314 30
.36041 28
.42887.106160
.47668
.3364I
29
.44363 .I14174
.49494
.45845
.122557
.51342
.29051 31
.47333
.131323
.53212
.26847 32
.48828
.140488
.55107
.26466 .032017
.24697 33
.50332.150067
·57028
I2
.24055 .026670
.22596 34
II
.21728 .021935
.20539 35
ΙΟ
.19478 .017764
.18524 36
456
.51844.160078
.58976
.53367
.170541
.60954
.54900
.181477
.62963
98 7 6 in t
.17298 .014114
.16546 37
.56444
.192907
.65006
.15181
.010950
.14603 38
.58001
.204854
.67084
.13123 .008240 .12691 39
.59572
.217346
.69199
.III18
5
4
.005955 .10808 40
.09163 .004071 .08953 4I
.07253 .002567 .07121 42
.61156
.230407
.71355
.62755
.244069
.73552
.64371 .258361
.75794
3 2
+I
.01761 .000154
.05385 .001424 .05312 43
.03555 .000625 .03523 44
.01753 45
.66003
.273319
78083
.67653 .288977
.80423
+
O
00000
000000
+ 46 .71010 .322553 .85264
47 .72718 ·340558
.87773
.74448 .359438 .90345
.00000
48
.69322 | 305375
.82815
69

y = 0.5
= 0.6
γ
9-
X
Y
T
9-
X
Y
T
0
575 575 57
49
50
51
52
•
79775
.76200.379244
.77975
.421860 .98481
.92984
.40003 I
.95695
.81599
444796
1.01348
53
.83449 .468910 1.04300
54
.85326.494277
1.07343
1 2 3
456
I
.01728
.000150
.01736
.03421
.000593
.03456
.05084 .001319
.05162
.06718.002318
.06854
.08326 .003583
.08534
.09909 .005107 .10204
10 10 10
55
.87230.520978
1.10482
56
.89163.549103
1.13725
57
.91124 .578748
1.17078
78 9
.11469
.006885
.11865
.13009 .008911
.13518
.14530 .011184
.15166
556
58
59
.93116.610016
1.20548 ΙΟ .16033 .013699
.16808
.95139.643023
I.24144 II
.17520 .016456
.18446
60
.97192 .677892
1.27875
12
.18993 .019452
.20083
13
.20452 .022687
.21717
14
.21899 .026161
.23352
= 0.6
15
.23336 .029876
.24987
16
.24763.033833
.26625
17
.26181 038034
.28266
9-
X
Y
T
18
.27592 .042483
.29911
19
.28996 .047181
.31562
26 .93344 .290751
.65330 20
.30395 .052135
.33219
25
.81202 .232737
.60239 21
.31789 .057348
.34885
ཝུསྶ སྶ
24
.72613 .193554
.55987 22 33180 .062826
.36559
23
.65771 | .163783
.52220 23
.34568 .068575
.38244
2,24
.60003 .139873
.48786 24 .35954 .074602
.39940
21
.54972 .I20045 .45602
20
.50484 .103259 .42614
19 .46416 .088847
.39788
22 2
25 .37339 .080915
.41649
26
.38724 .087522 .43372
27 .40110 .094432 .45109
18
H H
17
16
.42683 .076352 ·37096 28
.39225 .065444 .34520 29
.35997 .055877
.41498
.101655
.46864
.42887
.109202
.48636
.32044 30
.44280.117084
.50428
HHH
15
.32963
.047462
.29657
14
13
I 2
ΙΟ
96
7
4
.30098 .040049 .27347 32
.27378.033518
.24787.027773.22928
II .22310 .022731
.19934 .018327 .18735
.16711
.17649 .014502
.15446 .011208 .14728
.13317 .008403 .12783 39
.11255.006053 .10874
.09254 .004125 .08997 4I
.07309 .002594
.25IOI 33
333
1 2 3
31
.45677
.125314
.52240
.47079 .133905
•54075
.48487.142872 .55933
.20806
3 3 3
34 .49901
.152230
.57817
35
.51322 .161997 .59728
36
.52751 .172191
.61668
33
37
.54188 .182830
.63638
38
.55635.193936
.65642
.57093 .205531
.67680
40
.58562 .217638
.69755
.60042.230284
.71869
.07148 42
.61535 .243495 .74024
+I
321
+
.05415 .001435 .05327
.03568
.000628
.01764 .000155
+
43
.03530 44
.63042
.257303 .76224
.64563 .271737 .78470
.01755 45
.66099 .286833 .80766
+
46
.67651 | .302626 .83114
.00000 .000000
.00000
44
47
48
.69219 .319156
.70805 .336465 .87981
.85518
70

Y = 0.6
r = 0.7
9-
X
Y
T
X
Y
T
.72409
57 5
5I
ཀྱང་ང་
53
49
50 .74032 .373604
.75675 .393535
52 -77339 .414449
·79023 .436408
.354598 .90506
·93098
.95760
456
.06676
.002299
.06832
.08262
.003546
.08500
.09819
.005045
.10157
.98497 7
1.01314
54
.80730 .459477
1.04216
589
.11351 .006790
.11802
.12859.008775
.13439
.I4344 .010996
.15067
55 .82460
.483727
1.07208 ΙΟ
.15810
.013448
.16688
56
.84213
57
.509237
.85990.536090
58
60 .91472
.87792.564378
.89619 .594199
.625663
1.10297 I I
1.13488 12 .18687
1.16789
I.20207 14
1.23751
.17257 .016129
.18305
.019038
.19917
13
15
.20ior .022174
.21501 .025535 .23134
.22888 .029123
.21526
.24741
679
H
16
.24264.032938
.26349
Y = 0.7
17
18
.25629 .036981
.27959
.26985.041255
.29572
19
.28332.045763
.31189
20
.29672
.050509
.32811
9-
X
Y
T
21
.31006 .055495 .34440
23
.81319 .220114
.56922
22
22
.32334
.060728 .36076
23
•33658
.066211
.37722
22
.70121 .173643 .52145 24
.34978 .071952
.39377
21
.62208 .142434
.48153
20
19
.55903 .118842 .44613
.50586 .099998 .41382 27
222
25
26
ал
.36296 .077958
.41044
.37612 .084234
.42723
.38927 .090790
.44415
H H
18
.45945
.084464 .38381 28
.40241
.097634
.46123
17
.41804 .071399 .35562 29
.41557
.104776
.47847
16
.38047.060267 .32892 30
.42873
.I12225
.49589
15
.34598 .050697 .30346
HH
13
.28414
14 .31401 .042424 .27906
.035250 .25558
W W W
31
.44192
.I19994 .51350
32
33
23
.45514
.128094
·53131
.46839 .136539
.54935
12
.25606 .029022
.23291 34
.48169
.145341
.56762
II
.22951 .023619
.21094 35 .49504
.154518
.58614
ΙΟ
.20430 .018945
.18961 36 .50845
.164083
.60494
+I
987654
321
.18027 .01492I .16885
.15728 .011483 .14859
ww
37
.52193
.174056
.62401
38
.53548 .184453
.64340
.13521 .008577 .12880 39 .549II .195297
.66311
+
.I1397 .006156 .10942 40
.09348 .004182 .09042 4I
.07366 .002621
.07176 42
.05446 .001446
.03581 .000631
.01767 .000155
+
.56283.206607
.57664 .218407
.59056 .230721 .72440
.68316
.70359
.05342 43
O
.00000 .000000
+
.60459 .243577 .74562
.03536 44
.61874 .257001 .76728
.01756 45 .63301 .271025
+ 46 .64741 .285682
.00000 47 .66195 .301005 .83517
48
.78941
.81203

.67663 .317034
.85887
123
2
.01725 .000150
.03410 .000591
.05059 .001310
.01735 49
.0345I 50
.69147 .333809
.88316
.70647
.351373
.90807
.05149 51
.72164
.369773
.93365
71

Y = 0.7
Y = 0.8
X
Y
T
५
X
Y
T
0
52
53
54
234
.73698 .389060
.95994 ΙΟ
.75250 .409288
.15596 .013207 .16573
.98698 II .17005 .015819 .18168
.76821
5555
55
.78411
56
57
.430517 1.01481 12
.452812 1.04350 13
.8002I .476241 1.07310 14
.81651.500878 | 1.10366 15
.18395
.19768 .021690
.21125 .024947 .22926
.018647 .19758
.21343
.22467.028418
.24507
ir in
58
.83303.526806
59
84976 .554113
I.13527
1.16797
60 .86671 | .582897
1.20187
6 7∞
H H
16
.23795 .032103
.26087
17
18
.25112
.036002
.27668
.26418
.040119
.29251
19
.27713
.044454 .30836
20
.29000 .049012 .32426
Y = 0.8
21
.30280 .053795
.34022
22
.31552 .058808
.35624

23
.32819 .064055
.37233
9-
X
Y
T
24
.34081
.069541
.38851
20
19
.56945
.I17797
H H
76
H H H
18
17
16
.45205 .079501
.40632 .065947
15
14
.65361.147646
.50514.096255
.36587.054722
.32942.045287
.29609 .037282 .26064 33
456
.26531.030454 .23690 34
.024617 .21407 35
.019628 .19203 36
13
12
II
.23663
IO
.20973
.18435 .015379
.16028 .011780
.17069
www
.47637
.43575 27
222
25
26
56
.35339
.075274
.40480
.36593 .081258
.42119
.37846.087501
.4377I
.40043 28
.39096 *094012
.45437
.36852 29
.40346 .100799
.47118
.33906 30
.41596.107871
.48815
.31149 31
.28544
32
333
H 23
.42847
.115238
.50530
.44099 .122912 .52263
.45353 .130904 .54018
.46611 .139227 .55794
.47872 .147894
.57594
.49137 .156920
.59420
98 7 6 5 4
32H
.002649
.14997 38
.13737 .008761 .12980 39
.11546 .006264 .I1012 40
.09446 .004240 .09088 4I
.07426
ww
37
7∞
.50408
.166321
.61272
.51684
.176114
.63154
.52967.186317
.65065
.54256.196950
.67010
55554.208032
.68989
.07205 42
.56860 .219588
.71005
.05478
.001457
.05357 43
.58175 .231640
.73060
+I
.03595.000634
.01771
+
.03543 44
.59500 .244215
.75156
.000155
+
.01758 45

.60836.257340
.77297
+
46
.62182 .271045
.79484
.00000
.000000
+
.00000 47
48
.63541 .285361
.81721
.64911
.300323
.840II
1 2 3
456
7∞ a
.01722 .000150
.01734 49
.03399 .000588 .03445 50
.66295 .315968
.86357
.67693 .332335
.88762
.05035 .001302 .05137 5I
.69105.349466
.91230
.06634 .002279
.08199 .003510
.06810 52
.70532.367408 .93765
.08468 53
.71975 .386211
.96372
.09732 .004986 IOIII 54
.73434
.405928
.99055
.11236 .006699
.I1742 55
.12713 .008644 13361
9
.14166 .010815 .14971 57
50505
.74903
.426617 1.01818
56
.76402
.448342 | 1.04668
.77913
.471171 1.07611
72
Y = 0.8
Y = 0.9
γ

9-
X
Y
T
9-
X
Y
T

58
in
59
60
8000
0
.79442
.495177
1.10652 19
.27136
.043242
.30503
.80990.520441
1.13798
20
.28375 .047630
.32063
.82557 .547052
1.17057
21
.29605.052229
.33628
22
.30827.057044
•35198
23
Y = 0.9
.32043
.062078
.36774
24
.33252 .067337
.38359

9-
I had mad
10∞ 76
X
Y
T
56
222
25
26
.34456 .072825
.39952
.35656.078549
.41555
2,7
.36853 .084515
.43170
18
17
16
HHH
543
.31016
15
14
13
.58115
.50159
.44103
.39111.059967
•
116911
.42469 28
.38047
.090731
.44797
.091787
.38566 29
.39239
.097204
.46439
.073825
.35177 30
.40430 .103943
.48095
.34815.048848
.29286 32
.039721
.26638 33
www
.32114 3I .41621 .110957
•49768
.42812 .118256
.51460
.44004
.125851
.53170
I 2 .27592
.032124
.24134 34
.45198
.133754
.54901
II .24462,
.025753
.21749 35
.46395 .141977
.56655
ΙΟ .21571 .020391
.19464 36
.47595
.150534
.58432
9∞ 76
.18878
.015880
.16350 .012100
.13964 | .008958
.17266 37
.48798
159440
.60235
.15142 38 .50006 .168709
.62065
.13084 39
.51219 .178358
.63924
.11702
.006378
.II084 40
52438
.188406
.65815
5
.09547 .004301
.09136 4I
.53663
.198872
.67738
4
.07486
.002678
.07234 42
.54896
209775
.69696
+I
321
.05510
.001468
.05373 43
.56136
.221139
.71692
.03608.000637
.03549 44
-57384 .232987
.73727
.01774
.000156
.01760 45
.58642.245343
.75804
+
+
+
46
.59908.258236
.77925
O
.00000
.000000
.00000 47
.61185 .271695
.80094
48
.62473 .285751
1 2 3
456
789
I
.01719
+
.000149
.82313
.01732 49
.63773 300438
.84586
.03388.000586
.05011
.06593
.08137
.09647.004929
.II125
.03439 50
.65084
.315792
.86916
.001294
.05124 51
.66408 331853
.89306
.002261
.06789 52
.67745.348663
.91760
.003476
.08436 53
.69096 .366268
.94282
.10067 54
.70461.384717
.96877
.006612
.11683 55
.71840 .404062
.99550
.12573
.008518 13287 56
•
-73235 .424362
1.02305
.I3994 .010642
.14879 57
.74646.445681
1.05148
ΙΟ
.15391 .012979
II .16765 .015525
12 .18118 .018278
.16462 58
.76073 .468086
1.08086
.18037 59
.77517
.491651
I.III24
.19605 60
.78978
.516458
1.14271
JH H
13 .19452 .021235 .21168
14
.20769 .024395 .22727
15 .22069 .027758 .24283
16
.23354 .031323 .25837
17
18
.24627 .035091 .27392
.25887.039064
.28946
73
CO
9

Y = 1.0
Y = 1.0
X
Y
T
9-
X
Y
T
0
17
.59414 .116140
.41265
16
.49357
086268
.36907
223
28
29
.37080 .087740 .44199
.38220 .093933 .45805
15
.42541.067336
30
.39359 .100375
.47425
•3333 I
14
.37195 .053493
.30177
13
.32720
.042738
.27303
12
.28832 .034III .24635
www
31
.40496 .107075
.49060
32
.41633
.I14043
.50712
33
.42770 .121287
.52383
II
.2537I
.027063
.22127
ΙΟ
.22236
.021249
.19749
.19361 .016433
W w W
34
.43908
.128820
.54073
35
36
56
.45048 .136652
.55784
.46190 .144797
-57518
.17477
.16696 .012447
.15297 37
7
.14206 .009167
.13194 38
7∞
.47334 .153267
.59276
.48482
.162077
.61061
6
.11865 .006498
.III60
.49635
39
.171242
.62873
5
+I
321
.09652 .004365
4 .07549 .002708
.05543
.001480
.03622 .000640
.01777 .000156
42
.09185 40
.07263 4I .51954
.05389
.03556 43
.01761 44
.50791 .180780
.64714
.190708
.66588
.53122
.201044
.68494
.54297 .211810
.70436
.55479 .223027
.72417
+
+
+
45
.56669.234719
.74437
,00000
.000000
.00000 46
.57867 .246911
.76500
47
.59074
.259631
.78609
48
.60290
.272908
.80766
1 2 3
456
7∞
I
.01716
+
.000149
.000583
.01731 49
.61517 .286772
.82974
.001286
.11017
.03377
.04988
.06554.002243
.08078 .003442
.09565.004873
.006527 .11625 55
.12438 .008397 .13214
.03434 50
.62754.301258
.85236
.05112 5 I
.64002 .316402
.87557
.06768
52
.65262.332243
.89939
.08405 53
.66534 .348823
.92387
.10023 54
.67819.366188
.94905
9
.13830 .010477
555
56
.14790 57 .71755
ΙΟ
.15195 012762
.16355 58
со ал
.69117
.384389
.97497
.70429
.403479
1.00169
.423515
1.02926
.73095
.444561 1.05773
II
.16536 .015246
12
.17854 .017928
.17910 59
.19458 60
.7445I .466686 1.08717
.75822.489965 | 1.11765
13
14
15
HH H
78
16
.22938
17
18
.19152 .020805 .21000
.20431 .023876 .22537
.21693.027138 *24069
.030593 .25600
.24170 .034240 .27129
.25388.038081
γ I.I
.28657
19 .26594
.042117 .30187
15
21
20 .27789 .046349
.28975 .050781
.31720
•33255
H
14
13
9-
e in + m
X
Y
T
.47885
.079331
.35022
.40438
.060032
.31302
.34871.046645
.28098
22 .30151 .055416
-34796 I2
.30320.036544
.25212
23
.31320 .060258
.36342 II
.26422.028605
.22551
24
.32482 .065311
.37895 IO
.22985 .022228
.20060
225
25
.33639 .070580
•39456
.19892
.017049
.17704
26 -34790 .076070 .41026
.17069 .012826
.15460
27
-35936 .081788 .42607
7
.14463 .009392 .13309
74

Y = I.I
Y = I.I
9-
X
Y
T
9-
X
Y
T
+I
106 in +
321
.12037
.006625
.11238 40
.49293
.173917 .63695
5
.09761.004432
.09236 4I
.50400 .183370
.65523
4
.07613 .002739
.07294 42
.51512.193208
.67383
.05576
.001492
.05405 43
.52630.203449
.69277
.03636.000643
.03563 44
.53753
.214114
.71208
.01780.000156
.01763 45
.54884
.225224
.73178
+
+
+
00000
,000000
.00000
Y = 1.2
+
123
456
I
.01713 .000149
.01729
.03366.000580
.03428
.04965
.001278
.05100
9-
X
Y
T
.06515
.08020
.002225
.06748
78 9
.003409
.09485 .004819
.10913 .006446
.12308
.008281
.13671 .010319 .14703
08374 14
.09980 13
43
•45528
.070755 .32871
.37770
.052079 .29093
.11569 I 2
.32169 .039641
.25892
.13143 II
.27662.030459
.23031
ΙΟ
.23836 .023359
.20403
21
22
222
26
28
20 .27241 .045156 .31393
.28385.049436
22 .29521 .053907
23 .30647 .058574 .35933
24 .31767 .063441 •37457
25
.32879 .068511 .38989
.33986 .073790 .40528
27 .35088 079283 .42078
.36185 .084998 .43638
IO
.15007 .012554
.16251
H H
I I
.16317 .014981
.17788
86
12
.17602 .017596
.19317
7
13
14
15
16
I7
18
mt in 6 7∞
.18867 020398
.20839
.20III
.023385
22354
.026555
.21337
.22546.029907
.23739
.24919
.033443 26878
.037162 .28382 +1
.23865
64
.25372
32H
20
19
.26086
.041066
.29887
.20481 .017738 .17950
.17474 .013240 .15635
.14738 .009634
.13431
.12218 .006760 .11320
.09874 .004501
.09288
.07679
.002772
.07325
.05610
.001504
.05421
.03650.000647
03570
.01783
.000157
.01764
+
.00000
+
+
.32903
.000000
+
.00000
.34415
1 2 3
456
I
.01710 .000148 .01728
.03355
.000578 .03423
.04942
.001270
.05088
.06476.002208
.06728
.07963
.003377
.08344
.09407
.004767
.09939
29
.37280.090939
.452II
7∞
.10812 .006367
.11515
.12181
.008169
.13074
30
.38371 .097116
.46797
.13518 .010167 .14618
31
.39461.103535
.48397
ΙΟ
32
33
.41638
.40550 .I10206
.117138
.50014 II
.51648 12
www
www
35
36
37
38
39
456
34 .42726
78 a
.43815
.44905.139603
.45998.147688
.47093
.156092
.48191 .164829
.124340
.131825
•53301 13
.54973 14
.56668
15
345
.20997
.16150
14826: .012354
.16106 .014727 .17670
.17361 .017280 .19181
.18593 .020012 .20683
.19805 .022921 .22178
.026004 .23668
.58386 16
.60128
.22172
.029262 .25154
17
•
2333 1
.032694
.26637
.61898
H
18
.24474
.036300
.28119
75

Y = 1.2
Y = 1.3
γ
9-
0
19
20
21
22
23
24
.31095
.25605.040083
.26723 .044043
.27830 .048182 .32566
.28927.052504
.30015 .0570II .35545
.061706 .37042
X
Y
T
.29600
.31082
9-
X
Y
T
.00000 .000000
.00000
+
·34054
25
.32168
066595
.38546
26
.33235 .071682
.40058
27
.34296 .076973
.41578
123
456
I
.01707
.000148
.01726
.03345
.000576
.03417
.04919
.001262 .05077
.06438
.002191 .06708
.07907
.003346
.08314
.0933I
.004716
.09898
28
·35352
.082472
.43109
29
.36404
.088186
78
.44651
30
.37454
.094123
.46206
789
.10713
.006291
11461
.12059 .008061
.13007
.13370 .010021
.14536
32
33
333
34
35
36
x
auf www
.38501.100289
.47775 IO
.14651 .012164
.16052
.39546.106693
.49359 II
.15903 .014484
.17556
.40589 .113343
.50959 12
.17129 .016979
.19048
.41633 .120250
.42677 .I27423
.52577 13
.18332
.019645
.20532
.54215 14
.19513 .022480
.22009
.4372I
.134874
.55874 15
www
37
.44767
.142614
.57554 16
5 6
.20674.025483
.23479
.21817.028652
.24945
38
.45815 .150655
.59259 17
39
.46866 .159012
.60989 18
.00
.22943
.24054
.031988
.26407
.035490
.27867
40
.47919
.167699
4I
42
.48977 .176732
.50039 .186127
.62747 19 .25151 .039160 .29326
.64534 20 .26235 .042999 .30785
.66351 21 .27308 .047009 .32246
43
44
45
.51105 .195902
.52178.206078
.53256 .216674
.68202 22
.28370 .051193 .33710
.70088 23
.29423 .055553
.35177
.72012 24
.30467.060092
.36648
25
.31503 .064816
.38126
26
7 = 1.3
.32533 .069727
.39612
27
.33556.074832
.41105
28
.34575 .080135
.42608
X
Y
T 29
.35589 .085643 .44122
30
36600
0
13
.42223.060781
.30459
31
I2
.34604
.043839
.26731
32
II
.29172.032765
33
1 2 3
091361
.37608.097298 .47188
.38614 .103460 .4874I
.45648

.39618
.109856
.2359I
.50311
ΙΟ
24823 024693
.20790 34
.40621
.116496 .51898
9∞ 7 6 in +
.21141
.018522
.18220
35
.41623.123388
•53503
.013699
.17917
.15822
36
.42627.130543
•55128
.15032 .009896 .13558 37
•43630 .137973 .56775
.12408 .006903 .11405
38
.44636
.145689
.58445
39
5
4
.09993 .004574 .09342
.07748 .002805 .07357 40
.45644 .153704
.60140
.46654.162032
.61860
321
.05645 .001517
4I
.05438
.47667.170687
.63609
.03664 .000650 .03577
42
.48684 .179686
.65388
.01787
.000157
.01766
43 .49705 .189046
.67199
44
.50732 .198785
.69045
+
+
+
45
.51763 .208921
.70926
76

γ
y = 1.4
Y = 1.4
9-
X
Y
T
9-
12
.38169 .050213
II
ΙΟ
.31095 .035777
.25994 .026303
.27842 34
.24262
35
.21230 36
333
10456
X
-39678
.40644 .119671 .52834
Y
T
.II3034
.51259
.41610.126558
·54429
+
987654
32 H
.21890 .019424
.18517 37
.42576 .133707
.56044
.18405 .014209
.15350
.010180
.12610
.10116
.16024 38
.43543 .141129
.57682
.13695 39
.445II .148835
.59344
.007054
.I1495 40
45482.156838
.61031
.004650
.09399 4I
.46456 .165154
.62745
.07818 .002839
.07390 42
.47433
.173796
.64489
.05681
.001530
.05455 43
.48413 .182782
.66263
2
.03678.000654
.03584 44
.49398 .192127
.68070
I
.01790
.000158
.01768 45
.50387 .201853
.69913
+
+
+
.00000 .000000
.00000
Y = 1.5
+
2 3
I
.01705
.000148 .01725
.03334 .000573 .03412
.04897 .001255
.05065
❤
X
Y
T
456
78
.06401 .002174
.07853 .003316
.09257.004667
.10618
.06689
0
.08285 I I .33734 .040046
.25112
.09857
ΙΟ
.27427 .028320
.21743
.006217
.I1409
.I1940
.007958
.12941
86
.22753 .020482
.18848
.18946 014783
.16243
9
.13227
.009881 .14457 7
.15692
.010491
13839
ΙΟ
II
12
.14482 .011981 .15957 6
.15708 .014252
.16907 .016692
.12824
.007218
.11588
.17445
.18921
5 .10245 .004731
.09457
4
.07891 .002875 .07423
Кн
13
.18082 .019296 .20388
14
.19234 .022062 .21846
15
.20366 .024989
.23298 +1
32H
.05717 .001543
.000657
.03693
.05472
.03591
.01793 .000158 .01769
+
+
+
16
.21479
.028075 .24745
17
.22575
.031321
.26187
O
.00000
.000000 .00000
18
.23655.034726
.27627
19
.2472I .038292
.29065
20 .25774 .042019 30503
21
.26814 .045909 .31941
22 .27844 .049965 .33382
2 2
23
24
2 2
56 7
28
223
220
.34826
.28864 .054190
.29874 .058585 .36275
436
5
709
.39190
.40658
.11825
.007857
9 .13089 .009746
.12877
.14379
.30877 .063156 .37729
.31873 .067907
.32863 .072841
.33847 .077965 .42136 ΙΟ .14320 .011805 .15865
.34827 .083283 .43624
.15521 .014030 .17338
.35802 .088803 .45123 12 .16694 .016417 .18798
II
29
30
31
32
.37744
.36775 .094530
.100472
.46635
.48161
33
.38712.106637
.4970I 15
HHH
13
.17842 .018962
.20248
14
.18968
.021664 .21689
.20072 .024520 .23123
1 2 3
I
+
.01702 .000147 .01723
.03324 .000571 .03406
.04875
.06365
.002158 .06669
.07799 .003286 .08256
.09184 .004619
.09818
.10525 .006146 .11358
.001248 .05054
77

Y = 1.5
Y = 1.6
4
0
16
HH H
86 7∞
17
18
X
Y
T
9-
X
Y
T
.21157 .027529 .2455I
0
.00000 .000000
.00000
.22225
.23276
.030690
.25975
.034005 .27396
19
.24312
20
.037474
.25336 .041097 .30232
.28814
21
.26346.044876
.31650
1 2 3
+
I
.01699 .000147
.01722
.03313
.000569
.03401
.04854 .001240 .05042
2,2,
.27346 .048813
33069 4
23
24
36
222
25
26
.28336 .052912 •34492
.29316 .057175 •35918
.30288.061605
.37350
.31253 066207 .38787
27 .32211 .070985
.40232
+36
78 9
.06330
.002142 .06650
.07748
.003257 .08228
.09114 .004572 .09780
.10435
.006076
.11308
.11714 .007760
.12815
.12956
.009615
.14304
223
28
.33164.075944
41686 ΙΟ
.14164
.011636
.15776
29
.34III .081089
•43149 II
.15341
013817
.17234
30
.35054
.086426
.44624 12
.16490 .016154
.18679
www www
.35994
.091961
.46110 13
.36931
.097702
.47610
H
-37865
103655 .49124
34 in
H
.17613 .018644
.20113
14
.18713.021285
.21538
15
.19791 .024073
.22955
34
.38798
35
.39730
36
.109830
.116235
.40662 .122878
.50654 16
.52201
17
.53768 18
.20850 .027010
.24366
.21891 .030093
.22915 .033323 .27174
.25772
7∞
3 3 3
37
38
.41593
.129772 .55354
.42525 .136925 .56962
1 2
19
.23925
.036700
28573
20
.24921 .040226
.29972
39 .43459 .144350
58593 21
.25904
.043902
.31370
44
40
.44394
.152059
.60248 22
.26876
•
047730
.32770
41
42
.4533I .160066
.61931 23
.27837
.051712
.34172
.46271.168385
.63641 24
.28789
.055851
.35578
43
.47215 .177032
44 .48162 .186022
.65382 25
.67154
45
.49114 195375 .68961 27 .31598 .069249
222
567
·
29733 .060151
.36988
26
.30669 | .064616
.38404
.39827
Y = 1.6
223
28
.32522
.074055
.41258
29
.33440 .079040
.42699
30
.34353 .084209
.44150
9-
X
Y
T
www
31
32
33
1 2 3
.35263 .089568
.45612
.36169 .095123
.47088
37073 100883
.48577
100 7 6 int
.13051
5
ΙΟ
.29272.030982
.23770
.19554 .015433 .16483
.16064 .010830
.007392
.10380.004815
.021749
.19224
34
.22363 35
36
.37976 .106854
.50082
•
.38876 113046 .51603
.39777 .119466
.53142
.13994
ww
37
38
78
.40676.126125
.54702
.41577
.133033
.56282
.11686
.09517 40
.42478
39
.140201
.57884
.43380.147642
.595II
4
.07966 .002912 .07458 4I
44285 155368
.61163
+I
32 1
.05755
42
.03708 .000661
.01797
+
+
.001557 .05490
.03598 43
.000158 .01771 44
+
.45192
.163392
.62843
.46102 .171730
.64552
45
.47015
.47932 .189410
.180397
.66293
.68067
78

Y = 1.7
Y = 1.7
9-
X
Y
T
X
Y
T
0
0
IO
.31860.034841
.23157 37
.39815
.122733
.54082
86
.25006
.20245
.023321
19658 38
.40686 .129416 .55636
.016185
.16746 39
.41558.136349
.57212
7
.16472 .011206
.14158
40
.42430 .143543
.58812
6
.13294
.007578
.11788 4I
.43305 .151011
.60436
5
.10521
.004903
4
.08043
.002949
.09579 42
.07493
43
.44181
.158766
.62088
.4506I
.166821
.63768
+I
321
.05793 .001570
03723 .000664
.01800
.05507 44
.45943 .175192
.65478
.03605 45
.46828.183895
.67222
.000159 .01772
+
.00000
+
+
.000000
+
.00000
γ = 1.8
I
1 2 3
456
7∞ 9
.01696
.000147 .01721
.03303
.04833
.06295.002126
.07697
.000566
.03396
9-
X
Y
T
.001233 .05031
.06632
9
.26580.025369
.20180
.003229 .08201
.21048 .017071
.17042
.09046
.004527 .09742 7
.16925 .011628
.14338
.10347
.006009
.I1259 6
.13556.007781
.11897
8
.I1606 .007666
.12827 .009490
.12754 5
.10670 .004997
.09643
.14230 4
.08125
.002989
.07529
II
I2
IO .14012 .011474
.15167 .013613
.16292 .015903
.15689
.17132
32
.05833 .001585
.05525
.03739 .000668
.03612
.18563 +1
.01803 .000159
.01774
H
345
679
HH
13 .17392 .018340
.18468
14
.020923 .21391
15 .19521 .023648 .22792
16 .20556 .026516 .24186
.21571
17
.029525 .25575
18 .22571 .032675 .26960
19 .23555 .035968
.28342
.24525 .039403 .29722
.25482.042982
.31102
.32483
.19982
+
.00000
+
+
.000000
.00000
+
20
21
22
.26428.046707
23
.27363
.050581 ·33866
24
.28289 .054605
.35252
25
.29206.058784
.36642 ΙΟ
.13866 .011317 .15604
2 2
26
.30115 .063121 .38038 II .14999 .013416 .17034
27
28
30
.31018 .067620
.31914 .072285 .40850
29 .32805 .077122
.33691 .082135
.39440
I 2
.16102 .015661 .18450
13
.17180 .018049 .19855
.42269 14 .18233 .020577 .21249
.43697 15 .19263 .023242 .22634
31
32
1 33
34
www
456
35
36
333
1 2 3
.34573 .087330
.092715
.45137 16
.3545I
.36327.098295
.37201
.104079
.38073 .I10074
.49537 19
.51034 20 .24148
.38944 .116289 .52549 21 .25082
.46590 17
.48056 18
.20274 .026045 .24013
.21267.028985
.22242 .032060 .26753
.25385
.23202 .035272 .28118
.038622 .29482
.042III .30844
1 2 3
456
789
I
.01693
.000146
.01719
.03293
.000564
.03391
.04812
.001226 .05020
.06260 .002III .06613
.07647
.003201
.08174
.08979 .004483 .09705
.10262 .005944 .II2II
.11501 .007575 .12695
.12701 .009368 .14158
79

Y = 1.8
Y = 1.9
X
Y
T
9-
Y
0
22,
.26003 .045740
.32207
23
.26914.049512
.33572
222
567
24
25
26
.27815.053430
.34939
10 7∞ a
.005881
8
9
X
.10179
.I1399
.12580.009251 .14088
T
.11165
.007487 .12637
.28707.057496
.36310
ΙΟ
.13725 .011166 .15521
.29592
.061714
27
.30469
.066088
.37687 II
.39069 12
.14837
.013227 .16938
.15919
.015429 .18341
28
.31340
.070622
.40459
29
.32205
.075320
.41858
30
.33066
.080189
HH
345
13
.16975 .017770
.19732
14
.18007 .020245 .2IIII
.43266 15 .19016 .022855
.22482
31
www
32
33
34
123
.33922.085234
.44685 16
.34775
.090460
.46116 17
.35625 .095875
.47560 18
6 N∞
HHH
.20005
.025596
.23845
.20975
0.28470
.25202
.21928 .031475
.26555
35
36
456
.36473.101485
.49018 19
.22865 034612
.27904
.37319.107298
.38163 .113323
.50493 20 .23789
.51984 21
.037881
.2925I
.24699 .041285
.30596
www
37
38
78
.39007
.39851
.119569
.53494
22
.25598 .044825
.31942
.126044
.55024 23
.26486 .048502
•33290
39
.40695.132760
40
.4154I
.139727
4I
.42387 .146957
.56575 24
.58149
25 .28233
.59747 26 .29094
.27364 .052319
.34640
.056280
.35992
.060387
.3735I
42
.43236.154464
.61372 27
.29948 .064644
.38716
43
3333
.44087.162259
.63024 28
.30795 .069055
.40087
44 .44940 .170359
.64707 29 .31637 073626
.41466
45
.45797.178778
.6642I 30
.32474 078361
.42854
Y = 1.9
www
31
32
33
1 2 3
•33307 .083264 .44253
.34135 .088343 •45664
.34961 .093604 .47087
34
.35784.099053
.48525
9-
X
Y
T
35
.36606.0104699
.49978
36
.37426.110548
.51447
10966 76
.21997
.17426
.28737 .028267 .20836 37
.018142
.17377
.012104 .1453I 39
.13836.008003
www
38
78 9
.38245.116609
.52935
.39064 | .122892
.54442
.39883
.129407
.55969
.12012 40
.40703 .136164
.57519
5 .10826
.005098
.09710 4I
.41524 .143175
.59093
4
+I
321
.08207
.05872
•
.003032
.001600
.03619
03753.000672
.07566 42 .42346 .150452
.60693
.01807 .000160
+
+
.05544 43 .43170
44
.01776 45
+
.158008
.62320
.43997
.165857
.63976
.44827 .174013
.65664
O
.00000
.000000
.00000
|
9-
Y
2,0
X
Y
T
.23162.019478
.17994 .012646
.17769
.14744
108 7
+
.01691 .000146
.03283.000562
.01718
03385
.04791 .001219 .05009
.06227.002096
.07598.003174
.06595
.08147
.08913 .004440 .09669
123
456
80


y = 2.0
Y = 2,0
X
Y
T
५
X
Y
T
0
6
.14140
.008242
.12135 40 .39910
.132828
.56918
5
.10992 .005203
.09781 4I .40707 .139635
.58469
4
.08294 | .003074
.07605 42
.41506 .146700
.60045
321
.05914
.001615
.05563 43
.42306 .154033
.61649
.03770.000675
.03627 44
.43109.161650
.63280
+I
+
.01811
.000160
.01778 45
.43914 | .169564
.64942
+
+
.00000
.000000
.00000
+
Y = 2.1
|
123
.01688
.000146 .01716
.03273 .000559
.03380
456
7∞
.04771
.001212
.06193 .002081
.07550.003148
.08849
.004398
.10097 .005819
.04999
❤
X
Y
T
.06577
.08121 8
.24666 .021250
.18243
.09632
7
.18644
.013279
.14981
.III18
6
.1447I
.008508
.12267
.11300 .007402
.I2579 5 .11167
.005316
.09855
9 .12461 .009138
.14019 4 .08383 .003120
.07645
ΙΟ
II
.13587 .OIIO2I .15440
.14679 .013045 .16844
.15742 .015207 .18234 +1
32 1
.05955 .001630
.05582
.03785 .000679 .03635
191000'
I2
13
.01814
.01779
•
16777 .017502
.19611
+
+
+
14
.17788 .019928
.20977 O
.00000 .000000
.00000
15
.18776 .022484
.22333
+
16 .19744 .025168
.23682
17
.20693 .027979 .25024
18 .21625 .030917 .26362
19 .22542 .033983 .27695
.23444 .037177 .29026
.24333 .040501 •30356
20
21
22
222
234
23
24
.25210 .043955 .31686
.26076
123
456
.01685
.03263
.000145 .01715
.000557 .03375
.04751 .001205 .04988
.06161 .002067 .06560
.07503 .003122 .08095
.08787 .004357 .09599
.10018 .005760 .I1074
.047543 .33017
78
.11203 .007319
.12525
.26932 | .051266
25
26
2 2 2
567
.27780
.055128
.28619 .059131
ΙΟ
27
.29451 · .063279
.34350 9 .I2347
.35687
.37028 II
.38374 12
.009028
.13953
.13454
.010880
.15363
.14528
.012870
.16755
.1557I .014992
.18132
28
.39728 13
.16587 .017244
.41089 14 .17579
.19496
.019624 .20849
.22192
333)
1 2 3
31
32
33
34
HH
16
.19496 .024758
.23527
.30276 .067577
29 .31096 .072027
30 .31911 .076636 .42459 15
www
456
35
36
37
38
78
♡ ♡ ♡
39
.32721
.3433I
.35131
.35930
.18547 .022129
17 .20425 .027510 .24855
.21337 .030385 .26178
.081409 .43839
•33527 .086351 .45230
.091468 .46634 18
.096767 .48052 19
.102256 .49484 20
.36727 | .107941
.37523.113832
.38319 .I19937
.126265
.39114
.50933 21
.039755
.22233 .033384 .27497
.23115 .036507
.23984
.28813
.30127
.52400 22
.24841 .043129
.31442
.53885 23
.25687 | .046633
.32757
.55390 24
.26523 .050268
.34074
81
10

Y = 2.1
Y = 2.2
9.
X
Y
T
9.
X
Y
T
0
27
22 2
567
25
26
.27350.054036
.28168
.057942
.35395
.36719
ΤΟ
.13325 .010744
.15286
II
.14380 .012700
.16666
.28980 | .061987
.38049 I2
.I5405 .014785
.18031
223
28
.29785.066177
39385 13
.16403
.016997
.19383
29
.30584070516
.40729 14
.17376 .019332
.20723
30
.31377 .075007
.42081 15
H
.18326
.021788
.22053
www
31
32
33
34
35
36
333
1 2 3
.32167
.32952
079657
.43444
16 .19255
.024364
.23374
.084471
.44817 17
.20165
.027061
.24689
.33734.089454
.46202 18
.21058
.029876
.25998
456
.34514.094613
.35291 .099955
.36067.105488
.47601 19
.49014
.50443 2I
.21935 .032811 .27303
20
.22798 .035867
.23648
.28605
.039043
.29905
www
37
38
7∞
.36841
.III220
.51889 22 .24486
.042343
.31204
.37615.117158
.53354 23
.25312 .045767
.32505
39
.38389
.123313
.54839 24
.26129
.049319
.33807
40
.39163 .129694
41
.39938.136313
.56346 25
.57875
26
56
.26937 .052999
.35III
.27736.056813
.36420
42
.40714
.143179
.59429 27
.28529.060762
.37734
43
.41492
.150306
.61009
28
.29314
.064851
.39054
44
.42272
.157707
.62618
29
.30094
.069084

.40381
45
.43054.165395
.64256 30
.30868 .073465
.41717
Y = 2.2
9-
X
Y
T
333
456
31
32
33
34
35
333
1 2 3
.31638
.077999
.43063
.32403
.082692
.44419
.33166.087550
.45786
·33925.092578
.47167
.34683.097783
.48562
36
•35438
.103173
.49973
0
8
.26801 .023841
.18851
ww
37
38
7∞
.36193.108755
.51400
.36946
.I14539
.52846
7
.19404 .014029
.15246 39 .37700 .120531
.543II
6
+ I
54
32H
.I1352
.08477 .003166
.14834 .008800
.005436
.12407 40
.38453 .126743
.55798
.09932 4I
.39208 .133185 .57306
.07685 42
39963.139867
.58839
.05998 .001645 .05602 43
.03802 .000683 .03642
.40720 .146801
.60398
.01817
+
191000*
.01781
44 .41478 .154001
45.42239
.61985
.161479 .63600
+
+
.00000 .000000
.00000
Y = 2.3
+
I'
123
456
78 a
9
I
.01683
.000145 .01714
.03254 .000555 .03370
.0473I .001199 .04977
.06129 .002053 .06542
.07457 .003097 .08069
.08725 .004318 .09564
.09941 .005702 .11030
.II109 .007239 .12470
.12236 .008922
X
Y
T
0
654
321
.06043
.001662 .05622
.03818 .000687
.03649
.01821
.13888 +1
.000162
.01782
.009129
.I2559
.11550 .005566 .10012
.08575 .003216 .07727
.20318 .014950 .15552
.15236


82

y = 2.3
Y = 2.4
9-
X
Y
T
५
X
Y
T
0
.00000
.000000
.00000 7
.21464 .016130
.15914
+
123
456
789
I
.01680
.000145 .01712
.03244 .000553
.04712 .001192
.03365
.04967
.60 in 4
32
.15685.009501
.12726
5
.11760 .005706
.10098
.08675.003269
.07772
.06087 .001680
.05643
.03834.000692
.03658
+I
.01824 .000162
.01785
.06097
.002038 .06525
.07412 .003072
.08666 .004278 .09530
+
+
+
.08045 O
.00000
.09865 .005645
.10987
I
.01677
.000000
+
.000145
.00000
.01711
.I1018 .007161 .12417
.03235 .00055I .03360
9
.12127 .008819
.13824
ΙΟ
.13200
.010613
.15211
II
.14238
.012537
.16580
I2
.15245
.014586
123
456
.04693
.001186
.04957
·.06066
.002025
.06508
.07368 | .003048
.08020
.17934
.08607
.004241
.09498
13
.16225 .016758
.19273 7
.09792 .005590
.10945
14
.17180 .019050
.20601
8
.10928
.007085
.12365
15
.18112 .021460
.21918
9
12022
.008719
.13762
16
.23227
.19023 .023987
ΙΟ
•
13078 .010485
.15139
H H
17
.19916 .026631
.24529
II
.14099 .012378
.16497
18
.20791 .029389
.25824 I2
.15090 .OI4394
.17839
19
.21650.032264
20
.22495 .035256
.28404
27116 13
14
.16053
.016528
.19167
.16991
.018779
.20482
21
.23326.038365
.29690 15 .17906 .021145
.21787
22 .24146
.041593
30975
16
.18800 .023625
.23084
2 2
23
.24955 .044943
.32261
24
.25753
.048415
HH
17
•33549
18
70
.19675 .026217
.24373
.20533
.028922
.25656
222
25
.26543
.052013
19
-34839
.21375 .031740
.26934
26
.27324 .055740
27
.28098 .059599
.36133
.3743I
20
.22203
.034670
.28209
21
.23018 .037716
.29482
223
28
.28866 .063593
29
.29627.067726
.38736 22
.40048
.23820.040876
.30754
23
30
.30383 .072004
.41367 24
34
.24612
.044155
.32027
.25393
.047553
•33300
333
1 2 3
333
31
32
33
34
31134 .076430
.31881
.32625
35
36
.34105
.34842
333
78 a
37
38
39
40
.081010
.085749 .45387
.33366.090654 .46751 28
.095731
.100988 .49521 30
.35577 .106430 .5093I 31
.36312 .112067 .52358
.37046 .117908 .53805
.37781.123961
4I .38516 .130237
.39251.136747
416
♡ ♡ m
.42697 25
.44036
.26165 .051073 •34546
2 2
26
.26930 .054718
·35856
27
.27686 .058491
.37140
.28436 .062395
.38430
.48128 29
.29180.066435
.39726
.29919.070615
.41031
32
33
42
.55272
.56761 35
.58274
34
36
===
43 .39988 .143501
44
.40727
.59813 37
.61378 38
.62973 39
.35708
.150513
45 .41468 .157795
.104227
.50481
109727
.51891
.36424.115425
.53320
.34991
www
1 2 3
.30653
.074939
.31383 .079412
.32109 .084041
.43669
.45004
.32833 .088830 .46352
.33554 .093786 .47713
.34273 .098916 .49089
•
42345
83

y = 2.4
Y = 2.5
X
Y
T
9-
X
Y
T
40
.37140 .121330
•54769 28
.28025 .061254
.38134
41
.37857
.127451
.56240 29 .28752 .065206
.39417
42
.38575
.133799 .57734 30
.29475 .069293
.40707
: 43
.39293
.140384
.59253
| 44
.40013
.I47220
.60799
45
.40736.154318
.62373
m 3
1 2
m
3
Y = 2.5
31
32
33
34
35
36
.30192 .073520
.42006
.30906.077893
.43315
.31616 .082416
.44635
333
456
.32323
.087096 .45967
•33028 .091939 .47312
•33730.096950
.48672

X
Y
T
37
38
7∞
.3443I
.102138 .50049
.35131
.107509
.51442
4
O
765+
39
.35830 .I13073
.52854
.23004 .017756 .16361
.16193 .009927
.12907 40
.36530.118838
.54286
.11987 .005856 ,10188 4I
.37229 .124813
.55739
.08781 .003322
.07816 42
•37930 .131008
.57215
3
.06134 .001696
.05663 43
.38631.137435
.58715
2
.03851 .000695
.03665 44
.39334 .144105
.60242
+I
.01828 .000162
.01786
45
.40038
.151030
.61797
+
+
+
О
.00000
.000000
.00000
γ
r = 2.6
+
123
436
789
.01675
.000144
.01710
.03225.000548
.04673
.03355
9-
X
Y
T
.001179
.04946
13
15
.06035 .0020II .06491
.07324 .003025 .07996
.08549 .004204 .09465
.09719 .005536 .10903 3
.10841 .007012 .12314
9 .11919 .008623
ΙΟ
II
I2
•
.13701 +1
.12959 .010362 .15067
13964 .012226 .16415
.14939 .014208 .17746
.15886 .016307 .19062 I
14 .16808 .018519 .20367
.17706 .020843 .21660
6 in 4
.16780 .010425
.13110
5
.12232 .006019
.10283
.08892 .003379
.07863
32H
.06182
.001714
.05685
.03869.000699
.03674
.01832
.000163
.01788
+
+
+
O
.00000
.000000
.00000
+
1 2 3
.01672 .000144
.03216 .000547
.04655 .001173
.01708
.03350
.04936
16
.18584 .023277 .22945
18
17 .19443 .025821 .24222
.20285 .028475
456
.06005
.001999
.06475
.07282 .003002 .07972
.25493
.08493 .004168
.09433
19
.2IIIO .031238
.26759
20 .21922 .034III .28021 8
21 .22720 .037095 .29281
7∞ a
.09649 .005484
.10862
.10756 .006940
.12264
9
.11819 .008529
.13642
22
.23506 .040192 .30540 IO
.12844 .010243
.14998
2 2 2 2 2
24
597
26
27
.27291
.25047
25 .25803 .050176 .34322 13
.26550.053743 •35588 14
.057435 .36858
.24281 .043403 .31799 II .13834 .012078
.046730 .33060 12
.16335
.14793 .014029
.17655
.15724 .016093
.18961
•
16630 | .018268
.20254
15
.17514 .020552
.21536
84

Y = 2.6
Y = 2.7
9-
X
Y
T
9-
18
16
HHH
17
7∞
0
.18376.022943
.19219
.22810
.02544I .24075
.20045
.028045 .25334
456
X
Y
T
.05976 .001985 .06459
.07240
.002979 .07948
.08439
.004132 .09401
122
19
.20855 .030756
.26588
20
.21651 .033574
.27838
78
.09581 .005432
.10822
.10673
.006870
.12215
21
.22434 .036501
.29086
9
.11722 .008437
.13583
222
234
22
.23205 .039537 .30333
ΙΟ
.12732
.010127
.14929
23
.23965 .042684
.31579
I I
.13708 .011934
.16256
24
.24714 .045944
.32827
I2
.14652.013855
.17567
25
.25455 .049319
·34076 13
.15569 .015886
.18862
22
26
.26188
.052813
27
.26913 .056428
•35329
.36586 15
14
.16460 .018025
.20144
.17328
.020270
.21416
28
.27631.060167
.37848 16
.18175 .022620
.22678
29
.28343
.064035
.39117 17
30
.29050
.068034 .40393
18
78
.19004 .025074
.23932
*19815 .027631
.25180
333
3I
32
33
1 2 3
.29752 .072171 .41678 19
.20610 .030293
.26422
.30450 .076449
.42973 20
.21392 .033059
.27661
.31145 080873
.44278 21
.22160
.035930
.28897
333)
4556
34
35
36
.31836 .085450
.32525 .090184 .46926 23
.33212 .095084 .48271 24
.45595 22
.22916
.038907
.30131
.23661 .041993 .31366
.24396.045190
•32601
38
39
883
37 .33897
.100155
.49631 25
.34582
.105405 .51009
.35265.110842
40
.35948 .116475
4I
.36632 .122312
.52405 27
28
.53820
.55256 29
222
26
56
.25122
.048498
-33838
.25840 .051922 ·35078
.26550 .055464
.36323
.27254
.059127
.37572
.27952 .062915
.38828
42
.37316.128365 .56715 30
.28644.066832
.40091
43
.38001 .134642 .58198 31
44 .38687 .141156 .59707
45
.39376 .147919
.61244 33
www
32
1 2 3
.2933I
.070883
.41362
.30015 .075070
.42643
.30695
.079401
.43935
Y = 2.7
333
456
34 .31371 .083880 .45238
35
.32045
.088513
.46554
9-
X
Y
T
70
33
36
37
.32717 .093307
.47884
·
·33388.098267
.49230
38
.34057 .103402
.50592
39
.34725 .108720 .51972
64
·
5
321
+I
17470 .011021
.13339 40
.12496 .006198 .10386 4I
.09007 .003439 .07912 42
.06230 .001732 .05707 43
.03885 .000704 .03682 44
.01835 .000163
.3740I .131989
.38071 .138356 .59193
.38744 .I44965 .60712
.35394
.114228
.53372
.36062 .I19935
.54792
.36731 | .125852
.56235
.57701
O
+
+
.01790 45
+
.00000.000000
.00000
+
I
1 2 3
.01669 .000144 .01707
.03207 .000544 .03345
.04636 .001167 .04927
85

Y = 2.8
Y = 2.8
9-
X
Y
T
9-
X
Y
T
+I
P6 in 4
321
.18312 .011761
.13603 40
.34861.112089
.52940
5
.12786.006395
.10493 4I
.35515 .117674
.54345
.09130 .003502
.07961 42
.36170
.123463
.55772
.06282
.001751
.05728 43
.36825.129466
.57222
.03903
.000708
.03689 44
.37481.135694
.58698
.01839 .000163
.01791 45
.38139
.38139 .142158
.60200
+
+
+
O
.00000
.000000
.00000
+
Y = 2.9
123
456
789
.001973
.002957
.01667
.000143 .01706
.03197 .000542 .03341
.04618 .001161 .04917
.05947
.07199
.08385.004098
X
Y
.06442
T
.07925
0
.09370 6
.19388
.012730
.13921
.09513
.005383
.10783
.13105 .006616
.10611
.10592
.006802
.12167
4
.09258.003570
.08015
.11627 .008348
.13526
ΙΟ
.12623
.010014
II
.13584
.14863 +I
321
.06334 .001772
.05753
.03921 .000712
.03699
.011795
.16180
.01842 .000164
.01793
12
.14514 .013687 .17480
O
.00000 .000000
.00000
.17980.022309
13 .15416 .015687 .18765
345
14
.16293 .017791 .20037
15 .17147
16
HHH
6 700
17
18
.18794
.19591
.21298
.019999
123
.22550
.024721
.23793
.027234
.25030
19
.20373 | .029848
.26261
20
.21140 .032563
.27489
21
.21894
.035382
.28713
456
7∞ a
.01664 .000143 .01704
.03188 .000540 .03335
.04600 .001155 .04906
.05918 .001960 .06426
.07158 .002935 .07902
.08332 .004064
.09339
.I0744
9
.09448 .005335
.10513 .006737 .12119
.II534 .008263 .13469
22
.22636 | .038304
.29936 ΙΟ
.12517 .009906 .14797
23
.23367 .041332
.31159 II
.13464 .011661
.16104
24
.24088 .044468 32382 I2
·
14380.013525
.17395
22
26
.25504
27
28
29
30
31
32
123
25 .24800 .047713
.051070 ·34836
.26201 .054543 .36068 15
.26891 .058133 .37305 16
.27574.061846 .38548
.28253.065684
.39798
17
18
H H
•33608
13
14
.24883
.28926.069652
.29596
.41057 19
.073755 .42325 20
.20143 .029420 .26104
.20897 .032088
.27320
33 .30262 .077996 .43603 21 .21637 .034856
.28534
36
33 3
456
34
35
.30924.082383
.31584 .086919
.32242 .091612
.44892 2,2 .22366
.037726
.29746
.46195 23
.475II
24
34
.23083.040698
.30958
.23791
.043776
.32170
www
37
38
39
7∞ a
.32898 .096469 .48842 25
บา
.24490 .046960 •33383
.33553 .101495
.34208 .106699
.50190 26 .25181 .050254 .34600
.51555 27 .25864 .053660
.35820
mt in LO N∞
15269 .015494
.18670
.16132 .017566
.19933
.16972 .019738 .21183
.17791 .022010
.22424
.18592 .024381 .23657
.19375 .026851
86

Y = 2.9
Y = 3.0
9-
X
Y
T
9-
X
Y
T
28
.26541
.057182
.37045 16
.17608 .021720
.22303
29
30
.27211 .060822
.27876.064585
.38276 17
.39514 18
78
.18396 .024053
.23526
.19166 .026481
.24742
www
3I
32
33
34
35
333
1 2 3
.28537.068475
.40761 19
.1992I .029007
.25952
.29193
.29845
.072496
.42016 20
.20661.031629
.27158
076653
.43281 2I
.21389 .034349
.28361
36
456
.30495 .080951
.31141 .085396
.31786 .089994
.44558 22
.22104
.037168 .29563
.45847 23
.22810 .040088 .30763
.47150 24
.23505
.043110 .31965
33
70
37
38
39
40
4I
.32429 .09475I
.33070 .099674
.337II .104771
.3435I .110049
.34992 .115517
.48467 25
.24191
.046236
-33167
.49801 26
.24869 .049470
.34372
.51153 27
.25539 .052813
-35581
.52523 28
.26203.056269
.36794
.53913 29
.2686I .059841
.38014
42
.35632 .121185
.55325 30
.27513 .063533
.39240

43
44
45
.36274.127062
.36916 .133158
.37560.139485
.56759 31
.58219
.59705 33
www
32
1 2 3
.28161
.067348
.40474
.28805 .071292 .41717
.29444 .075368
.42970
y =
= 3.0
33
34
35
36
37
38
333
456
.30081.079583
.44234
.30715
.083941
.45510
.31347
.088448
.46800
4
31977 .093110
.48105

.32606.097936
.49426
9-
X
Y
T
39 .33234 .102930
.50763
40
.33861.108102
.52119
5
4
.13457 .006862
.09393.003641
.10736 4I
•34489 .113460
.53495
.08068 42
.35116 .I19012
.54893
32
.06387
.001791
.05775 43
.35745 .124769
.56313
+I
.03939 .000717
.01846 .000164
.03707 44
.36374 .130741
-57758
.01795 45
.37004
.136938
.59229
+
+
+
.00000
.000000
.00000
+
I
.01662 .000142
.01703
Y = 3.1
9-
X
Y
T
.06411
.07880 5
.09310 4
.13855
.007143
.10875
.09536.003716
.08126
.10706
.12074
.13415
+I
32 H
.06442
.001812
.05799
.03958
.00072,I .03715
.01850 .000165 .01797
123
456
7∞
.03179 .000538 .03331
.04582
.001149 .04897
.05890.001948
.07119 .002914
.08280
.004031
.09383.005287
.10436 .006672
.008178
.009799 .14733
.13347 .011530 .16032
.14250.013367
9
.11444
HHH
IO
.12413
II
12
.17313
13 .15125 .015307
14 .15975 .017346
.18579
.19831
15
.16802 .019485 .21072
O
I
1 2 3
+
+
+
.00000
.000000
.00000
+
.01659
.000142 .01701
.03171 .000536 .03326
.04564 .001143 .04887
87


Y = 3.I
Y = 3.2
Ф
X
Y
T
9-
0
456
.05862 .001936
.06395
.07080 .002893
.07857
54
.08229.003999
.09280
7∞
.09320 .005241
.10669
8
.10360 .006610
.12028 +I
32 1
X
Y
T
.14309.007467
.I1027
.09689
.003798
.08185
.06500 .001835
.05824
.03977
.000726
.03724
.01853 .000165 .01799
9
.11356 .008097
.13361
+
+
+
IO
.12312.009697
.14670
O
.00000 .000000
.00000
II
.13233
.011404
.15960
+
12
.14123
.013214
.17232
13
.14985
.015126
.18489
14
.15823
.017135
.19732
1 2 3
I
.01656
.03162
15
.16637 .019240
.20963
16
.17431 .021441
.22184
17
18
19
20
21
.18206 .023736 .23397
.18964 .026125 .24603
.19706 .028609 .25803
.20434 .031187 .26999
.21149 .033861 .28192 ΙΟ
456
7∞ a
.000142 .01700
.000535
.04547 .001137 .04878
.05835
.001924 .06380
.07041 .002872 .07836
.08179 .003967 .09251
.03321
.09258 .005196 .10632
.10286 .006548
.11983
9
.11269 .008017
.13308
.12213 .009596
.14609
22
.21852 .036632
.29383 II
23
.22545 .039501
24
.23228 .042470
.30572 12
.31764
.13122 .011280 .15890
.14000 .013066 .17153
13
.14850 .014950
.18401
2 2 2
567
25 .23902 .045541
.24567 .048717
26
.32956 14
.15675 .016929
.19635
.34150 15
.16477
.019003
.20857
27
.25226 .052000
.35348 16
.17258 .021170
.22069
28
29
.25878 .055393
.26523 .058900
.36550 17
.18021 .023429
.23272
.37758 18
.18767 .025781
.24468
30
.27164.062524
www
31
32
33
1 2 3
.27800.066268
.28431 .070138
.29059 .074138
2 I
.38973 19
.40195 20 .20213 .030760
.41427
.42668
.19497
.028224
.25659
.26845
.20917 .033390 .28028
22
.21608 .036115
.29209
333
456
34
.43920 23
35
36
37
38
www
78 9
.29684 .078273
.30306 .082548
.30925.086969
.45184 24
.46462
25
.31543 .091542 .47754 26
.32160 .096273 .49061 27
.32776 .101171 .50386 28
.33391.106242 .51729 29
.34006 .III495
.34622 .116937
39
40
41
•53091 30
42
.54475
31
43
.35238.122580
.55881
32
44
.35854.128433 .573II 33
1 2 3
45
.36472 .134506 .58767
www
34
35
36
456
333
37
38
39
7∞ ∞
.23623 .044872 .3275I
.24277 .047993 .33935
.24924 .051218 .35122
.25564.054551
.36314
.26199 .057995 .375II
.26828 .061554
.38715
27452
.065231
.39927
.28072.069030
.41147
.28688
.072957
.42376
.2930I .077016
.43617
.29912 .081212
.44870
..22289.038935
.30389
.22961 .041854
.31569
.30520 .085550 .46135
.31127 .090038 .47415
.31732 094680 .48710
.32336 .099485 .50022
88

= 3.2
Y = 3.3
9-
X
Y
T
9-
40
4I
.32939 104460
.33543 .109613
.51352 28
X |
.25262.053742
Y
T
.36083
.52702 29
.25885 .057126 .37270
42
.34146
.I14951
.54072 30
.26503 .060623
.38464
43
.34750
.120486
44
.35355
.126225
45
.35961.132181
.55464 31
.56881 32
.58323
33
123
.27117 .064236
.39665
.27726.067968
.40874
.28331.071825
.42093
γ
Y = 3.3
333
34
35
456
.28933
.075810
.43322
.29533 .079930
•44564
36 .30130 .084190
.45818
9-
X
Y
T
333
37
38
39
78
.30725 .088596
.47086
.31320
.31913
.093154 .48369
.097871
.49669
40
.32505 .102754 .50987
.14836.007850
.II199 4I
.33097
.107811
.52324
4
.09850 .003885
.08249 42
33689
.I13051
.53681
+I
321
+
+
.06559 .001857 .05849 43
.03996 .000730 .03733 44
.01857 .000165 .01800 45
+
.118482
.34282
.34875 .124114 .56464
.35470 .129957 .57892
-55061
O
.00000 .000000
.00000
+
y =
= 3.4
1 2 3
I
456
78
9
ΙΟ
I 2
13
14
34i
7
.01654
.000142 .01698
.03153 .000532 .03317
.04530 .001132 .04869

.05808
.001912 .06365
.07004 .002852 .07814 5
.08130 .003936
.09223
.09198 .005151 .10596
.10214 .006488 .I1940
.11185 .007940
.12117 .009499 .14549
II .13014 .011161 .15822
.13880 .012922
.17076
.14718 .014779 .18315
.1553I .016731 19540
15 .16321 .018774 .20753
16 .17091
.00
.020909 .21956
.023134 .23150
.24337
X
Y
T
£4
.15467.008315
.I1392
.10023
.003979
.08314
3 .06619
.001881
.05875
2
.04015 .000735
.03742
.13256 +1
01861 .000166
.01802
+
+
+
O
.00000
.000000
.00000
1 2 3
456
.01651
-+
.000141
.01697
.03144 .000531 .03312
.04513 .001127
.05781 .001901
.06967 .002833 .07792
.04859
.06349
17
.17842
18
.18576.025449
.08082 .003906
.09194
19
.19295
.027854
.25518
20
.20000 .030350 .26694
718
.09138 .005108 .10560
.10143
21
.20692 | .032937
.27868 9 .II103
.006430
.007864 .13205
.11896
22
.035617
IO
.29039
.21372
.12023 .009404
.14490
22
23 .22042 .038391 .30210 II
.12908 .0I1044
.15754
24
.22702 .041261
.31380 I2
.13763 | .012782
.17000
222
25
26
26
.23353
.3255I
.044229
13
.14589 .014614
.18230
.23997 .047296
.33725 14
.15391.016538
.19446
27
.24633 .050466 .34902 15
.16170 .018552
.20651
89
11

Y ร
Y = 3.5
= 3.4
9-
X
Y
T
X
Y
T
16
17
.16928 .020655 .21845
.17668 .022847
.23030
18
.18391 .025127
.24208
167∞ a
.09080.005066
.10525
.10074 .006374
.11854
.II022
.007791
.13155
19
20
21
.19099 .027496
.19793 .029953
.20474 .032499
.25380 ΙΟ .11931 .009311 .14432
.26548 II .12806 .010931
.15688
.27712 12 .13649 .012646 .16926
22
.21144035136
.28873 13
.14464 .014453
.18148
23
.21803 .037866
•30035 14
.15255 .016351
.19356
24
.22452.040689
.31196 15
.16023 .018337
.20552
25
.23093 .043608
.32357 16
.16770 .020410
.21737
26
.23725.046625
.33522 17
.17500 .022570 .22914
27
.24351.049742
•34689 18
.18212 .024816
.24083
28
.24969.052963
•35860 19
.18910
.027149
.25246
29
.25582 .056290
.37037 20
.19593 .029569
.26405
30
.26190 .059727
.38220 21
.20264.032076
.27560
www
31
32
33
123
.26792 .063277
.394II 22
.20923 .034672
.28713
.27391 .066945
.40609 23
.21571 .037359
.29864
27986 .070735
.41818 24 .22210 .040137
.31016
333
456
www
39
78 9
34
35
36
37
38
.28577.074652
.29166 .078700
.29753 .082885
.43036
.44267
2 2
25 .22841 .043009
.32169
26
.23463.045978
.33323
.45510 27
.24078 .049045
.3448I
.30338 .087213 .46766 28
.30921 .091690 .48038 29
.31503 .096322
.24687 .052213
.35643
.25289.055485
.36810
.49327 30 .25887 .058865
.37983
40 .32085 .IOII17
.50632 31
4I
42
.32667 | .106083 .51957
.33248 .111228 .53302 33
333
32
1 2 3
.26479.062356
.39164
.27068
.065962
.40352
.27652
.069688
.41550
43
.33830.116560
.54669
44
.34412 .122089
.56060
35
45
.34996.127826
.57475 36
333
456
34
.28234.073538
.42759
.28813
.077517
.43978
.29389 .081631
.45811
Y = 3.5
38
3 3
78
37
.29964 .085884
.46456
.30537
.090283 .47717
39 .3IIIO .094835
.48994

X
Y
T
40
.31681 .099546
.50289
41
.32252 .104425 .51602
2
+I
HNWA
4
10210
.004080
.08383
42
.32823
.109478
.52935
3 .06683 .001905
.05901 43
.33395
.114716
.54290
.04036
.000740
.01865 .000166
.03750 44
.01804 45
.33967 .120147
.55668
.34540
.125781
.57070
+
+
+
о
.00000 .000000
y = 3.6
.00000

+
X
Y
T
1 2 3
456
I
.01649 .000141 .01696
.04496
.03135 .000529 .03307
.001121 .04850
4
3
.10410 .004190
.06748 .001931 .05928
.08457
.05755 .001890 .06335
.06930 .002813 .07771
.08035 .003876 .09166
2 1
.04056 .000745 .03760
.01868 .000167
.01806
+
+
+
90
7 = 3.6
Y = 3.7
9.
X
Y
T
9-
X
Y
T
.10627
.0043II
.08536
,06816
.001957 .05956
.04076
.000750 .03769
.01872 .000167
.01808
+
+
+
I
.00000 ,000000
.01644
.03119
.04463
.000525 .03299
ΟΙΙΙΟΟ
.00000
+
.000140 .01694
.00000 .000000
.00000
+
+I
432 H
1 2 3
456
I
.01646
.000141
.01695
.03127
.04479
.000527 .03303
.001116
о
.04840
.05729 .001878
.06320
.06894 .002794
.07750
.07989 .003847
.09139
1 2 3
.04832
78 9
.09023 .005025 .10491
.10005 .006318
9 .10943 .007719 .13105
.11812
IO
.11841 .009221 .14375
II
.12705 .010821
.15623
12
.13537 .012514
.16853
456
789
.05704 .001868
.06306
.06859 .002776
.07730
.07943 .003819
.09112
.08967.004984
.10458
.09939
.006264
.I1771
.10866 .007648
.13058
13
.14342 .014298
.18067 ΙΟ
.II754 .009133
.14320
14
.15122 .016169
.19267 II
.12607 .010713
.15560
15
.15879 .018128
.20454 12
.13429
.012385 .16782
16
.16616 .020172
.21631 13
.14223
.014146
.17988
17
.17335 .022301
.22800 14
.14993
.015993
.19180
18
.18037 .024515
.23960 | 15
.15740 .017925
.20360
19
.18724 | .026813
.25115 16
.16467 .019941
.21529
20
21
.19397 .029197
.20058 .031667
.26265
17
.17176 .022040
.22689
.274II 18
.17868 .024223
.23841
2 2 2
22 .20707 .034224
.28555 19
.18545 .026488
.24988
23
.21346.036869
.29698 20
.19209 .028837
.26129
24
.21975 .039604
.30840 21
.19859 .031271
27267
56
222
25
26
27
.22595
.23208
.23813.048371
.042432 .31984 22
.045353 .33130 23
.34278 24
.20499 .033790
.28402
28
.24412.051488
20
29
.25005.054708
30
.25592 .058033
.21128 .036396 .29537
.21748
.3543I 25 .22359
.36588
26 .22962
.37752 27 .23558
.039090 .30671
.041874
.31805
.04475I
.32942
.047722
.34082
32
33
333
123
31
.26175 .061467
.38923 28 .24147
.26754 .065014
33
35
36
37
333
456
34
.27901
.28471
.29038.080422
.27329 .068679
.072465
.42488 31
.076378 .43697
.44919 33
333
1 2 3
32
.46155
.29603.084604
38
.30166.088928
39
.30729 .093403
40
.31291.098034
4I
.31852 .102829
.51256
42
.32413.107796
43
.32975 .II2944
44 ·33537 .118281
.5392I 40
.30915.096582
.49631
45
.55287 4I
.34101 .123817 .56677 42 .32019
.31467 .JO1297
.50921
.106181
.52232
4556
.40101 29 .2473I .053960
.41289 30 .25309 .057232 .37528
.25883 .060612 .38690
.26453 .064103 .39859
.27019 067708 .41037
34 .27581 .071433 .42226
.47405 35 .28141 .075282 .43426
.48671 36 .28699 .079261 .44638
.49954
37 29255 083374 .45863
38
www
.29809.087627
.47103
.52578 39 .30363 .092027 .48358
.050791 .35225
.36374
91
Y = 3.7
7 = 3.8


9-
X
Y
T
9-
X
Y
T
43
.32572
44
45
.33678
.III242
.53564
.33125.116489 •54918
34
35
.121932.56296 | 36
mm m
456
.27271 .070436
.41971
.27822.074224
.43161
.28371 .078139
.44363
y = 3.8
www
37
38
39
789
.28917.082186
.45578
.29463 .086371
.46808
.30007 .090701
.48053
9-
X
Y
T
40
•3055x .095181
.49336
4I
.31094 .099820
.50596

32
4 .10864 .004444
.06886 .001984
.08619
.05985 43
42
.31637.104624
.51896
.32180.109602
.53217
+I
.04097 .000755
.01876 .000168
.03778 44
.32723 .114763
.54560
.01809
45
.33268
.120116
.55927
+
O
123
456
689
+
.01641
.03III
.000000
+
.000141
.000523 .03294
.04823
.04448 .001105
.05678 .001857
.06824 .002757
.07898 .003791
.08912.004945
+
00000
% =
= 3.9
.01693
9-
X
Y
T
.11126
.004592
.08709
.06291
3
.06960 .002013
.06015
.07710 2
.09085 +1
.10424
.09874 .006210
.11731
+
.00000 .000000
.04118 .000761
.01880 .000168 .01811
+
.03788
+
.00000
.10791
.007580
.13010
+
|
1 2 3
456
I
.01639 .000140
.01691
.03102 .000521
.03289
.0443I .001099
.04814
.05654 .001846
.06277
.06790 .002739
.07690
15
.07854 .003763
.09059
ΙΟ 11668 .009048 .14264
II .12511
.010609 .15498
12 .13322 .012260 .16712
13
14
16
17
18
234
23
24
men On 222 222
H
19
20
21
.14107 .013998 .17911
.14866 .015821 .19095
.15604 .017728 .20266
.16321 .019717 .21428
.17020 .021787 .22580
.17703
.023940 .23724
78 9
.18370.026173 .24862 ΙΟ
.08858 .004906 .10392
.09810 .006159
.10717 .007513
.11691
.12963
.11584 .008964
.142II
.19024 .028489 .25996 II .12417 .010507 .15437
.19666 | .030888
.27125 12
.13219 .012138
.16644
.20296
033370
.28253
.20916 .035938 .29378
H H
13
.13993 .013855
.17835
14
.14743
.015655
.19011
.21527
.038592 .30504 15 .15471 .017537
.20176
22265
567
2
.22128
27
28
.23890
29
30
.041334 .31630 16
.22722 .044168 .32758
.23309 .047094 .33889
.050116
.24465.053236
.25034.056458
670
H H
.16179 .019499
.21329
17
.16869 .021542
.22473
18
.17542 .023665 .23610
31
32
33
en en
1 2 3
en
.25599 .059785
.26160 .063221
.26717.066770
.35024 19 .18200 .025868
.36164 20 .18845 .028152
.37310 21 .19478 .030517
.38462 22
.20099 .032964 .28107
.39622 23 .20710 .035494
.40792 24 .21312 .038110
.24740
.25866
.26987
..29224
.30342
92

Y = 3.9
y = 4.0
9-
X
Y
T
9-
X
Y
T
2 2
25
.21905
.040813
.31460 16
.16040 .019288
.21232
26
.22490
.043604
.32580 17
.16721 .021304
.22369
27
.23068.046487
.33703 18
.17385 .023399
.23498
www w N N
28
23640 .049464
.34829 19
.18035 .025572 .24621
29 .24206 .052537
.35960 20
.18671 .027824
.25739
30
.24767.055711
.37097 21
.19295 .030156
.26853
123
..25323
.25875
.058987
.38241 22
.19907.032569
.27964
.062371
.39392 23
.20510 .035065
.29074
.26424.065865
.40552 24
.21103 .037643
.30183
35
mmm
456
34
.26969
.069474
.41722 25
.27512
.073204
.42903 26
56
.21688
.040307 .31293
.22264
.043059 .32405
36
.28052.077057
.44096 27
.28834 .045900
•33520
37
38
39
789
.28590
.081041
.45302 28
.23398.048833
.34638
.29127
.085161
.46522 29
.23955 .051862
.35761
.29663 .089422
47758 30
.24508 .054988
.36889
40
.30198 .093832
.49010 3I
.25056 | .058216
.38024
4I
.30732 .098396
.50280 32
.25600 .061549
.39167
42
.31266 .103124
.51570 33
.26140 | .064991
.40318
43
.31801 .108023
.52880 34
44
•32336
.113IOI .54212
45
.32872
.118367 .55568
www
35
36
456
.26677.068546
.41479
.27212
.072219
.42651
.27744
.076014
.43835
37
r = 4.0
38
78
.28274.079937
.45032
.28802
.083993
.46242
39
.29330 .088189
.47469

9-
+I
432 H
123
456
789
X
.11417
.07036 .002044
.000766
.04140
.01884 .000169
Y
T
40 .29856 .092530
.48711
4I
30382 .097024
.4997I
.004759 .08807
42
.30908 .101678
.51251
.06046 43
.31435 .106500
.5255I
.03797 44
.01813
.31961 .111498
.53872
45 ||
.32489
.116682
.55218
+
+
+
.00000 .000000
.00000
=
Y = 4.1
+
.01636
.000140 .01690
.03094 .000519. .03285
.04415 .001094 .04805
.05629 .001836 .06263
.06756 .002722 .07670
.07811 .003737 .09033 +I
.08805 .004868 .10359
.09747 .006108 .11652
.9 .10644 .007448 .12917
ΙΟ .11502 .008883 .14158
II .12325 .010408 .15377
I2 .13117 .012019 .16577
.13882 .013715 .17760 4
13
14
15
•
.14623 .OI5493 .18929
15342 .01735I .20086
9-
X
Y
T
0
432 1and
.11746 .004949
.08914
.07116 .002075
.06078
.04162 .000772
.03807
.01888
.000169
.01815
+
+
+
.00000 .000000
.00000
+
1 2 3
I
.01634
.000139 .01689
.03085 .000517 .03281
.04399 .001089
.04797
.05605 .001825
.06250
.06723.002704.
.07651
.07769.003710
.09008
93

Y = 4.1
Y = 4.2
9-
O
X
Y
T
X
Y
T
78 9
.08753
.004831
.10328
O
.00000 .000000
.00000
.09686 | .006058
.11615
.10573.007383
.12873
+
ΙΟ
.11421 .008802
.14107
1
II
.12235 .010310
.15319
I2
.13018
.011903
.16512
123
.01631 .000139
.01687
.03077
.000516 .03276
.04384 .001085
.04788
13
.13774
.013578 .17688
14
.14506
.015334 .18850
15
.15215 .017169
.20000
16
.15905.019082
.21138
17
.16577 .021072
.22268
18
.17233 .023139
.23389
456
789
.05581.001816
.06236
.06691 .002687 .07632
.07727
.003684
.08983
.08703 .004795
.10296
.09626 .006009
.11576
.10504
.007321 .12828
19
.17874 .025283
.24505 IO
.II343
.008725 .14055
20
21
.18501 .027505
.25615 II
.12148
.010216
.15260
19117 .029806
.26722 12
.12922 .011790
.16446
2 2 2
22
.19721 | .032186
.27826 13
.13669 .013446
.17616
23
.20315
.034647
.28928 14
.14392
.015181
.18771
24 .20900
25
.21477
.037190
.039817
.30030 15
.15092
.016993
.19913
•31132 16 .15774 .018882
.21045
26
.22045 .042529
·32236 17
.16437 .020847
.22167
27
.22607.045330
•33342 18
.17084 .022888
.23282
28
.23163.048222
29
.23712
.051207
30 .24257 .054288
7∞
33
31
32
33
34
35
36
37
38
39
333
1 2 3
.24797.057469
.25333 .060754
.19540
.20126
.25865.064145
.34452 19
·35567 20 .18337 .027197
.36687 21 .18944 .029468
.37814 22
.031816 .27688
.38948 23
.40091 24 .20703 .036752
.17717 .025004
.24390
.25493
.26592
.034243 .28783
.29877
333
456
.26395
.067648
.41244 25
.26921
.071266
.42407 26
56
.21272
.039342
.30972
.21833
.042018
.32068
.27445 .075005
.43582 27
.22387.044780
-33167
.28488.082865
.29008.086998
40
.29526.091273
4I
.30044.095699
.27967 078870
.44770 28
.45971 29
.47188 30
.48421
.49672
.24013
.22934.047631

.34269
.23476.050574
•35376
.053612
.36488
42
.30562 .100282
.50941 33
333
1 2 3
31
.24546.056748
.37607
32
.25074.059985
.38733
.25599.063328
.39868
43
.31080.105031
44
45
.32118
.31599 .109953
.I15057
.52231 34
.53543
.54878 36
333
35
456
.26120.066780
.41012
.26639.070346
.42167
.27155
.07403I
.43333
γ
4.2
38
70
33
37
27670.077839
.44512

.28183
.081775
.45705
9-
X
Y
T
39
.28695 .085847
.46913
40
29206 .090060
.48137
+I
432 1
0
.12122
.005169
4I
.09032
.29717 .094420
··49378
.07199
.002109
.06110
42
.30227.098935
.50638
.04185.000778 .03817 43
.30737.103612
.51919
.01892
.000170 .01817 44
.31248 .108460
.53221
+
+
+
45
.31760.113488
.54545
94

Y = 4.3
y =
= 4.3
X
Y
T
X
Y
T
+
.12561 .005431
.09165 43
.30403 .102241
.51614

.07286
2
+I
.002143
.04207 .000783
.01896
.06144 44
.30907
.107018
.52906
.03827 45
.31411 .111972
.54222
.000170
.01819
+
+
+
O
.00000
.000000
.00000
Y = 4.4
+

1 2 3
456
I
.01629
.000139
.01686
.03070 .000514
.03271 Ф
X
Y
T
.04368 .001079
.04779
70
.05558 .001805
.06222
.06658 .002670 .07612
.07686 .003659 .08958
.08652 .004759
.06566 .005962
.10436 .007260 .12784
32 H
.07378 .002180
.06180
.04231 .000789
.03837
I
.01900
.000171 .01821
+
+
.10265
.11539
+
O
.00000
.000000
.00000
+
9
IO
II
.11266
.12061
008649 .14004
.010123 .15203
12
13
14
15
16
17
18
.12827 .011680 .16383
.13565 .013317 .17545
.14279 .015031 .18693
.14972 .016821 .19829
.15644 .018687 .20953
.16300 .020628 .22069
.16939 .022643 .23176
123
436
I
.01626 .000139 .01685
.03061 .000512 .03268
.04353 .001075 .0477I
.05535 .001796 .06209
5
.06627 .002654
.07594
.07646
.003634
.08933
78 9
.08603.004724
.10235
.09508
.005915
.11502
.10369.007200
.12741
19
.17563 .024733
.24277 ΙΟ .II190
.008574
.13955
20
.18175 .026898
.25373 II .I1977
.010033
.15147
21
.18774 .029138
.26465 I2 .12734
.011572
.16320
22
.19363
.031456
.27554 13
.13464
.013190
.17476
23
.19941
.033851
.28642 14 .14170 .014884
.18618
24
.20510.036326
25
.21071
.03888 2
.29729 15
.30816 16
.14854 .016654
.19747
.15519
.018497
.20865
26
.21625 .041521
.31905 17 .16166 .020414
.21973
27
.22171.044246
.32996 18
.16798 .022404
.23074
28
.227II .047058
.34091 19
.17414 .024468
.24168
29
.23246.049960
.35190 20
.18018 .026605
.25257
www www wwww.
.23775 .052956
.36295 21
.18610 .028817
.26342
.24300
.056048
.24821
.059240
.37406 22
.38524 23
.25338
.062536
.39651 24
.25852
.065940
.40787 25
.26364.069455 .41933 26
.26873
.073087
.43091 27
.19191 .031105 .27424
.19762.033469
.28504
.20323 .035912 .29584
.20877.038434 .30664
.21423 .041038 .31745
.21962 .043727 .32829
.27380
.076840
.44262 28
.22495
.046501 .33917
.27886.080721
.45446 29
.23022
.049364 •35009
.28391.084733
.46645 30
.23545
.052319 .36106
40
.28894
.088885
.47860 31
.24062
.055370
.37209
4I
.29397 .093182
.49093 32
.24576
.058518
.38320
42
.29900 .097631
.50343 33
.25086.061768
.39439
95

Y = 4.4
y = 4.5
9-
X
Y
T
X
Y
T
0
36
33 3
456
34
35
.25593 065125
.26098 .068592
.26600 .072173
.40567 25
.20687 .038001
.30516
.41705 26
.21226 .040572 .31590
.42855 2.7
.21758 .043224
.32667
333
37
38
39
789
.27100.075874
.27599 .079700
.44017 28
22284 .045962
.33747
.28096 .083656
.45193 29
.46384 30 .23320
.22805
.048788
.34831
.051704
.3592I
40
.28593.087749
.47590
41
.29089.091985
42
.29584 .096371
43
.30080.100914
44
.30576.105623
45 .31073 .I10505
.48814
.50056 33
.51317 34
.52600 35
.53906 36
ww
31
32
1 2 3
.23831 .054713
.37017
.24338 .057819
.38120
.24841 .061026
.39232
333
456
•
25341 064337
.40352
.25839 .067756
.41483
.26334.071289
.42625
ww
37
38
78
.26828 .074939
-43780
.27319
.078712
.44947
Y = 4.5
39
.27810 .082614
.46130
40
.28300 .086650
.47328

4I
.28789 .090828
.48543
+I
9-
32 H
X
Y
T
42
.29277 .095153
.49776
43
.29766 .099633
.51029
о
123
456
789
345 6 NO
15
16
17
18
H
IO
II
12
13
14
I
.07474 .002219
.04255 .000795
.01904 .000171
+
+
.00000 .000000 .00000
.01624 .000138 .01683
.03054 .000510 .03263
.04338 .001069 .04762
.05512 .001785 .06196
.06596 .002638 .07576
.07606 .003609 .08910 +1
.08555 .004690
.09452.005870
.10303
.III16
.J0205
.I1467
.007142 .12700
.008502 .13907
.11895 .009945 .15093
.12644 | .011468
.13366
.16259
.013067 .17409
.14063 .014742 .18544
.14740 .016491 .19666
.06216 44
.03848
.01823
+
.30256
.104275
.52303
45
.30745.109089
.53599
7 = 4.6
9-
X
Y
T
321
.07575 .002260
.06255
.04280 .000802
.03858
.01908 .000172
.01825
+
+
+
.00000
.000000
.00000
+
1 2 3
456
I
.01622 .000138
.01682
.03046 .000509
.03259
.04324 .001065
.04754
.16036 .020207 .21879
•
15397 018313
.20778
.16660 .022173
.22973
19
.17269 .024212
.24061
ΙΟ
.I1043 .008431
.13859
20
.17865.026323
.25143
II
.11814 .009859
.15038
21
.18449 .028507
.26221 12 .I2555 .011365
.16198
22
.19023.030766
23
.19587 .033100
24 .20141
.035511
.27296 13 .13268 .012947
.28370 14 .13958
.29442 15
.17342
.014603 .18470
.14627 .016332
.19586
76
78 9
.05490
.001777 .06182
.06565 .002621 .07557
.07566 .003585 .08885
.08507 .004656 .10175
.09395 .005825 .11430
.10239 .007085 .12657
96

† = 4.6
Y = 4.7
9
Χ
Y
T
9
X
Y
T
0
16
.15276 .018133
.20691 7
.08460 | .004623
.10146
17
.15908 .020005
.21786 8
.09340 .005781
.11396
18
.16525
.021947
.22874 9
.10176 .007029 .12617
19
.17127.023961
.23954 IO
.10972
.008362 .13812
20
.17716 .026047
.25030 II
.11735 .009775 .14985
21
.18293
.028205
.26101 12
•
.12468
.011265
.16139
22
•
18859.030435
23
.19416
.032741
.27170 13
.28236 14
24
19964 .035121
.29302 15
34in
.13174 .012830
.17276
.13856
.014468
.18399
.14517 .016177
.19508
25
2 2
26
56
.20503.037580
.30369 16
.15159 .017958
.20607
.21035 .040117
.31437 17
.15784 .019808
.21696
27
.21560.042736
.32507 18
.16393
.021728
.22777
28
.22079 .045439
.33580 19
.16988 .023718
.23851
29
.22593 .048228
.34657 20
.17570 .025779
.24920
30
.23102 .051106
.35740 21
.18140
.027911
.25985
31
.23606.054076
.36829 22
.18700
.030114 .27047
32
.24106 .057141
.37924 23
.19249
.032391
.28108
33
.24603.060305
.39028 24
.19790
.034743
.29167
34
35
36
333
236
.25097.063572
.25588 .066946
.40142 25
.41265 26
56
.20323
.037170
.30227
.20848
.039676
.31288
.26076 .070431
.42399 27
.21367.042263
.3235I
333
37 .26563 .074033
•43546 28
.21879 .04493I
.33417
38
39
.27048 .077755
.27532.081604
.44706 29
.22386.047684
.34488
.45880 30
.22888 .050526
•35563
40
.28015.085586
.47070 31
.23386
.053457
.36645
4I
.28497
.089706
.48277 32
.23880.056483
.37734
42
.28979
.093972
.49502 33
.24370 .059606
.38831
43
.29461
.098391 .50746
44
.29944 .102970
.520II
45
.30427 .107717 •53298
Y = 4.7
w w w
34
.24858.062831
.39937
35
.25342
.066161
.41053
36
.25824
.069600
.42180
33
37
38
78
.26305
.073154
.43319
.26783.076828
.4447I
39
.27261
.080626 .45637
40 .27737 .084554
.46819

9-
X Y
T
41
.28213 .088620
.48018
42
.28689 .092828
.49234
3
.07682
2
+I
.002304
.06295 43
.04304 .000808 .03869 44 .29641 .101704 .51727
.01912 .000172 .01827 45
.53005
+
+
+
.29165
.097187
.50470
.30117 .106388

00000 .000000
.00000
Y = 4.8
+
I
.01619 .000138
.01681
9-
X
Y
T
1 2 3
456
2
.07539 +I
.03038 .000507 .03255
.04309 .001060 04745
.05467
.001766 .06170
.06534 .002606
0
3 .07795 .002351 .06336
.04330 .000814 .93880
.01916
.000173
.01828
.07528.003562 | .08862
+
+
+
97
12

7 = 4.8
4.9
7
9.
x Y
T
X
Y
T
.00000
.000000
.00000 3
2 .04356
+I
.01921
.07915 .002400
.000821
.000173 .01831
.06380
.03892
+
I
1 2 3
456
I
.01617 .000138 .01679
.03030 .000506 .03250
.04294 .001056 .04737
.05445
.001758 .06157
.06504 .002590 .07521
.07490 .003539 .08838
+
+
+
.00000 .000000
.00000
+
1 2 3
I
.01615
.000137
.01679
.03022
.000504
.03247
.04280 | .001051
.04730
718
.08414 .004591
.10116
.09286 .005738
.11361
9
.10114 .006974
.12576
456
.05423 .001748
.06145
.06475
.002575
.07504
.07453 .003516
.08816
IO
.10902 .008294
.13765
II
.11657 .009692
7 08369
.08369.004559
.10089
.14933
.09233 .005696
.11328
I2
.12382 .011167
.16081 9
.10053 .006920
.I2537
13
14
15
16
•
•
.13081
.012715
13756 .014336
.14410 .016026
.17212 ΙΟ
.10834 .008227
.13721
.18328 II
.11581 .009611
.14883
.19431 12
.12299 .011071
.16025
17
18
15044 .017787
.15662 .019616
.16264 .021514
.20524 13
.12990 .012603
.17150
.21606 14
.13658 .014206
.18260
.22681 15
.14305 .015878
.19358
19
.16852 .023481
.23749 16
.14933 .017619
.20444
20
.17427 .025517
.24812 17
.15543 .019428
.21521
21
.17991 .027624
.25871 18
.16139 .021305 .22589
22
.18544
.029801
.26927 19
.16720 .023249
.23652
23
.19087 .032051
.27981 20
17288.025262
.24708
24
.19621.034374
.29034 21
.17845 .027344
.25761
25
26
56
.20147
.036773
.20666 .039248
27 .21178 .041802
22
.30087
.18392
.31141 23 .18928
.32198 24 .19456
.029496
.26811
.031719 27858
.034015
.28905
28
.21685 .044437
.33258 25
.19976 .036384
.29952
29
.22185 .047156
.3432I 26
.20489 .038829
.31000
30
.22681 .049962
•35390 27
.20995 .041352
.32050
333
123
333
456
3 3 3
789
.26054
31
32
·
23173 052857
.23660 .055844
33 .24144 .058927
34 .24625
.25104.065398
35
36
37
38
.25580.068793
.26526 .075927 .44240
39 .26997 .079675
40 .27468.083553
.36465 28
.21495
.37547 29
38637 30
.043955 •33103
.21990 .046641 •34160
.22480 .049412 .35222
.062111
.39736 3I
.40845 32
.22965 .052271
.23446 .055222
.36291
-37366
.41964 33
.23924 .058267 .38449
.072301 .43096 34
.45399 36
www
35
456
.24399 .061410 .3954I
.24872 .064656 .40642
.25342 .068008 .41755
41
.27938 .087565
42
.28407
.091719
.46573 37
.47764 38
.48973
39
78 a
.25810
.26276
.071472 .42879
.075051 •44016
.26741.078752 .45168
43
.28877 .096021
.50201 40
.27206
.082580
.46334
44
.29346 .100479
.51449 4I
.27669
.086541
.47517
45
.29817 .105100
.52719 42
.28133.090641
.48718
98.

γ
y = 5.0
= 4.9
X
Y
T
9-
43
44
45
.28596.094887
.29060 | .099287
.29524 .103848
.49938
16
.51178 17
.52440
18
6 7∞
X
.14822 .017456
.15426 .019245 .21434
Y
T
.20363
.16015 .021IOI
.22497
19
.16590 .023024
.23553
20
.17152 .025014
.24604
y = 5.0
21
.17703
.027072
.25650
X
Y
T
222
22
.18243 .029200
.26694
23
.18773
.031397
.27735
24
.19295 .033665
.28776
+
3 3 W
3
2
.08044 .002455
.04383 .000828
.01925
+
.06425
.03902
.000174 .01832
2 2 2
25
26
27
567
.19809.036007
.29816
.20315 .038423
.30858
.20816
.040916
.31902
+
+
28
.21310
.043488
.32949
O
.000000
.000000
.00000
29
.21798 .046142
.34000
1 2 3
I
45 A
+
.01612 .000137
.03015 .000502 .03242
.04266 .001046 .04721 33 .23709 .057626
.05401 .001740 .06131 34
.06446 .002560 .07486
30
.22282
.048879
·35056
.01677 31
www
32
123
.22762 .051703
.36118
.23237 .054618
.37186
.38263
35
.07416
.003493 .08793 36
333
456
.24179
.060730
.39348
.24645 | .063936
.40443
.25109 .067247
.41548
7∞
.08324.004527
.10060
8
.09181 .005654
.I1294
ww
78
37
.25571
.070667
.42666
38
.26032
.074202
.43796
9
.09993
.006868
.12497 39 .26491
.077857
.44940
IO
.10767 .008162
.13675 40
.26950
.081637
.46099
II
.11506 .009533
.14831 4I
.27408 .085548
.47275
I2
.12217 .010977
.15967 42 .27865 .089596
.48468
13
.12901 .OI2494
.17087 43
.28323
093789
.49680
14
•
13562 .014080
.18191 44 .28781 .098133
.50913
15
.14201 .O15734 .19283 45 .29239.102636
.52167
h = 1 gt²
VII. TABLE OF VALUES OF
193.1447 t2 INCHES.

t
h
t
h
t
h
t
h
"I
0.10
Inches.
1.9314 0.21 8.5177
Inches.
Inches
"1
Inches.
0.32
19.778
0.43
35.713
.II
.12
.13
.14
2.3371
2.7813 .23
3.2641 .24
3.7856
.22
9.3482
.15
.16
4 9445
.17
.25
4.3458 .26
.27
5.5819 .28
.33 21.034
10.217 .34 22.328
11.125 .35 23.660 .46
12.072 .36 25.032
13.057 .37
26.442
.44 37.393
.45
39.112
40.869
.47
42.666
.48
14.080 .38 27.890 .49
44.50I
46.374
15.143 .39 29.377
.50
48.286
.18
.19
.20
6.2579 .29 16.244 .40 30.903
6.9725 .30 17.383
7.7258
.4I
.3I 18.561 .42
.5I 50.237
32.468
.52
34.07I .53
52.226
54.254
99
VIII. A GENERAL TABLE OF VALUES OF S
FOR OGIVAL HEADED SHOT.
W

V
0
| 1
2
со
3
4
5
6
7
89
F-s.
54
55
55555
56
Feet.
I 7303
1 6780
6276
6227
1
57
58
59
I 5790
I 5320
I 4866
5742
5274
60
6I
62
없​엉엉 ​없었
​63
64
65
66
67
68
69
677
70
71
72
73
74
75
76
77
22N
567
∞
72∞
78
79
80
81
82
∞ ∞ ∞
123
83
84
∞ ∞ ∞
456
85
86
87
7∞
88
∞ ∞ ∞
89
90
222
91
92
93
94
95
96
97
98
4822
Fect. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet.
7250 7197 7144 7092 7039 69876935 6883 6832
6729 6678 6627 6577 6526 6476 6426 6376 6326
6178 6129 6080 6031 5982 5934 5886 5838
5695 | 5647 | 5600 5553 5506 5459 5413 5366
5228 5182 5137 5091 5046 5001 4956 4911
4777 4733 4689 4645 4601 4558 4514 4471
I 4428 4385 4342 4299 4256 4214 4171 4129 4087 4045
I 4003 3962 3920 3879 3838 3796 3755 3714 3674 3633
I 35933552 3512 3472 | 3432 | 3392 3353 3313 | 3274 | 3234
I 3195 3156 3117 3079 3040 3001 2963 2925 2886 2848
I 2810 2772 2735 2697 2660 2622 2585 2548 2511 2474
I 2437 2400 2364 2327 2291 2255 2218 2182 2146 2111
I 2075 2039 2004 1969 1933 1898 1863 1828 1793 1758
I 1724 1689 1655 1620 1586 1552 1518 1484 1450 1417
I 1383 1349 1316 1283 1250 1216 1183 1150 1118 1085
I 1052 1019 0987 0955 0922 0890 0858 0826 0794 0762
1 0731 0699 0667 0636 0605 0573 0542 0511 0480 0449
1 0418 0387 0357 0326 0296|0265|0235 0205 0174 0144
I 0114 0084 0055 0025 9995 9966 9936 9907 9877 9848
9819 9790 9761 9732 9703 9674 9646 9617 9588 9560
9531 9503 9475 9447 9419 9391 9363 9335 9307 9279
9252 9224 9197 9169 9142 9115 9087 9060 9033 9006
8979 8952 8926 8899 8872 8846 8819 8793 8766 8740
8714
8688 8662 8636 8610 8584 8558 8532 8507 8481
8455 8430 8404 8354 8379 8329 8303 8278 8253 8228
8203 8179 8154 8129 8104 8080 8055 8031 8006 7982
7958 7934 7909 7885 7861 7837 7813 7789 7766 7742
7718 7694 7671 7647 7624 7600 7577 7554 7531 7507
7484 7461 7438 7415 7392 7369 7347 7324 7301 7279
7256 7234 7211 7189 7166 7144 7122 7100 7078 7055
7033 7011 6990 6968 6946 6924 6902 6881 6859 6837
6816 6794 6773 6752 6730 6709 6688 6667 6646 6625
6604 6583 6562 6541 6520 6499 6478 6458 6437 6417
6396 6375 6355 6335 6314 6294 6274 6254 6233 6213
6193 6173 6153 6133 6113 6093 6074 6054 6034 6014
5995 5975 5956 5936 5917 5897 5878 5859 5839 5820
5801 5782 5763 5744 5725 5706 5687 5668 5649 5631
5612 5593 5575 5556 5538 5519 5501 5483 5464 5446
54285410 5392 5374 5356 5338 53215303 5285 5268
5250 5232 5215 5198 5180 5163 5146 5129 5111 5094
5977 5060 5044 5027 5010 4993 4976 4960 4943 4927
4910 4894 4847 4861 4845 4829 4812 4796 4780 4764
4749 4733 4717 4701 4686 4670 4654 4639 4624 4608
45934578 4562 4547 4532 4517 4502 4487 4473 4458
4443 4429 4414 4399 4385 4371 4356 4342 4328 4314
100

V.
9
•
со
0
7
1 2
6
3
5
4
F-s.
99
100
ΙΟΙ
102
103
I04
105
106
107
108
109
ΙΙΟ
III
II2
113
114
115
116
117
118
II
119
120
I2I
I22
123
124
125
126
127
128
129
130
Feet.
Feet. Feet. Feet. Feet. Feet. Feet. l'eet. Feet. Feet.
4258 4244 4230 4216|| 42034189 | 4176
4123| 4110 4097 4084. 4071 4058 4045
3995 3983 3970 | 3958 3946 3934 3921
4300 4285 4271
4162 4149 4136
4033 4020 4008
3910 3898 3886
3795 3784 3773
3687 3677 3666
3874 3863 3851 38403829 3817 3806
3762 3751 3740 3730 3719 3708 3698
3656|| 3646 3636 3626 3616 3606 3596
|
3586 3576 3567 3557 3547 3538 3528 3519 3510 3501
3491 3482 3473 3464 3455 3446 3438 3429 3420 3411
|
3402 3394 3385 3377 3368 3360 3351 3343 3334 3326
3318 3310 3301 3293 3285 3277 3269|3261 | 3252 3244
3236 3228 3220 3213 3205 3197 3189 3181 3173 3165
3158 3150 3142 3134 3127 3119 3111 3103 3096 3088
3080 3073 3065 3058 3050 3043 3035 3028 3020 3013
3005 2998 2990 2983 2976 2968 2961 2953 2946 2939
2931 2924 2917 2910 2902 2895 2888 2881 2874 2867
2859 2852 2845 2838 2831 2824 2817 2810 2803 2796
2789 2782 2775 2768 2761 2754 2747 2740 2733 2727
2720 2713 2706 2699 2692 2686 2679 2672 2665 2659
26522645 2638 26322625 2618 2612 2605 2598 2592
2585 2579 2572 2566 2559 2552 2546|2539 2533 2526
2520 2513 2507 2500 2494 2488 2481 2475 2468 2462
2456 2449 2443 2436 2430 2424 2417 2411 2405 2399
2392 2386 2380 2374 2367 2361 2355 2349 2342 2336
2330 2324 2318 | 2312 2306 2299 2293 2287 2281 2275
2269 2263 2257 2251 2245 2239 2233 2227 2221 2215
|
2209 2203 2197 2191 2185 2179 2173 2167 2161 2155
2149 2143 2138 2132 2126 2120 2114 2108 2102 2097
2091 2085 2079 2073 2068 2062 2056 2050 2045 2039
2033 2028 2022 2016 2011 2005 1999 1994 1988 1982
1977 1971 1965 1960 1954 1948 1943 1937 1932 1926
|| |
1921 1915 1909 1904 1898 1893 1887 1882 1876 1871
1865 1860 1854 1849 1844 1838 1833 1827 1822 1816
131
1811 1806 1800 1795 1789 1784 1779 1773 1768 1762
132 1757 1752 1746 1741 1736 1730 1725 1720 1715 1709
133 1704 1699 1693 1688 1683 1678 1672 1667 1662 1657
134 1651 | 1646 1641 1636 1631 1625 1620 1615 1610 1605
1599 1594 1589 1584 1579 1574 1569 1564 1558 1553
1548 1543 1538 1533 1528 1523 1518 1513 1508 1503
1498 1493 1488 1483 1477 1472 1467 1462 1457 1452
135
HHH
www
56 7
136
137
138
139
140
14I
142
143
144
145
146
147
148
149
|
1447 1442 1437 1432 1427 | 1422 | 1418 | 1413 1408 1403
|
1398 1393 1388 1383 1378 1373 1368 1363 1358 1353
1348 1344 1339 1334 1329 1324 1319 1314 1309 1304
1300 1295 1290 1285| 1280 1275 1270 1266 1261 1256
1251 1246 1242 1237 1232 1227 1222 1217 1213 1208
1203| 1198 1193 1189|| 1184 1179 1174 1170 1165 1160
1155 1151 1146 1141 1136 1132 1127 1122 1118 1113
1108 1103 1099 1094 1089 1085 1080 1075 1070 1066
1061 1056 1052 1047 1042 1038 1033 1028 1024 1019
1014 1010 1005 1001 996 991
991 987 982 977 973
968 963 959: 954 950 945 940 936 931 927
922 917 913 908 904 899 894 890 885 881
101
v. |
6789
0
5
1
| 4
2
3
F-s.
150
151
867 862
Fect. Feet.
876 871
Feet.
Feet.
Feet. Feet. Feet.
830826 821 817 812 808
Feet. Feet. Feet
858 853 849
844 840 835
803
799 794 790
14
1
မာ
1
111
101104
750
753 | 749
744
153
740 735 731
726
722 717
713
708 704
699
154
695
690
686
681
676
672
668 663
659
654
155
650 645
645
641
637
637
632
628
623 619
614 610
156
605 601
596
592 588
583
579
574 570
565
157
561 556
552
548 543
539
534
530 525
521
158
516 512
508
503
499
494
490
486 | 481
477
159
472 468
468
464
459
455
450
446 | 442
437 433
160
428 424
420
415
4II
406
402 398
393
389
161
385 380
376
371
367
363
358 354
350
345
162
341 337
332
328
324 319
315
310
306
302
163
297 293
289
284
280
276
271
267
263
258
164
254 250
245
241
237 232
228
224 220 215
165
211 207
202
198
194
189
185
181
166
168 164
160
155
151 147
142
138
167
126
121
117
113
109 104
100
H ∞
39
177 172
134
130
96
92
88
168
83
79
75
71
67 62
169
41
37
33
29 25 21
ня
58
54
17 12
42
50
46
4
d2
IX. A GENERAL TABLE OF VALUES OF t
FOR OGIVAL HEADED SHOT.
W
V.
0
1 2 3
5 6
7
со
9
F-s.
Seconds. Sers. Secs. Secs. Secs. Secs. Secs. Secs Secs. Secs.
54 22.078.980 *.882 *.784 *.688 *.592*.496 *.401 *.306 *.212
55 21.118 .025 *.933 *.841
*.841 *.749 *.658 *.567 .477.388
.388.299
56 20.210 .122
.034.947.860.774 *.688.603.518.433
57 19.349.265 .182
•
58 18.532 453 374
59 17.756 .681 .606
.455 .386
.781.715
.099 .017*.935.854773 .692.612
.295 .217
.217.139.062.985
*.908*.832
.531.457 .383 .309
.457.383 .309 .236
.236.163.091
60 17.019.947.876*.805.734.664 *.594.524
61 16.318 .249 .182 .114 .047.980.913.847
.847.781
62 15.650.585 .520 .456
.520 .456.392 .328 .265 .201 .139.076
63 15.014.952.890 *.829*.768 *.707 *.647.586.526 *.467
64 14.407.348 .290 .231 .173.115 .057.999
.999.942.885
65 13.829.772 .716 .660
.716 .660.605 .549 .494
.605.549 439
.439 .385.330
66 13.276 .222 .168 .115 .062 .009 *.956
*.956.904 *.852 *.800
.904.852.800
67 12.748.696 .645 .594
.645.594 .543 .493
.493.442 .392
.392 342 .292
68 12.243.194.145 .096
.145 .096.047 *.999 *.950
.047*.999*.950.903 *.855 *.807
*.855.807
69 11.760 .713 .666 .619
.666 .619.572.526 .480
.572.526.480.434 .388 .343
.252.207.162
.162.118 .073 .029
.029.985
.985 *.941 *.898
10.854.811 .768 .725
.768.725 .682 .639.597 .555 .513 .471
70
11.297
71
•
102


V.
9
0 1 2 3
со
4
6 7
100
5
72
F-s. Seconds. Secs. Secs. Secs. Secs. Secs. Secs. Secs. Secs. Secs.
10.429| .388.346.305 .264 .223 .183.142 .102 .062
10.022 *.982 *.942 *.903 *.863
.903 *.863 *.824.785.746
´.824 *.785 *.746 *.708 *.669
9.631.592.554 .516 .478 .441 403 .366 .329 .292
73
74
75
76
557
77
9.255.218 .182.145.109
8.894 859.824 .789 .754
8.547.513 .480 .446 .412
78 8.214 .181 .148 .116.084
79 7.893.861.830 .799 .768
7.584.553.523.493 .463
778
80
∞ ∞ ∞
82
83
.170
.073 .037 .001 *.965 *.930
.719 .684 .650 .616 .581
.379 .346 .312 .279 .246
.052 .020*.988 *.956 *.924
.737 .706.675 .645 .614
433404 .374 .345 315
.141.113.084
113 .084
.859.832.804 .777 .750
.588 .561| .535
.326 .300 .275 .249
.056
.027 |
.508
.482
.224
81 7.286 .257 .228.199
6.999 .971 .943 .915.887
6.722.695.668 .641 .615
6.456.430.404 .378 .352
6.198 173 .148 .123 .098
.048.024 *.999*.974
5.950 .926.901.877.853 .829 .805.781 757 734
84
∞∞∞
456
85
86
87
88
∞∞∞
89
90
91
92
93
94
95
96
97
98
129
99
100
ΙΟΙ
·
5.710 .686 | .663 .640 .616
5.478 .455 .433
5.254.232 210
.410 .388
.188.167
5.038 .016 *.995 *.974*.953
4.829.808.788 768 747
4.628.608.589 .569 .550
.073
.593 .570 .547.524 .501
.365 .343 .321 .298 .276
.298.276
.145.123 .102 .080 .059
.932 *.911.890 *.870 *.849
.727 707 .687 .667 .648
.531.511 .492
.492.473 .454
4.435.417.398 .379 .361 .342 324.305 .287 .269
.197.179 .161 .144 .126.109.091
.022 .005.988.971.955.938.921
4.251.233 .215
4.074 .057 .039
•
•
3.905 .888.872 .856 .839 .823.807 791 775 759
3.743 .728 .712 .697 .681 .666 .650.635 .620.605
3.590.575 .560 .546 .531 .516 .502 .487.473 .459
3.444.430
.430 | .416 .402 .389
3.307.294 .281 .267 .254
3.178.165.153 .141.128
.375 .361.347 .334 .320
.241.229.216 .203 .190
.116 .104.092.080.069
I02 3.057 .045.034 .022 .OII .999*.988 *.977.966 *.955
2.944 .933 | .922 .911 .901.890 .880.869.859 .849
2.839 .829 .819 .809 .799 .789 .780 770 .761 .751
103
104
105
106
107
108
109
IIO
III
II2
113
114
115
116
117
118
119
•
·
•
.696.687
2.742 733 724 714 705 .696 .687 .679 .670 .661
2.652.644.635 .627
.618 .610 .602
2.569.561.553 545 537 .529 .521
.444
.371
2.490.482.475 .467 .459 .452
2.415.407 .400 .393 .386 .378
2.343 336.329 .321 .314 .307 .301
2.273 .266
.266 .259.252.246.239.232
·
.593 .585.577
.513.505.498
437 .430.422
.364 .357 .350
.294 .287 .280
.225 .219 .212
.159 .153 .146
.095 .089 .083
2.205.199 .192 .186.179 .172.166
2.140.134.127 .121 .114 .108 .102
2.076 .070.064 .058 .052 .045 .039 .033
.033.027 .021
2.015 .009 .003.997.991.985.979.973 *.967 *.961
1.955.949.943 .937 .931 .926 .920 .914.908 .902
1.897.891.885 .880.874.868.863.857.851.846
1.840 .834.829 .823.818 .812 .807 .801 .796 .790
1.785.779.774 .768|| .763 •758.752.747 742 736
103

V.
9
0 1
8
2
со
دن
5 6 7
4
F-S.
120
121
I22
Seconds. Secs. Secs. Secs. Secs. Secs. Secs. Secs. Secs. Secs
1.731 726 720 715 .710 .705 .699 .694 .689 .684
1.678.673.668 .663 .658 .653 .648 .642 .637 .632
1.627 .622 .617 .612 .607 .602 .597.592 .587.582
123 1.577 .572.568.563.558 .553.548.543.538.533
124 1.529 .524.519 .514 .509 .505 .500 .495 .490.486
125 1.481 .476 .471 .467 .462 .457 .453.448.444.439
1.434 .430.425 .420 .416 .411 .407 .402 398 393
127 1.389.384 .380 .375 .371 .366 .362 .357 .353 349
.353.349
128 1.344 340 335 .331 .327 .322 .318 .314 .309 .305
|
129 1.301 .296 .292 .288 .284 .279 .275.271 .267 .262
.271.267.262
130 1.258 .254 .250 .245 .241.237.233.229.225 .220
131
1.216 .212 .208 .204 .200 .196.192 .188 .184 .179
1.175.171.167.163.159 .155 .151.147 .143
126
132
.139
133 1.135 .I3I .127 .123 .120 .116 .II2 .108 .104 .100
134 1.096 .092 .088.084.081 .077 .073.069|.065
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
.061
1.058 .054 .050 .046 .042.039 .035 .031 .027 .023
1.020.016
.012 .008 .005 .001
0.982 .979 .975 .971 .968 .964
0.982.979
.997.994.990 *.986
.961
.961 .957 953 .950
.946.942.939 .935 .932 .928 .924 .921 .917
.917 .914
.910 907 .903 .899 .896 .892
.896.892 .889
.889.885 .882.878
.875 871.868.864.861 .857 .854.850.847
.847.843
.840.836.833.830.826| .823
.830.826.823.819
.819 .816 .813| .809
.806.802 799 .796 .792 789
.796.792.789 .785
.785782
.772.769.765
.772 769 765 .762.759 755 .752 749 745
.752
.739.736.732.729 .726 .722
.729.726.722 .719
779 .775
.719 .716 713
742
.709
.706 703 700
.700 .696 .693 .690 .687 .683 .680 .677
.674.671 .667 .664.661 .658
.655 .651 | .648
.642 .639 .636.632.629.626 .623.620.617
.610.607 .604 .601 .598.595
.579 .576.573 .570 .567 .564
.549.546.543.540 .537 534
.645
.592.589| .586
.589.586 .583
.561 | .558 .555 .552
|
.540.537.534 .531.528 .524 .521
.518 .515 .512 .509 .506 .503
.509.506.503 .500
.500 .497.494 .491
.488 485.482.480.477.474.471| .468 .465| .462
.459.456.453 .450 .447 .444 .441.438.435 .432
.429 .427 .424 .421 .418.415
.412 .409 .406 .403
.392| .389 .386| .383| .380 .377 .375
.335| .332 | .329
332.329
.307
.307.304 .301
346
.355 .352 .349
.326| .324 .321 .318
.298
151
152
153
154
155
.400 .398 .395
156
.372 .369.366
.363 .360 .358
157
.343 .341 .338
158
.315 .312 .310
.298.296
.296 .293| .290
159
.287.285 .282 .279 .276 .274
.271
.268|| .265
.268.265 .263
160
161
.260 .257 .254.252.249 .246
.232.230 .227.224 .222 .219 .216 .214 .211
.243
.241 .238
.238.235
.208
162
163
164
165
166
167
168
.155
.129
.205 .203 .200 .197.195.192 .189 .187 .184
.179.176 .173 .171
.171 .168.165
.168 .165.163 .160.157
.152.150 .147 .144 .142 .139.136| .134 .131
.126.123 .121 .118.116.113 .IIO .108 .105 .103
.097.095.092.090.087.085 .082.080
.100
.075.072 .070.067 .064 .062 .059 .057.054
|
|
.049.047.044 .042.039 .037 .034 .032 .029 .027
169 .024 .022 .020.017.015' .012 .010.007 .005! .002
.181
.052
104
X. A GENERAL TABLE OF VALUES OF
FOR SPHERICAL SHOT.
d²
W
√.
S

0
1
2
3
4
5 | 6 |
7
8
9
F s.
50
012
51
555
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
ྗg
67
68
69
70
71
72
73
74
75
76
7∞
300 H NM IM AD822
77
78
79
80
81
83
∞∞∞
84
85
∞ ∞ ∞
86
87
88
89
90
91
Feet.
Feet. Feet.
Feet. Feet. Feet.
Feet. Feet. Feet. Feet.
10649 0620 0592 0563 0535 0506 0478 0450 0422 0394
10366 0338 0310 0283 0255 0228 0201 0174 0147 0120
10093 0066 0040 0013 9987 9961 9935 9909 9883 9857
9831 9805 9779 9754 9729 9703 9678| 9653| 9628|| 9603
9578 9553 9529 9504 9480 9455 9431 9407 9383 9359
9335 9311 9287 9263 9240 9216 9193 9169 9146 9123
9100 9077 9054 9031 9008 8986 8963 8941 8918 8896
8851 8829 8807 8785 8763 8741 8719 8698 8676
8633 8612 8591 8569 8548 8527 8506 8485 8464
8873
8655
7665
7486
8443
8423 8402 8381 8361 8340 8320 8300 8279 8259
8239 8219 8199 8179 8159 8139 8120 8100 8081 8061
8041 8022 8003 7983 7964 7945 7926 7907 7888 7869
7850 7832 7813 7794 7776 7757 7739 7720 7702 7683
7647 7629 7611 7593 7575 7557 7539 7521 7504
7468 7451 7433 7416 7398 7381 7364 7346 7329
7312 7295 7278 7261 7244 7227 7210 7194 7177 7160
7144 7127 7110 7094 7078 7061 7045 7029 7012 6996
6980 6964 6948 6932 6916 6900 6884 6868 6853 6837
6821 6806 6790 6775 6759 6744 6728 67136698 6682
6667 6652 6637 6622 6607 6592 6577 6562 6547 6532
6503 6488 6473 6459 6444 6430 6415 6401 6386
6358 6343 6329 6315 6301 6287 6273 6259|6245
6217 6203 6189 6175 6161 6148 6134 6120 | 6107
6079 6066 6052 6039 6026 6012 5999 5986 5972
5959 5946 5933 5920 5907 5894 5881 5868 5855 5842
5829 5816 5803 5790 5778 5765 5752 5740 5727 5714
5702 5689 5677 5665 5652 5640 5627 5615 5603 5591
6517
6372
6231
6093
5578
5566 5554 5542 5530 5518 5506 5494 5482 5470
5446 5434 5423 5411 5399 5387 5376 5364 5352
5341 5329 5318 5306 5295 5283 5272 5260 5249 5238
5458
5226
5215 5204 5193 5181 5170 5159 5148 5137 5126
5115 5104 5093 5082 5071 5060 5049 5038 5027 5017
5006 4995 4984 4974 4963 4952 4942 49314921 | 4910
4900 4889 4879 4868 4858 4847 4837 4827 4817 4806
4796 4786 4776 4765 4755 4745 4735 4725 4715 4705
4695 4685 4675 4665 4655 4645 4635 4625 4615 4605
4596 4586 4576 4566 4557 4547 4537 4528 4518 4509
4499 4490 4480 4471 4461 4452 4442 4433 4423 4414
|
4405 4395 4386 4377 4367 4358 4349 4340 4331 4321
4312 4303 4294 4285 4276 4267 4258 4249 4240 4231
4222 4213 4204 41954186 4177 4169 4160 4151 4142
4134 4125 4116 4107 4099 4090 4081 4073 4064 4056
105
13

V.
9
0
8
1
7
2
6
3 4
5
Feet.
F-8.
Feet. Feet.
Feet. Feet. Feet.
Feet.
92
93
94
95
96
97
98
99
100
ΙΟΙ
I02
103
104
105
106
107
108
109
IIO
III
II2
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
133
Feet. Feet. F'eet.
4047 4039 4030 4022 4013 4005 3996 3988 3980 3971
3963 3954 3946 3938 3930 3921 3913 3905 3897 3888
3880 3872 3864 3856 3848 3840 3832 3823 3815 3807
3799 3791 3784 3776 3768 3760 3752 3744 3736 3728
3721 3713 3705 3697 3689 3682 3674 3666 3659 3651
3643 3636 3628 3621 3613 3606 3598 3591 3583 3576
3568 3561 3553 3546 3539 3531 3524 3516 3509 3502
3495 3487 3480 3473 3466 3458 3451 3444 3437 3430
34233416 3409 3402 3395 3388 3381 3374 3367 3360
3312 3305 3298 3291
| |
3245 | 3238 | 32313225
3179 3173 3166 3160
3353 3346 3339 3332 3325 3319
|
3285 3278 3271 3265 32583251
3218 3212 3205 3199 3192 3186
3154 3147 3141 3135 3128 3122 3116 3109 3103 3097
3091 3084 3078 3072 3066 3060 3054 3048 3041 3035
3029 3023 3017 3011 3005 2999 2993 2987 2982 2976
2970 2964 2958 2952 2946|2941|2935|2929|2923 | 2918
2912 2906 2900 2895 2889 2883 2878 2872 2866 2861
2855 2850 2844 2838 2833 2827 2822 2816 2811 2805
2800 2794 2789 2784 2778 2773 2767 2762 2757 2751
2746 2741 2735 2730 2725 2719 2714 2709 2704 2698
2693 2688 2683 2678 2672 2667 2662 2657 2652 2646
26412636| 2631 | 2626| 2621 2616 2611 2606 2601 2596
2591 2586 2581 2576 | 2571 2566 2561 2556 2551 2547
2541 2536 2531 2526 2522 2517 2512 2507 2502 2497
2492 2487 2483 2478 2473 2468 2464 2459 2454 2449
2444 2440 2435 2430 2426 2421 2416 2411 2407 2402
2397 2393 2388 2383 2379 2374 2369 2365 2360 2356
2351 2346 2342 2337 2333 2328 2323 2319 2314 2310
2305 |
2301 2296 2292 2287 2283 2278 2274 2269 | 2265
2260 2256 2252 2247 2243 2238 2234 2229 2225 2220
2216
2212 2207 2203 2199 2194 2190 2185 2181 2177
2172 2168 2164 2159 2155 2151 2146 2142 2138 2134
2129 2125 2121 2116 2112 2108 2104 2099 2095 2091
2087 2082 2078 2074 2070 2066 2061 2057 2053 2049
2045 2040 2036 2032 | 2028 2024 2020 2015 2011 2007
2003 1999 1995 1991 | 1986 | 1982 | 1978 | 1974 1970 1966
1962 1958 1954 1949 1945 1941 1937 1933 1929 1925
129
1921
1917 1913 1909 1905 1901 1897 1893 1889 1885
130
1881 1877 1873 1869 1865 1861 1857 1853 1849 1845
131 1841 1837 1833 1829 1825 1821 1817 1813 1809 1806
132
1802 1798 1794 1790 1786 1782 1778 1774 1770 1766
1763 1759 1755 1751 1747 1743 1739 1736 1732 1728
1724 1720 1716 1713 1709 1705 1701 1697 1694 1690
1686 1682 1678 1675 1671 1667 1663 1660 1656 1652
1648 1645 1641 1637 1633 1630 1626 1622 1618 1615
1611 1607 1603 1600 1596 1592 1589 1585 1581 1578
1574 1570 1567 1563 1559 1556 1552 1548 1545 1541
|
1537 1534 1530 1526 1523 1519 1516 1512 1508 1505
1501 1497 1494 1490 1487 1483 1479 1476 1472 1469
1465 1461 1458 1454 1451 1447 1444 1440 1437 1433
134
135
136
137
138
139
140
141
106

V.
01
9
8
2
| 7
3 4
6
5
149
150
151
152
153
F-8.
142
143
144
145
146
147
148
Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet.
1429 1426 1422 1419 1415 1412 1408 1405 1401 1398
1394 1391 1387 1384 1380 1377 1373 1370 1366 1363
1359 1356 1352 1349 1345 1342 1338 1335 1331 1328
1324 1321 1318 1314 1311 1307
1290 1287 1283 1280 1276| 1273
1256 1253 1249 1246 1242 1239
1304
1300 1297 1293
| |
1270|| 1266 | 1263 | 1259
1236 1232 | 1229| 1225
1202 | 1199 1195 1192
1169 1165 1162 1159
1222 1219 1215 1212 1209 1205
1189 | 1185 | 1182 1179 1175 1172
1155 1152 1149 1145 1142 1139 1135 1132 1129 1126
II22
1119 1116 1112 1109 1106 1103 1099 1096 1093
1090 1086 1083 1080 1077 1073 1070 1067 1064 1060
1057 1053 1051 1047 1044 1041 1038 1034 1031 1028
154
155
993
156
961
958 955 952
1025 1022 1018 1015 1012 1009 1006 1002
990 987 983 980 977 974 971
949
999 996
968 964
945
942 939
936
933
HHH
i ir i
157
158
7∞
930
927
924 920
899
895 892 889
917
886 883
914 911 908
880 877
159
160
868
864
861
858
855 852 849
674
900 ∞
905
902
874
871
846
843
840
837
834 831
828
825 822 818
815
812 809
161
162
776 773
806 803 800 797 794
770 767
791
764 761 758
788
785
782
782 779
755
752 749
163
746
743
740 737 734 731 728
725
722 719
164
716
713
710 707 704
701
698
695
692
692
689
165
686
683
680
677
674
672
669
666
666
663
660
166
657
654 651 648
645
642
639
636
633
633 630
167
628
625 622 619
616
613
610 607
604
601
168
598
596 593 590
587
584
581
578
575 572
169
569
567 564
561
558
555
170
541 538 535 532
529
552 549 546 544
526 524 521 518 515
171
512 509
506 504
501
498 495
492 489 487
172
484 481
173
478 475 472
447 444
174
470 467 464
467 464 461
456 453 450
442 439 436 433
428 425 422 419 416 414 4II 408 405
458
430
402
175
400 397 394 391
389
176
372
177
344
369 366 364 361
342 339 336
178
317
179
290
314 311 309
287 284
282
180
263 260 257
255
HHH
∞ ∞ ∞
181
182
183
∞ ∞ ∞0
184
386 383 380 377 375
358
355 353 350
347
333 331 328 325 322 320
306 303 301 298
279 276 273
252 249 246
236 233 230 228 225
209 206 204 201 198
182 180 177 174 172
156 153 150 148
295
271
268
244 241
222 220
217 214
196
169
193
166
190 188
164
161
1 ∞ 1
~H
H
238
212
185
21000
222
292
265
158
25000
185
186
129 127 124 122
103 ΙΟΙ 98 95
145 143
119 116 114 III 108
140 137 135 132
106
93 90 88
85
82
80
187
188
189
752
96
77
75
72! 69
67
64
62
59
51
26
49
46
44
41
39
36
33
w i
57
31
52
54
28
23 21
18 15 13 IO
7
5
3
107
XI. A GENERAL TABLE OF VALUES OF
t
FOR SPHERICAL SHOT.
d2
W

V.
0
1
2
3 4 5
6
7
8
9
F-s.
Secs. Secs. Secs. Secs. Secs.
Seconds. Secs.
=
+
Secs. Secs. Secs.
.017*.962 *.907
.531 .478 .426 .374
.020
*
*.970*.921 *.872
.537.490 .443 397
50 T3.4T4 356 299 .242 .185.129 .073
51 12.852.798 .744 .690 .637 .584
52 12.323 .272 .221 .170.119 .069
11.823.775 .726 .679 .631 .584
II.351.305 .259 .214.169.124 .080 .036 *.991 *.948
10.904 .861.818 .775 732 .690 .647 .605.564 .522
53
54
55
56
57
58
59
60
62
10.481.440 .399 .358 .318 .278 .238 .198 .158 119
10.080.041 .002 *.964 *.926 *.887 *.849 *.812 *.774 *.737
9.700 .663.626.589 .553 .517 .481 .445 .409 .374
9.338.303.268.234.199.165.130.096.062.029
8.995 .962 .929 .895 .863 .830.797 .765 .733 .700
61 8.669 .637 .605 .574 .542 .511 .480.449 .419 .388
8.358 .327 .297 .267 .237
.237.208.178 .149 .120 .ogo
63 8.061.033 .004.975
.004.975.947 *.919 *.890 *.862 *.834 *.807
.947.919 *.834.807
7.779 752 724 .697 .670 .643 .616.589 .562.536
7.510 .483 457 .431 .405 379 354 .328 .303 .277
7.252.227 .202 .177 .153 .128 .103 .079 .055.030
7.006.982.958.935 *.911 *.887 *.864.841 *.817 *.794
6.771.748.725 703 .680 .657 .635 .613 .590.568
6.546.524.502 .481
6.331.310 .289 .268
6.124.104 .084 .064
64
65
66
67
68
69
70
72
777
1 2 3
73
74
456
75
76
77
78
24
789
.459.437 .416 .394 .373 .352
.247 .227.206 .185 .165.145
.044 .025 .005.985 *.966 *.946
5.927.907.888.869 .850 .831 .812
.850 .831 .812.793 .774 | .756
.663 .645 .627 .609 .591 .573
| |
5.737 718 700 .681
5.555 537 .519 .502
5.380 .363 .346 .329
5.212.195.179 .163
.484 .466 .449
.484.466.449 .432
.432 .414 .397
.312 .295 .278 .261
|
.245 .228
.146 .130 .114 .098 .082 .066
5.050 .034 .019 .003 *.987 | *.972 *.956 *.941 *.926 *.910
4.895.880.865 .849 .834 .819 .805 790 775 .760
4.745 .731 .716 .702 .687 .673.659.644 .630 .616
4.602.587 .573 .559 545 .532 .518 .504 490 .476
81 4.463 .449.436 .422 .409 .395 .382.369 .356 .342
82 4.329 .316 .303 .290 .277 .264 .251 .239.226.213
79
80
83
∞∞∞
84
85
86
∞∞∞
78
87
88
89
90
91
4.200 .188 .175.163
.175 .163.150.138
4.076 .064 .052.040.028 .016
3.957 945 933 921 .910 .898
3.841.829 .818 .807 .796.784
3.729 .718 707 .696 .685 .675
3.621.611 .600 .590.579 .569
3.517.507.496 .486 .476 .466
3.416 .406 .396 .386 .377 .367
.309 .299.290.280.271
3.318
.125 113 .IOI .088
.004.992 *.980 *.968
.887 .875 .864 .852
773 762
.664 .653
.558.548
751 .740
.642.6321
.642 .632

|
.537 .527
.456.446 .436 .426
.357 .347 .338 .328
.252.243.233
.261
108

V.
89
0 | 1
7
23
6
45
F-s.
92
93
94
95
96
99999
97
98
99
ΙΟΟ
ΙΟΙ
I02
103
104
105
106
107
108
109
IIO
III
II2
Seconds. Secs. Secs. Secs. Secs.
3.224.215.206 .196 .187
3.133 .124 .115.106 .097
3.044.036
.036.027.019 .010
2.959.951
.951.942
.942 .934.926
2.876.868 .860.852.844
2.797 .789
.789.781.773 |
.781 773 .765
Secs. Secs. Secs. Secs. Secs.
.178 .169 .160 .151 .142
.088 .079
.088 .079 .071.062 .053
.001 *.993 *.984 | *.976 *.967
.917 .909
.917 .909 .901.893.885
.836.828
.836.828.820.812.804
.758 .750 .742 734 .727
2.719 .712 704 .697 .689 .682 .674.667.659.652
2.645 .637
.637 .630.623 .616 .608.601.594 .587 .580
2.572.565.558 | .551 | .544| .537| .530| .523 .517 .510
2.503.496.489 .482 .476
2.436 .429.423 .416 .409
2.371.365 .358 .352 .346
2.308 .302 .296 .290
2.248 .242 .236.231
2.190.184 .179 .173
.469 .462.456| .449 .442
403 397.390 .384 .377
.339 .333 .327 .321 .315
.278 .272 .266 .260
.278.272.266| .254
.284
.225
.167
.219 .213 .207
.162 .156 .151
.202.196
.145.140
2.134 .129.123 .118 .I12 .107 .102.096 .091| .085
2.080.075 .070.064.059 .054 .049 .043 .038 033
2.028.023
.023 .018 .013 .008 .003 *.998 *.993 *.988 *.983
1.978.973 .968 .963.958.953.948.943.938
1.929
.933
.924 .919 .914.910 .905 .900 .895 .891.886
1.881 .877
.877.872.867 .863.858.854.849
.858.854.849 .845 .840
113
1.835 .831 .826.822.817 .813.808.804 .800.795
114 1.791 .786.782.778.773 .769 .765 .760 .756 .752
115 1.747 .743 739 735 730 726 722 718 714 .709
116 1.705 .701 697 .693 .689 .684 .680 .676 .672 .668
1.664 .660.656 .652.648 .644.640| .636|.632 .628
1.624 .620.616.612 .608 .604.600.596.593.589
117
118
119
120
121
122
123
124
125
1.585.581 | |
.581 .577 .573 .569 .566.562 .558 .554 550
1.547.543
.543.539.535 | .532 | .528 .524 .520 .517 .513
1.509 .506.502
.506.502498.495 .491 .487.484 .480 .476
1.473.469
.469 .466 .462 .459 .455 .451.448
.455 .451.448 .444 .441
1.437.434.430.427.423 .420 .416 .413 .409 .406
1.402 .399 .395 .392 .388.385 .382.378 .375 .371
.348 .344 .341 .338
1.368 .365.361 .358 .355 .351
126
I.334 .331 .328 .325 .321 .318 .315 .311 .308 .305
127 1.302
.298.295.292 .289 .285 .282 .279 .276 .272
128 1.269 .266 .263 .260 .257 .253.250 .247 .244.241
1.238 .234 .231
.231.228.225 .222 .219 216 .213 .210
1.206 .203 .200 .197.194.191 .188 .185 .182 .179
1.176 .173
.173 .170.167| .164| .161.158.155.152 .149
1.146 .143 .140 .137 .134 .131 .128 .125 .122 .I20
1.117 .114 .III .108.105
129
130
131
132
133
.102 .099 .096 .093 .091
134 1.088 .085.082.079 .076 .073 .071.068.065 .062
135 1.059 .057
.057 .054 .051.048 .045 .043
136 1.032 .029 .026| .023 .020 .018 .015
•
100'
*.999*.996 *.993 *.991
.040.037.034
.012
.007
ΟΙΟ
*.988 *.985 *.983 *.980
0.977 975 972 .969 .967 .964 .961| .959 .956| .954
0.951.948 .946 .943 .940 .938 .935 .933 930 .927
0.925 .922 .920.917 .915 .912| .909 .907 .904 .902
0.899.897.894 .892 .889 .887.884.882.879 .877
137
1.004
138
139
140
141
109


V.
9
| 0 | 1
2 3 4 5 6 7 8
F-s.
142
Secs.
Seconds. Secs. Secs. Secs. Secs. Secs. Secs. Secs. Secs.
0.874.872.869 .867.864.862.859.857 .854.852
0.849 .847.844.842 .839.837 .835 .832 .830.827
.822.820.818 .815 .813 .810 .808 .806 .803
0.825
143
144
145
0.801
146 0.777 .775
147 0.754
148
149
150
151
152
153
154
155
156
157
.722.720
.782 780
759 756
740
.738
736 .733
.717
695
715 713 .711
·
•
.693 691 .688
.798 796 794 .791 .789 .787 .784
773 .770
.770.768.766 .763 .761
752 749 747 745 742
0.731 .729 .727 .724
0.708.706 .704
.706 704 702 700 .697
0.686 | .684.682 .680 .677 .675 .673 .671 .669| .666
.669.666
0.664 .662 .660.658.656 .653 .651
.656.653.651 .649
.649 .647 .645
0.643
.641.638 .636.634 .632 .630 .628.626 .623
0.621.619 .617 .615 .613 .611 .609 .607 .605 .602
.607
0.600 .598.596 .594 .592 .590 .588.586 .584 .582
0.580.578 .576 .574 .572 .570 .568 | .565
.565.563.561
0.559 .557 .555.553.551| .549 .547
.551.549.547 545 .543 | .541
0.539
.537 .535 .533
.533 .531 529.527.525
.511.510 .506
.506 .504 .502
.523.521
158 0.519
159 0.500
.517 .515 .513
.511 .510 .508
.498.496.494
.498 496 494
492 .490.488
.492 .490.488.486
.484.482
160
161
0.481 .479.477 .475 .473.471.469
.473.471.469.467.465 | .463
0.462 .460 .458 .456.454 .452 .450
0.462.460
.454.452.450.448.446 .445
162 0.443 .441 439.437.435.433
.439.437 435 .433 .432.430
.430.428.426
0.424 .422 .421 .419 .417 .415 .413 .411 .410 .408
0.406 .404.402 .401 .399 .397.395
0.388.386 .384 .383 .381
163
164
165
166
167
168
0.335
.381
393 .392 .390
.379 .377
.379 .377 375 374 372
.361 | .360
0.370.368 .367 .365 .363
0.353
.363 .361 .360| .358 .356 .354
.346.344.342.340 .339 337
.351 349 347 346 .344 342
.333 .332.330.328 .327 .325
.328 .327 .325 .323 | .321 .320
169 0.318 .316 .315 .313 .311 .309 .308 .306 .304.303
.293| .291
.294 293 .291.289.288 .286
.276 .274 .273 .271.269
.273
170
171
172
173
.260 | .258
.260 .258
.243 .242
.256 .255.253
.240.238.237
0.301 .299 .298.296 .294
0.284.283 .281 .279
.279 .278
0.268 .266.264
.266 .264 .263 .261
.261
0.251 .250 .248 .247 .245
.245
174 0.235 .234 .232.230.229 .227.226 .224 .222 .221
175 0.219 .218 .216 .215 .213 .2II .210 .208.207.205
0.203 .202 .200 .199 .197.196 .194
196 .194.192
.192.191 .189
0.188 .186 .185.183.182 .180.178
178 .177 .175.174
178 0.172 .171.169.168 .166 .165.163
.166| .165| .163| .162 .160
.162.160.159
.156 .154.152 .151 .149 .148 .146 145 .143
.140.139.137 .136 .134 133 .131 130 129
.126 .124.123 .121 .120 .118.117
176
177
II
.117 115 114
•
•
.III .109.108.107.105 104 .IO2 .IOI .099
0.098.096 095.093
.096.095.093 .092 .090.089
.092.090.089 .088 .086 .085
179 0.157
180 0.142
181 0.127
182 0.112
183
184
0.083
185
0.069.068 .066 .065
186
0.055
187
0.041
188 0.027
.082.080.079 .078 .076 .075
.078.076 .075
.068 .066.065| .063 .062| .061
.063 .062 .061
.073.072.070
.059 .058 | .056
.054 .052.051.049 .048 .047
.049.048.047 .045
.045 .044 .042
.040.038 .037 .035 .034 .033 .03 .030 .029
.026.024 .023 .022 .020.019 .018 | .016| .015
.009 .008 .007.005.004.003

189 0.014 .012
.OII
ΙΟΟ
110
XII. TABLE OF VALUES OF
h = 1 gt² - 16.0954t2 FEET.
t
0
1 2 3 4 5
6
7
со
9

1.0
I.I
I.2
1.3
Feet.
27.20
Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet.
16.10 | 16.42 16.75 17.08 17.41 17.75 18.08 18.43 18.77 19.12
19.48 19.83 20.19 20.55 20.92 21.29 21.66 22.03 22.41 22.79
23.18 23.57 23.96 24.35 24.75 25.15 25.55 25.96 26.37 26.78
27.62 28.04 28.47 28.90 29.33 29.77 30.21 30.65 31.10
32.00 32.45 32.91 33.38 33.84 34.31 34.78 35.26 35.73
36.70 37.19 37.68 38.17 38.67 39.17 39.67 40.18 40.69
41.20 41.72 42.24 42.76 43.29 43.82 44.35 44.89 45.43 45.97
46.52 47.06 47.62 48.17 48.73 49.29 49.86 50.43 51.00 51.57
52.15 52.73 53.31 53.90 54.49 55.09 55.68 56.28 56.89 57.49
1.4 31.55
1.5 36.21
1.6
HHH
1.7
1.8
1.9
2.0
2.I
મેં હેં હેં હેં હેં
2.2
2.3
2.4
2.5
2.6
567
2.7
ää Òä ભેં
2.8
2.9
3.0
حب حب حب
1 2 3
3.I
3.2
3.3
3.4
3.5
www
56
3.6
3.7
78
3.8
3.9
4.0
4.I
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
58.10 58.72 59.33 59.95 60.58 61.20 61.83 62.46 63.10 63.74
64.38 65.03 65.68 66.33 66.98 67.64 68.30 68.97 69.64 70.31
70.98 71.66 72.34 73.02 73.71 74.40 75.09 75.79 76.49 77.20
77.90 78.61 79.32 80.04 80.76 81.48 82.21 82.94 83.67 84.41
85.14 85.89 86.63 87.38 88.13 88.89 89.64 90.41 91.17 91.94
92.71 93.48 94.26 95.04 95.83 96.61 97.40 98.20 98.99 99.79
100.6 101.4 102.2 103.0 103.8 104.7 105.5 106.3 107.2 108.0
108.8 109.6 110.5 111.3 112.2 113.0 113.9 114.7 115.6 116.4
117.3
118.2 119.1 120.0 120.8 121.7 122.6 123.5 124.4 125.3
126.2127.1 128.0 128.9 129.8 130.7 131.6 132.6 133.5 134.4
135.4 136.3 137.2 138.2 139.1 140.1 141.0 142.0 142.9 143.9
144.9 145.8 146.8 147.8 148.8 149.7 150.7 151.7 152.7 153.7
154.7 155.7 156.7 157.7 158.7 159.7 160.7 161.7 162.8 163.8
164.8 165.8 166.9 167.9 169.0 170.0 171.1 172.1 173.2 174.2
175.3 176.3 177.4 178.5 179.6 180.6 181.7 182.8 183.9 185.0
186.1 187.2 188.3 189.4 190.5 191.6 192.7 193.8| 195.0 196.1
197.2 198.3 199.4 200.6 201.7 202.8 204.0 205.1 206.3 207.4
208.6 209.7 210.9 212.1 213.3 214.4 215.6 216.8 218.0 219.2
220.3 221.5 222.7 223.9 225.1 226.3 227.5 228.8 230.0 231.2
232.4 233.6 234.9 236.1 237.3 238.6 239.8 241.0 242.3 243.5
244.8 246.1 247.3 248.6 249.9 251.1 252.4 253.7 255.0 256.2
257.5 258.8 260.1 261.4 262.7 264.0 265.3 266.6 267.9 269.2
270.6 271.9 273.2 274.5 275.9 277.2 278.5 279.9 281.2 282.6
283.9 285.3 286.6 288.0 289.3 290.7 292.1 293.5 294.8 296.2
297.6 299 0 300.4 301.8 303.2 304.6 306.0 307.4 308.8 310.2
311.6 313.0 314.4 315.9 317.3 318.7 320.2 321.6 323.0 324.5
325.9 327.4 328.8 330.3 331.8 333.2 334.7 336.2 337.6 339.1
340.6 342.1 343.5 345.0 346.5 348.0 349.5 351.0 352.5 354.0
355.6 357.1 358.6 360.1 361.6 363.2 364.7 366.2 367.8 369.3
370.8372.4 373.9 375.5 377.1 378.6 380.2 381.7 383.3 384.9
386.5 388.0 389.6 391.2 392.8 394.4 396.0 397.6 399.2 400.8
402.4 404.0 405.6 407.2 408.8 410.5 412.1 413.7 415.3 416.9

I
111

t
| 8 | 9
1
7
2 3
6
4
5
5.I
5.2
5.3
5.4
456
5.5
5.6
5.7
72
5.8
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.I
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
1 2
8.2
8.3
8.4
456
8.5
8.6
∞ ∞ ∞ ∞ ∞ ∞
8.7
8.8
789
8.9
9.0
9.I
9.2
9.3
94
+
Feet.
638.8
659.3
680.0
Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet.
418.6 420.3 421.9 423.6 425.2 426.9 428.5 430.2 431.9 433.5
435.2 | 436.9 438.6 440.3 442.0 443.6 445.3 447.0 448.7 450.4
452.1453.8 455.5 457.3 458.9 460.7 462.4 464.1 465.9 467.6
469.3 471.1 472.8 474.6 476.3 478.1 479.8 481.6 483.4 485.1
486.9 488.6 490.4 492.2 494.0 495.8 497.6 499.4 501.2 503.0
504.8 506.6 508.4 510.2 512.0 513.8 515.6 517.5 519.3 521.1
522.9 524.8 526.6 528.5 530.4 532.2 534.0 535.8 537.7 539.6
541.4 543-3 545.2 547.549.0 550.8 552.7 554.6 556.5 558.4
560.3 562.2 564.1 566.0 567.9 569.8 571.7 573.7 575.6 577.5
579.4 581.4 583.3 585.2 587.2 589.1 591.1 593.0 595.0 597.0
598.9 600.9 602.9 604.8 606.8 608.8 610.7 612.7 614.7 616.7
618.7 620.7 622.7 624.7 626.7 628.7 630.7 632.8 634.8 636.8
640.9 642.9 644.9 647.0 649.0 651.1 653.1 655.2 657.2
661.3 663.4 665.5 667.5 669.6 671.7 673.8 675.9 677.9
682.1 684.2 686.3 688.4 690.5 692.6 694.8 696.9 699.0
703.5 705.4 707.5 709.7 711.8 713.9 716.1 718.2 720.4
724.7 726.8 729.0 731.2 733.4 735.5 737.7 739.9 742.İ
746.4 748.6 750.8 753.0 755.2 757.5 759.7 761.9 764.1
768.5 770.8 773.0 775.2 777.4 779.7 781.9 784.2 786.4
790.9 793.2 795.4 797.7 800.0 802.3 804.5 806.8 809.1
813.7 816.0 818.2 820.5 822.8 825.1 827.5 829.8 832.1
836.7 839.0 841.3 843.7 846.0 848.4 850.7 853.0 855.4
860.1 862.4 864.8 867.1 869.5 871.9 874.3 876.6 879.0
883.8 886.2 888.5 890.9 893.3 895.7 898.1 900.6 903.0
907.8 910.2 912.6 915.1 917.5 919.9 922.4 924.8 927.2
929.7 932.1 934.6 937.0 939.5 941.9 944.4 946.9 949.3 951.8
954.3 956.8 959.3 961.8 964.2 966.7 969.2 971.7 974.2 976.7
979.2 981.8 984.3 986.8 989.3 991.8 994.4 996.9 999.4 100.2
1005 1007 1009 1012 1015 1017 1020 1023 1025 1028
1030 1033 1035 1038 1041 1043 1046 1048 1051 1053
701.1
722.5
744.3
766.3
788.7
811.4
834.4
857.7
881.4
905.4
1056 1059 1061 1064| 1066| 1069| 1072 1074 1077 1080
1082 1085 1088 1090 1093 1095 1098 IIOI 1104 1106
1109 1112 1114 II17 1120
1122 1125 1128 1130 1133
1136 | 1138| 1141 1144 1147
1163 1166 1158 1171 1174
I 190 1193 1196 1199 1202
1149|| 1152 1155 1157 1160
1177 1179 1182 1185 1188
1204 1207 1210 1213 1216
1218 1221 1224 1227 1230 1232 1235 1238 1241| 1244
1246
1249 1252 1255 1258 1261| 1264. 1266|| 1269| 1272
1275 1278 1281 1284 1286 1289 1292 1295 1298 1301
1304 1307 1310 1312 1315 1318 1321 1324 1327 1330
1333 1336 1339 1342 1345 1348 1350 1353 1356 1359
1362 1365 1368 1371 1374 1377 1380 1383 1386| 1389
1392
1422
9.5 1453
1483
1514
1546
9.6
9.7
9.8
9.9
1578
1395 1398 1401 1404 1407 1410 1413 1416| 1419
1425 1428 1431 1434 1437 1440 1444 1447 1450
1456 1459 1462 1465 1468 1471 1474 1477 1480
1486 1490 1493| 1496 1499 1502 1505 1508 1511
1518 1521 1524 1527 1530 1533 1536 1540 1543
1549 1552 1555 1558 1562 1565 1568 1571 1574
1581 1584 1587 1590 1593 1597 1600 1603 1606
112
Fig. 1
R
R
Fig.3
Fig. 4
D
D
B
Fig. 8 a
Fig 7
P
ŽInches.
...
P
У
୧୯
Fig. 8
N
P
Pr
Q
n
R
A
N
M
T
Pigr
Rigb
M
n
X
A
I


M
H Fig5 M R
X
th
B
22 02
x
I
dd
C
ne
n'
n's
باره
A
223
M
na
ni
B
x
Fig. 9
Fig Q...
G
C
d
W
Figll
PA
·MEDOST
Bell
m
Fig.15.
九
​E
Marker
h
B
+
୪୪+
Fig. 16
B₂
Fig. 12
A
B
A
Хотоо
Fig. 13
Fig. 14
D
E
a
a
A
Fig. 18
TA
B
B
Fig.19
D
]]

Fig.17
SUPPLEMENT
TO A
MATHEMATICAL TREATISE
ON THE
MOTION OF PROJECTILES,
FOUNDED CHIEFLY ON THE
RESULTS OF EXPERIMENTS
MADE WITH
THE AUTHOR'S CHRONOGRAPH.
BY
FRANCIS BASHFORTH, B.D.,
LATE PROFESSOR OF APPLIED MATHEMATICS TO THE ADVANCED CLASS OF
ROYAL ARTILLERY OFFICERS, WOOLWICH, AND OFFICIAL REFERER
TO THE LATE ORDNANCE SELECT COMMITTEE; AND
FORMERLY FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE.
LONDON:
ASHER & CO.,
13, BEDFORD STREET, COVENT GARDEN;
BERLIN: 11, UNTER DEN LINDEN.
1881.
All rights reserved.
:
CAMBRIDGE:
PRINTED BY W. METCALFE AND SON, TRINITY STREET.
SUPPLEMENT
TO
A TREATISE ON THE MOTION OF PROJECTILES,
BY
FRANCIS BASHFORTH, B.D.,
1881.
91. Preliminary Remarks.] My work, commenced in the
Spring of 1864, is now brought to a conclusion. When the
first part of this Treatise was published in 1873, the Advanced
Class of Royal Artillery Officers was in a state of abeyance,
as already explained, because there was not a sufficient
number of candidates qualified for admission to form a new
class. Some time before this, the Advanced Class had been
frequently attacked in the newspapers, and a very distin-
guished Fellow of the Royal Society, who had nothing
whatever to do with the matter, thought proper to join in
the fray. Having attacked the Professor of Applied Mathe-
matics, he concluded his diatribe with the remark "It is
"high time that the guardians of the public purse should
set their faces against such vicious reproduction of fresh
"nuclei of misapplied prodigality, surely destined to eventuate
(if I may apply the words of an eloquent member of the
deputation of which I formed a part) in endowed and
"decorated idleness.”—The Times, Feb. 11, 1870. The letter
was characteristic and required no answer. If the reader
be desirous of knowing more about the objections to the
Advanced Class, or the schemes propounded for bettering
it, he is referred to Parliamentary Paper 1872 [c. 589]. In
spite of all opposition the Advanced Class was revived after
an interval of upwards of two years, and I believe is now
being conducted at Woolwich in very much the same
manner as before the interruption.
cc
(6
""
It must be confessed that this turmoil, coming on so
soon after the printing of my Reports of experiments made
1865-1870, did not afford me much encouragement in
pursuing my self-imposed, and almost thankless work of
making laborious experiments. And as I did not see that
B*
92
SUPPLEMENT TO A TREATISE
I could succeed better than I had already done, I retired
from my office of Professor of Applied Mathematics, because,
if I was not to be engaged on original research, the reason
for my connection with the Advanced Class ceased.
Subsequently I was informed that my results were being
"tested" at Shoeburyness, and that it was intended to
complete my work by making experiments with low velocities
by the help of such chronoscopes as the Government possessed.
Although I had given up experimenting I continued to
prepare tables, &c., at my leisure, with a view to render
the application of my results already obtained to the calcu-
lation of trajectories, &c., as easy and simple as possible.
In 1877 I proceeded to print the Supplementary Tables A.
to G. But when my work had proceeded thus far, much
to my surprise, I received an invitation to lend my chrono-
graph and give my assistance in completing the experiments
commenced about ten years before. It appeared to me that
I ought to comply with this request because I had the means
ready for the performance of the work, and because the
invitation itself, under the circumstances, seemed to be a
sufficient recognition of the practical value of my previous
labours, and also a confession that, for that particular work,
my chronograph was superior to all those possessed by
Government and used more or less in their experiments.
These were the new Ballistic Pendulum, the Navez, the
Leurs, the Boulengé, and the Schultz Chronoscopes.
The
My instrument was therefore sent back to Shoeburyness
and set up in June, 1878, having been removed from that
place in May, 1870. These recent experiments were carried
out with very high as well as very low velocities.
projectiles were ogival-headed, excepting three rounds of
flat-headed and three rounds of hemispherical-headed shot,
which gave very satisfactory results for each form of head
through a considerable range of velocity. The reader is
referred to the final Report on Experiments made with my
chronograph 1878-1880 for all particulars respecting these
late experiments. The chronograph has now been returned
to the South Kensington Museum. In addition to the
Officers of the Royal Artillery already mentioned, my thanks
are due to the following Officers, who most carefully carried
out all these recent experiments at Shoeburyness, and sent
me the records for reduction, namely, Major C. Jones, R.A.;
Captain Bainbridge, R.A.; Captain Morley, R.A.; Captain
McClintock, R.A.; Captain O'Callaghan, R.A.; and Captain
White, R.A.
ON THE MOTION OF PROJECTILES.
93
After I had satisfied myself respecting the capabilities
of my instrument in 1865, I was careful to refrain from doing
anything which might prevent it from being applied to that
purpose for which it was designed. Time has gradually
removed obstructions and softened down asperities, and
brought forward better instructed officers capable of forming
an independent opinion in scientific matters. Thus, by
working and waiting, the desired result has been arrived
at at last. Every possible assistance has been most readily
afforded me, so that for any deficiency I am alone respon-
sible. I wish also to express my great obligation for the
official publication of all my reports just as they were
written* at the specified dates, for these speak for themselves.
And it is a curious question why, when an instrument much
wanted in the service had proved uniformly successful, the
use of it was neglected, so that important results, which might
have been easily obtained in three or four years, were delayed
for fifteen years. These original Reports of my Experiments
and my complete "Treatise on the Motion of Projectiles "
being now before the world, I have every confidence that
my labours will receive just that appreciation which they
may be found to deserve.
92. The best evidence of the real opinion entertained of
the value of my experiments will be found in the use that
has been made of the results at home and abroad. After
the trial of my instrument, the first real experiment was
made to determine the resistance of the air to elongated
projectiles provided with various forms of heads. The Report
of this experiment was dated October, 1866.† By minute
23,351, September 21, 1867, the Ordnance Select Committee
adopted my result as the produce of their own experiments,
so far as Service Shot was concerned. The omitted acknow-
ledgment was afterwards fully supplied in the Proceedings of
the O. S. C. vol. VI. 397, 1868. My General Tables were
reprinted in the new edition of the Treatise on the Construction
of Ordnance, by Major John F. Owen, R.A., Captain In-
structor, Royal Gun Factories; and again in Principles of
Gunnery, for the use of the R. M. Academy, Woolwich, by
* "Reports on Experiments made with the Bashforth Chronograph to deter-
mine the Resistance of the Air to the Motion of Projectiles, 1865-1870.”
London: W. Clowes and Son, Harrison and Sons, &c., &c.
And "Final Report on Experiments made with the Bashforth Chronograph to
determine the Resistance of the Air to the Motion of Elongated Projectiles, 1878-
1880." London: W. Clowes and Sons (Limited), Harrison and Sons, &c., 1880.
Price Two Shillings and Threepence.
+ Reports, fc., p. 10, and Phil. Trans., 1868.
94
SUPPLEMENT TO A TREATISE
Major Sladen, R.A., Professor of Artillery. Captain Ken-
sington, R.A., contributed to the Proceedings of the Royal
Artillery Institution some useful explanatory notes on the
former part of this treatise. And more recently application
has been made to me for assistance in the production of a
Naval Manual of Gunnery. In Russia, General Mayevski,
Professeur de Balistique à l'Académie d'Artillerie, adopted my
results, so far as they were known to him, in his Balis-
tique Extérieure, published in Russian in 1870 and in French
in 1872. The following table has been copied from this
work, concerning which General Mayevski remarks: "Aussi
66 pour compléter les données se rapportant aux projectiles
"de forts calibres nous avons profité des tableaux des vitesses
'décroissantes déduites par M. Bashforth de ses expériences
"faites en 1868 au moyen de son chronographe; ces tableaux
comprennent les vitesses de 518m.s à 283m.s qui correspondent
แ aux trajets de 305 (30·5) en 305 (30.5) mètres des projectiles
"oblongs de 178mm, 203mm et 229mm, et qui sont obtenues
แ pour le cas où le mouvement des projectiles peut être con-
"sidéré comme rectiligne. Nous avons calculé d'après les
"résultats insérés dans ces tableaux les valeurs de la résistance
(( correspondantes à différentes vitesses."* Those determi-
nations of p' marked "angl." were derived from my tables.
Projectiles Oblongs.†
66
Bouches
à feu
Vitesses Valeurs
Bouches
Vitesses Valeurs
v
de p'
à feu
v
de p'
C. de
4¹
172m.s 0.0151 C. de 203mm
329m.8
0.0338
C. de 203mm
207
0.0137 || C. de 203mm angl. | 332
0.0327
C. de
41
239
0.0148 C. de 229mm angl.
334
0.0332
C. de
12¹
247
0.0170 C. de 41
337
0.0341
C. de
24¹
266
0.0160 || C. de 178mm angl.
340
0.0334
C. de 203mm
282
0.0163 C. de 203mm angl.
345
0.0354
C. de 203mm angl.
287
0.0184 || C. de 229mm angl. | 355
0.0364
C. de 229mm angl.
291
0 0247 C. de 178mm angl.
358
0.0382
C. de 203mm angl.
300
0.0230 C. de 203mm
360
0.0384
C. de 178mm angl.
302
0.0218
C. de 203mm angl.
360
0.0393
C. de 12¹
304
0.0221
C. de 41
401
0.0450
C. de 4¹
1
307
0.0158 C. de 203mm
409
0.0430
C. de 229mm angl. 316
0.0305|| C. de 203mm angl.
419
0.0433
C. de
41
317
0.0259 || C. de 229mm angl.
420
0.0427
C. de 203mm
319
0.0174 || C. de 203mm angl.
460
0.0419
C. de 203mm angl.
320
0.0277 C. de 203mm angl.
508
0.0440
C. de 24¹
320
C. de 178mm angl.
322
0.0299 C. de 178mm angl.
0.0270
512
0.0443
* Mayevski, Balistque Extérieure, 1872, p. 38. † Пb. p. 39.
ON THE MOTION OF PROJECTILES.
95
General Mayevski plotted these results, taking the veloci-
ties for abscissæ, and the values of p' for ordinates, and then
from the curves drawn through the points so found he
deduced various formulæ connecting v and p'.
93. My experiments with elongated shot, made in 1868,
were subsequently reduced more carefully, after which process
new coefficients were obtained, differing somewhat from those
used by General Mayevski. Therefore Didion's p' has been
calculated from the coefficients given in Tables II. and H.
for both spherical and elongated shot for comparison with
the results of other experimenters. Didion's p' p÷πR'v',
where p denotes the resistance of the air in kilogrammes
P
to a shot of radius R moving with a velocity v, R and v
being expressed in metres. Hence it appears from my
experiments, both with spherical and ogival-headed projec-
tiles, that p' is nearly constant for velocities above 1300 or
1400 f.s, or, in other words, for those and higher velocities
the resistance of the air varies very nearly as the square of
the velocity.
Spherical Shot.

Velocity
f.s
m.s
p'
900 274.3 0.0521 0.0440
1000 304.8 ·0545 ⚫0459
'
Hutton¹ Didion² Virlet3 Mayevski Bashforth
p'
p'
p
0.0389 0·0380 0.0373
⚫0433 .0442 ⚫0422
1100 335.3 *0569 ⚫0478
1200 365.8 ⚫0590 ⚫0497
1300 396.2 ⚫0609
⚫0476 ·0510 0499
⚫0519 *0584
•0551
•0516
•0563 ⚫0610
*0575
1400 126.7 •0623
•0535
•0606
•0610
*0592
1500 457.2 •0634
1600 487.7 ⚫0639 *0573
1700 518.2 ⚫0640
1800 548.6 ⚫0636
1900 579.1 ⚫0628
2000 | 609.6
2100 640·1
·0554
•0649
•0610
*0602
*0692
•0610 •0611
⚫0592
*0736 •0610
•0615
•0611 ⚫0779
•0615
·0630 •0822
•0618
:
'0622
:
:
⚫0623
•0616 ⚫0649 *0866
•0668 ⚫0909
1 Hutton's Experiments made with the Ballistic Pendulum at Woolwich,
1783-1791. Tracts, vol. III. p. 218.
2 Didion's Experiments made with the Ballistic Pendulum at Metz, 1839, 1810.
Didion's Traité de Bal. p. 66.
3 Virlet's Experiments made with Navez E.B. Pendulum at Metz, 1856-1858.
See Mayevski's Traité de Bal. p. 36.
4 Mayevski's Experiments made with Boulengé's Chronograph at St. Peters-
burg, 1868, 1869. See Traité de Bal. p. 41.
5 Bashforth's Experiments made with his Clock-Chronograph at Shoeburyness,
1868. See Reports &c., p. 114.
96
SUPPLEMENT TO A TREATISE
Ogival-headed Shot.
Hélie Mayevski Bashforth

Velocity
fs
m.s
'p
p'
100
30.5
0.0120 0.0174
200 61.0
0.0122 0.0174
400 121.9
0.0127 0.0177
600 182.9
0.0137 0.0180
800 243.8 0.0146
0.0150
0.0183
1000 304·8 0.0182
0.0224
0 0224
1200 365.8
0.0437
0.0393
1400 426.7
0.0437
0 0439
1600 487.7
::
0.0437
0.0430
1800 548.6
0.0416
2000 609.6
2200
670.6
•
2400
731.5
2600 792.5
::
:::
0.0412
0.0441
0.0428
0.0408
0.0436
2800 853.4
94. It has been shewn (54) that
Z
3
-
▲t, − {▲³t,+ §Ã³t¸− ‡A*t, + &c. '
༡༡.3
and that f=
(A³t, — ▲³t, + ††4*t, − 19▲³t, + &c.).
z-nh
1+nh
If now we develope u in the same manner as u
developed, and equate coefficients of n and n', we obtain
and ƒ.=
V₂
3
7
3
At₂-¿ + 1 A² t₂-21 + §A³t9-91 + {A*ts-41 + &c.
-1
• 2 ( {}A°t‚„1+
A²te_2¿ + A³te_32 + ‡‡A*ts-41 + ¦ ¦ A³í»-¿¿+ &c.).
で
​was
The formula (2) page 40 should have a term 'At‚‚²
instead of 14° -3°
The expressions for the retarding force should have been
written
and
fs=
5
V
(A³t
ƒ. = – ½ (4¾½‚¸ — 1'44't¸) at page 42,
fs:
V
3
1½" (▲˚t, — ▲ºt¸) at page 44.
• Hélie's Experiments made at Gâvre 1859. Hélie's Traité de Bal. p. 416.
7 Mayevski's Experiments made with Boulengé's Chronoscope at St. Petersburg,
1868-9. Traité de Bal. p. 41. § 27.
8 Bashforth's Experiments made at Shoeburyness 1867-8 and 1878-80. See
Final Report.
ON THE MOTION OF PROJECTILES.
97
12 27
19
1929
95. The example (57) given to illustrate the mode of
reducing an experiment formed round 341 in the published
Reports, &c., 1865-1870. The above formulæ have been
used to find v₁, V2, V3, &c., the velocities with which the
projectile passed each of the ten screens; and also to find
f₁, ff, &c., the retarding forces which were acting upon
it at the same points. This round was fired October 28, 1868.
The barometer stood at 30-44 inches, and the wet and dry
bulb thermometers at 44° and 48° at 10 o'clock A.M. These
observations give 556.2 grs. as the weight of a cubic foot
of air. And at 3 o'clock in the afternoon the barometer stood
at 30-38 inches, and the wet and dry bulb thermometers at
48° and 50°, which observations give 551.5 grs. as the weight
of a cubic foot of air. And the mean of these two is 553.9 grs.
which has been taken as the weight of a cubic foot of air at
the time of the experiment. All the experiments were
originally reduced to the standard weight of 530-6 grs., which,
for all round shot, remains unaltered, = 150 feet.
150
∙10417 — 100367 + 1·00011
150
= 1465·3 f.s,
•10237
⚫00356
3
(150)2
2b′v‚³.
3
fr
V7
7
(0036700011)
(150)2
W 530.6
7
and K₁ = 26' (1000)³×
3
553.9
•00356
(150)*
44.094 530.6
× (1000)³ ×
X
= 139.6.
(6.92)* 553.9
And in the same manner the rest may be found as stated
in the following Table:

No. of
Screen.
Ꮴ
AV
f.s
Kv
AK,
1
1811.4
116.1
63.6
2 8
1747.8
+3·9
120.0
61.1
3
1686.7
+ 3·9
123.9
58.7
4 1628.0
+ 3·9
127.8
56.5
5 1571.5
+3·9
131.7
-
54.2
6
1517.3
135.6
+ 3·9
52.0
7
1465.3
139.6
+4·0
50.1
8 1415.2
143.9
+ 4·3
47.9
9
1367.3
+ 4·3
148.2
- 46.3
10
1321.0
+ 4·3
152.5
98
SUPPLEMENT TO A TREATISE
From the above Table we easily find the values of K
corresponding to velocities 1800, 1780, 1760...1340 f.s. The
values of K for corresponding velocities have been also taken
from Table II. for comparison.
V
K
Kv
Ꮴ
K₂
K
f.s
Round Table Error
341
II.
Round
Table Error
f.s
341
II.
1800 116.8 114.2
1780 118.0 115.5
+ 2·6
1560
132.5
130.1+2·4
+2.5
1540
134.0
131.5
+2.5
1760 119.2 116.8
+ 2.4
1520
135.4 132.8
+ 2·6
1740 120.5 118.1
+2.4
1500
136.9 | 134·1
+2·8
1720 121.8 119.4
+ 2.4
1480
138.4 135.5
+2.9
1700 123.0 120.8
+2.2
1460
140.1
137.0
+3.1
1680 124.3 122.1
1660 125.7 123.5+ 2·2
1640 127.0 124.8 + 2.2
+ 2.2
1440 141.8
138.4
+ 3.4
1420
143.5
139.8
+3.7
1400 145.3
141.3
+ 4·0
142.7
+4·4
144.1 +4·8
66
1380 147.1
1360 | 148.9
1620 128.4 | 126·2 | + 2·2
1600 129.7 | 127.5 + 2·2
1580 131·1 128.8+2·3 1340 150.7 145·3 +5°4
On reference to the Report it will be found that the
coefficients of resistance derived from the above round, and
actually made use of, corresponded to velocities between 1700
and 1400 f.s. The reasons for making this limited use of
each experiment were given as follows:-"A considerable
part of the extremities of the range in each round has been
"rejected in finding the values of 26', because the records
"of the first and last screens are not so trustworthy as those
"of the intermediate screens, and at the two extremities of
"the range the direction of the motion of the shot is not
"so nearly horizontal as at the intermediate points," p. 59.
96. Still more remarkable was round 399, used as an
example at page 57 of Reports, &c. In this case the round
shot was hollow and weighed 7.894 lbs. Its diameter was
4.92 inches, and the density of the air on the day of the
experiment was 1.0335. The values of the coefficients de-
duced from this round, and actually used, corresponded to
velocities 2150 f.s to 1200 f.s. Values of K have been calcu-
lated from this single round corresponding to velocities
between 2500 f.s and 1200 f.s, for comparison with the values
of K given in Table II. for corresponding velocities.
ON THE MOTION OF PROJECTILES.
99

V
K₁
Round
Kv
Table
V
K
Kv
Error
Round
Table
Error
f.s
399
II.
f.s
399
II.
2500 82.5
1750
117.8 117·4
+ 0·4
2400 86.6
1700 120·6 | 120·8
0.2
2300
90.8
:
1650 123.7 | 124.1
0.4
2200
95.2
1600 126.8 127.5
0.7
2150 97.5
2100
99.8 99.2
96.9 +0.6
+0.6
1550
129.9 | 130.8
0.9
1500
133.1 134.1
1.0
2050 102.2
101.5 +0.7
1450
136·4 | 137·7
1.3
2000 104.6
|
103.9
+0.7
1400
139.8 | 141·3
1.5
1950 107.1
106.3
+ 0.8
1350
143·4 | 144·7
1.3
1900 109-7 108.7
+1.0
1300
147.3 147.8
0.5
1850 112.4 111.3
+ 1·1
1250
151.3 | 151·1 | + 0·2
1800 115.1 114.2
+0.9
1200
155.5 153.4 + 2.1
97. As a further illustration of the power of this chrono-
graph I give below the values of K for flat-headed and
hemispherical-headed elongated shot derived from three rounds
of each form of head for comparison with the value of K for
ogival-headed shot.*
Hemi-
Flat
Ogival spherical
V
V
Hemi-
Head
Head
Head
Ogival spherical
Flat
Head
f.s
Head
f.s
Head
1520 | 96.2
173.3
1700 83.0 113.6
173.5
1530
95.3
173.6
1710
|
82.4 113.1
173.2
1540 94.4
173.8
1720 81.8 112.6
172.9
1550 93.6
173.9 1730 81.2 112.0
172.5
1560 92.8
174.0 1740 80.6 1114
172.1
::
174.1 1750 80.0 110.8
|
171.7
1570 92.0
1580 91.2
1590 90.4
1600 89.7
1610
89.0
1620 88.3
1630 87.6
174.2 1760 79.5 110.2 171.2
174.3 1770 78.9
109.5 | 170·7
174.4 1780 78.4
108.8 170.2
•
:
174.5 1790 77.8
108.0
169.7
174.5 1800 77.3
107.1
169·1
174.5 1810 76.8
106 1
168.6
1640 86.9 115.8 174.5 1820 76.2
105.1❘ 168.0
1650 86.2 115.5 174·4
1660 85.5 115.2
1670 84.8 114.8 174.2
1680 84.2 114-4
1690 83.6 114.0 173.8
1830 75.7
104.0 167.4
174.3
1840 75.2
102.9 166.8
1850 74.7
101.7 166.2
174.0
1860 74.2
100.5 165.6
1870 73.6
99.3 165.0
* See Final Report, pp. 37, 40, and 43.
Crate
100
SUPPLEMENT TO A TREATISE
98. If now we increase the values of K, given in
Table II. by 2-300ths of their values to render the supposed
density of the air there the same as in the above table, we
shall find that the value of K, for a hemispherical-headed
elongated shot is a little smaller than that for round shot.
Thus, for velocities 1650 f.s 1700 f.s 1750 f.s 1800 f.s and 1850 f.s
K, for-spherical 115.5
101.7
113.6 110.8 107.1
heads is
K₁ for spherical
shot is
124.9
121.6 118.2 115.0
112.1
Difference
9.4
8.0
7.4
7.9
10.4
The value of K, for hemispherical-headed shot was found
to be 133.1 for a mean velocity of about 1130 f.s, while that
for spherical shot for the same velocity was 153′1, as given
in Table II.
99. It is remarkable through what a wide range of
velocities a very correct law of resistance to spherical shot is
given by each of the rounds 341 and 399. But the like
favourable result cannot be expected from any single round
of ogival-headed elongated shot. For the resistance of the
air to a pointed elongated shot is less than the resistance to a
spherical shot of the same diameter, while the former is
heavier than the latter. On both accounts the variation of
velocity of an elongated ogival-headed shot, in a given range,
will be much less than that of a spherical shot under like
conditions. If the round shot be truly spherical, it will
always present the same surface to be resisted, while this
is perhaps seldom the case with an elongated shot through
any considerable range.
With high velocities the shot moves in an approximately
straight line, it cuts the threads of the screens promptly, and
the retarding force of the air, being great, causes great
variation in the velocity of the shot. So there is no difficulty
in finding the resistance of the air to projectiles moving at
high velocities. But when we get below a velocity of about
1000 fs our difficulties go on increasing as the velocities
decrease. The trajectory becomes so curved that we are
limited in range, or have to leave out the central screens. An
ogival-headed projectile, 3 inches in diameter will then often
pass between the threads of the screens, only one inch apart,
without breaking one of them, and the resistance of the air
* Reports, &c., p. 13, and Phil. Trans. p. 426, 1868.
ON THE MOTION OF PROJECTILES.
101
is so reduced that it produces only a slight variation of the
velocity of the shot in a single round.
100. After the coefficients of resistance had been found in
the usual manner down to a velocity of 450 f.s, the trajectories
and times of flight were carefully calculated for the 6.3-inch
howitzer, and compared with the range tables given in the
Instructions for Field Service, 1879, and also with numerous
German range tables, where the value of d÷w was nearly
the same as for the English howitzer. As these comparisons
appeared to be as satisfactory as could be reasonably expected,
the coefficients were plotted, making v the abscissa and K, the
ordinate. The average curve passing among the points so
determined was produced so as to give approximate values
of K for still lower velocities. Special experiments were
afterwards made with the 6.3-inch howitzer with low charges
of 1½ lb., 1 lb., 2 lb., ½ lb. and 1 lb. in order to test the value
of these coefficients, when employed to calculate the ranges
and times of flight of projectiles moving with very low
velocities. The result appeared to be satisfactory.†
3
4
101. In these comparisons of the results of calculation
and experiment great discrepancies appeared in the cases of
"final velocities" and "angles of descent." This shews how
much the final velocities and the angles of descent of ordinary
range tables need correction, for it is manifest that these are
not matters which admit of accurate measurement. But if a
set of coefficients give tolerably correct ranges and times of
flight for various low muzzle velocities and various elevations,
there can be no doubt about the accuracy of the terminal
velocities and angles of descent calculated by the help of those
coefficients.
It is probable that the coefficients adopted for low velocities
are somewhat greater than those which would have been
obtained if the projectiles had moved with their axes strictly
in the direction of motion during the experiments. Still, so
far as they have been derived from experiment, they are such
as may be expected in practice, for at low velocities projectiles
must move with their axes inclined to the direction of motion,
because their trajectories are much curved. In the experi-
ments made with low charges care was taken to measure the
'jump" for elevations up to 30°, but these were found to be
so small and irregular that they could be neglected.‡
* See Final Report, 1880, Tables XII. (4) and (B), pages 45–47.
† Final Report, p. 7, and Table XIII., p. 49.
Final Report, p. 6.
102
SUPPLEMENT TO A TREATISE
102. I have not had an opportunity of testing the value
of the coefficients of resistance for high velocities by a
comparison of calculated and experimental ranges and times
of flight. And there does not appear to be much necessity
for this, as the experiments with high velocities are SO
satisfactory. It is probable that the projectiles in these
experiments moved with their axes more nearly in the
direction of their motion than would be the case throughout
a long range. That would lead us to expect the calculated
range to be too great. With high charges the "jump" of
the gun might be expected to give too great an experimental
range. And if the elevation of the gun was considerable, a
great part of the horizontal motion of the shot would take
place in air much less dense than that for which the coefficients
hold good. This would cause the experimental ranges to
exceed the calculated ranges. But in these cases the "jump
may be measured by careful experiment, and in calculating
the trajectory, account may be taken of the decreasing
density of the air as the projectile rises above the earth.
""
103. The new values of K for ogival-headed elongated
shot, given in Table H., have been calculated on the sup-
position that the weight of a cubic foot of air is 534-22 grs.,
which is the weight of a cubic foot of dry air at a temperature
62°F., under a pressure of 30 inches of mercury. The values,
therefore, of K in Tables I. and C. have been increased in
the ratio 530-6: 534-22, or the old values of K have received
an increment of about 2-300ths of their former values. But
as no new experiments have been made with round shot, the
values of K in Table II. remain unchanged.
When calculating a trajectory, &c. suppose that a cubic
foot of air weighs (534.22 ± ▲) grs., this may be allowed for
in the following manner. K, is generally found multiplied
by
d2
พ
d2
៩ | ន
In this case
X
When then
พ
1±
(1-
d²
W
K becomes
い
​A
× K。°
K.
534-22) K-} - { (12634-22)}
Δ
พ
534-22) is once found, it can be used in the
calculations in combination with the tabular values of K.
This will be found a simpler process than correcting the
tabular values of K, for in the case of the General Tables,
we shall have
d²
20
Δ
xt =
(1±53423) × Ty- T
?
ON THE MOTION OF PROJECTILES.
103
and
or,
ď
੪॥੩
(1+
Δ
534-22) xs = Sv - S₁;
534.22
if we suppose the normal height of the barometer to be
30 inches, and the observed height 30 ± ß inches, then we
shall have
d²
and
W
30
(1+) - Ty-T₂
t=Tv-
ď
B
S=
W
(1±8) = Sy - 8,
30
104. In the careful calculation of the trajectory of a shot,
it will be necessary to take into account the diminution in the
density of the air corresponding to the probable height above
the surface of the earth of the middle of each successive arc
into which the trajectory is divided.
The following Table calculated for a temperature of 62°
will be useful in taking account of this diminution of the
density of the air as the shot rises. The height of the shot is
expressed in feet, and the height of the barometer in inches.
Thus, suppose that the height of the barometer at the surface
of the ground is 29 inches. This corresponds to a height
2700 feet in the Table. When the shot is 2200 feet above
the ground, the height of the barometer corresponding to
And the proper
2700 +2200=4900 feet is 26.84 inches.
value of K will here be K× 26·84 ÷ 30, approximately.


H.
Ft.
0
0 100
In. In.
In. In. In.
In.
32.00 |
31.89 31.78 31.66 31.55 31.44
1000 30.88 30.77 30 65 30.54 | 30·43
2000 29.78 | 29.67 | 29.57 | 29.46 | 29.35
200 300
400
00 | 500
700
600
800
900
Mean
Diff.
In. In.
In.
In.
| |
31.33 31-22 31.11 31.00-
•111
30.32
30-21 30.10 30.00
29.89
•110
29.25
29.14 29.04
28.93 28-83
•106
3000 28-73 28.62 28.52 28′42 | 28-32 28.22
4000 27.71 27.61 27.52❘ 27·42 | 27.32 | 27-22
5000 26.74 26-65 26.55 26-46 26.36 26-27
28.12 | 28.02
27.92
27·81
•102
27-13 27.03
26.93
26.84
⚫097
26.17 | 26.08 | 25.98
25.89
⚫094
25.61 | 25·51 | 25.42
25.51 25.42
20·64 | 20:57 | 20:49
19.84 19.77
19.14 19.07
6000 25.79 25.70
7000 24.88 24.79 24-70 24.61 | 24·53
8000 24.00 | 23.92 | 23·83 | 23·75 | 23.66 | 23.58 | 23:49 | 23·41 | 23·32 | 23.24
22.91 22.83
9000 | 23.16 23.07 22.99 | 22.91
10000 22:34 | 22:26 | 22∙18 | 22∙10 | 22 02
11000 21-55 | 21-48| 21-40 | 21-33 | 21.25
12000 20·79 | 20-72
13000 20.06 19.98 19.91
14000 19-35 19-28 19:21
15000 18.67 18.60 18.53 18:47 18:40 18:34 18.27 18-21 18:14 18.08
25-33
24.44
| |
25-24 25-15 25.06
24.97
⚫091
24.35 | 24.26 | 24-18
24-09
•088
⚫085
22.74
21.94
|
22.66 22-58 22.50 22.42
|
⚫082
21.86 | 21-78 | 21·70| 21·63
21·17 | 21·09 | 21-02 | 20.94 20.87
⚫079
⚫076
20.42❘ 20.35 | 20-27 20·20❘ 20·13
⚫073
19-70 19-63 19.56
19:49 19:42
⚫071
19.00 18.94 18.87 18.80 18.73
|
⚫069
⚫066
104
SUPPLEMENT TO A TREATISE
105. In the groups of values of K, for low velocities.
(see Final Report) there is often a greater variation in these
values than might have been expected, and in consequence
of the shot having so frequently passed through the screens
without giving a record, it is difficult to decide whether these
irregularities be entirely due to the unsteadiness of the shot.
All that I could do was to reduce the records sent to me as
carefully as possible and state the result. It must be borne
in mind also that the unit in the value of K, depends upon the
second differences of the times of passing each screen
expressed to the 1-100000th of a second. It is not pretended
that any chronograph can measure time practically to this
fabulous nicety, but in reducing the observations, where the
records were read off to the 1-3000th of a second, it was
found insufficient for the purposes of calculation to express
time in seconds to four places of decimals only, and con-
sequently five places are always used now. The experiments
were nearly completed before either of the two following
plans suggested themselves as likely to secure a good record
for every screen passed through by the shot.
106. First, drill three or more equidistant holes normally
to the surface of the head of the shot, not far from the
cylindrical part, and plug these holes with soft wood. Just
before loading fix pieces of softened watch spring wire in each
of these plugs, sharpened on the front edge, so that they may
clear the gun and yet catch the string of the screens. I have
tried a wooden model dropped from the height of 3 or 4 feet
on strong threads stretched horizontally
with perfect success.

C
Fig 20
d
b
ел
W
a
The second method depends on an
alteration of the screen.
A strong linen
thread is to be attached to a fixed point a
(fig. 20) at the top of the screen, and
carry a sufficient weight W attached
at the other end. The arrangement for
breaking circuit by wire springs ch is
much the same as that which has been
used hitherto, only the springs act down-
wards here. Each string of the screen
is turned once about the round spring
wire at b, so that the weight W will keep
the string ab stretched, and when a sudden
pressure comes against the thread ab, the
spring cb will be jerked upwards sufficiently
to break contact at e; d is a stop.
ON THE MOTION OF PROJECTILES.
105
107. Robins and Hutton endeavoured to determine the
resistance of the air to the motion of bodies of various
forms for very low velocities by the use of the whirling
machine. When the motion of the machine appeared to be
uniform they found the average time of a revolution,
and thence deduced the resistance of the air. My chrono-
graph is well adapted for experiments of this kind, because
it would give the exact time of each successive revolution,
beginning from rest, and so afford means for calculating the
retarding force for all velocities below the greatest attained.
Yet there is not much encouragement to pursue this method
of experimenting as it has only been used for velocities
below 25 f.s, and there seems to be a liability to error arising
from friction and from the stretching and rigidity of the
string. It therefore appears that it would be preferable to
experiment with bodies of the required forms, constructed of
some suitable light material, falling freely under the action
of gravity. The arrangement shewn at fig. 11 would be very
suitable for dropping the bodies. If the greatest available
height were H, then I would first find the time of falling
from rest through a height H, taking the mean of 8 or 10
experiments. Afterwards I would find the time of falling
freely through heights of H, H, &c. When the experi-
ment was completed these times might be differenced, and
hence the retarding force of the air might be deduced as in
(55) and (57).
3
108. When y is large, 1−y (3p+p³) varies rapidly, and
therefore it is not convenient to calculate the values of X, Y
and T'in such cases by quadratures in the usual manner from
formulæ (6), (7) and (8) in (63).
In (62) we have found
1
3
u³
น
U
3
9
1
26
(3p+p³)
1
3
{1 − y (3p + p³)}.
When =,, or p=p₁, suppose u= ∞, and then we have
19
therefore
dp 1
3
u³
u
3
(-1 2/2)
3
dt
Y
3
1
9
3
3p, + p = = = 2b, ;
هرات
1
Y
26
(3p+p³)} = ²² ((³p,+p,'")−(3p+p')},
9
106
or
therefore
dt
dp
*
SUPPLEMENT TO A TREATISE
1
X
1
(2bg³)} ^ {(3p,+p‚³) − (3p+p³)}}
ΦΓΦ' (1 + p²) dp
T
1
1
t=
•T"',
(2bg²)}
{(3p,+p¸³) − (3p+p³)}}
(2bg2)
and, similarly,
1
(1 + p²) dp
X
11
(46³g)
{(3p, +p¸³) − (3p+p³)}
1
•X'•',
(4b³g) 3
and
1 $
Φ Φ
Y
3
1
¢Y'p'.
(4b²g)}
p (1+p³) dø
{(3p₁+p,³) − (3p+p³)}}
3
1
The above formulæ have been used in calculating tables of
values of X', Y' and T", for 3p, +p, equal to 0.01, 0.001,
0.0001 and 0, or for values of y=100, 1000, 10000 and
infinity.
109. Let PPAB (fig. 21) be the trajectory of a given
shot, on the supposition that the resistance of the air varies as

t
C
A
Ꮽ
P
b
a
N
q
N
Fig 21
B
the cube of the velocity.
If now the given shot be pro-
jected from any point P in the direction of the tangent Pt at
ON THE MOTION OF PROJECTILES.
107
that point, which is inclined to the horizontal line PN at an
angle 4, with the velocity vo, it will describe the path PAB,
if v be determined from the equation
1
(vs cos 4.
3
1
{1-y (3 tan + tan³p)}.
As the point of projection approaches P₁, will increase
and 1-y (3 tano+tan p) will decrease, and, therefore, vo
will increase. At P₁, when = 4₁, suppose that
$
1-y (3 tano, +tan³,) = 0, then v¿₁ = ∞ .
If we suppose to be larger than ₁, 1−y (3 tan + tan³y)
is negative, and, therefore, v is negative. That is, if sQq=Ÿ,
then the shot must be projected from Q downwards in the
direction Qr.
1
1
110. Thus the velocity of projection upwards decreases
from o at P, to u, at the vertex A. And the velocity of
projection downwards decreases from ∞ at P, towards C, till
it becomes a minimum at a point where the resolved part of
gravity in the direction of the tangent is equal to the
retarding force of the air, or where g sin 2bv.
↓ =
substituting for v
vy
its value from
-3
-3
·
(vy cos¥)˜³ = µ³ {y (3 tan ¥ + tan³) — 1},
3
And
we arrive at the equation, tan 2=-2y for determining y.
3
111. PAB and P, QC are independent branches, having a
common tangent at P,. If we supposed the shot to move
under the action of gravity and an accelerating force 2bv³ in
the arc CQP, and that the force + 2by suddenly changed to
-2bv at P, when the velocity was infinite, then a single shot
could be made to describe the whole arc CP,AB continuously.
3
3
112. The velocity of the shot moving in the neighbour-
hood of P is very great, so that arcs Pa and Pb are not
important because we do not possess any practical means of
communicating the corresponding high velocities to the shot.
Tables of values of X, Y and T have been calculated for all
practically useful values of between + 60° and 60°, from
y=0 to y= 10; and after that for values of o down to - 45°,
from y=1.1 to y=x, where 1-y (3 tano+tan") is positive.
These tables, therefore, apply to the branch aAB only.
If it was desired to depress guns very much it would
D*
108
SUPPLEMENT TO A TREATISE
be necessary to calculate tables where y (3 tan + tan³) – 1
is positive, which would apply to the branch bQC.
113. We can find the extreme depression of each gun for
which the accompanying tables enable us to calculate the
trajectory. When a shot is fired from a gun elevated above
the horizon it is known that it will pass the vertex of the
trajectory with a velocity u, less than the velocity of pro-
jection. But when a gun is depressed below the horizon, the
theoretical velocity u, at the vertex A of the trajectory will
generally be greater than the velocity of projection. If the
velocity of the shot would pass through infinity between the
vertex A and the point of projection B, then the tables
already calculated do not apply to the case. But if the
velocity would be infinite at the vertex A, or on the side AP,
then the accompanying tables are applicable. Let the shot be
fired at an angle ' below the horizon with an initial
velocity v.
Then we have
(1000) * = (1090
cos
3 K ď²
Po
I w
= ∞ =
19
Suppose now, that u and that v cos p' 1500 f.s, then
we obtain,
Por = (1000) *
P&
K d²
=
w/
9 го
from which p' can be found by the help of Table V. or D.
after numerical values have been assigned to K, d and w.
Example 1. Take the 11.6-inch gun, where d = 11·52
inches, w700 lbs., and therefore d÷w=0'1896, and
K÷g=3·134.
Po
=
3
(1000) + (8·1340-1896)=-49965,
which gives ' = 9° 20′ nearly.
Hence the tables already calculated apply to all elevations
of this gun from + 60° to 9° 20'.
Example 2. Take the 3-inch gun, where d= 2.92 inches,
w= 12 lbs., and therefore d÷w=0·7105.
2
Here P= (1900) + (3·134 × 0·7105) = 0·13306,
which gives p' = 2° 32'.
Hence the accompanying Tables apply to all elevations of
the 3-inch gun from + 60° to - 2° 32'.
ON THE MOTION OF PROJECTILES.
109
VB
β
114. Let va, v be the velocities of a shot moving in the
ascending branch of its trajectory at points where the tangents
are inclined to the horizon at angles a, ẞ respectively, and
K÷g the mean coefficient through the intervening arc (a-B).
Suppose that va is given, then in order to find v we must
make use of the equation
'ß°
3 K d²
1000) = (1000) + (P. - P.) .. (4).
(1000) * =
a
9 го
Pg)...
But we cannot assign a correct value to K÷g till we have
found u
uB. On referring to Table I. or H. giving correspond-
ing values of v and K÷g, it will be found that sometimes
K÷g is varying slowly with v, while at other times it is
constant or nearly so. It will, therefore, be advisable to begin
by assuming a value for K÷g equal to that given in the
table opposite to va, or, preferably, to a velocity something
less. Substitute this value of K÷g in the above equation,
and then v a good first approximation to v will generally be
found, if the arc (a – B) be not too large.
Now find the mean value of K÷g between va
the first adopted value of K÷g require a small positive
correction then v will require a small negative correction.
The way in which this correction may be introduced will be
best seen from an example.
5
β
β
β
and v
B°
If
Taking the example worked out at p. 60, we have given
va = v = 1335 ƒ.s; u=1329·9 f.s;
u=1329·9 ƒ.s; d²÷w=56935.
And on
referring to Table I. we find for a velocity 1335 f.s the value
3-322 for K÷g. Substituting in the above equation, we have
3
3
1000
(1000) * = (119909)*
4
1329.9
+3·322 × 0·56935 × 0·05301
= 0·42515 +0·10026 = 0.52541,
which gives u=1239.3 f.s;
and
V
v₁ = 1239•3 sec 4° = 1242·3 f.s.
4
The value of K÷g corresponding to the mean of the
velocities 1335 f.s and 1242-3 will be found by Table I. to be
3·358 = 3·322 +·036 = K÷g + SK÷g. Now
K÷g=3·322 gave 10026,
therefore +&K÷g=0·036 will give + ·00109,
and from Table F. it appears that for a velocity about 1239 f.s
a correction of +00127 applied to (1000÷v)³ gives a
110
SUPPLEMENT TO A TREATISE
correction
1.0f.s to be applied to v. Therefore, a correction
+ '00109 will give a correction of −0·9 f.s to the velocity, or
= u₁ = 1239.3 – 0·9 = 1238·4 f.s.
u,
β 4
uß
115. If we proceed to calculate first approximate values
Изд
u, u₂, u₁ and u, using the same value 3.322 of K÷g,
we find
of
and
u,
3
U2
1169.5 f.s instead of 1167-8,
= 1113·3…..
1113 3......
u₁
= 1066·8....
u = 1027·1..
...1111.6,
1066 0,
..1031.9.
0
B°
Thus it appears that a somewhat erroneous value of K÷g
will generally give a very fair first approximation to v And
this arises from the fact that when the arc (a-B) is properly
assigned, K÷g is always multiplied by a very small quantity.
For d÷w varies from about 3 for a small arm bullet to 0.25
for the 10-inch gun, and becomes still less for larger guns.
When d÷w is large aẞ must be small, say about 0°5,
then d÷wx (P-P) = 3 × 0.03 0.09 nearly.
=
lating trajectories for the 6.3-inch howitzer for low velocities
(below 800 f.s), where d²÷w=0.5616, it was found very
satisfactory to divide the trajectory into arcs of 4° or 5°.
β
In calcu-
B
116. Some difficulty has been experienced in calculating
v from the equation (A) where K generally varies slowly
with v.
The trajectory has to be broken up into a series of
small arcs through each of which K has to be supposed
If
constant, and equal to the mean value between v。 and v
the arc (a-B) be properly chosen K may have its mean
value derived from the table of values of Σ (K÷g), or it may
be the mean of its values corresponding to v。 and v
VB or it
may be the value of K corresponding to the mean velocity
1
(vv). For if the results differed sensibly it would
appear that the arc (a-B) was too large. As the problem
does not admit of a direct and precise solution it is necessary
to resort to approximations. But if these approximations be
intelligently carried out, any required degree of precision may
easily be obtained in the result; and the smaller the arcs are
taken the more nearly exact becomes the result.
117. But to meet this difficulty I have calculated an
auxiliary Table J., where
บ
3
R₂ = 100 [{(1000) * - (1000)},
g
K
m
ON THE MOTION OF PROJECTILES.
111
By this we find
3
Ro₂- Rep = {(1000)" - (1000) } }
Rv
Rva
α β
β
K'
where K is supposed to vary continuously with v, from v to
But in the equation we have to deal with, va
replaced by u. and u
UB.
(A) under the form
1000
cos ẞ seca
3
and v are
'β
We must, therefore, put the equation
9 d
P¸)
)' - (1000) } // = = (P. - P₂) cos'a,
K
W
d2
3
where
Rv Rv cosẞ seca
W
(Pa- Pg) cos³α......(I),
α
β
for the ascending branch, or
or
3
(1000)² - (0/0
β
α
1000
cosa secß,
3
ď
(Pß - Pa) cos³ß,
W
+Rv' cosa secẞ- Rv' = (PË-Pa) cos³ß (II),
α
β W
for the descending branch, where, the smaller cosa secß
is, the nearer the calculated result will approach the true
result.
118. Suppose that a Martini-Henry bullet is projected at
an elevation of 1° above the horizon, with a velocity of
1353 f.s. Then d=0·45 inch; w = 480 grs.
· 480 grs. = 0·06857 lbs. ; and
d² ÷ w = 2·953. As the velocity of the bullet will vary
rapidly in consequence of the high value of d÷w, it will be
advisable to employ the general Table J. to calculate the
velocities when the bullet is moving in directions inclined at
0°•5, 0, — 0°·5, — 1°.0 and -1°3 to the horizon. We thus obtain
0.5
v0.5=1145.3 f.s; u=10347 f.s; v'o.s =973'6 f.s;
v'1.0=924'9 f.s; v'1.3900'0 f.s.
And using the tabular values of X, Y and T, we find:
120.5
0.5000
=
-416.8 feet;
= 321·0
1x=737·8
"}
0.530
= 1·463
0.55553 feet;
Y0.5
1t0.5
""
0":3357
0.5to=0·2948
"}
0
13% +7·016
22
10
ito:
= 0·6305
112
SUPPLEMENT TO A TREATISE
and for the descending branch:
0.5
0·5t10
t
0 2571
1.01.30.1488
охо
0° 0.5
=272·7 feet; oyo.5=1164 feet; otos=0"-2723
0.5X10
= 244.3
""
0.591
= 3.191
""
0.5 1 0
1.41.g
= 135·3
""
Y 1.3
= 2·726
99
X1.3
=+652·3
11 Y 1.3
Y1.3
7.081
""
Ο
x=+737·8
""
19%=+7·016
22
0
1390·1
- 0·065
11
0*1.3=0·6782
to =+0·6305
1. 3087
A correction of -0.065 applied to y will give a correction
-0.065 cot 1°.3=-2.9 ft. to be applied to x, and a correction
- 2.9900 x 1" -0".0032 to t. Hence the range on the
horizontal plane passing through the point of projection is
1390.1 2.9 feet 1387 2 feet. The greatest height to which
the bullet rises above this plane is 7'016 feet. And the time
of flight is 1"-3087-0" 0032 = 1"-3055.
Or, if we divide the trajectory into two parts at the vertex
we obtain u=10345 f.s; v1.3=9000 f.s; and
1x0
X1.3
= 652.3
x=747·0 feet;
0
1399.3
10
190
+7·114 feet;
and t =0"-6344
99
Y 1.3
-7.080
""
0
ot1.3
0 6781
+ 0.034
1 .3125
Hence the range on the horizontal plane is 1399.3 + 1·5
=1400-8 feet. The greatest height is 7.114 feet, and the
time of flight is 1"-3125 + 0"·0016 = 1″·3141.
119. When, as in the above example, a shot is moving in
a direction so little inclined to the horizon that the resolved
part of gravity in the direction of its motion is very small,
the general Tables L. and M. may be used to calculate the
trajectory in the following manner. Suppose a, B, to be the
0
Fig 22
ť
t
A
m
n
P
p
N
M
m
inclination to
the
horizon of the tan-
gents to the trajectory
at the points O, P.
Then the velocity of
s the shot v at O being
supposed to be known,
its velocity vs at P
And by the help of

N'
P'
can be found by the help of the Table J.
a
ON THE MOTION OF PROJECTILES.
113
the general Tables L. and M. it can be found in what space s
and in what time t, the velocity of the shot, unaffected by
gravity, will be reduced from v. to vg. Then take the angle
NOt=a; Ot=s; and draw tN perpendicular to the hori-
zontal line passing through 0. Then
and
ax
β
a
= ON= Ot cosa =s cosa;
y=PN=Nt−tP= s sina — ½gť².
The value of gt" will be found from Table G. when t is
known. In the descending branch we have
pN' = m'p' = m's + sp' =ps sin spm' +1gť".
In the above example we find
10.5=416.5 feet; 0.5 = 3188 feet; 0.5=272.9 feet;
S 244.2 ; 1$1.3 = 135.1
0.5 1
1
""
"" ;
。to
and t=0"-3359;t=0" 2940; 50"-2722;
0-st₁ =0"2579;t=0"1483.
0.5 1
=
416.4 feet; 10.5=5452 feet;
120.5
o.5 = 318•7
0
1x=735·1
19
= 1.391
1 ; 0.590
。。.5=272·9 feet;
o•C =
X
1.3
1
= 244.2
= 135·1
652.2
DX1.3
1200
= 735·1
1387.3
G
19% = 6·843
=
1
to.50"-3359
;
to
";
=0·2940
0.5°0
to=0·6299
t0.5
= 0".2722
999
;
t
1.193 feet;
Yo
90.5
22
;
0.531
""
;
191.3
- 7.105
= 3.201
= 2·711
Y 1.3
19% =+6·843
- 0.262
0.5*1=0 •2579
11.3 = 0·1483
0t1.3=0·6784
t =0·6299
1°0
1 -3083
Correction for y is 0.262, for x is 0.262 cọt 1°•3——11·5,
— –
and for t is 0"-0128. Therefore, the range is 1387.3 - 11.5
= 1375-8 feet; the greatest height is 6.843 feet; and the time
of flight is 1"3083 - 0"·0128 = 1″·2955.
Or, if we divide the trajectory into two parts at the
vertex, we find u= 10345 f.s; v1.3=900 f.s; s=736.0 feet;
081.3=651-3 feet; and
S.
0
651′2,, ; oY 1.3
12=735-9 feet; 19%
X1.3
1387.1
+6:440 feet; t = 0"-6308
-7.390
0.950
t
= 0".6776
013
1"-3084
114
SUPPLEMENT TO A TREATISE
as B
cos (a+B), and
t = 0"-3359
120. If we suppose OP to be the arc of a circle, then
L PON=(a+B), and therefore a = a³ß
ay 3 = as sin (a + B).
β β
10.5=416.5 feet;
0.5 = 3188 „ ;
x=735·3
22
and=272.9 feet;
x, = 244·2
0.5
0.5 1
21.3
1.3
, хо
= 135.0
= 652.1
= 735.3
ах в
Thus we find
9%=+6·842
to.5
1 0.5
0
0.5 0
=0·2940
t = 0"-6299
1955.451 feet;
0.5 Yo =1·391 „ ;
Yo-5
090.5=
1.191 feet; 0.5=0″•2722
t
0
0531-
;
191.3
19 1.3 =
3.196
2.711
22 ;
0.5t₁ = 0·2578
t
"" 2
₁₁.9=0·1483
OY 1.3
-7.098
" ;
22
1
01.30"-6784
t=0·6299
1387'4
19%=+6·842
· 0.256
1":3083
Hence the range on the horizontal plane through the
point of projection is 13874 – 0·256 cot 1°·3 = 1387·4 – 11·3
= 1376.1 feet; and the time of flight 1"3083-0" 0126-1" 2957.
Or, if we divide the trajectory into two parts at the
vertex, we obtain
120
DX1.3
=736·0 feet; 19%
+ 6.423
0
=6513 „ ; Y1.3
=-7.390
10
0"-6308
0*1.8=0·6776
1387.3
- 0.967
1 .3084
Mr. W. D. Niven has shewn how to apply corrections to
the angle (a+B), so that by the help of the general
Tables L. and M. trajectories may be approximately calcu-
lated in the absence of the tabular numbers X, Y and T.
121. The Tables of values of X, Y and T, or X', Y'
and T', for y=130 to y = ∞ have been calculated for the
descending branch only of the trajectory. But if it should be
necessary to calculate the ascending branch in any of these
cases, then, as the inclination of the tangent of the trajectory
to the horizon cannot much exceed 1° 30', one of the above
simple methods would apply and must be used up to the
vertex. After that the tables of values of X, Y and T, or of
X', Y' and T may be used to complete the calculation of the
trajectory from the vertex downwards.
ON THE MOTION OF PROJECTILES.
115
122. Suppose it to be required to calculate the trajectory
of a 70 lb. ogival-headed shot fired from the 6.3-inch howitzer,
with a muzzle velocity of 751 f.s at an elevation of 30°. Here'
d=6•27 in. and ď÷w = 5616. We will divide the trajectory
into arcs 30°-28°, 28° - 24°, 24° -20°, &c. Calculate the
velocities v287 249 20, &c., by the help of Table J.
Generally Ru sec a
Բ
Ry
α
ing branch (117),
or
d²
พ
(P-P) cos³a for the ascend-
Rugs sec 30°=R751-5616 (P-P) cos³30°
= 7·6109 – 0·0653 = 7·5456 = R794.
734-57
30
or
28
Ugg sec30°=734.5, 636.1, and v₂ =
==
=
28
28
= 720·4 f.s.
ď²
W
or
Again Ru sec28° = Rv₂ (P28-P24) cos³28°
728
=7·4864 0·1243 = 7.3621= R
693.47
u₂ sec 28° = 693·4, u=612.2, and v₁ = 670.2 f.s.
24
24

ย
$
น
γ
f.s
A₁
A₂ f.s
Δι Δη
A₁
30° 751.0
650'4
28 720.4
*223
636.1
- 50.2
23.9
24 670.2
*231
+10.5 612.2
39.7
+4.2
19.7
+11
20 630.5
•242
+ 8.1
592.5
31.6
-
16 598.9
16.8
+2.9
+11
•253
+ 6.2
575.7
25.4
+2.1
+11
14.7
12 573.5
•264
+5.0 561.0
20.4
+1.5
13.2
+10
•274
8
553.1
+ 4·3547·8
16.1
+1·1
+ 8
12.1
+ 1
+ 4
537.0
-282
+ 3.7 535.7
12.4
+1.0
+8
11.1
•290
0
524.6
+ 3.3
524.6
9.1
10.2
+0·9
+ 6
•296
4
515.5
+2.9 514:4
6.2
+0.1
+ 5
8 509.3
10'1
•301
+2.6 504·3
3.6
+0.4
+ 3
9.7
•304
12
505.7
+ 2.6
494.6
1.0
+0.2
+ 1
9.5
⚫305
16
504.7
+ 2.5 485.1
+ 1·5
+0·1
0
9.4
•305
20
506.2
+ 2.5 475.7
0.2
+ 4.0
2
9.6
⚫303
24
510.2
+ 2.6 466.1
0.2
+ 6.6
3
9.8
28
516.8
⚫300
+2.5 456.3
-0.5
+ 9.1
6
- 10.3
•294
32
525.9
+2.9 446.0
-0.5
+12.0
9
- 10.8
36 537.9
•285
+ 2.9435.2
+14.9
12
•273
40
552.8
E*
116
SUPPLEMENT TO A TREATISE
Again Ru sec 24° =
Rv24
(P24-P₂0) cos³24°
พ
or
=7·2435 −0·1215 = 7·1220 = R.
648-67
U20 sec 24° = 648.6, u 592.5, and v₂-630.5 f.s.&c.
=
20
20
To test the accuracy of the calculations as we proceed we
difference the values of v, u, and also y when calculated, as
in the above Table.
123. It is convenient to find by interpolation values of v
corresponding to p=26°, 22° &c. Thus we obtain
φ
Y
f.s
30° 751.0
28
720.4
- 30.6
-
26
693.8
24
670.2
22
649°2
20
630.5-18.7+2·3 −0·3
-21·0+2·6
26.6 +4.0
23.6+3·0-1·0
-0.4
These interpolated velocities may be used as guides to the
proper values of K÷g to be used in calculating Ax, Ay and
At for each arc.
124. We must now proceed to find u, and thence y,
Ax, Ay and At for each arc. Here
(1000)* = (1000) *
Also
y=
ор
(10000)
8
228
28
1000
636-1
K d2
9
g w
3
3
+
Ꮶ d
9 го
P₂
2
+2.519 × 5616 x 1.74545, &c.
and then ▲x, ▲y and At may be
found for the arc (30°- 28°), and so on for the remaining arcs.
of the trajectory. We thus obtain the columns headed Ax,
Ay and At, which are also to be differenced as the work
proceeds to test their correctness. Afterwards the columns
x, y and t are found by addition commencing from zero at the
vertex in each case, as in the table following:
ON THE MOTION OF PROJECTILES.
117
8
$
୫
Ax
A²x
A³x
У
Ay
feet
A²y
Aჰყ
t
At
A2t
A³t
feet
30°
6074
1882
593
10".413
28
5481
329
1553
- 0"-916
1055
9 *497
24 4426
516
+135
1037
1 .682
920
- 35
+ 143
7 815
20
3506
373
+159
+ 100
37
664
· 1 ·523
- 39
820
+ 106
25
6 .292
16
2686
267
+120
+ 75
25
397.3
1 403
27
745
18
+ 81
4 ·889
12
1941
186.3
+ 93
+ 57
211.0
16
1 .310
688
+ 64.7
23
12
3 .579
8 1253
121.6
+ 45
11.0
+ 70
89.4
- 1 240
18
643
+ 53.7
2 ·339
12
1 +
+ 4
610
67.9
+ 52
+ 33
+ 21.5
7.3
1 .188
15
610
9
+46·4
21.5
+1 ·151
+ 37
0
0
+ 24
4.6
1 .151
0
14
586
+41.8
4 586
7
0
+
20·3
+
23
+ 17
+ 20.3
2.3
- 1 128
12
569
6
+ 39.5
-1 128
8 1155
+ 11
+
59.8
80.1
0.9
+
11
•
1 117
Going
558
+ 38.6
11
2 .245
6
12 1713
+ 98.4
0
553
16
2266
+1
5
178.5
1
554
1 1
6
+ 39.5
+ 0·9
1
117
12
3 ·362
12
+ 137.9
316'4
+ 2·6
1 .129
12
+ 42.1
4 ⚫491
20
2820
6
+ 180.0
24
496'4
+ 4.2
· 1 ·153
12
560
+46.3
5 .644
24
3380
+226.3
36
11
722.7
+ 6.1
1 .189
14
571
+ 52.4
6 .833
7
50
28
3951
+278.7
18
1001.4
+ 9.3
1 ·239
589
+61.7
17
8 .072
7
32
4540
+ 340·4
67
25
1341.8
+ 12.3
1 .306
614
36
8
+74.0
19
9 .378
86
5154
+ 414·4
32
1756.2
+17·1
1 ·392
646
+91.1
25
10 •770
40
5800
+ 505.5
111
2261.7
-1 ·503
12 ·273

118
SUPPLEMENT TO A TREATISE
125. We might now find the range, time of flight, final
velocity and angle of descent on a horizontal plane, a given
distance above or below the muzzle of the gun. But we can
derive other important results from the calculations already
made, which will be of great service when we come to the
systematic calculation of Range Tables. Any point in the
trajectory may be considered to be the point of projection,
provided the shot be supposed to be projected with the
velocity, and in the direction of the tangent to the trajectory,
at that point. We shall, therefore, suppose that shot are fired
at elevations of 0, 4°, 8°, 12°, &c., with muzzle velocities
537-0, 553·1, 573.5, &c., f.s respectively, and find their
ranges, &c., on horizontal planes 6'5 feet below the points of
projection. It will be advisable to find by interpolation the
values of x', y', t' and v' in the descending branch for values.
1º, 2º, 3º, &c. of p'. The following is small portion of such
a table:
x'
φ'
B’
f.s
A
feet
A
feet
A
ť
Δ
20°❘ 506.2
2820
496.4
+0.8
+ 139
5" 644
+52·0
2 I
507'0
0.9 2959
140
548.4
22
507'9
1.1
3099
140 603.4
55.0
3.0 5 937 ·296
+293
58.1
23 509'0
24 510.2
1.2
3239
3380
141
661.5
61.2
3.1 6 532 301
3.1 6 233 .299
722.7
6.833
126. When the gun is supposed to be elevated 20° above
the horizon the corresponding muzzle velocity of the shot is
630-5f.s (122). The shot rises 664.0+6.5=670·5 feet above the
horizontal plane on which we desire to find the range, &c.
By the help of the above table we find that when the shot is
moving in the descending branch of the trajectory in a direc-
tion inclined to the horizon at 23°, it has fallen through a
vertical height of 661.5 feet, and when at 24°, it has fallen
through a vertical height of 722-7 feet. We desire to find
o', x', t', v' corresponding to y' = 670.5. Hence
$' = 23° + (670·5 — 661·5) ÷ 61·2 × 1° = 23° + 9°÷61•2 = 23°15;
x′ = 3239 + (9÷61·2) × 141 = 3239 +21 = 3260 feet;
t'
ť
6"532+(9÷61.2) × 0"-3016"-532 +0"0446"•576;
v' = 509′0 + (9 ÷ 61·2) × 1•2 = 509′0 + 0·2 = 509·2 f.s.
ON THE MOTION OF PROJECTILES.
119
Hence
v=630·5f.s; R=3506 +3260=6766 feet = 2255 yards; Y=-6·5 feet;
T= 6″•292 + 6″•576 12" 868; p=23°15; and v' = 509-2 f.s.
=
In the same way we proceed to calculate the ranges, times of flight, &c., for elevations of 0°, 4°, 8°...28°
and 30° and obtain the following results:
Ø
V
R
Y'
f.s
yards
A₁ Δ2
feet
A₁
Δ2 A3
T Δι
A 2
p' Δι
f.s
A₁ A₂
0° | 524.6
109
6.5
4 537.0
+ 322
431
+85
+ 21.5
0".63
2°.22
28.0
+ 407
8
553.1
838
+31
+ 67.9
95.9
12
+438
573.5 1276
+ 32
16
+ 470
598.9 1746
+
+ 39
403.8
20
+ 509
630.5 2255
+ 266.7
+ 54
670.5
24
+ 563
670.2 2818
+70
1043.5
28
720.4 3451
+633
+ 106.3
+373·0
+143·0+ 36·7
1559.5+ 516·0
217.5+ 121.6+ 53.7+ 7.3
186.3+ 64.7+11.0
+15.7
+25.9
+ 46.4
2".47
+2·84
519.1
+ 2.46
4 ·79+2·32
-52
4.68
+18
8.75
+ 80.4
7 .29
9·97+2·68
+ 2.50
+4.50
+1813 25
+22 18 06
12 ⚫87
+2.90
+38
+ 3.57
19 •63
30
751.0 3802
1888.5
21 ⚫59
28 .53
34 ⚫20
37.12
+ 43 508.5
3·4+2·4/
•0+3·4/
+4·1
+ 4·1
16 06+3·19+29 23 15+ 5'09+ 28 505 1,
509-2
+8·7+4·6
+14.2
517.9 + 5.5
532.1
541.8
+4·07+1·61 514-37
4.8
|
1.0
5.8
+4·81+31 505.1
+ 5.38
+5.67+ 29
+ 29
127. By interpolation we can now find the values of V, R, Y', T, Þ' and v′ for Þ= 1º, 2º, 3º...28°, 29°, 30°,
as follows:
120
SUPPLEMENT TO A TREATISE
V
Φ
R
f.s
yards
Y'
feet
સ
T
Φ
f.s.
24° 670.2
2818
+11.5
1043.5
16"-06
25
681.7
2968
+12·1
+ 150
28°.53
26
693·8
27
706-7
+13.7
+12.9 3124 +161
+ 156
1157'5
16 '91
+ ·85
517.9
1.39
+ 3·0
12805 17 79
+ 788
29 '92 1·41
520'9
3285
28
720.4
+166
1414'0
18 70
+ ·91
3451
+14·7
+172
1559.5 19 ·63
+
·93
32 '76
31 33 1·43
+3·3
524°2
+3·7
1.44
527'9
+4.2
34 ·20
532.1
29
735'1
30 751.0
+15.9
3623
+
·96
1.45
+ 4.6
+ 179
1717'5
20 '59
3802
1888.5 21 59
+1·00
35 65
1.47
536'7
37.12
541.8
+ 5·1
128. In the next place suppose that we wish to calculate the Range, &c. of the shot for a muzzle velocity
of 751 f.s, and an elevation of 30° in a more expeditious manner. After having found by calculation as
above, u=524.6 f.s and v', 541-4 f.s, we will divide the trajectory into two parts at the vertex.
37
= =
The mean
value of K÷g found by the help of the table of values of (K÷g) between 750 and 525 f.s, is 2.961. This
gives y = 0.2401; 30=5947 feet; 30-1836-7 feet; and set = 10" 304.
t=
In the descending branch the shot has to fall a vertical height of 1836.7 +6.5=1843.2 feet. The mean
value of K÷g between velocities 525 and 540 f.s is 3.555, which gives y = 0.288 and

Hence
0038 = 5179.8 feet; 03-1768-2 feet;
86
X 37
03
37
s=10"-797,
; 0Y 3
""
036.64-1843-2 feet; 0.0411″·030,
3000+1836.7
5338.6
0236.64 = 5281 feet;
30xC0
= 5947
=
";
1885.8 " ;
0 87
otз7 = 11 ·162.
99 9
30% = 10 ·304.
Therefore, R=11228 feet 3743 yards. Y=-6.5 feet. And T=21"-33.
ON THE MOTION OF PROJECTILES.
121
0
36
129. Lastly we will find u and v' by the formula A (114).
The initial velocity is 751 f.s, and we are certain that this will
be very much reduced when it reaches the vertex of the
trajectory. We will assume K÷g= 2.905, which corresponds
to a mean velocity of 640 f.s. Substituting in formula A (114)
we find u 528 5 f.s a first approximation. By the help of
the table of values of Σ (K÷g), we find the mean value of
K÷g between velocities 750 and 530 ƒf.s = 2·946. This gives
u' 527·4 f.s. a second approximation. The mean value of
K÷g between 751 and 527.4 f.s will be found to be 2.951.
And it has been found that
0
and
Ο
K÷g=2.905 gave u = 528.5 f.s,
Ο
K÷g=2·946 gave u'= 527·4 f.s,
or
+·041 gave
-
– 0·9
Therefore
+⚫005 would give – 0·1,
or K÷g=2.951 would give u=527.3. This gives y=0.2428,
302-6023·0 feet, y= 1862.2 feet, and t = 10" 369.
30
0
For the descending branch mean value of K÷g between
velocities 527 and 540 f.s will be found to be 3.554, which
gives y = 0·2925 and
0238=5218.3 feet; Ye=1779.8 feet; and t=
86
X37
36
=5378′0,, ; 0Y37
0 36
10"-837,
= 1898·0,, ; 0*37=11.203.
The vertical height through which the projectile has to
descend is 1862·2 + 6·5 = 1868·7 feet, from which consideration
we obtain,
08.75=5338-4 feet; 338:75-1868.7 feet; and 75 = 11" 112,
and
0°36'75
30%
30¹² = 6023·0,, ; 30% =+1862-2,, ; and
Hence x=11361 feet=3787 yards; Y=-6.5 feet;
and
T=21" 481.
10 369.
130. A somewhat more accurate result would have been
obtained if the descending branch had been divided into two
sections at the point where v' is a minimum. Differentiating
= u(1+ y (3 tan o' + tan³p')}
or
-3
2
3
3
v'¯³ = u˜³ {cos³p' + y (3 sin p' cos²p' + sin³¿')},
122
SUPPLEMENT TO A TREATISE
we find tan 24' = 2y = 0.582=tan 30° nearly. Or the descend-
ing branch may be divided into two arcs, 0 to -15°, and
15° to 37°.
131. On sending in my report on the additional ex-
periments 1878-9, undertaken to determine the resistance of
the air to projectiles moving at very high and very low
velocities, some range tables of the 6.3-inch howitzer were
forwarded to me from Woolwich because they appeared to
shew that my coefficients did not give the same final velocities
as were recorded in those range tables. But as those were
cases were the muzzle velocities were low, and the elevations
were often high, the trajectories of the shot were mostly
much curved, so that the general tables connecting v, s and t
did not apply. I therefore decided to calculate the tra-
jectories carefully, so as to compare the results of calculation
and experiment for low velocities in a thoroughly satisfactory
manner in this case, which was not of my own choosing.
The range tables were arranged to shew what elevation must
be given to the howitzer in order to obtain ranges of 400,
500, 600, &c., yards with charges of 1, 2, 3, 31, 31, 32 and
4 lbs. of powder. But it was easy to deduce from these
tables the ranges, times of flight, final velocities and angles
of descent for elevations of 5°, 10°, 15°, 20°, &c., corresponding
to the various muzzle velocities given with each of the
specified charges.
w=
132. In these calculations w = 70 lbs.; d=6·27 inches
and ď² ÷ w = 0·5616. The calculation of the velocities com-
menced from the vertex with values of u=100, 140, 180,
220...700, 740 f.s. The formula used was
d
Rv = Rv_cosa secẞ ±(P-Pa) cos³ß,
β
d²
ur β
3
where it is manifest that (P-Pa) cos³ß having been cal-
พ
culated for one trajectory at intervals of 5°, the result would
be applicable to all the other trajectories to be calculated. As
has been explained (126) the ranges, &c., were calculated for
each of the above specified values of u for elevations of
5°, 10°, 15°...45°.
5
5
133. Afterwards values of V, R., T, and ' were arranged in
tables where u was the argument. Then seven intermediate
values of these quantities were found by interpolation. The
same thing was done with V₁o, R., To and ', and so on for
elevations of 15°, 20°...45°.
109 107 10
109
ON THE MOTION OF PROJECTILES.
123
The following is a specimen of the table for an elevation of = 30°:
Цо
A
B
f.s
f.s
yards
I'
feet
A
Φ
v
A
f.s
300 367.2
1122
+ 6.9
305 374'I
1161
+39
509.5 | 11″-21
32°06
334.6
+⚫20
+05
+ 5.1
6.9
39
II '41
32.11
310 381'0
20
6339'7
5.1
I 200
7.0
41
II '61
315 388.0
20
32 '17
6
344.8
7.0
1241
5.1
41
11 ·81
320 | 395'0
1282
20
32.23
7 349'9
5.1
7.1
42
12 'OI
20 32°30
325
402 I
7.2 1324
7 355'0 5.1
43
330 409'3
7.3
1367
12 '21 21
32°37
7
360'1
5.0
44
335 416.6
12 '42 21
32'44
7
365'1
1411
7.3
45
12 '63
5.0
340 423.9
1456
7.4
667.5
21
32°51
8370'1
46
12 84
5.0
32.59
375.1
345 | 431*3
7.4 1502
21
8
5.0
46
13 '05
350 438.7
7.5
1548
21
32·67
9
380'1
48
13 .26
5.0
355 446'2
7.6
1596
21
32 76
9
385.1
48
13 '47
360 453.8
7.7
1644
22
32.85
5.0
9390'1
4.9
50
13 '69
365 461'5
7.7
1694
21
32 '94
395'0
9
4.9
50
13 '90
370 469*2
22
33 '03
7.8 1744 51
10 399'9
4.8
14 '12
22
33 '13
375 477.0
10 404'7
7.9 1795 52
4.8
14 '34
380 484.9
1847
856.5
8.0
53
14 .56
22
33 23
10
409'5
4.8
385 492'9
22
33°33
1900
14 '78
10
414'3
4.7
33°43
419'0
540 | 784-4
4067
2038-5) 22-42
37.61
554.2

124
SUPPLEMENT TO A TREATISE
For
134. It is easy to find from this Table the range, &c. of the
6.3-inch howitzer corresponding to any muzzle velocity
between 360 and 500 f.s for an elevation of 30°.
instance, suppose the muzzle velocity given is 450 fs, a
velocity not found in the Table. We must, therefore, make
use of proportional parts.
The muzzle velocity is
The range R is
(446·2 + 3·8) ƒ.s.
1596 +48 × (3·8 ÷ 7·6) = 1596 + 24 = 1620 yards.
The time of flight T is
13″:47 + ·22 × (3·8 ÷ 7·6) = 13″·47 +0″·11 = 13″-58.
The angle of descent Þ' is
32°.85+09 (3.8÷7.6) 32°•85+0°05=32°.90.
And the final velocity v' is
=
390·1 + 4·9 (3·8 ÷ 7·6) = 390·1 + 2·5 = 392·6 ƒ.s.
And the like may be done with the other tables for
elevations of 5°, 10°, &c., which will be sufficient when the
object is merely to compare the results of calculation with
those given in Range Tables.
135. But if the object be to calculate the elevations to be
given to the gun, with a known muzzle velocity, in order to
obtain ranges of 400, 500, 600, &c. yards; first find, as above,
from the Tables calculated, the Ranges, &c. for elevations of
0, 5°, 10°, 15°, &c. corresponding to the given muzzle velocity.
Then find the like quantities by interpolation for elevations

R
Φ
A
T
A
Φ
A
v' A
yards
15° 2462
11":56
17°.64
593
16 2585
+ 123
+72
+1.31
6
12 28
118
71
1895
1.31
587
6
17
2703
113
12 '99
20°26
70
1.31
581
6
18 2816
108
13 '69
70
21 57
575
1.31
5
19 2924
103
14 '39
22.88
69
1.31
570
20 3027
15 ·08
24.19
566
98
68
1.31
21 3125
94
15 76
25'50
67
1.31
562
22
3219
89
16 43
26.81
67
1.31
558
3
23 3308
85
17 '10
28 12
66
1.31
555
3
24 3393
80
1776
66
29 43
1.31
552
2
25
3473
18 42
30.74
550
ON THE MOTION OF PROJECTILES.
125
of 1º, 2º, 3º, &c. It will then be easy by the use of pro-
portional parts to find the elevations to be given in order to
obtain any possible range. Thus suppose the muzzle velocity
of the 6.3-inch howitzer to be 751 f.s, then we obtain from the
Tables already calculated the values of R, T, Þ' and v' corre-
sponding to ... 15°, 20°, 25°.
15°, 20°, 25°….., and interpolate, as in the
preceding table.
136. We can now calculate ♣, T, Þ' and v', corresponding
to ranges of 2500, 2600, &c. yards, and we obtain for v=751 f.s,

R
V'
Φ
A
T
Δ
Φ'
Δ
A
yards
f.s
2500 15° 31
11"-78
18°.05
591
+82
2600
16.13
12 .37
+0"-59
+1.07
5
19.12
586
•84
•60
1.11
5
2700 16.97
12 ·97
20·23
581
.89
⚫62
1.15
2800 17.86
13.59
21 38
576
.92
.64
1.21
5
2900 18·78
14 ·23
22.59
571
•96
.66
1.26
4
3000 19.74
14.89
23.85
567
1.00
•70
1.31
4
3100
20.74
15.59
25.16
563
1.06
71
1.38
3200 21 ·80
16.30
26.54
559
1.11
•74
1.46
4
3400
3300 22 91
24.11
•
17 .04
28.00
555
1.20
•80
1.58
3
17 .84
29.58
552
which may be compared with the Range Table for a charge
of 4 lbs. given in Instructions for Field Service, 1879, p. 103.
137. In this way numerous comparisons of calculated and
experimental ranges, &c. for low muzzle velocities, have been
made by the help of Range Tables, both English and
German, (100). Considering all the uncertainties which
attach to such Tables, the results appear to be as satisfactory
as could be reasonably expected.
138. If it had been desired to provide means for the ready
preparation of theoretical Range Tables for any practical
value of d÷w whatever, then it would be necessary to
calculate sets of Tables, for values 0.1, 0.3, 0.5, &c. of d²÷w,
such as we have calculated for dw=0.5616; or, with a
view to utilise the work already done, d'÷w might be taken
equal 0.5616 ± 0·2000 × n. The trajectories were calculated
by arcs of 5°, but the results would have been a trifle more
precise, and the interpolations much simplified, if arcs of 4°
had been used.
126
SUPPLEMENT TO A TREATISE
139. Inasmuch as the unit of the value of K depends upon
something like the 1-100000th of a second (105), it was,
perhaps, needless to express its value to one place of decimals.
This affords an additional explanation of the observed fact,
that a small error in the numerical value of K does not
produce errors of much importance in the calculation of
trajectories in practical cases. In the reduction of the
experiments and in the calculation of the Tables, the Arithmo-
meter of Thomas (de Colmar) has proved a most valuable
assistant.
140. In making systematic calculations of trajectories
printed forms were found of great assistance; and the recent
invention for producing manifold copies of writing renders it
easy for any one to prepare his own forms. Tables of five
figure logarithms are quite sufficient for these calculations,
while Tables of four figures may be used for many purposes;
and some proportions can be worked by the slide rule with
sufficient accuracy.
Supplementary Tables.
Tables printed 1877.
A. Continuation of Table VI. for values of y = 5.1 to y= 100.
B. Continuation of Table A from y = 100 to y=x.
C. Tables of values of K, K÷g, &c. (This Table, as well
as Table I. are now superseded by Table H.)
D. Table of values of P for every tenth of a degree from
0 to 21°.
u and
E. Table of values of (v versin 4) useful in calculating u from
a given value of v and p, or v from a given value of
without the use of logarithms, thus
u = v cos &=v-v verso....
v=u+v vers $=u+ (u+v verso) vers &
=u+u versp+(u vers o) verso nearly... (2).
and
Example (71)
u₁ =
(1),
- 1335 – 1335 vers 5° = 1335.0 — 4·95 — ·13
= 1335·0 — 5·08 — 1329·9.
Again
"₂ =
= 1329·9 + 1329.9 vers 5°
+(1329·9 vers5°) vers 5°
1329·9+ 5·06 +5.06 vers 5°
= 1329·9 + 5·06 + 0·02 = 1335·0.
ON THE MOTION OF PROJECTILES.
127
u₁= 1450-1450 vers 15°=1450 - 49′4=1400·6.
15
Again v 1400·6 + (1400·6) vers 15°
15
Example.
=
+(1400.6 vers 15°) vers 15°
1400·6 +47·7 +47·7 vers 15°
= 1400·6+47·7 + 1·6 = 1449·9.
F. Table of values of [1000÷v]³ for velocities 500 to 2000 f.s.
G. Table of heights in feet, through which a body would
descend when falling freely from rest under the action
of gravity in the latitude of London in the specified
times
Tables printed in 1881.
H. Tables of values of K, K÷g, log K÷g, and Σ (K,÷g)
for velocities 100 to 2800 f.s for ogival-headed projec-
tiles, to supersede Tables I. and C.
-
J. Table of values of R, = (1000) {[(v-1)-v³]g÷K₂_4},
where the variation in the density of the air may be
ď
ď²
taken into account by writing (1) for in
W
equations I. and II. (117) just as in (103).
W
K. Table of values of [1000÷v]³ in continuation of Table F.
L. General Table of values of T, for v100 to 2919 f.s
to supersede Table IX.
M. General Table of values of S, for v = 100 to 2919 f.s
to supersede Table VIII.
In Tables L and M the tabular numbers T, and S, are now
arranged to increase with v, for it is a convenience to have
Sv, ST, and SS, all of the same sign. t=T-T, and
d
W
d²
พ
s=S-S, where the velocity of the projectile falls from
v to v in the time t and space s (79).
-
N. Table of values of log (P- Pr-1), log (P¿ − P¿-2) and
log (P$ — P$-4)•
O. Table of values of log cos³p.
128
SUPPLEMENT TO A TREATISE
141. The Committee on Explosives appear to have
brought their labours to a conclusion. Two former members
of this Committee, Messrs. Abel and Noble, have contributed
to the Transactions of the Royal Society (1875) some
Researches on Fired Gunpowder, based on results said to
have been arrived at by the Committee. But I believe
that the details of the experiments from which these results
were obtained have not yet been published. On this point
the authors remark that they have not space within the
limits of their paper to enter upon a discussion of the
methods of calculation and correction necessary to arrive at
the results tabulated. Notwithstanding this missing link,
the Royal Society have considered the paper of such high
importance that they have honoured it with two royal
medals, one to each of its joint authors. When conferring
the second medal, the President of the Royal Society
referred to the beautiful invention of the Chronoscope,
"by which intervals of time as small as the one-millionth of
a second can be measured,"! (Nature, Dec. 9, 1880, p. 139).
A reference to the paper commended would have shewn
that its authors allowed that in reading a single record
there would be a probable error of 4 or 5 millionths of a
second, even when the instrument was in good working
order. But this high degree of accuracy could only be
attained by cutting all the primaries as nearly simultaneously
as possible, which is no test whatever of the accuracy of the
instrument when applied to any useful purpose, as I have
already shewn (9) to (19). The cutting plugs, the toothed
wheels, and the stop clock are all left out in that method.
of "testing," but they are all essential parts of the
Chronoscope. The only way to test the power of such an
instrument is to employ it in the actual experiments for
which it was designed, to apply the necessary corrections,
and to give the results of numerous trials. It is obvious
that the cutting plugs would fail grievously during the
initial motion of the shot, which is the most trying time
to the gun.
142. Messrs. Abel and Noble direct attention to two
consecutive rounds fired from the 10-inch gun. In both
cases the shot quitted the gun with the same velocity.
The chronoscopic records were nearly identical for the two
rounds. But the pressures indicated by the gauges differed
widely. The pressures on the plug A, were 634 and
28.0 tons per square inch; at B, 41·6 and 29·8; at C, 37′0
ON THE MOTION OF PROJECTILES.
129
and 30; at 1, 419 and 29.8; and at 4, 25.8 and 19.8
respectively. Although it may be readily admitted that
the pressure of the gas at any moment would be greater
against the fixed parts of the gun than against the moving
shot, it is difficult to understand how it was that the shot
did not show, in some more marked manner, the excessive
variation of the pressure inside the bore. But the obvious
conclusion is, that the pressure of the gas on the base of
a shot is no measure of the pressure it exerts on the bore
of the gun. Therefore, as the Chronoscope, at the most,
can only determine the motion of the shot, and through
that the pressure on the shot, it can be of no use in finding
the stress of fired gunpowder on the gun, except in simple
cases of no real importance.
143. Under these circumstances it might have been hoped
that this powder question, after an agitation of some fifteen
years, was at last satisfactorily settled. But Sir W. Armstrong
complains that his 100-ton gun had failed, although its
calculated strength at the point of fracture was far in excess
of what a legitimate pressure would demand. And on the
16th March, 1880, he declares that "Nothing, in fact, wants
investigation so much as this powder question, and until
we arrive at more definite conclusions, we cannot hope to
be altogether free from accident." (Proceedings of the
R. A. Institution, July, 1880, p. 197).
144. We may accept the doctrine that there is a
considerable variation in the pressure exerted by exploding
powder on the bore of a gun, according to the nature
of the powder used, or to the manner in which it is fired.
Sir W. Armstrong says that "with only 5 lbs. of this smaller
powder (R.L.G) mixed with 100 lbs. of pebble, the pressure
on the crusher gauges rose to 30 tons, the velocity of the
projectile remaining almost the same; so that the extra
12 tons was all wave action," (p. 196). Without entering
into speculations about "wave action" the simple fact seems
to be that the use of a small quantity of fine grained powder
causes the pebble powder to explode more rapidly at first,
and consequently the pressure rises much higher than when
pebble powder is used by itself. Anything which retards
the initial motion of the shot, or promotes the rapid explosion
of the charge, must commonly have the effect of increasing
the stress upon the gun.
130
SUPPLEMENT TO A TREATISE
145. Now it appears that, to be perfectly satisfactory,
guns should be constructed of strength sufficient to resist
not only normal but all abnormal pressures, except those which
arise from some such cause as double loading. The experi-
ments should be therefore simplified and confined to attempts
to measure the maximum pressures exerted by the various
powders in and about the powder chamber; and it seems
probable that pressure gauges, used with new precautions,
would suffice for this object, while the Chronoscope should
be rejected as a troublesome and useless encumbrance.
146. The copper cylinders used in the crusher gauges
should, in the first instance, be compressed up to at least
one-half of the pressure expected at the various points where
they were to be used. This would test their quality, and
save much useless work that otherwise would have to be
done by the powder. After the gun had been fired once
I would take out the coppers and carefully measure the
length of each, and then return them to their former places,
and fire the same charge and shot a second time, and
afterwards measure the lengths of the coppers as before.
This operation might be repeated till the coppers ceased to
contract in length. In order to determine the pressures to
which the coppers had been subjected, I would submit them
to actual pressures, beginning with something below what
their lengths seemed to indicate, and then test their lengths.
If they did not contract, I would go on augmenting the
pressure by equal increments till the copper plugs began to
shew signs of further compression. In this way very near
approximations to the maximum pressures in the bore of the
gun would probably be obtained. Afterwards the same
coppers might be used in the same plugs in experiments with
a mixture of a small quantity of more rapidly exploding
powder, or with charges altogether more violent. When a
crusher gauge is inserted in the base of a shot care should be
taken to arrange it so that the piston would not strike a blow
on the copper cylinder when the shot was brought suddenly
to rest.
147. As a matter of course the velocity of the shot should
be measured near the gun by some perfectly trustworthy
instrument. The Boulengé Chronoscope appears to be found
convenient in use, and when in good working order, it gives
satisfactory results. For during the progress of our recent
experiments the velocities of the shot were measured by the
ON THE MOTION OF PROJECTILES.
131
Boulengé instrument. For three comparisons in rounds 469,
470 and 472 the differences of the velocities given by the two
instruments were 0, 0, and +1. But in round 468 two com-
parisons were obtained with differences +28 and -61 f.s.
I am disposed to think that the working of the Boulengé
chronoscope would be made more uniform (1) if some kind of
a galvanometer was used, (2) in conjunction with the form of
screen shewn in fig. 20, which would secure galvanic circuits
of constant resistance. (Final Report, p. 8).
148. Of late, guns of large calibre have been provided
with a powder chamber of greater radius (R) than that of the
bore of the gun (r). It is evident that, for a given charge
and shot, this arrangement would give the shot a longer
portion of the bore to be propelled through, and that the
powder, being formed into a thicker and shorter cylinder,
would explode more rapidly. The consequences to be
expected would be an increased muzzle velocity and a greater
strain on the gun, even supposing that the gaseous pressure p,
denoted by the crusher gauge, was the same in a chambered
and in an unchambered gun. For the ratio of the pressure of
the gas tending to burst the powder chamber to that tending
to burst the bore of the gun would be as R : r. In a gun of
uniform bore there is little longitudinal tension on the steel
tube except that due to rifling and the friction of the shot
which is but trifling. But when a gun is chambered the
longitudinal tension on the sides of the powder chamber
becomes π (R) p, in addition to that which existed in the
unchambered gun. And the scooping out of the powder
chamber causes a great variation of thickness in the steel just at
that point where the forces acting upon it are likely to vary
so much.
149. It has been stated that the Italian 100-ton gun failed
because "the steel tube parted at the base of the conical
portion connecting the powder chamber with the bore," which
is just the place where the gun might have been expected
to fail, if it failed at all. In this case R = 9·85 and r= 8.86
inches. So that the ratio of the force tending to burst the
powder chamber to that tending to burst the bore would
be as 9.85 8.86, supposing the gaseous pressure p to be
the same in both places. And the increased longitudinal
tension on the powder chamber, entirely due to chambering
would be π (R-r²) p=58.2p=1164 tons, if p= 20 tons per
square inch, and so on in proportion to p. This is quite
G*
132
SUPPLEMENT TO A TREATISE, &c.
sufficient to explain the tearing asunder of the coils. The
chamber is reported to have been extended as if by an
extraordinary pressure.
150. In consequence of recent improvements in the
manufacture of steel shot, Krupp has succeeded in firing
an ogival-headed shot through 20 inches of iron plates,
the shot preserving its form, and retaining sufficient velocity
to carry it 2000 metres beyond the iron plates. It therefore
appears, that now experiments might be undertaken with
some prospect of success, to determine the laws which govern
the perforation of iron plates by steel shot with any given
form of head, provided only the velocity of the shot could
be measured satisfactorily just before and just after impact.
If the material of the shot and of the iron plates could be
considered uniform, there would remain as variables the
weight, diameter, and velocity of the shot, and the thickness
of the iron plates. Experiments on a small scale could be
made to yield valuable results.
151. The investigations of Trajectories, &c., contained in
this work, suppose that the shot moves in the vertical plane
passing through the direction of projection, and therefore
they apply strictly only to the motion of spherical shot.
One effect of the resistance of the air on rotating elongated
projectiles is to draw them slightly from the vertical plane
of projection as explained already (73), and also to affect, in
some measure, their ranges, &c. I must refer the reader
for information on these points to Saint-Robert, Mémoires
Scientifiques. Balistique, p. 173; to Mayevski, Traité de Balis-
tique, p. 154; and to several papers by Professor Greenhill
in the Proceedings of the Royal Artillery Institution, Woolwich,
vol. x, p. 577, and vol. XI, pp. 119, 124, and 131.
ERRATA.
Page 102, seventh line from bottom, for "K," read K, and page 122, eleventh
line, for "were" read where. Also in Table (J), opposite velocity 1031 f.s, for
"2238" read 2538; and opposite velocity 2016 f.s, for "4546" read 5546. Also in
Final Report, p. 59, opposite velocity 505 f's, for "3.6608 read 3.6408.
March, 1881.
(A) VALUES OF X Y & T.

Y
5.I
y = 5.2
X
Y
T
Ф
X
Y
T
0
0
3.6
3.5
.13106 .005395
.11783 .004577
.08719 0.4 .00674
.000023
.00686
3.4
.10805 .003987
3.3
.10008
.003520
3.2
.09325
.003132
.08239 0.6 .00995
.07825 0.8
.07452 I .01607
.07106
.00005 I
.01021
.01306
.000089
.01350
.000137
.01675
3.I
.08723
,002801
.06781
322 2
3.0
.08181 .002512
.06473
2.9 .07687 .002258
.06179
2.8
.07231 .002030 .05896
2.7
.06806
.001826 .05624
2.6
.06408 .001642 .05360
2.5
.06033 .001475
.05104
2 co ti N∞ a
.03000
.000499
.03234
3
.04238 .001037
.04705
4
.05359 .001722
.06107
.06389 .002531
.0745I
.07344
.003449
.08748
7
8
.08237
.09079 .005574 .11228
.004466 .10005
.09876
.006765 .12420
2.4
.05677
.001323
.04855 ΙΟ
.10635 .008035 .13587
2.3
.05339
.001184
.04611
I I
.11361
.009380
.14732
2.2
.05016
.001057
.04374
12
.12057
.010796
.15857
13
.12728
.012282
.16965
γ 5.2
14
•
13375
.013836
.18058
15
.14002 .015456
.19138
Ф
X
Y
T
16
.14610
.017142
.20207
17
18
78
.15201 .018893
.21267
17777 .020708
.22318
0
3.6
.14255
.006067
.08961 19
.16339 .022589
.23362
3.5
.12385
.004906
.08392 20
.16889
.024536
.24402
3.4 .I1208
.004195
3.3
.10304
.003666
.07939 21
.07541 22 .17955
.17427
.026548
.25436
.028628
.26468
3.2
.09554
.003239
N NW w
3.1
.08905
.002882
.07179 23
.06842
24
3.0
.08329
2.9 .07809
.002576 .06524
.002307
.06223
2 2
25
26
2.8
2.7
2.6
2.5
.06892
.06480
.06094
ི་
30
2.4
2.3
2.0
1.4
1.2
.05729 .001339
.05383 .001197
2.2 .05054 .001068
.04438 .000842 .03926 34 .23753
1.8 .03869 .000653
.03480 35 .24209 .062556
1.6 .03340 .000496 .03050
.02844 .000366 .02634
.02376 .000260
1.0 .01934 .000175
.02229
.01836
www www
.07332 .002070 .05934 27 .20469 .040078 .31617
.001858 .05656 28
.001668 .05388 29 .21429 .045182
.001496 .05128
.21902 .047855
.04875 3I .22370 .050613
.04630 32 .22835 .053459 .36839
.04390 33 .23295 .056395 .37902
+367
.035281 .29555
.19981 .037642 .30585
.18474 .030776 .27498
.18984 .032993 .28527
.19486
.20952 .042590
.32652
•33690
.34734
.35783
.059426 .38974
36
.24662
.065787
.40056
.41148
.25113
.069125 .42252
38
.25562
.072575
.43368
39
.260II
.076141
.44499
0.8
.01512 .000108
.01453
40
.26458
.079829
.45644
+.2
0.6 .OIIIO .000059
0.4 .00725 .000026
.00356 .000006
.01078
41 .26905
.083645 .46805
.00352
43
.0
--.2
.00000 .000000
.00343
.00000 44
.000006 .00346
45
ABBR.
.00712 42 .2735I
.087595 .47984
.27798
.091684 .49181
.28245
.095922 .50398
.28692
.100315 .51636
1
I

ge
y = 5.4
5.3
Ф
X
Y
T
Q
X
Y
T
0
0
लंलंग लंलं लं तं तं
3.4
.11705
.004456
.08071 0.4
.00673
.000023
.00686
3.3
3.2
3.I
321
.10647
.003836
.07641 0.6
.00993
.00005 I
.01020
.09810
.003360
.07258 0.8
.01303
.000089
.01348
.09105 .002972
.06907
3.0
.08489
.002644
.06578
2.9
.07939 .002361
.06268
1 2 3
I
.01603
.000136
.01672
.02984 .000495
.03226
.04210 .001028
.04689
2.8
.07440
.002112
.05973
2 2 2
2.7
.06982 .001892
.05690
2.6
.06556 .001695
.05417
2.5
.06158 .001517
.05153
2.4
.05784 .001357
.04897
2.3
.05429 .001212
.04648
456
789
.05317
.001704
.06082
.06333
.002502
.07417
.07273
.003407
.08704
.08153 .004408
.09951
.08980
.005496
.11163
.09763
.006667
.12346
2.2
.05093 .001079
.04406 10
.10509
.007913
.13502
2.I
.04773 .000959
.04169 II
.I1221
.009233
.14636
2.0
.04466 .000849
.03938 12
.11904
.010623
.15750
1.9
.04172
.000749
.03711
13
343
.12562
.012080
.16847
.13196 .013602
.17929
.13810 .015190
.18999
y = 5.4
16
.14406 .016841
.20057
17
.14985
.018556
.21105
18
.15548
.020334
.22145
Ф
X
Y
T
19
.16099
.022175
.23179
20
.16637 .024080
.24207
0
21
3.4
3.3
.12361
.11060
.004806
.08231
.17164 .026050
.25231
.004044
3.2
.10104 .003501
.07754 22
.07345 23
.17680
.028084
.26251
.18188
.030185
.27269
3.I
.09327
.003073
.06977 24 .18686
.032354
.28287
3.0
.08664
.002719 .06636
2.9
.08080
.002418 .06317
2.8
.07555 .002157
.06014
567
.19177
.034592
.29304
.19661
.036901
.30323
.20139
.039282
•31343
2.6
2.5
2.7 .07077 .001927
.06635 .001723
.05446 29
.06225 .001540 .05178 30
.05724 28 .206II
.041738
.32366
.21077 .044271
.33393
2.4 .05840
.001375 .04919
2.3
2.2
G
2.0
.05477 .001226 .04667 32 .22451
.05134 .001ogi
.04495 .000857 .03950
1.8 .03911 .000663 .03497
.04422 33
1.6 .03369
.000502 .03063
1.4 .02864
.000370 .02643
1.2
.02390 .000262 .02236
1.0
.01942 .000176 .01840 39
www wwww wwww.
.21539
.046884
.34424
.21997
.049580
-35461
.052360 .36505
34 .23348
.22901 .055230
.058191
-37556
.38616
35
.23793
.061248 .39685
36
.24236
.064405 .40765
37
.24676
.067666 .41856
38
.25115 .071035 .42959
.25553
.074518 .44076
0.8 .01517 .000109
0.6 .01113 .000060
0.4 .00726 .000026
.01455 40
.25990
.078120 .45208
.01079 41
.26426
.081846 .46355
.00712 42
.26862
.085703 .47520
+.2
.00356 .000006
1
.2
.0 .00000 .000000
.00343
.000006
.00353 43
.089696
.00000 44 .27734 .093833
.00346 45 .28170 .098122
.27298
.48703
.49905
.51129
2

y =
5.6
5.5
Y
Ф
X
Y
T
Ф
X
Y
T
0
3.3
3.2
0
3 W
.11576 .004308
.07887 0.4
.00673
.000023
.00685
3.I
.10447
.09577 .003187
.003666
.07443 0.6
.00991 .00005 I
.01019
.07053 0.8
.01300
880000*
.01347
3.0
.08856.002803
.06698
2.9
.08232 .002481
.06367
2.8
.07678 .002205
.06056
123
.01598 .000135 .01670
.02971 .000492 .03218
.04184 .001020
2.7
.07177 .001965
.05760
2.6
.06719 .001753
.05477
2.5
.06294
.001563
.05205
456
.04674
.05278 .001687
.06058
.06279 .002474 .07384
.07206 .003366
.08662
2.4
.05898
.001394
.04941
2.3
.05527 .001241
.04686
2.2 .05175
.001103
.04439
78 9
.08071
.08885
.00435I
.09899
.005421
.IIIOI
.09655
.006572 .12273
2.I
.04842
.000978
.04197 10
.10387
.007796
.13419
2.0 .04525 .000865
1.9 .04222
.03962 II
.11087
.009092
.14542
.000762
.03732 12
.11757
.010456
.15646
13
5.6
14
15
16
op
X
Y
T
3 4 5 6 7
.I2402
.011885
.16733
.13025
.13627
.013379 .17805
.014935 .18864
.14210
.016554
.19911
.14778
.018234 .20949
18
.15330
.019976 .21979
www.
3.3
.12276
.004673
.0805 I
19
.15870 .021780
.23002
3.2
.10860
.003869
.07555
20
.16396 .023646
.24019
3.I
.09863 .003320
.07137 21
.16912 .025574
.25032
ôહું હૂં
3.0
.09069 .002896
.06765
22
.17418
.027567
.26042
2.9
2.8
.08397 .002550
.07810 .002258
.06422 23
.06101 24
.17915 .029623
.27049
.18403
.031746
.28056
2.7
.07285 .002005
.05798 25
.18883 .033936
.29062
2.6
.06807
2.5
.06367
.001784
.001588
.05509
26
.19357 .036195
.30070
.05232 27
.19824
.038525
.31079
2.2
2.3 .05578
.05218
2.4 .05959 .001414
.001257
.04965 28
.20286
.040928
.32091
.04706 29
.20742
.043406
.33107
911100*
.04456
30
.21194 .045962
.34127
2.I
1.9
1.8
.03953
1.7
1.6
.04879 .000988
2.0 .04556 .000873
.04248 .000768
.000673
.03671 .000586 .03294
..03399 .000508
1.4 .02885
.04212 31
.21642 .048598
.35152
.03975 32
.22086 .051318
.36185
.03743 33
.03516
.03076 36
333
34
.22526
.22963 .057019
.054123
.37224
.38272
35
.23398 .060008 .39329
.23831
.063094
.40397
.000373
.02652 37 .24262
.066282
.41475
1.0
1.2 .02404 .000264
.01951 .000177
0.8
'01522 .000110
0.6 .01115 .000060
0.4 .00727 .000026
+.2 .00356 .000006
.02242 38 .24691
.069576
.42566
.01844 39
.25119
.072980
.4367I
.01458 40
.01081 41
.00713 42
.25546
.076500 .44790
.25972 .080142
.26398 .083911
.45924
.47076
.00000 .000000
.00343 .000006
.00353 43 .26824
.00000 44 .27250
.00346 45 .27677
.087813 .48245
.091856 .49434
.096047 .50643
3
I

y = 5.7
5.8
γ
Ф
X
Y
T
Ф
X
Y
T
0
0
3.2 .11382 .004130
.07687 0.4 .00672 .000023
.00685
3.I
.10198
.003477
.07232 0.6
.00989
.000051
.01018
3.0
2.9
2.8
0 000
.09308
.003003
.06837
0.8 .01296
.000088
.01345
.08579
.002627
.06480
.07952
.002315
.06149
1 2 3
.01 594
.000135
.01667
.02956
.000489
.03210
2.7
2.6
2.5
765
.07399
.002049 .05838
.04157
.001012 .04659
.06900 .001818 .05543
.06444 .001615
.05260
456
.05238
.001671 .06035
.06226
.002448 .07352
2.4
.06023 .001434
.04989
.07140
.003326
.0862I
2.3
.05631 .001273
.04727
2.2
.05263
178
.001129
.04473
789
.07992
.004296 .09848
.08793
.005350 .I1040
2.I
.04916
.09550
.006481
.12202
.000999
.04227
2.0
1.9
.04587 .000881
.04274
.03988 ΙΟ .10269
.007684
.13339
.000774
.03754 II
.10956
.008957
.14452
12
.11615
.010296
.15546
13
.11248
.011699
.16623
Y
5.8
14
.12858 .013165
.17685
15
.13449 .014692
.18733
16
.14022
.016279
.19771
X
Y
T
17
.14578
.017927
.20798
18
.15120
.019635
.21818
19
.15648
ww.
.021403
.22831
3.2
3.1
2 -
.12111
.10602
.004501
.07852
20
.16164
.023231
.23838
.003670 .07340
21
.16670
.025121
.24841
3.0
.09582
.003126
.06918
22
.17165
.027073
.25840
2.9 .08780 .002712
2.8 .08107 .002377 .06200 24
.06543 23 .17652 .029088
.26837
.18130
.031167
.27833
2.7
2 2
2.6
.07521 .002096 .05880 25
.06998 .001854 .05578
2.5
.06525 .001643 .05290
56
2 2 2
.18601
.033312
.28829
26
.19064 .035524 .29826
27
.19522 .037805
.30825
2.4
.06089 .001456 .05014 28 .19974 .040158
.31826
2.3 .05686
2.2 .05309
.001291 .04748 29 .20421
.001143
2.I
.04955 .001009 .04243
2.0 .04620
1.9 .04302
.000781 .03765
.03535
1.8
.000683
.03999
1.7 .03709 .000595 .03310
.03431 .000515 .03089
1.6
1.4 .02907 .000377 .02662
.02418 .000266 .02249
.01960
1.2
1.0
0.8 .01528
0.6 .01118
.000178
.000889 .04001
.01849
www www www ww
.042584
.32831
.04492 30 .20863
.045085
•33840
.21301
.047665
.34855
.21735
.050327
.35876
.22166
.053073
.36904
.22595
.055906 .37941
.23020
.058831 .38987
.23443
.061850
.40043
.23865
.064969
.41109
.24285
.068191
.42189
.24703 .071521
.43281
OI 1000'
,000060
0.4 .00729
.000026
.01460
.01082 41
.00713 42
+.2
.00356
.000006
.00000
·2
.00342
,000000
.000006
40 .25121
.25538
.25955
.00353 43 .26371 .086029 .47804
.00000 44 .26788 .089982 .48980
.00346 45
.50176
.074964 .44387
.078526
.45509
.082212
.46648
.27205 .094080
4

y = 5.9
Y
y = 6.0
Ф
X
Y
T
ዎ
X
Y
T
0
0
3.0 .09901
.003272
.07007 1.0
.01589
.000134
.01665
2.9
.09005
.002809
.06611 1.2
.01876
.000189 .01981
2.8
.08275
.002446
.06254 1.4
.02154
.000252 .02292
2.7
2.6
2.5
765
.07652
.002146
.05924 1.6
.02424
.000323 .02599
.07103 .001892
.05614
.06609
.001672
.05320
23
.02942
.000486 .03202
.04132
.001003 .04644
2.4
.06159
.001479
.05040 4
2.3
.05744 .001308
2.2
.05357
.04770
.001156 .04510
456
.05200
.001655 .06012
.06175
.002422 .07321
.07076
.003288 .08580
2.1
.04995
.001020 .04259
2.0
.04653 .000898
.04014
1.9 .04329
.000788
.03776
789
.07915
.004243
.09798
.08703
.005280
.10981
.09448
.006393
.12134
1.8
.04022
.000688
.03545 ΙΟ .10156
.007576
.13261
1.7
.03728
.000599
.03318 II
.10831
.008827
.14365
1.6 .03447
.000518
.03096 12
.11478
.010142
.I 5449
13
.12099
.011520
.16516
γ
6.0
14
.12699
012959
.17568
15
.13278
.014458
.18607
Ф
X
Y
T
16
.13840
.016016 .19634
17
.14386
.017633
.20652
18
.14918
.019308
.21662
0
3.0
.10285
.003448
.07109 19
.15436 .021042
.22665
2.8
2.9 .09260 .002920
.08461
.06686 20
.15942 .022835
.23662
.002522
.06312 21
.16438 | .024687
.24655
2.7
2.6
765
.07794
.07214
.002201
.001933
.05971 22
.16924 .026601
.25645
.05653 23 .17400
.028576
.26632
2.4
2.3
32
2.5 .06699 .001703 .05352 24
2.2
.06232 .001503 .05066
.05803
.001327
.04793
.05406 .001171 .04529
2.I .05035 .001032
2.0 .04687 .000907
1.9 .04358 .000795
.17869 .030613
.27618
2 2 2
25
26
567
.18330
.032715
.28604
.18785 .034883
.29591
27 .19233 .037118 .30579
.31570
.04275 28 .19676 .039422
.04028 29 .20113
.03788 30 .20546
.041799 .32565
.044249 •33565
1.8
.04045 .000694 .03554
1.7
.03748 .000603 .03326
1.6
.03463 .000521 .03103 33 .21823
www
.20975
.046776
•34568
.21401
.049383
.35579
.052072
.36596
1.0
+.2
900000*
.000000
.00000
1
.00346
.00684 43
0.6
1.4 .02929 .000381 .02671 34
1.2 .02433 .000268
.01969 .000179
0.8 .01533 .000IIO
0.6 .01121 .000060 .01083
0.4 .00730 .000026 .00714 39
.00357
,00000
.2 .00342 .000006
0.4 .00671 .000023
.00988 .00005 I
0.8 .01294
.000088
.01017 44 .26347 .088201 .48543
.01344 45 .26755 .092211 .49726
.22242
.054846
.37622
333
333
38
.02255 35 .22658
.01853 36 .23073
.01463 37 .23486 .063719
.23897 .066873
.057710 .38657
.060666
.39701
.40757
.41825
.24307 .070133
.42905
.00353 40 .24716
.073503
.44000
4I
.25124
.076990 .45110
7744
42
.25531
.080597 .46236
.25939
.084332 .47380
5

y = 6.1
y = 6.2
ዋ
X
Y
T
X
Y
T
0
0
3.0 .10774
.003678
.07227
1.0
.01585
.000134
.01663
~ 2
2 2
2.9
.09556
.003050
.06769 1.2
.01870
.000189 .01978
2.8
.08668
.002607
.06375 I.4
.02146
.000252 .02288
2.7
.07948
.002261
.06021 1.6
.02414 .000322
.02594
2.6
.07334
.001977
.05693
2.5
.06793
.001736 .05385
2 3
.02928
.000483
.03194
.04106
.000995
.04629
2.4
.06308
.001528 .05094
2.3
.05865
2.2
.05457
981100*
.001346 .04816
.04549
456
.05163
.001640 .05990
.06125
.002396 .07290
.07014 .003251 .08540
2.I
.05077
.001044 .04291
2.0
.04722
.000916
.04042
78
.07840
.004192 .09750
.08616
.005213
.10923
1.9
.04387
.000802
.03800 9
.09349
.006308
.12067
1.8
.04069
.000699
.03564
ΙΟ .10045
.007472
.13184
1.7
.03768
.000607
.03335
II
.10709
.008702
.14279
1.6
.03480
.000524
.03110
12
.11345
.009995
.15354
13
.11956
.011349
.16411
14
.12545
.012762
.17454
γ
= 6.2
15
.13114
.014234
.18484
16
17
g
X
Y
T
18
678
.13666 .015764
.19502
.14202 .017351
.20510
.14723
.018995
.21511
19
0
.15232
.020696
.22504
2.9
.09911
.003208
.06864
20
.15728
.022456
.23492
2.8
.08901
.002705
.06444
21
.16215
.024273
.24476
2.7
.08117
.002328
.06074 22
.16691
.026150
.25456
2.6
.07462
.002025
.05735 23 .17159 .028087
.26434
2.5
.06893
.001771
.05420 24
.17618
.030085
.27410
2.4
.06388
2.1
2.3 .05930
2.2 .05510
.05121
.001202
.001555 .05123 25
.001367
26
.04840
.04569 27 .18956
~~
.18071
.032146
.28386
.18516
.034271 .29363
.036462 .30342
1.9
2.0 .04758
.04416
.001056 .04308
.000926 .04056 29 .19819
.000810 .03812 30 .20243
28
.19390
.038721 .31323
ннн
1.8
.04094
.000705 .03574
1.7
.03788
1,6 .03497
,000612 .03343
.000528 .03118 33
wwwww
.041051 .32308
.043452
.33297
.20663
.045929
.34291
.21080
32
.048483
.35291
.21494
.051117
.36299
1.4
1.2
.0295I
.02447
1.0 .01978
.000385 .02681 34
.21905
.053836
.37314
.000270
,000180 .01857
0.8
.01538
,000III .01465
0.6
.01124
.000060 .01085
0.4
.00731
.000026 .00714
www ww
.02262 35 .23313
.056642
.38338
36
.22719
.059538
.39372
.23123
.062529
.40417
.23526
.065618
.41474
.23927
.068811
.42543
+.2
.00357
.000006
.00353 40
.24328
.072113
.43627
.00000
2 .00342
.000000
.00000 4I
.24728
.075527
.44725
.000006
.00345 42
.25127
.079060
.45840
0.4
.00670
.000023
.00684
43 .25526
.082718
.46972
0.6 .00986
.00005 I
.01016
44 .25926
.086507
.48123
0.8
.01290
.000088
.01342
45 .26325
.090433
.49294
6

γ
6.4
6.3
X
Y
T Ф
X
Y
T
0
2.9
.10358 .003412
.06975 1.6
.02404
.000320
.02589
2.8
.09171 .002820
.06520 1.8
.02663
.000396
.02890
2.0
.02915
2.7
.08304
.002403
.06131
.000480 .03187
2.6
.07601
.002077
.05780 2.2
.03160 .000570
.03480
2.5 .07001
.001810
.05457 2.4
.03398 .000666
.03769
2.4
.06473
.001583
3.0
.04082
.000987
.04615
.05153
2.3
.05998
.001389
.04865
2.2
.05565 .001219
.04589
2.I
.05166
.001069 .04325
2.0
.04795
.000936
.04071
1.9 .04447
.000817
.03824
456
78 9
.05126
.001625 .05967
.06077
.002372 .07259
.06953
.003214
.08502
.07768
.004142 .09702
.08532 .005148
.09254
.10867
.006226 .I2002
1.8
.04119
.0007II
.03585
1.7
.03809 .000616
.03352
ΙΟ
.09939 .007371
.13110
1.6
.03514 .000532
.03125
II
.10592
.008581
.14196
12 .11217
.009852
.15262
Y
6.4
13
.11817
.011183
.16310
14
.12396 .012572
.17344
15
.12955 .014018 .18364
ф
X
Y
T
16
.13497
.015521
.19374
17
.14023
.017080
.20373
0
2.8
.09490 .002957
.06606
18
.14535
.018694
.21364
2.7
.08515 .002488
.06193 19
.15035
.020365
.22349
2.6
.07753 .002135
.05828 20 .15523
.022092
.23327
2.5
.07116 .001851
.05495 21 .16000
.023876
.24302
2.4
.06563 .001614
.05184 22 .16467
.025717
.25272
2.3
.06070 .001411
.04890 23
.16926
.027618
.2624I
2.2
.05623 .001236
.04611 24
.17377
.029579
.27209
2.I
.05213
.001082 .04343 25 .17821
.031600
.28176
2.0
.04834 .000946
.04086 26
.18258
.033685
.29143
1.9
.04479
.000825
.03837 27
.18689
.035835
.30113
1.8
.04145 .000717
.03595 28
.19115
.038050
.31084
1.7
.03831
.000621
.03361 29
.19536
.040335
.32059
1.6
.03532 .000535
.03132 30 .19952
.042690
.33039
1.4 .02974
1.2
.000389
.02463 .000273 .02268
.02691
1,0
.01988
0.8
.000181 .01861 33
.01544 .000112 .01468
0.6 .01127 .000061 .01086
0.4
+.2
.00732 .000026 .00715
.00357 .000006 .00353
.0
-.2
.00000 .000000
.00342
.00000
.000006
.00345
www www www.
.20364 .045118
.34023
32
.20773 .047623
.35014
.21178
.050206
.36011
.21581 .052871
.37016
.21981
.055622 .38031
.22379 .058461
.39054
.22775
.061392
.40089
.23170 .064421
.41135
.23564
.067551 .42194
0.4
.00669
.000023
.00684
40 .23956
.070786
.43267
0.6 .00984
.00005 I
.01015
41 .24348 .074132
.44354
0.8
.01287
.000088
.01341 42 .24739
.077595
•45458
0.0
.01580
.000134
.01660 43 .25130
081179
.46579
0.2
.01863 .000189
0.4
.02137 .00025I
.01975 44 .25521
.02284 45 .25913
.084892 .47718
.088740 .48877
7

y = 6.5
6.6
γ
Ր
S
X
Y
T
Ф
X
Y
T
0
0
2.8
.09884 .003126
2.7
.08755
.002585
2 2
2 2 2
3 2
2.6
.07920
.002199
.06705 1.6
.02394
.06262 1.8 .02651
.05880 2,0
.000318
.000394 .02884
.02584
.02901
.000477 .03179
2.5
.07240
.001895
.05535 2.2
.03144
.000566
.03470
2.4
2.3
.06658
.001646
.05216 2.4
.03380
.000661 .03757
.06145 .001435
.04917 3.0
.04057
.000980
.04600
2.2
.05683
.001254
.04633
2.I
.05262 .001096
.04362
2.0
.04873 .000957 .04101
1.9
.045II
.000833
.03850
1.8
.04172
.000724
.03606
1.7
.03852
.000626
.03370
456
78 a
.05090
.001610 .05945
.06029
.002348
.07229
.06893
.003179
.08463
.07697
.004094
.09655
.08450
.005085
.10812
9
.09161
.006146 .11938
1.6
.03549
.000539
.03140
IO
.09835
.007274
.13037
II
.10477
.008464
.14114
12
.I1092
.009715
.15171
6.6
13
.11683
.011024
.16211
14
.12252 .012390
.17236
15
.12802
.013811
.18248
Փ
X
Y
T
16
.13334
.015288
.19248
17
.13851
.016819
.20239
0
18
2.6
2.5
2.8
.10393 .003344
2.7 .09033 .002700
.08105 .002270
.07373
.06825
.14354
.018406
.21222
.06338 19
.14845
.020046
.22197
.05936 20 .15324 .021743
.23167
.001944
.05578 21
.15792
.023494 .24133
2.4 .06759
.001681
.05251 22
.16251
.025303 .35095
2.3
.06223
.001461
.04945 23
.16702
.027169
.26055
2.2
.05746 .001273
.04656 24
.17145
.029094
.27013
2.I
1.9
1.8
1.7
.03875
1.6
.03568
.05312 .00IIIO
2.0 .04914 .000968
.04544
.000842
.04199 .000730 .03617 28
.00063I .03379 29 .19263
.000543 .03147 30
.04380 25
.17580
.031079
.27972
.04117 26 18009
.033125
.28930
.03863 27
.18432
.035234
.29890
.18850 .037409 .30853
.039650 .31819
1.4
.02998
.000393 .02702
1.2 .02478 .000275 .02275
www
.19671
.041961 .32789
1.0 .01997
0.8
.01549
0.6 .01129
.000183 .01866
333
23
.20076
.044344
.33764
32
33
.20477
.046801 .34745
.20875
.049335 .35733
.000112
0.4 .00733
.000026 .00715
+.2
.00357
.000006 .00353
о
.00000 .000000
-.2 .00341
0.4 .00668 .000023
.00000
.000006 .00345
www wwww
35
.01470 34 .21270 .051950 .36729
.000061 .01087
.054648 .37733
.21662
36
.22053
.057433 .38747
.22441
.060308
.39772
.22828
.063278
.40808
.23214
.066348
.41857
.00683 40 .23599 .069520
.42919
0.8
0.6 .00982 .000050
.01285 .000087
.01014 4I .23983 .072802
.43996
1.0
1.2
.01576 .000133
.01858 .000187
1.4 .02130 .000249 .02280 45 .25518
343
.01658 43 .24751 .079712
.01972 44 .25134
.01339 42 .24367 .076197 .45088
.083352
.46198
.47326
.087125 .48474
8

7 = 6.7
y = 6.8
Փ
X
Y
T
Ф
X
Y
T
0
0
2.7
.09365
.002839
.06425 2.2
2.6
.08313
.002351
.05997 2.4
.03129
.03363
.000562
.03461
.000657 .03747
2.5
.07519 .001997
.05624 2.6
.03592
.000756 .04030
2 2
2.4
.06867 .001718
.05287 2.8
.03815
.000861
.04310
2.3
.06307
.001488
.04974 3.0
.04033
.000972
.04586
2.2
.05811
I
.001293
.04679 3.2
.04246
.001088 .04859
2.1
.05365
.001125
.04400
2.0
.04956
.000979
.04133
1.9
.04579 .000850
.03876
1.8
.04227 .000737
.03628
1.7
.03897 .000636
.03388
1.6 .03586
.000546
.03155
456
789
.05054
.001596
.05924
.05982 .002324 .07200
.06835
.003145
.08426
.07627
.004047 .09610
.08370
.005024
.10758
.09070
.006069
.11875
ΙΟ
.09733
.007179
.12967
6.8
II
.10366
.00835I
.14035
12
.10971
.009582
.15084
Փ
X
Y
T
13
14
15
34 in
.11552
.010870
.16115
.12112 .012213 .17131
.12652 .013611 .18135
0
2.7
2.6
2.5
765
.09787 .003017
.06528
16
.07680
.08552 .002445 .06064 17
.002056
.05674 18
678
.13176
.015063 .19127
.13684
.016568 .20109
.14179
.018127 .21083
2.4 .06984 .001758
.05325 19
.14661
.019740 .22050
2.3
.06395 .001516
.05004 20
.15131
.021406 .23012
2.2
.05880 .001314
.04704 21
.15592
.023127 .23969
2.I
.05419 .001141
.04420 22
.16043
.024904 .24923
2.0
.05000 .000991
.04150 23
.16485 .026737
.25874
1.9
.04614 .000859
.03890 24
.16920
.028627
.26824
1.8
.04256 .000744
.03640 25
.17348
.030577
.27773
1.7 .03921 .000641
.03398 26
17769
.032586
.28724
1.6
.03605 .000550
.03163 27
.18185
.034657
.29675
1.4
1,0
.03023 .000398
1.2 .02494
.02007
.02712 28
.18595
.036793
.30629
.000277
.02282 29 .19000 .038993
.31586
.000184
.01870 30 .19401 .041262
-32547
0.8
.01555
.000113
.01473 31
0.6
.01132
.00006 I
.01089
32
0.4 .00734
.000026
.00716
33
1 2 3
.19798 .043601 -33513
.20192
.046013 .34485
.20582
.048500 •35464
+.2
.00358 .000006
.00353 34
.0
.00000 .000000
.00000
―.2
.00341 .000006 .00345
www.co
.20970
.051067
.3645I
35
.21355
.053714
.37446
36
.21738
.056447
.38450
0.4
0.8
.00667 .000023 .00683
0.6 .00980 .000050 .01013
.01281 .000087 .01337
www
.22120
.059269
•39465
.22500
.22878
.062184
.40491
.065196 .415.30
1.2
I.4
1.0 .01571 .000132 .01656 40
.01851 .000186 .01969 4I
,02122 .000248 .02277
☆☆☆
.23256 .068309
.42582
.23633
.071528
.43649
42
.24010
.074860 .44731
1.6
1.8
2.0
.000474
.02385 .000317 .02580 43
.02640 .000392 .02878
.02888
+++
.24386
.078308
.45830
44 .24762 .081879
.46948
.03171 45 .25139
.085580 .48085
9
Y
= 7.0
6.9
γ
Φ
X
Y
T
Ф
X
Y
T
0
2.6
.08832
.002557
.06138 2.2
.03114 .000559
.03453
2.5 .07858
.002123
.05727 2.4
.03347 .000653
.03738
2.6
2.4 .07110
.001802
હૂં હું
2.3
.06489 .001547
.05365
.05036 2.8
.03574 .000752
.04019
.03795
.000856
.04297
2.2
.05952
.001336
.04730 3.0
.04010
.000965
..04572
2.I
.05476
3.2
.04221
.001079
.04844
.001157
.0444I
2.0
.05045
.001003
.04167
1.9
.04650
.000869
.03904
1.8
.04285
.000750
.03651
1.7
.03944 .000646
.03407
1.6
.03624
.000554
.03171
456
78 9
.05020
.001581
.05903
.05937 .002301 .07171
.06779
.003111 .08389
.07561
.004001
.09565
.08293
.004964
.10705
.08983
.005995
.11814
ΙΟ
.09637
.007089
.12898
I I .10260
.008243
.13958
γ
y = 7.0
12
.10855
.009454
.14998
13
.11427
.010722
.16022
P
X
Y
T
14
.11978
.012044 .17030
15
.12510
.013419
.18025
16
0
.13025
.014847 .19009
2.6 .09169
.002693
.06224
17
2.5
.08058
.002198
.05784
18
78
.13525
.016328 .19983
.14011
.017860 .20949
2.4
.07247 .001850
.05408 19
2.3
.06589
.001580
2.2
.06028 .001360
2.1
1.9
1.8
.04315
1.7
1.6
.05535 .001174
2.0 .05091
.001016
.04687 .000878
.000758
.03663 25
.03968
.000652 .03417 26
.03644 .000558 .03179
.14485
.05069 20
.14948 .021084
.04756 21
.15400 .022776
.04462 22 15844
.04184 23 .16279
.03918 24 .16706
.019446
.21908
.22862
.23811
.024522 .24756
.026324
.25699
.028181
.26641
1.4 .03048
1.0
.000402
1.2 .02510
.000280
.02017 .000185
.02723 28
.02289
.01875
0.8 .01560 .000113 .01475
0.6 .01135
.000061
.01090
0.4 .00735
+.2 .00358
.000026 .00716
.000006
.00354 34
-.2
0.4
0.6 .00979
.00000
.00341
.000000
.00000
.000006 .00345
.00667
.000023 .00682
37
0.8
.01279
.000050
.000087 .01336 39
.01012 38
www www www w☺☺ NUN
.17126 .030097
.17540
.17949
.27582
.032071 .28524
.034106
.29467
.18352 .036203
.30412
29
.18750
.038365
.31361
30
.19144
.040594
.32314
.19534
.042891
.33271
.19920
.045260
.34234
.20304
.047702 .35204
.20684
.050222
.36182
35
.21063
.052822 .37168
36 .21439
.055506
.38163
.21813
.058277
.39169
.22186
.061138
.40186
.22558
.064095
.41215
1.0
.01567 .000132 .01653 40 .22929
.067151
.42258
2.0
1.2 .01845 .000186
1.4 .02114 .000247
1.6 .02375 .000315
1.8 .02629 .000390
.02875 .000471
.01965 41
.23299
.070312
.43315
.02272 42
.23669
.073582
.44387
.02574 43 .24038
.02871 44 .24407 .080473
.03164 45 .24777 .084105
.076967 .45476
.46583
.47709
10

y =
7.2
7.I
y =
Ф
X
Y
T
X
Y
T
0
2.5
.08288
.002285
.05848
2.2
.03099
.000556
.03445
2.4
.07397
.001904
.05454 2.4
.03329
.000649
.03728
2.3
.06696
.001616
.05104 2.6
.03554
.000747 .04008
2.2
.06109
.001385
.04784 2.8
.03773 .000850
.04285
2.I
.05597
.001193
.04484 3.0
.03987
.000958 .04559
2.0
.05140
.001029
.04202 3.2
.04196
.001071
.04830
1.9
.04726
.000888
.03933
1.8
.04346
.000765
.03675
1.7 .03993 .000657
.03427
1.6
.03664 .000562
.03187
1.5
.03354 .000479
.02955
1.4
.03061 .000404 .02728
456
789
.04987
.001568
.05882
.05892 .002280
.07143
.06724
.003079
.08353
.07496
.003958
.09521
.08218
.004907
.10653
.08898
.005923
.11755
1.3
.02783
.000339
.02508
ΙΟ
.09542
.007001
.12830
II
.10156
.008138
.13883
12
.10742
.009331
.14915
y = 7.2
13
.11305
.010579
.15931
14
.11847
.011880
.16931
15
.12371 .013234
.17918
Ф
X
Y
T
16
.12877 .014639
.18894
17
.13369 .016095
.19860
18
.13848
.017603
.20818
2.5
.08558
.002389
.05920
2.4
.07565
.001963
.05503 19 .14314
.019163
.21770
2.3
.06813
.001655
.05141
20
.14769
.020774
.22715
2.2
.06194
.001412
.04812 21
.15214
.022438
.23656
2.I
.05662
.001212
2.0
1.9
.04766
.001043
.05191
.000898
.04507 22
.04220
.03948 24
.15650
.024155
.24593
.16077
23
.025926
.25529
16497
.027752
.26462
1.4
1.2
Ι.Ο
1.8 .04378
1.7 .04019 .000663
1.6 .03685 .000567
.000407
.03075
.02527 .000282
.02027
.000773
.03688 25
.16910
.029635
.27395
.03437 26
.17317 .031576
.28329
.03195 27 .17718
.033576
.29264
28
.000186
0.8
.01566
.000114
.01478
0.6
.01138
190000*
0.4
.00737
.000026 .00717 33 .20033
+.2
.00358 .000006
-.2
.00000
.00341
.000000
.000006
.00000
.00345 36
0.4
.00666 .000023
.00682
0.6
.00977 .000050
ΟΙΟΙΙ
0.8 .01276 .000086 .01334
www www wwww☹
.02734
.02297 29 .18506
.01879 30 .18893
.18114
.035637
.30201
.037762
.31142
.039951
.32086
.19276
.042209
.33035
.01091 32 .19656
.044536
.33990
.046936
.34952
.00354 34 .20407
.049412
.35921
35
.20778
.051966
.36898
.21148
.054603
.37884
.21516
.057324
.38881
.21882
.060136
.39889
.22247
.063040
.40909
1.0
.01563
1.2
.01840
1.4
.02107
.000132
.000185
.000246
42
.01651 40
.01962 41 .22975
.02267
.23338
.22611
.066042
.41943
.069146 .42990
.072358
.44053
1.6
.02366 .000314
.02568 43
.23700 .075682
.45132
1.8
.02618 .000388
.02865 44
.24063 .079125
.46229
2.0
.02862 .000469
.03157
.24426
45
.082692
.47345
Į
11

γ
y = 7.4
y = 7.3
Ф
x
X
y Y
T
gp
X
Y
T
0
0
2.5
.08886
.002518
.06001 2.2
.03084 .000552
.03436
2.4 .07752
.002031
.05557 2.4
.03312 .000644
.03719
2.3 .06939
.001697
.05180 2.6
.03535 .000741
.03998
2.2
.06285
.001440
.04843
2.1
.05730
.001232
.0453I
2.0 .05244
.001058 .04240
तं लल
2.8
.03752
.000844 .04273
3.0
.03964
.000951 .04545
3.2
.04171
.001063
.04814
1.9
.04808
.000909
.03964
1.8
.044II
.000781
.03700
1.7
.04045
.000669
.03448
456
.04954
.001554
.05862
.05849
.002258
.07116
.06670
.003048
.08318
1.6
.03705
.00057I
.03204
1.5
.03387
.000485
.02968
1.4
.03088
.000409
.02740
689
.07432
.003914 .09479
.08144
.004851
.10603
.08815
.005853
.11697
1.3
.02804
.000342
.02517
IO
.09450
.006915
.12765
II
.10054
.008035
.13810
12
.10632
.009211
.14834
y = 7.4
13
.11187 .010440 .15842
14
.11721 .011721
.16835
15
.12236 .013054
.17814
Ф
X
Y
T
16
.12735 .014437
.18783
17
.13219 .015871
.19741
0
18
2.5
.09305
.002685
.06099
.13690
.017355
.20692
2.4
.07962
.002109
.05616 19
.14148 .018889
.21635
2.3
.07076 .001744
.05222 20 .14596 .020475
.22573
2.2
.06382
.001471
.04874
21
.15034 .0221II
.23506
2.1
1.9
.05802
2.0 .05299
.04850
.001253
.04556 22
.15463 .023800
.24436
.001073
.000920 .03980 24
.04260 23 .15883 .025542
.25363
.16296 .027338
.26289
1.8
1.4
1.2
1.0
.02037
.03102 .000412
.02544 .000285
.000188
0.8
.01572
.000114
.01481 31
0.6
.0I 141
.000062
0.4
.00738
.000026
.01093 32
.00718 33
+.2
.00359
.00000
.000006
.000000
.00354
.00000
35
-.2
.00340 .000006 .00345 36
333
456
34
.20505
.20868 .053735 .37615
.20140 .048633
.35668
.051144
.36637
0.4
0.6
.00665 .000023 .00681
.00975 .000050
0.8 .01273 .000086
ΟΙΟΙΟ
70
33
37
38
.21230 .056409
.38603
.21590 .059172 .39602
.01333 39 .21949 .062026 .40613
1.0
.01559 .000131
.01649 40 .22308
.064976
.41637
1.2
.01835 .000184
.01959 41 .22664
.068027
.42676
1.4
.02101 .000245
.02264 42
.23020
.071183
.43729
11 2
1.6
.02358 .000312
.02563 43
.23377
.074449
.44799
1.8
.02607 .000386
.02858 44
.23733
.077832
.45886
2.0
.02849 .000466
.03149 45
.24090 .081336
.46993
.04445 .000789 .03714
1.7 .04072 .000675 .03458
.03727 .000576 .03213
1.6
.17103 .031098 .28140
.17497 .033065 .29068
.02746 28 .17886 .035092 .29997
.02304 29 .18271 .037180 .30930
.01884 30 .18652 .039333
1 2 3
.31866
.19028 .041552 .32807
.043840 .33754
.19772 .046200 .34707
.19402
2 2 2
567
25
.16703 .029190
.27215
26
27
12

y = 7.5
7.6
γ
Ф
X
Y
T
Ф
X
Y
T
0
0
2.4
.08209 .002201
2.3
.07227
.001796
.05682 2.8
.05268 3.0
.03732
.000838
.04261
.03942 .000944
.04532
2.2
.06486
.001504
.04908 3.2
.04147
.001055 .04800
2.I
.05878
.001276
.04582 3.4
.04347
.001171 .05065
2.0
.05356
.001089
.04280 3.6
.04542
.001290
.05327
1.9
.04895 .000932
.03996 3.8
.04733 .001413
.05586
1.8
.04479
1.7 .04100 .000681
.000797
.03727
.03469
1.6
.03749
.000580
.03222
456
.04921
.05807
.001541
.05842
.002237 .07089
.06618
.003017 .08284
1.5
.03422 .000492
.02983
1.4
.03115 .000414
.02751
1.3
.02826 .000346
.02527
78 a
.07370
.003873 .09437
.08073
.004797 .10555
9
.08734
.005785
.11641
ΙΟ
.09360
.006833
.12701
y = 7.6
II
.09956
.007937
.13738
12
.10526 .009095
.14756
Ф
X
Y
T
13
14
15
345
.11072
.010306
.15756
.11598
.011569
.16741
.12106
.012881 .17713
0
2.4
.08513
.002313
.05758 16
2.3
.07396
.001854
.05316 17
2.2
.06598
.001540
.04943 18
678
.12597
.014243
.18674
.13074 .015655
.19626
.13537 .017116
:20569
2.I
.05959 .001300
.04609 19
.13989
.018626 .21505
2.0
.05416 .001106
.04301 20
.14430
.020187
.22435
1.9
.04941 .000944
.04013 21
.14861
.021797
.23361
1.8
.04515
.000806
1.7
.04128
.000688
.03740 22
.03480 23
.15283 .023459
.24283
.15696 .025173
.25203
1.6
.03771 .000585
.0323I 24
.16103
.026941
.26122
I.4 .03129 .000417
.02757
1.2
.02561 .000288 .02312
1.0 .02048 .000189 .01888
2 2 2
25
26
27
0.8
0.6
+.2
.01578 .000115 .01484 28 .17667 .034568
.01144 .000062
.01094 29 .18046
0.4 .00739
.000026 .00718 30 .18420
.00359 .000006 .00354
5678
.16503 .028762
.27040
.16896 .030640
.27958
.17284 .032574
.28877
.29799
.036622 .30724
O
.00000
.000000
.00000 32 .19157
.00340 .000006 .00345 33 .19522
0.4
.00664 .000023 .00681
0.6
.00973 .000050
ΟΙΟΙΟ
0.8
.01270
.000086 .01331
1.0
.01555
.000131 .01647
1.2
.01829 .000184 .01957
1.4
.02093 .000244 .02261
1.6
.02349 .000311 .02560 40
2.0
1.8 .02597 .000384 .02854
.02837 .000463
2.2 .03070 .000549
2.4
.03296 .000640
2.6 .03517 .000736
AAA www www wwww.
.038740
.31653
.18790
.040922
.32586
.043172
.33525
.045492 .34470
.19883
.047885
.35423
.20242
.050354
·36383
.20599 .052902 .37353
.20955
.055532 .38333
.21309
.058248 .39324
.21661
.061054 .40326
.22013
.063954 .41342
4I .22364
.066953 .42371
.03143 42 .22715
.070055 .43416
.03429 43
.23065
.073266 .44476
.03710 44
.23416
.076591 .45554
.03987 45
.23766
.080036
.46651
13

y =
7.8
= 7.7
=
y =
Ф
X
Y
T
Ф
X
Y
T
0
0
2.3
2.2
.07587 .001921
.05369 2.8
.03713
.000832
.04249
.06720 .001580
2.I
.06044
2.0
.05479
.04981 3.0
.001326
.04637
.001124 .04323 3.4
.03920
.000937 .04519
3.2
.04123
.001047 .04785
1.9
.04989
.000956
.04030 3.6
1.8
.04552
.000815
3.8
468
.04321
.001161 .05048
.04515 .001280 .05309
.04705 .001402
.05567
.03754
1.7
.04157 .000694
.03491
1.6
.03794
.000590
.03240
456
.04890
.001528
.05823
.05765 .002216
.07062
1.5
.03458
.06567
.002987
.08251
.000499
.02997
1.4
.03144
.000419
.02763
1.3
.02849
.000350
.02536
789
.07309
.003832
.09396
.08003
.004745 .10506
.08656
.005720 .11585
IO
.09273 .006753
.12638
II
Y
7.8
.09861
.007842
.13668
12
.10423
.008984
.14678
13
.10961
.010177
.15671
Ф
X
Y
T
14
.11479 .011421
.16649
15
.11979 .012714 .17614
16
.12463 .014056 .18568
2.3
.07806 .001998
2.2 .06853 .001623
.05428 17
.12933 .015446 .19512
.05021 18
13389
.016885
.20448
2.I
.06135 .001354
.04667 19
.13834
.018372
.21376
2.0
.05546 .001142
.04346 20
.14268 .019909
.22300
1.9
.05039 .000970
.04048 21
.14692 .021495
.23218
1.8
.04591 .000825
.03769 22
.15108
.023131
.24133
1.7
.04187 .000701
.03503 23 .15515
.024818
.25046
1.6
.03817 .000595
.03249 24
.15915
.026558
.25957
1.4
.03158 .000422
.02769 25
.16308
.02835I
.26868
1.2
.02579 .000290
.02320 26
.16696
.030198
.27779
1.0
.02059 .000190
.01893 27 .17078
.032101
.28691
0.8
.01584 .000116
.01486 28
.17454
.034064
.29605
0.6
.01147 .000062
.01096 29
.17827
.036086
.30523
0.4
.00740 .000026 .00719 30
.18195 .038169
.31444
+.2
.00359 .000006
.00354 31 .18559 .040316 .32370
.2
0.4
O .00000 .000000
.00340 .000006
.00663 .000023
.00000 32
.18921
.042530
.3330I
.00345 33
.19279 .044812
.34238
.00681 34
.19635 .047166
.35183
0.6
0.8
.00972 .000050
.01267 .000086
.01009 35
.19988 .049595
.36136
.01330 36 .20339 .052101
.37098
1.4
1.6
I.O .01551 .000130
1.2 .01823 .000182
.02086
.02340 .000309
1.8
.02586 .000382
2.0 .02824 .000461
.01644
.01953
78
33
37
.20689 .054688 .38070
38
.21037 .057359 .39052
.000242
.02256 39 .21384
.060119
.40047
.02554 40 .21730
.062972
.41054
.02847 4I .22075
.065921
.42075
.03136 42
.22420
.068972
.43III
2.2 .03056 .000546
2.4 .03281 .000636
2.6 .03500 .000732
.03420 43
.03700 44
.03976 45 .23454
.22765
.072130
.44163
.23109
.075400
.45232
.078788
.46320
14

Y
= 8.0
y = 7.9
Υ
&
X
Y
T
S
Փ
X
Y
T
0
0
2.2
.07000
.001672
.05064 3.4
.04295
.001152
.05034
2.1
.06233 .001383
.04698 3.6
.04486
.001270
.05293
2.0
.05615 .001162
.04369 3.8
.04674
.001391
.05549
1.9
.05091
.000983
.04067 4.0
1.8
.04630
.000834
.03783 4.2
1.7 .04217
.000708
.03515 4.4
1.6
.03841
.000600
.03258
4.6
1.5 .03495
.000506
.03012
1.4
.03173 .000425
.02775
1.3
.02872 .000354
.02546
024 656
78 9
.04858
.001516
.05803
.05038 .001645
.06054
.05214 .001778
.06303
.05387
.001914
.06550
.05724
.002196 .07036
.06517
.002958 .08218
.07250
.003793 .09356
.07935
.004694
.10459
.08579
.005656
.11531
y = 8.0
γ
IO
.09188
.006675
.12577
II
.09768
.007749
.13599
12
.10322
.008875
.14603
Ф
X
Y
T
13
.10853 .010052
.15588
14
.11363 .011278
.16559
0
2.2
.07165 .001726
15
.11856
2.I
.06338 .001416
2.0
1.9
1.6
1.4
I.2
.05689 .001183
.05145 .000998
1.8 .04671 .000845
1.7 .04249 .000715
.03866 .000605
.03188
.000427
.02597 .000293
1.0 .02069 .000191
.05110
.04731 16 .12333
.04394 17 .12796
.04086 18
.013874
.012552 .17517
.18464
.015244 .19401
.13245
.016662
.20330
.03798
19
.13683
.018127
.21252
.03526
.03268 21
20
.14III
.019640 .22168
.14529
.021202
.23080
.02781 22
.02327 23 .15339 .024475
.01898 24 .15733
.14938 .022814
.23988
.24894
.026188
.25798
0.8
.01590
911000*
.01489
0.6 .01150 .000062 .01097 26
0.4 .00741 .000026 .00719
2 2 2
567
25 .16120 .027953
.26702
.16502
.029772
.27606
27 .16878 .031646 .28511
+.2
.00359
O .00000 .000000
.000006 .00354 28 .17249 .033578
.00000 29
.00340 .000006 .00344
30
៩៖
0.4
0.6 .00970 .000050 .01008
.00662 .000023 .00680
0.8 .01264 .000085 .01328 33 .19045
1.0 .01546 .000130 .01642
1.2
.01817
.000182 .01950
1.4 .02078 .000242
.02252
1.6
.02330 .000308 .02549
w www www.
.039732
.29418
.17615 .035568 .30328
.17978 .037618
.18336
.31242
.32160
32 .18692
.041911 .33084
.044157
.34015
.19395
.046474
.34952
.19742
.048863
.35897
.20088
.051329 .36851
37 .20432
.053875
.37815
1.8
2.0 .02812
2.2
.03041 .000542 .03412
2.4 .03264 .000632 .03691
.03481 .000727 .03966
.03692 .000826 .04238 43
.03898 .000930 .04507 44
.04099 .001039 .04772 45
2.6
2.8
3.0
3.2
.067930 .42815
.22474 .071037 .43859
.22813 .074253 .44919
.23152 .077586 .45998
4I
2 9 7 7 * $4
39
.02575 .000380 .02841 38 .20774 .056504 .38790
.000458 .03129
.21116 .059220 ·39776
40
.21456 .062026 .40775
.21796 .064928
.41788
42
.22135
15

Y
8.2
8.1
Y
φ
X
Y
T
Ф
X
Y
T
0
2.2
.07352
.001788
.05161 3.4
.04270 .001144
.05019
2.I
.06453
.001451
.04766 3.6
.04460 ,001260
.05277
2.0
.05768
.001206
.04420 3.8
.04646
.001380
.05532
1.9
.05202
.001013
.04106
4.0 .04828 .001503
.05784
1.8
.04714
.000855
.03814 4.2
.05006
.001630
.06034
1.7
.04281
.000723
.03539 4.4
.05180
.001761
.06281
1.6
.03891
.000610
.03278
4.6
.0535I
.001896 .06526
1.5
.03534
.000514
.03028
1.4
.03203
.000430
.02788
1.3
.02895
.000357
.02556
56
78 9
.05684
.002176
.07010
.06467
.002930
.08185
.07192
.003754 .09317
.07868 .004644
.10413
.08504 .005594
.11478
γ
8.2
10
.09106 .co6600
.12517
II
.09677
.007659
•
13532
12
.10224
.008770
.14529
Ф
X
Y
T
13
.10747
.009930
.15508
14
.11251
.011139
.16472
0
15
2.2
.07568
.001861
.05218
.11737
.012396
.17423
2.I
.06578
.001490
.04803 16
.12207
.013699
.18363
2.0
.05851
.001230
.04447 17
.12663
.015049
.19294
1.9
.05261
.001029
.04126 18
.13106
.016446
.20216
1.8 .04758
.000866
.03830 19
.13537
.017890
.21131
1.7 .04315 .000731
.03551 20
.13958
.019381
.22040
1.6
.03917
919000*
.03288
21
.14370
.020920
•
22945
1.4
1.2
I.0
0.4
.000433
.03219
.02616 .000296
.02080
0.8 .01596
0.6 .01153
.000062
.00743 .000026
.02794
22
.14773
.022507
.23847
.02335 23
15168
.024144
.24745
.000193
.01903 24
.15556
.025831
.25643
.000117
.01492 25
.15937
.027570
.26540
.01098 26 .16313
.00720 27
.029361
.27437
.16683
.031207
.28335
+.2
.00360
-.2
0.4
O .00000
.00340
.00662
0.6 .00968 .000049
0.8 .01261 .000085
.000006
.000000
.000006
.00354 28
.17048
.033109
.29235
.00000 29
.17409 .035068
.30138
.00344 30
17766 .037088
.31045
.000023
.00680 31
.18120 .039169
.31956
.01007 32 .18470 .041314
.01327 33 .18817 .043525
.32873
•33796
1.0
1.2
I.4
2 4
2.2
1.6
.02322 .000306
1.8 .02565 .000378
2.0 .02800 .000455
.03028 .000538
2.4 .03249 .000627
2.6 .03464 .000721
.02544 37
.02835 38
.03122 39
.01542 .000129
.01812 .000181
.02071 .000240
.01640 34 .19162
.01947 35 .19504
.02248 36
.19844
.20183 .053092 .37567
.20520 .055680 .38534
.20856.058353
.39512
.045806 .34726
.048159 •35664
.050586 .36611
.03404 40
.21191 .061116 .40504
.03682 41
.21525
.063972
.41508
.03956 42
.21859
.066927 .42528
2.8
.03673
.000820
.04227 43
.22193
.069984 .43563
3.0
.03877 .000924
.04494 44
.22527
.073150 .44615
3.2
.04076 .001032
.04758 45
.22860
.076431 .45685
16
γ
8.4
8.3
γ
Ф
X
Y
T
P
X
Y
T
0
2.I
2.0
0
.06715
.001534
.04843 4.0
.04798
.001491
.05765
.05940 .001256
1.9
.05324 .001046
.04475 4.2 .04974
.04147 4.4
.001617
.06013
.05147
.001747
.06259

1.8
.04804 .000878
.03846 | 4.6
1.7
.04350
.000739
.03564 4.8
1.6
.03943
.000622
.03298 5.0
.05316 .001880
.05482
.05645 .002157
.06503
.002017 .06745
.06985
1.5
.03574
.000522
.03044 5.2
.05805
.002300
.07223
1.4
.03235
.000436
.02801 5.4 .05962
.002446
.07458
1.3
.02920
.000362
.02566 6.0
.06420 .002902
.08153
7
y =
8.4
8
3 .07136
.003717
.09278
.07804
.004596
.10367
9
.08432
.005533
.II426
IO
Ф
X
.09025
.006526
.12458
Y
T
I I .09590 .007572
.13467
12
.10129
.008668 .14456
0
2.I
2.0
.06869 .001583
.06036 .001284
.04886
13
.10645
.009812 .15429
.04505 14
.II142
.011005
.16386
1.9
.05390 .001064
.04169 15
.11621
.012244
.17331
1.8
.04852 .000890
.03863 16 .12085
.013529
.18264
1.7
.04385 .000748
.03577 17 .12534 .014860
.19188
1.6
.03971 .000628
.03308 18
.12971
.016237
.20104
1.4
.03251 .000439
.02807 19
.13396
.017660
.21012
1.2
.02635
1.0 .02092
.000299
.02343 20
.13811
.000194
.01908 21
.14217
.019130 .21915
.020647
.22813
0.8
.01602 .000117
0.6
.01156
.000063
0.4
.00744
.01495 22 .14614 .0222II
.01100 23
.000027 .00720 24
.23708
.15003
.023824 .24600
.15385
.025486 .2549I
+.2
.00360
.000006 .00354
O
-.2
.00000
.00339
.000000
.00000 26
.000006 .00344
2 2 2
25
27
0.4
0.8
8888888
.00661
.00967
.000023
.00679 28
567 co
.15761
.027199 .26381
.16131 .028964 .27272
.16496 .030782
.28163
.16856 .032656 .29057
.01259
.000049
.000085
.01006 29 .17212 .034586 .29953
.01325 30 .17563 .036575
.30853
1.0
.01538
.000129 .01638
1.2 .01807 .000180
1.4 .02065 .000239
.01944
.02244
1.6
1.8
.02314 .000305
.02555 .000376
2.0
.02788 .000453
2.2
35 .19274 .047479
.19609
.03014 .000536 .03396 37 .19943
.02539 34
.02829
.03115 36
2.4 .03233 .000624 .03673
2.6
.03446 .000717 .03946
2.8
.03654.000815
A www www www.
.17911 .038625
.31758
32
.18256 .040738
.32667
33
.18598 .042916
•33583
.18937 .045162
•34506
•35436
.049870
.36376
.052338 .37326
38
.20275
39
.054886 .38285
.20606 .057518 •39256
40
3.0
.03856 .000917
3.2
.04053
.001024
.04215
.20936 .060238
.04481 41 .21265 .063051
.04744 42 .21594 .065960
.40240
.41237
.42248
دبا دبا دیا
3.4
3.6
3.8
498
.04245
.001135
.04433
.001250
.04617 .001369
.05004 43 .21922 .068970 .43275
.05260 44 .22251 .072088 .44319
.05514 45 .22580 .075317 .45381
17

γ
8.6
8.5
Y
Ф
X
Y
T
g
X
Y
T
ง
2.I
2.0
0
.07042
1.9
.001639
.06139 .001315
.05460 .001083
.04933 4.0
.04769
.001480
.05747
.04537 4.2
.04943 .001605
.05994
.04193 4.4
.05114
.001734
.06239
1.8
.04902 .000903
.03880 4.6
.05281 .001866
.06482
1.7
.04422 .000756
.03591 4.8
1.6
.03999 .000634
.03319 5.0
o ∞
.05445
.002001
.06723
.05607
.002139
.06961
1.5
.03617
.000530
.03061 5.2 .05766
.002280
.07197
I.4
.03268
.000442
.02814
1.3
.02945
.000366
.02577
56
5.4 .05921
.002425
.07431
6.0
.06373
.002875
.08122
Y
8.6
789
.07081
.003681
.09241
.07741 .004549
.10323
.08361
.005475
11375
Փ
X
Y
T
ΙΟ
.08947
.006456
.I2400
II
.09504
.007488
.13403
I 2
.10036
.008569
.14386
0
2.I
.07243
.001703
.04986
13
.10546
.009699
.15352
2.0
1.9
1.8
.06252 .001348
.05534
.04954
1.7 .04461 .000765
.04571 14 .I1036
.010875
.16303
.001103
.04217 15
.11508
.012097
.17241
.000916
.03898 16
.03605 17
1.4
I.2
642
1.6 .04028 .000640
.03285
.03330 18
678
.11965
.013365
.18168
.12409 .014677 .19086
.12839 .016035
.19995
.02655
.000445
.000302
.02821 19
.13259 .017439
.20897
0.8
0.6
0.4
+.2
20
.000000
.02352 20
.01913 21
.01609 .000118 .01498 22
.01159 .000063 .OIIOI
.00745 .000027 .00721
.00360 .000006 .00355
1.0 .02103 .000196
.00000
2 2
23
24
.13668 .018888 .21794
.14068 .020383
.14460 .021925 .23574
.14843 .023515 .24460
.15220 .025153 .25345
.22686
.00000
2 2
25
.15591
.026841 .26228
26
.15955 .028581
.27112
.2
.00339 .000006
.00344 27
.16315 .030373
.27997
0.4 .00660 .000023
0.6 .00965 .000049
0.8 .01256 .000085
.00679 28
.16669
.032219
.28884
.01005 29
.17020
.034121
.29774
.01324 30
.17366
.036081 .30667
1.0 .01534 .000128 .01636
1.2
.01801
1.4 .02058 .000237 .02241 33
1.6 .02305 .000302 .02535
.000179 .01941
www.
.17709 .038101
.31565
32
.18049
.040182
•32468
.18386 .042328
•33378
1.8
2.0
.02544 .000373 .02824
.02776 .000450 .03108
333
35
36
34 .18720 .044541 .34294
.19052 .046824
.19382 .049179 .36150
-35217
2.4
2.2 .03000 .000532 .03388 37 .19711
.03217 .000620
.20038
2.6 .03429 .000713 .03936 39 .20364
.03664
www.
.051610 •37092
38
.054120
.38044
.056713
.39008
www www
2.8
3.0
∞ O
.03635 .000810 .04204 40
.03836 .000911 .04469 41
.20689
.059392
.39984
.21013
.062162
.40974
3.2
.04031 .001017
.04731 42
.21337
.065028
.41978
3.6
3.8
3.4 .04222
.04409
.04591
.001127
.001241
.04989 43
.21661
.067993
.42997
.05244 44 .21984
.071063
.44033
.001359
.05497 45
.22308 .074243
.45087
18

γ
Υ 8.8
8.7
φ
X
Y
T
Ф
X
Y
T
0
2.0
0
.06375
.001385
.04607 4.0 .04740
.001468
.05729
1.9
.05612 .001125
.04242
4.2 .04912
.001592
.05975
1.8
4.4
.05009 .000930
.03917
1.7
.04500 .000774
.03619
4.6
1.6
.04057 .000647
.03341
4.8
1.5
1.3
.03661
1.4 .03302
.02972
.000539
.000448 .02828
.000370
.03078
.02587 5.4
ம் ம்ம்ம்
5.0
46∞ O
.05081 .001719
.06219
.05247
.001849
.06460
.05410 .001983
.06699
.05570 .002120 .06936
5.2
.05727
.002260 .07171
.05881
.002403 .07404
6.0
.06327
.002848
.08091
7178
γ
8.8
78 9
.07027
.07679
.003645 .09203
.004503
.10279
.08292
.005418 .11325
ΙΟ
.08871
.006386
.12344
Ф
X
Y
T
I I
.09421
.007405
.13340
12
.09946
.008473
.14317
0
13
.I0449
.009587
.15277
2.0
.06511
.001426
.04645
14
.10932
.010748
.16222
1.9
.05696
.001148
.04268 15
.11399 .011954 .17153
1.8
.05066 .000944
1.7 .0454I .000780
1.6 .04088 .000653
I.4 .03319 .000452
.03936 16
.11850 .013205 .18074
.03634 17
.03352 18
78
.12287
.014500
.18985
.12712
.015839
.19888
.02834
1.2
.02675 .000305
.02360
1.0 .02115
.000197
0.8
0.6
.01615
.01162
611000*
.01918
.01500 22
21
19 .13126
20 .13529 .018652
.13924 .020127
.017223 .20784
.21675
.22560
.14310 .021647
.23443
.000063
0.4
.01103 23 .14688
.00746 .000027 .00722 24
.023215
.24322
.15060 .024830
.25201
+2.
O
-.2
.00360.000006
.00000 .000000
.00339 .000006
.00355
.00000
.00344
2 2 2
25
26
56
.15425 .026495
.26078
.15784 .028210
.26956
27
.16139 .029976
.27834
0.4
0.6
.00659 .000023
.00963.000049
.00678 28
.16488 .031796
.28715
0.8
.01253 .000084
1.0
1.8
2.0
2.2
.03101
.02987 .000529 .03380
2.4 .03203 .000616 .03654
2.6 .03413 .000708 .03925
2.8
2 3
∞ O N
3.0
3.2
3.4
3.6
333
469
3.8
.03816 .000904 .04457
.01530 .000128
I.2 .01796 .000179
1.4 .0205I .000237
1.6 .02297 .000302 .02529
.02535 .000372 .02817
.02765
.000448
*‡ ‡ ‡‡‡ www www www.
.18180
.01004 29 .16834 .033671 .29598
.01322 30 .17175 .035603
.01633 31 .17513 .037593 .31377
.01938 32 .17848
.02236 33
.30485
.039645 .32273
.041760 .33176
.18509
.043941 .34085
.18836 .046190 .35002
.19162
.048511 .35928
.19486
.050907 .36863
.19808
.053380 .37809
.20129
.055935 .38765
.03617 .000804 .04193 40 .20449
.058575 .39734
.20769
.061305
.40716
.04010 .001009 .04718
42
.21088
.064128
.41713
.04199 .001118 .04975
43
.21407
.067050
.42725
.04383 .001231
.04563 .001348 .05480 45
.05229 44 .21726 .070074
.22045
.43753
.073208
.44799
19.
2

y =
y = 9.0
8.9
P
X
Y
T
Ф
X
Y
T
0
0
2,0
.06664 .001473
.04687 4.0
.04711
.001457
.05711
1.9
.05786 :001173
.04296 4.2
.04882
.001580
.05956
1.8
.05126 .000960
4.4
.05049
.001706 .06198
.03957
1.7
.04584
.000794
.03649 4.6
.05214
.001835
.06438
1.6
.04119
.000660
.03364 4.8
.05375
.001968 .06676
1.5
.03708
.000549
.03096
5.0
.05533
.002103
.06912
1.4
.03337
.000455
.02842 5.2
1.3
.02998
.000375
.02599
5.4
6.0
240
.05688
.002241
.07146
.05840 .002383
.07378
.06282
.002823 .08061
Y 9.0
78 9
.06974
.003611
.09166
.07619 .004459
.10236
.08224
.005363
.11275
IO
.08796 .006319
.12288
Ф
X
Y
T
I I
.09339
.007326
.13279
12
.09857
.008380
.14249
0
13
.10354 .009480
.15203
1.9
2.0 .06838 .001526
.05883 .001200
.04733 14 .10831 .010626
.16142
1.8
1.7
.04629
.05189 .000976
.000805
.03978 16
.03665 17
1.6
.04151 .000667
.03376 18
678
.04326 15 .I1292 .011816
.11737 .013051 .17982
.17067
.12169
.014328 .18887
.12588
.015650
.19784
1.4
.03355 .000458
.02849 19
.I2996 .017015
.20674
1.2
1.0
.02696 .000308
.02369 20
.13394 .018425
.21558
.02127 .000199
.01923 21
.13783
.019879
.22438
0.8
.01622
.000119
.01503 22
.14164 .021379
.23314
0.6
+2
.01165
0.4 .00747 .000027
.00361 .000006
.000063
.01104 23
.14537
.022925
.24188
.00722 24
.14903
.024518 .25060
.00355 25 .15264
.026160
.2593I
.00000
000000
.00000 26 .15618 .027851
.26803
.2
1.0
.00339 .000006
0.4
.00658 .000023
0.6 .00962 .000049
0.8
.01250 .000084 .01321 30 .16989
.01526 .000128 .01631
.00344 27
.00678 28
.15968
.029593
.27675
.16312 .031387
.28550
.01003 29 .16653 .033236
.29427
1.2 .01790
.000178 .01935 32 .17653
1.4 .02044 .000236
.02233 33
www.c
.035141
.30308
.17323
.037103
.31193
.039126
.32083
.17980 .041211 .32979
1.6
.02289 .000300 .02525
1 2
1.8
.02525 .000370
.02812
35
2.0
.02753 .000446 .03095
456
333
34
.045579
.18305 .043361 .33882
.18627
.34793
36
.18948 .047867
.35712
તે હું જે છે
∞ O
.03796
2.2 .02974 .000527 .03373 37
.03188 .000613 .03647 38
2.4
2.6 .03396 .000704 .03916 39
2.8
3.0
.19267
.050228
.36640
00
.19585
.052666
.37579
3.2 .03988
.001003
3.4
3.6
.04175
IIII00*
.04358
3.8 .04537 .001338
.001223
.19902 .055185 .38529
.03599 .000799 .04182 40 .20217 .057787
.39491
.000899 .04444 41 .20532 .060477 .40466
.04703 42 .20847 .063260 .41455
.04959 43
.21161
.066139
.42460
.05212 44 .21475 .069120 .43480
.05463 45 .21789
.072209 .44519
20

y = 9.1
y = 9.2
S
X
Y
T
Ф
X
Y
T
0
0
1.9
.05989
.001230
1.8
.05257 .000994
1.7 .04675 .000816
1.6
.04185
.000675
1.5
.03757
.000558
.04357 4.0
.03999 4.2
.03681 4.4 .05018
.03388
4.6
.03115 4.8
.04683
.001446 .05693
.04852
.001567 .05937
.001692 .06178
1.4
.03374
.000462
.02856 5.0
.05180
.05340 .001951
.05497
.001820
.06417
.06653
.002085
.06888
1.3
.03026
.000380
.02610
5.2
.05651 .002222
.07121
5.4
.05802 .002362 .0735I
6.0
.06238
.002798
.08031
y = 9.2
४
78 9
.06922
.003577 .09131
.07560 .004415 .10194
.08158
.005308
.11227
Փ
X
Y
T
10
.08723
.006254
.12234
I I .09259
.007248 .13219
0
1.9
1.8
.06105 .001264
.05328 .001012
12 .09771
.008289
.14183
.04390
.04022
13
.10262
.009376
.15131
1.7
.04724 .000828
.03697 14
.10733
.010507 .16064
1.6
.04220 .000682
.03401 15
.11188
.011682
.16983
1.5
.03782 .000564
.03124 16
1.4
.03393 .000465
.02864 17
1.3
.03041
.000382
.02615 18
678
.11627
.012900
.17892
.12053 .014161
.18791
.12466 .015466
.19682
1.2
.02718
.000312
.02378 19
.12869 .016813
.20566
I.I
.02418
.000252
.02149 20
.13262
.018204 .21445
1.0
.02139
.000200
.01928 21
.13646
.019639
.22318
0.8
.01628 .000120
.01506
22
.14021
.021119
.23189
0.6 .01168 .000064
.01106
23 .14390
.022644
.24056
0.4
.00749 .000027
.00723
24
.14751
.024216
.24923
+.2
.00361
.000006
.00355
25
.15106
.025835
.25788
O
.00000
.000000
.00000 26
.15456
.027503
.26654
-.2
.00338 .000006
.00344 27
.15801
.029221 .27520
0.4
.00658
.000023
.00678 28
.16141
.030991
.28389
0.6 .00960
.000049
.01002 29
.16476
.032815 .29260
0.8
.01248
.000084
.01319 30
.16808
.934693 .30135
1.0
1.2
.01522
.01785 .000177
1.4 .02038 .000235
.000127
.01629 31
.01932 32 .17463
.02229 33
.17137
.036629
.31014
.038624
.31898
.17785
.040680 .32787
.02520 34
.18106
.042800 •33684
233
2.8
3.0
3.6
3.8
2.2 .02961 .000523
2.4 .03174 .000609
2.6 .03380 .000699
.03581
.03776 .000893
3.2 .03966 .000996
3.4 .04152 .001103
.04333 .001214
.04510 .001328
1.6 .02281 .000299
1.8 .02515 .000368 .02807 35 .18424
.03088 36 .18740
1 2
2.0 .02742 .000443
.044987 ·34588
.047243 .35501
.19055 .04957I .36423
.19368 .051975 -37355
.03906 39 .19680 .054458
73
.03365 37
.03637 38
.38298
.000794
.04171 40 .19991 .057024
.04433 4I .20302
.39253
.059676
.40221
.04691 42 .20612
.062420
.41204
.04946 43 .20922
.05198 44 .21232
.05447 45 .21541
.065259
.42201
.068198 .43215
.071242 .44246
21

y = 9.3
y = 9.4
❤
X
Y
T
Ф
X
Y
T
0
0
1.9
1.8
1.7
1.6
9876
.06233 .001301
.04426 4.0
.04656 .001435
.05676
.05404 .001033
.04046 4.2
.04824
.001555
.05919
.04774
.04256 .000690
.000840
.03714 4.4
.04988
.001679
.06159
.03413
1.5
.03808.000569
.031344.8
1.4
.03413 .000469
.02871
1.3
.03055
.000385
.02621
OMU MAA
4.6
.05149
.001806
.06397
.05307
.001936 .06632
5.0
0
.05462
.002068
.06865
5.2
.05614
.002203 .07096
5.4
.05763
.002342 .07325
6.0
.06196
.002773 .08002
γ 9.4
78 9
.06872
.003544 .09096
.07502 .004373
.10153
.08094
.005256
.11180
Փ
X
Y
T
IO
.08652
.006190
.12181
II
.09182 .007172
.13160
0
12
1.9
.06377
.09688 .008200
.14118
.001343
.04465
1.8
.05485
.001054
.04071 13
.10172 .009274
.15060
1.7
.04828
.000853
.03732 14
.10638
.010391
.15987
1.6
.04293
.000699
.03427 15
.11086
.011552
.16901
1.5
.03835
.000575
.03144
16
.11520
.012754
.17804
1.4 .03433 .000473
.02879 17
.11941
.014000
.18698
1.3
.03070 .000388
.02627 18
.12349
.015287
.19583
1.2
.02740 .000315
I.I
1.0
.02435 .000254
.02151 .000202
20
.02387 19 .12747
.02156
.13134
.01934 21 .13513
.016617
.20461
.017990
.21334
.019406 .22202
0.8
.01635 .000121
0.6
.01172
.000064
.01509 22
.01107 23
0.4
.00750
.000027
.00723 24
.13884
.14247
.14604
.020867
.23066
.022372 .23929
.023923
.24789
+.2
.00361
.000006
.00355
25
O .00000
.00338
.000000
.000006
.00000
.00344 27
N N
26
567
.14955 .025521
.25649
.15300 .027167 .26508
.15640 .028862
.27369
0.4
0.6
0.8
.00657 .000022
.00958
.000049
.01245 .000084
.00677 28
.15975 .030608
.28232
1.0
1.2
.01002 29
.01318
.01519 .000127 .01627
.01781 .000177 .01929
1.4 .02032 .000234 .02225
.16307
.032407
.29097
30 .16634
.034260 .29966
2.0
.000520 .03358
1.6 .02273 .000297 .02515 34
.17914 .042257 •33491
1.8 .02506 .000366 .02801 35 .18228 .044414
•34389
.02731 .000441 .03082
36
2.2 .02949
2.4
.03160
.000605
.03629
2.6
.03364.000695
.03897
www wwN
2.8
3.0
3.2
3.4
3.6
∞ 0 2
.03563
.000789
.04161
3.8
468
.03757 .000887 .04421
.03946 .000989 .04678
.04130 .001095 .04932 43
.04309 .001205 .05183
.04484 .001318 .0543I 45
42
44
£ £ £ £ £ £ www wwww www.
.16958 .036170
.30839
32
.17279 .038138
.31717
33
.17598 .040166
.32601
.18539
.046640 •35296
.18850
.048936
.36211
.19159
.051307
-37137
.19467
.053756
.38074
40
.19774 .056287 .39022
.20080
.058903 .39984
.20386 .061608
.40959
.20691
.064408
.41950
.20997
.067306 .42956
.21302
.070309
.43980
22

9.5
y = 9.6
Ф
X
Y
T
X
Y
T
0
1.9
.06541
.001391
.04508 4.0
.04629
.001424
.05658
1.8
.05573
.001078
.04097 4.2
.04795
.001 543
.05899
1.7
.04884 .000867
.0375I 4.4
.04958
999100'
.06138
1.6
.0433I .000707
.03440
4.6
.05117 .001792
.06375
1.5
.03863.000581
1.4
.03453
.000477
1.3
.03085
.000390
.03154 4.8
.02887 5.0
.02633
.05273
.001920
.06610
.05427 .00205I
.06842
5.2
.05577
.002185
.07072
5.4
.05725 .002322
.07300
6.0
.06153 .002749
.07973
γ
9.6
•
78 9
.06823
.003511 .09061
.07446
.004331
.10113
.08031
.005204
.11134
Ф
X
Y
T
ΙΟ
.08582 .006128
.12129
II .09106
.007098
.13101
1.9
1.8
1.7
09856
I2
.09606
.06732
.05668
.001448
.04556
.008115
.14054
.001103
.04126 13
.10085
.009175
.14990
.04942
188000*
.03770 14 .10544
.010279
.15912
1.6 .0437I
.000716
.03454 15
.10988
.011425
.16820
1.5
.03891 .000586
1.4 .03474 .000481
.03165 16
.02895
17
1.3
.03101
.000393
.02640 18
678
.11416 .012613
.17717
.11831 .013842
.18605
.12234
.015113 .19485
1.2
.02763
.000319
.12627
I.I
.02452
.000256
1.0
.02164
.000203
.13384
0.8
.01641
.000121
.01512 22
.13750
.02397 19
.016427 .20358
.02164 20 .13010 .017782 .21225
.01939 21
.019180 .22087
.020622 .22946
0.6
.01175
.000064
.01109 23
.14108 .022107
.23803
0.4
+.2
O .00000
.00751 .000027
.00362 .000006
.000000
.00724 24
.14460 .023638
.24658
.00355 25
.14806
.025215
.25512
.00000 26 .15147
.026840
.26366
.2
.00338 .000006
.00344 27
.15483
.028513
.27221
8888888
0.4
0.6
.00656 .000022
.00957
0.8 .01242
.00677 28 .15814
.030236
.28078
.000049
.000084 .01316
ΟΙΟΟΙ 29
1.0
1.2
1.6
1.8
.01515
.000126 .01625
.01776 .000176
1.4 .02026
.000233
.02266 .000296
.02497 .000364 .02796
2.0
.02720 .000438 .03075
2.2
.02936 .000517 .03350 37
2.4 .03145 .000602 .03620
2.6
.03348 .000691 .03886 39 .19258
به لب لب لب لب
2.8
.03546 .000784 .04149
3.0
3.2
.03738
.03925
188000*
.04409
3.4 .04107
.000982 .04665
.001088 .04918
43
: £ £ £ £ www www wwww☺
.16140
.032011
.28938
30
.16464
.033840
.29801
.16784
.035724 .30668
34
.01926 32 .17100
.02221 33 .17415
.025II
.17726
.037666 .31540
.039667 .32418
.041731
•33303
35 .18036
.043859
.34195
.18344
.046054
.35095
.18650
.048320 .36004
38
.18955
.050659
.36924
.053075
•37854
40
.19561
.055571
.38796
41 .19863
.058152
.3975I
42
.20165
.060821
.40720
.20467 .063583
.41704
3.6
.04285
3.8
.04459
.001197 .05167 44 .20768
.001309 .05414 45 .21069
.066442
.42703
.069403
.43720
23

Y
y = 9.7
9.8
γ
op
S
X
Y
T Ф
X
Y
T
0
0
1.8
.05771 .001131
.04156 4.0
1.7
.05005
.000896
.03790 4.2
.04603 .001414
.04768
.05642
.001532 .05882
1.6
.04413
.000726
.03469 4.4
.04929
.001653
.06120
1.3
I.I
SES SEA
1.5
.03921
.000592
.03175 4.6
.05087
.001778
.06356
1.4
.03495 .000485
.02903 4.8
.05242
.001905
.06589
.03117
.000395
.02646 5.0
.05394
.002035
.06820
1.2 .02775
.000321
.02401 5.2
.05543
.002168
.07049
.02461
.000258
.02167 5.4
.05689
.002304
.07276
.02170
.000204
.01942 6.0
.06113
.002726
.07945
γ
9.8
789
.06775
.003480
.09027
.07391
.004291 .10073
.07969
.005155 .11089
Ф
X
Y
T
IO
.08515
.006067
.12078
II
.09033
.007027
.13045
12
.09527
.008031
.13992
0
1.8
1.7
1.6
978
.05885 .001161
.05071 .000913
.04457 .000735
.04187
.03811 14
.03483 15
13
1.5 .03952
.000599
.03186
16
J
1.4 .03517
.000489
.0291I
17
1.3
.03133
.000398
.02652 18
678
.10000 .009079
.10454
.10892
.11315 .012474 .17633
.14923
.010169 .15839
.011301
.16742
.11725
.013689 .18516
.12123
.014945 .19390
1.2
.02787 .000322
.02406 19
.12511
.016241 .20257
I.I
1.0
.02470
.02177 .000205
.000259
.02171 20
.12889
.017580
.21119
.01945 21
.13258
.018960
.21976
0.8
.01649 .000122
.01515 22 13619
.020384
.22830
0.6
.01178 .000064
.OIIIO 23
.13974
.021851
.23681
0.4 .00752
.000027
.00725 24
.14321
.023362
.24530
+.2
20
.00362 .000006
.00355 25
.14663
.024919
.25379
.00000 .000000
.00000 26
.14999
.026523
.26227
-.2
.00338 .000006 .00343 27
.15330
.028174
.27077
0.4 .00655 .000022
0.6 .00955 .000049
0.8 .01240
.00676 28
.15657 .029875
.27929
.01000 29
.15980 .031628
.28783
.000083 .01315
I.0 .01511
.000126 .01623
1.4
1.2 .01770 .000176 .01924 32
.02018 .000233 .02218 33 .17237
1.6
2.0
.02257 .000295 .02506 34
1.8 .02487 .000363 .02790
.02710
.000436 .03069 36
2.2
.02925 .000514 .03343
∞ O N
મેં મેં હું હું
2.4
.03132 .000598 .03612
www www wwww w
30
.16299 .033433
.29640
.16615
.035292
.30501
.16927
.037208
.31368
.039184
.32240
.17545
.041220 .33119
35
.17851
.043320
.18154 .045487
.34005
•34900
37
38
73
.18457 .047723
.18757
-35803
.050031 ·36717
2.6
.03333 .000687 .03877 39 .19057
.052416 .37641
2.8
3.0
3.2
.03529 .000779 .04139 40
.03720
.000875 .04398 4I
.03906
.000975 .04653 42
دب کی دی
3.4
3.6
3.8
+68
.04087
.001079 .04905 43
.04263 .001187
.05153 44
.04435 .001299 .05399 45
.19356
.19654 .057425 .39525
.19952 .060059 .40488
.20249 .062784 .41465
.20546 .065605 .42458
.20844
.054879 .38577
.068527 .43468
24
y = 9.9
= 10.0
Y
Գ
X
Y
T
&
X
Y
T
0
I.I
1.0
SER SIA
0
1.8 .060II .001196
1.7 .05141 .000930
1.6
.04502 .000746
1.5 .03983 .000605
1.4
1.3
.03540 .000493
.03150 .000401
1.2 .02799 .000324
.02479 .000260
.02184 ,000206
.03197 4.6
.02919 4.8
.02658 5.0
.04221 4.0 .04577
.03833 4.2 .04740
.03499 4.4
.001404
.001521
.05625
.05864
.04900
.001641 .06101
600 0
.05057
.001764 .06335
.05210
.001890
.06567
.05360
.002019
.06797
.02411 5.2
.02175 5.4
.01948 0.0 .00072
2 4
.002151
.05508
.07025
.05652
.002285 .07251
.002703
.07917
y = 10.0
78 9
.06728
.003450
.08994
.07337
.004252
.10034
.07909
.005106 .I1044

S
X
Y
T
ΙΟ
.08449
.006008
.12028
II
.08961
.006957
.12990
12
.09449
.007950
.13932
0
1.8
.06154 .001236
.04259 13
1.7 .05216 .000949
.03855 14
1.6 .04550
.000757
.03514
15
is w
.09916
.008985
.14857
.10365
.010063
.15767
•
10798
.011181
.16664
н
1.5
.04015
.000612
.03209
16
1.4
.03563
.000498
.02928 17
1.3
.03166 .000404
.02665
18
679
.11216
.012341
.17551
.11621
.013541 .18428
.12015
.014781 .19297
I.2
I. I
1.0
0.8
0.6
.02811 .000326 .02416 19
.02488 .000262 .02178 20 .12771
.000207 .01950 21
.02190
.13136
.01655 .000123
.01518 22 .13493
.01181 .000065 .01112 23 .13842 .021602
12398
.016062 .20159
.017384 .21015
.018748
.21867
.020153
.22715
.23561
0.4
.00754 .000027 .00725 24
.14186 .023095
.24405
+.2
-.2
.00362 .000006 .00356
,00000 .000000
.00338 .000006
.00000
.00343
2 2 2
25
.14523 .024632
.25248
26
.14855 .026215
.26092
27
.15182 .027846
.26936
0.8
0.4 .00654 .000022
0.6 .00954 .000048
.01237 .000083
.00676 28 .15505 .029526
.27782
.00999 29 .15823 .031256 .28630
.01314 30
.29482
.16138 .033038
1,0
.02012
.01507 .000125 .01621
1.2 .01765
1.4
1.6 .02249
1.8 .02478
2.0 .02699
.000175 .01921
.000231
.000293 .02502
.000361 .02785
.000434
.02214 33
.03063
2.2
.02912
2.4 .03118
.000512
wwww www
.16450
.034874
.30338
32
.16759
.036766
.31199

.17065
.038716
.32066
34
.17369
.040726 .32939
35
.17671 .042800 .33820
.000595
2.6
.03318
.000683
2.8
.03513
.000775
3.0
.03702
3.2
.03886
.000870
.000969
3.4
.04065 .001073
ww
3.6
.04239 .001180
3.8
.04410 .001290
.03336 37
.03605 38
.03869 39
.04130 40 .19157
.04387 4I .1945I
.04641 42
.04891
.05138 44 .20332
.05383 45 .20625
7x
•
.17970 .044938
18269 .047146 ·35606
.18566 .049424 36513
.18862
.051778 .37432
•34708
.054209 .38361
.056722
.39304
.19745 .059322
.40260
43 .20038 .062011
.41231
.064796
.42217
.067680
.43021
25

y = 10,1
10.4
γ
Ф
X
Y
T
Ф
X
Y
T
0
0
1.7
1.6
I.4
1.3
.05297 .000970
.04600 .000769
1.5 .04049
.03587
.03183
1.6
.000619
2.4
3.0
.03879
.03531 1.8 .02460 .000359 .02774
.03221 2.0 .02677
.000431 .03050
.000502 .02937 2.2 .02887 .000508
.02671
.000407
.03321
.03090 .000590 .03587
.03665 .000860 .04364
.02235
.000292
.02493
Y
10.2
0
1.7 .05384 .000992
.03905
1.6
.04652
.000781
.03548
456
78 9
.04526
.001385
.05592
.05295
.001988
.06754
.05993
.002660 .07863
.06635
.003391
.08928
.07232
.004176
.09958
1.5
.04083
.000627
9
.07792
.005012
.10957
.03233
1.4
.03611 .000507
.02946 IO
.08320
.005895
.11930
1.3
.03201
.000410
.02678 II
.08821
.006823
.12881
12
.09298 .007794
.13812
13
y = 10.3
14
15
i3 W
.09755
.008806
.14727
.10193 .009859
15627
.10616
.010952
.16514
0
16
1.7
.05478 .001016
.03932 17
1.6
.04707
.000793
.03565 18
678
.11024 .012084
.17390
.I1420 .013256 .18257
.11804
.014467
.19115
1.5
.04120
.000634
.03245
1.4
.03636
.000512
.02955
19
.12178
.015718
.19967
1.3
.03219
.000413
.02685
20
.12543 .017009
.20813
21
.12899
.018340
.21655
22
.13247
.019712
.22493
γ
= 10.4
23
.13588
.021126
.23328
24
.13923
.022582
.24162
0
1.7
25
.14253
.024083
.24995
.05582
.001043
.03961 26
.14576
.025628
.25828
1.6
.04765 .000807
.03584 27
.14896
.027219
.26662
1.5
.04157 .000642
.03258 28
.15210
.028857
.27498
1.4
.03661 .000517
.02964 29
.15521 .030545
.28336
1.3
.03237 .000416
.02692 30
.15828 .032283
.29178
1.2
.02862 .000335
.02436
I.I .02526
1.0 .02218
.000267 .02194
.0002II .01962 33
0.8
0.6
.01670 .000124 .01524 34
.01188
.000065 .01115
0.4
.00756
35
.000027 .00726 36
+.2
.00363 .000006 .00356
-.2
0.4
O N
.00000
.000000
.00000
.00337
.000006 .00343 39
wwww wwww www.
.16133 .034074
.30023
32
.16434 .035920
.30873
.16733
.17028 .039782 .32591
.17323
.041804 .33461
.17615 .043890 •34338
.037822 .31729
37
.17906
.046042 .35225
38
.18196
.048264
.36121
.18484
.050558 .37027
0.6
0.8
468
.00653 .000022 .00675 40 .18772
.052929 .37945
.00950 .000048
.01232 .000083
1.0 .01500 .000125
1.2 .01755 .000175
1.4 .02000 .000231
.00997 41 .19059 .055379
.01311 42
.38876
.19345
.057914
.39820
.01617 43
.19631
.060536
.40779
.01915 44
.19917
.063250
.41753
.02207 45
.20203
.066061
.42743
26
γ
10.8
10.5
γ
Ф
X
Y
T
Ф
X
Y
T
0
1.7
.05697 .001072
.03992 1.6
.02219
.000288
.02485
1.6
.04826 .000821
.03603 1.8
.02442
.000354 .02764
1.5
.04196
.000650
.03271 2.0
.02657
.000425 .03038
1.4
.03688
.000522
.02974 2.2
.02864
1.3
.03255 .000420
.02699
.000501 .03307
2.4
.03065
.000582 .03572
3.0
.03631
.000848
.04343
y = 10.6
0
1.7
.05824 .001105
1.6 .04892 .000836
1.5 .04237 .000659
I.4 .03715
1.3 .03275
.04026
.03623
.03285
456
78 9
.04477
.001365 .05560
.05232
.001957
.06711
.05917
.002616
.07809

.06547
.003333
.08864
.07132
.004103
..09884
.07680
.004921
.10873
.02983
.000527
.02706
.000423
ΙΟ
.08197 .005786
.11835
II
.08687
.006694
.12776
12 .09154
.007643
.13697
13
09601
.008634
.14601
14
.10030
.009663
.15491
γ 10.7
15
16
1.6 .04962
17
1.3
1.2
.000853 .03644
1.5 .04279 .000668
.03299
1.4 .03744 .000532
.03294 .000426
.02903 .000341
18
567∞
.10443 .010732
.16369
.10842
.011839

.17235
.11229 .012985
.18092
.11604 .014168
.18941
.02993 19
.11970 .015391
.19783
.02714 20
.02452
.12326 .016652
.20619
21
.12674 .017952
.21451
22 .13014
.019293
.22280
23
.13348
.020674
.23105
Y
10.8
24
.13675 .022096
.23929
0
1.6
.05037
1.5
1.3
.000871
.04323 .000678
1.4 .03772 .000538
.03314 .000430
.03666
2 2 2
567
25 .13996 .023561
.24753
26
.14313 .025070
.25576
27
.14624
.026624
.26400
.03314
28
I.I
0.8
0.6
990000*
.03004 29
.02721 30
I.2 .02918 .000343 .02458
.02565
1.0 .02247
.000214 .01974
.01684 .000125 .01531
.01195
.14932
.15235 .029872
.028224 .27226
.28054
.15535
.031569
.28885
.000272
.02210
.01118
0.4 .00759 .000027
+.2 .00363 .000006
.00728
333
www www
.15832
.033318
.29720
.16126 .035119
.30560
.16417 .036975
.31406
34
.16707 .038889
•32258
35
.16994 .040863
.33117
36
.17279 .042899
•33984
.00356
37
.00000 .000000
.00000 38
.2
.00337 .000006
0.4
.17846 .047168 .35745
.00343 39 .18127 .049407 .36640
.00651 .000022 .00674 40 .18408 .051720 .37547
789
.17563 .044999 -34859
0.6
0.8
.00947 .000048
.01227 .000082
.00996 41 .18688 .054112 .38467
.01308 42 .18967
.056584 .39399
1.0
.01492 .000124
1.2
1.4
743
.01612 43 .19246 .059143 .40346
.01745 .000172 .01910
.01987 .000227
44 .19525
.02200 45
.19805
.061792 .41308
.064535 .42287
27
Υ
= 11.2
= 10.9
y
g
X
Y
T
Ф
X
Y
T

0
0
1.6
.05118
.000891
1.5
.04370
.000688
1.4
.03802
1.3
1.2
.000544
.03335 .000433
.02932
.03690 1.6 .02204
.03329 1.8 .02425
.03014 2.0 .02637
.02729
2.2
.02842
.02464
.000346
2.4 .03040
.000285
.02476
.00035I
.02754
.000421 .03026
.000496
.03293
.000576
.03556
3.0
.03597
.000838 .04321
y = 11.0
456
.04430
.001346 .05529
.05172 .001929
.0667I
.05844
.002575 .07759
0
1.6
.05207 .000912
1.5 .04419 .000698
.03715
.03345
78 9
76
.06462
.07036
.003279
.08804
.004033
.09813
.07573
.004836 .10792
1.4
.03834 .000550
.03025
1.3
.03356
.000437
.02736
ΙΟ
.08079
.005682
.I1745
1.2
.02947
.000348
.02469
I I
.08559
.006572
.12676
12
.09016 .007501
.13588
13
.09453 .008471
.14482
14
.09873 .009478
.15363
γ = II.I
15
16
0
17
1.6
.05304
.000936
.03742 18
5678
.10278
.010524 .16231
.10668
.011607 .17088
.I1046
.012728
.17935
.11414 .013885
.18774
1.5
.04470
.000709
.03361
1.4
.03866 .000556
.03036 19
.I1771
.015081
.19607
1.3
.03378
.000441
.02744 20
.12119
.016314
.20434
1.2 .02962
.000350
.02475 21
.12459
.017585
.21257
22
.12792 .018896
.22076
23
.13118 .020246
.22892
24
.13438
Υ
== 11.2
.021636
.23707
25
.13752 .023069
.24521
26
.14061
.024543 .25335
0
1.6
.05410
.000962
27
.14366
.03770
.026062 .26150
1.5
.04524 .000721
I.4
.03899 .000563
.03048
.03378 28 .14666 .027625
29 .14963
.26966
.029236
.27785
1.3
.03400 .000445 .02752
1.2
1.0
.02977 .000353 .02481
I.I .02608
.02277
0.8 .01700
.000279 .02227
.000127 .01538
.000218 .01987
0.6
.01202
.000066 .OI121
0.4
.00761
.000027 .00729
www www w
30
.15256
.030894 .28606
.15546 .032602
.29432
.15833
.034362
.30262
.16118 .036176 .31098
.16401
.038046 .31940
.16681
.039974 •32789
.16960
.041963
.33646
+.2
.00364
.000006 .00356 37 .17237
.044015 .34512
.00000
.000000
.00000 38
.17513 .046133
-35386
-.2
.00336 .000006
.00343 39 .17788 .048320
.36272
0.4 .00650 .000022
0.6 .00944 .000048
0.8
.01222 .000082
.00673 40 .18062
.050580
.37168
.00994 4I .18336
052915
.38077
.01305 42
.18609
.055331
.38998

1.0
.01485 .000123
.01608 43 .18882
.057829
.39934
1.2
.01736 .000171
.01904 44 .19154
.060416
.40885
1.4
.01975 .000225
.02193 45 .19427
.063095
.41852
28

y = 11.3
γ
y = 11.6
Ф
X
Y
T
Ф
X
Y
Τ
0
0
1.5
.04582
.000734
.03395
1.6
.02190
.000283
.02468
1.4
.03934
.000570
.03059 1.8
.02408
.000347
.02744
1.3
.03423
.000449
.02761 2.0
.02617
.000417
.03015
1.2
1.I
.02993 .000355
.02619 .000280
.02487 2.2
.02819
.000491 .03281
.0223I 2.4 .03014
.000570 .03542
3.0
.03564
.000828
.04301
γ
II.4
ала
4
.04384 .001328
.05499
.05113
.001901
.06631
6
.05774
.002536
.07709
Ο
1.5 .04643
.000748
.03414
1.4 .03970
.000576
78 a
.06380
.003226
.08745
.06943
.003967
.09745
.03071
9 .07470
.004753 .10714
1.3
.03446
.000454
.02769 ΙΟ
.07966
.005582
.11658
1.2
I.I
.03009
.000358
.02493 II
.08436
.006455
.12579
.02630
.000282
.02236 12 .08884
.007366
.13481
13
14
γ
11.5
15
345
.09312
.008315
.14367
.09724
.009302
.15238
.10120 .010326
.16097
16
17
0
18
67∞
.10502
.011387
.16944
.10873
.012484 .17783
1.5
.04708 .000762
.11232
.013617
.18613
1.4
1.3
.03470
1.2
.04007 .000584
.000458
.03026 .000361
21
.03433
.03084 19
.02777 20 .11923
.02499
.11582
.014787 .19437
.OI5994
.20255
.12255
.017238
.21069
I.I
.02641 .000284
.02240
22
.12581
.018520
.21879
23
.12900
.019841 .22686
24
.13213
.021201
.23492
γ
11.6
2 2
25
.13520 .022602
.24297
26
.13823 .024044
.25102
27 .14120 .025529
.25908
0
1.5
.04778 .000778
1.4
I.I
1.3
1.2 .03042
.02653
.03453 28 .14414
.04046 .000593 .03096 29 .14704 .028633
.03495 .000462 .02786 30 .14991
.000364 .02505
.027059
.26715
.27525
.030255
.28337
.000285 .02244
1.0 .02309 .000222
.01999
0.8
0.6
.01715 .000129 .OI544 34
.01209 .000067 .01124
0.4
.00764 .000028 .00730 36 .16657
www www.
.15275 .031926
.29154
32
.15555 .033647
.29975
.15834 .035420 .30801
.16110
.037248 .31634
35
.16384
.039133 .32473
.041077 .33321
+.2
o .00000 .000000
.00364 .000007 .00357
.00000
୭୭
37 .16928 .043083 .34176
38 .17198 .045154
.35041
-.2
.00336 .000006
.00342 39 .17467 .047292
.35917
0.4
.00648 .000022
0.6
.00941
.000048
.00673 40 .17735 .049501
.00992 41 .18002 .051784
.36803
.37701
0.8
.01217
.00008 I
.01303 42 .18269
.054145
.38612
I.0
.01478
.000122
.01605 43 .18535
.056587 .39537
1.2
.01726 .000170
.01899 44 .18802
.059115
.40477
1.4
.01963 .000224
.02187 45 .19068
.061734
.41433
29

บ
y = 12.0
= 11.7
X
Y
T
Ф
X
Y
T
0
0
1.5
.04854 .000795
1.4
.04087
109000*
.03475 1.6
.03109 1.8
.02176
.000281
.02460
.02391
.000344 .02734
1.3
.03521
.000467
.02795 2.0
.02598
.000413 .03003
1.2
.03059
.000366
.025II 2.2
.02798
.000486
I.I
.02665 .000287
.03267
.02240
2.4 .02990
.000564
.03527
3.0
.03532
.000819
.04281
γ.
11.8
0
1.5
.04938
.000814
.03497
I.4
.04130
.000609
.03123 9
456
78 a
.04340
.001311 .05470
.05057
.001874 .06592
.05706
.002498 .07661
.06301
.003176
.08687
.06854
.003903
.09678
.07371
.004675
.10638
1.3
.03547 .000472
.02804
IO
.07857 .005489
.11572
1.2 .03077 .000369
.02518
II
.08318 .006343
.12485
I.I
.02676
.000289
.02254
12
.08758 .007236
.13378
13
34i
14
.09177 .008167
.14255
.09580 .009134 .15117
y
= 11.9
15
.09968
.ΟΙΟΙ37
.15967
16
17
18
678
.10343 .011176 .16806
.10706 .012250
.17636
1.5
.05031
.000835
.11058
.013360
.18458
.03522
1.4
.04175 .000618
.03137
19 .11400
.014506
.19273
1.3
.03574
.000477
.02813
1.2
.03095
.000372
20 .11734
.02525 21
.015687 .20083
.12060
.016906 .20888
I.I
.02688
.000291
.02258 22
.12378
.018161
.21690
23
.12691 .019454
.22489
24
.12997
.020785
.23286
2
12.0
25
.13298
.022157
.24083
26
.13594
.023569 .24879
27
.13885
.025022 .25676
1.5
.05127 .000858
.03549 28 .14173
.026519
.26475
1.4
1.3
I.I
1.2 .03113
.02701 .000292 .02263 32
1.0 .02342 .000227
0.8 .01731 .000130 .01551
0.6 .01216 .000067 .01127
.04222 .000628
.03602 .000482
.000375 .02531
.02013 33
0.4
.00767 .000028
.00731 36
+.2
.00365 .000007 .00357
.2
.000006
0.4
.00000 .000000
.00335
.00647 .000022 .00672
.00000
.00342
A www www wwww
.03152 29 .14457 .028060
.02824 30
.27275
.14737 .029647
.28079
•
15015 .031282
.28887
.15290 .032966 .29699
.15562
.034701 .30517
34
.15832
.036490
.31341
35
.16100
.038334
.32171
.16367
.040236
.33009
.16632
.042198
•33855
.16896
.044224
.34711
.17159
.046316
-35577
40
.17421
.048477
.36453
180000*
0.6 .00938 .000047 .00991 41
0.8 .01212
.17683
.050710 .37342
.01300 42
.17944 .053019
.38243
I.2
1.4
1.0 .01471 .000121
.01717 .000169
.01951 .000222
.01601 43
.18204 .055409
.39158
.02180
.01894 44
45
.18465 .057882
.40088
.18725 .060443
.41033
30

Y
y = 13.0
y = 11.0
Ф
X
Y
T
S
X
Y
T
0
0
0.0
.00000 .000000
0.2
.00337
.000006
0.4
.00651
.000022
.00000 0.0
.00343 0.2 .00334
.00674 0.4 .00643
.00000
.000000
.00000
.000006 .00342
.000022 .00670
0.6
.00946
.000048
.00995 0.6
0.8
.01224
.000082
.01307 0.8
.00930 .000047
.01200 .000080
.00987
.01293
1.0
.01489
.000123
.01610 1.0
.01454 .000120 .01591
1.2
.01741
.000172
.01907 I.2
.01694
991000*
.01881
1.4
.01981
.000226
.02197 I.4
.01923 .000218 .02164
1.6
.02212
.000287
.02482 1.6
.02142
.000275
.02440
1.8
.000352
2.0
.02433
.02647 .000423 .03032 2.0
3.0 .03615 .000843 .04332 3.0
.02759 1.8
.0235I
.000337
.02710
.02552
.000404
.02975
.03455
.000796
.04232
456
.001355
.05881
789
.04455
.05203 .001943 .06691
.002596 .07784
.06505 .003306 .08834
.05545
.07085 .004068
.09848
.07627 .004878
.10832
456
789
.04233 .001270
.05399
.04922 .001811 .06499
.05545
.002410 .07546
.06115
.003059 .08550
.06644 .003754 .09519
.07138 .004492
.10458
ΙΟ
.08139 .005734
.11790 ΙΟ
.07603
.005270
.11371
II
.08624 .006632
.12726 II
.08043
.006085
.12262
12
.09086 .007572
.13642 12
.08462
.006937
.13135
13
14
34
.09527 .008551
.14542 13
.08862
.007824
.13991
.09952 .009570
15 .10361 .010627
.15427 14 .09246
.008746
.14833
.16299 15 .09615
.009701
.15662
16
.10756
.011722
.17161 16
17
18
78
.11138 .012855
.18013 17
.11509 .014025
.18857 18 .10653
678
.09972 .010691
.16481
.10318
.011714
.17291
.012770
.18093
21
19 .11871 .015234
20 .12223 .016481
.12567 .017767
22 .12903 .019092
23 .13233 .020457
24 .13556 .021863
.19694 19 .10979
.013860
.18888
.20526 20
.11296 .014984
.19678
.21353 21
.11606
.016143
.20463
.22177 22 .11909
.22998 23
.017337
.21245
.12206
.018567
.22024
.23818 24
.12497
.019833
.22801
222
567
26
25 .13874 .023312
.14187 .024803
.24636 25
.25455 26
.12783 .021137
.23578
.13065
.022479
.24354
27
.14495 .026339
.26274 27
.13342
.023860
.25131
28
.14799 .027921
.27095 28
.13615 .025283
.25910
29
.15099 .029549
.27918 29 .13885 .026747
.26691
30
.15395 .031227 .28745 30 .14151
.028255
.27474
333
1 2 3
31
.15688
.032955 .29575 31 .14415
.15979 .034735 .30410
.16267 .036570 .3125I
333
456
.16553 .038462 .32098
78
33
.17678
32
33
34
35
36
37
38
39
.16836 .040412
.32952
.17118 .042424 .33814
.17399 .044500 .34684 37
.046643 •35564 38
.17956 .048856 .36455 39
.16451 .044085 .34780
40 .18234 .051142 -37356 40 .16700
.046137 .35635
ww www www.m
.029808
.28261
32
.14676 .031407 .29053
33
.14935
.033056 .29850
.15191
.034755
.30652
.15446
.036506
.31462
.15699 .038312 .32278
.15951
.040176 .33103
.16202
.042099 .33937
31

γ
15.0
y = 14.0
γ
Ф
X
Y
T
Ф
X
Y
T
0
0.0
.00000
.000000
.00000 0.0
.00000
.000000
.00000
0.2
.00333
.000006
.00341 0.2
.00332
.000006
.00341
0.4
.00639 .000022
.00668 0.4
.00636
.000022
.00666
0.6
.00923
.000046
0.8
.01188
.000079
1.0
.01437
.000118
.00983 0.6
.01287 0.8 .01177
.01582 1.0 .01421
.00916
.000046
.00979
.000078
.01280
.000116 .01573
1.2
.01673
.000163
.01869 1.2
.01652
.000161 .01857
1.4
.01896
.000214
.02148 1.4
.01870
.000210 .02133
1.6
.02109 .000269
.02421 1.6
.02077
.000364 .02402
1.8
.02313 .000330
.02688 1.8
.02277
.000324
.02666
2.0
.02508
.000395
.02949 2.0
.02467
.000386 .02923
3.0
.03384
.000775
.04186 3.0
.03317
.000756
.04142
4
.04135
.001233
.05333
.04799
.001754
.06412
6
.05398
.002329
.07439
456
.04043
.001 199
.05270
.04684
.001701
.06330
.05260 .002255
.07338
7
.05945
.002953
.08423 7
8
.06452
.003619
.09372
9
.06926
.004326
.10291
689
.05787.002856
.08303
.06275 .003496
.09234
.06730
.004176
.10134
IO
.07371
.005071
.11184
I
ΙΟ
.07158
.004891
.IIOII
II
.07792
.005852
.12057 II
.07562
.005640
.11865
12
.08193
.006667
.12910 12
.07947
.006422
.12701
13
.08576 .007515
.13747 13
.08314
.007236
.13521
14
.08943
.008396
.14571 14
.08666
.008080
.14327
15
.09296 .009310
.15382 15 .09004
.008956
.15122
16
.09637
.010255
.16182 16
.09332
.009862
.15905
17
.09967
.011232
.16974 17
.09648
.010798
.16679
18
.10287 .012241
.17758 18
.09954
.011765
.17447
19
20
21
.10599 .013282
.10902 .014356
.11198 .015462
.18535 19
.10252
.012762 .18207
.19306 20 .10543
.20074 21
.10826
.013790
.18962
.014849
.19713
22
.11487 .016601
.20837 22 .11103
.015940
.20460
23
.11770 .017775
.21599 23
.11374
.017063
.21205
24
.12048 .018983
.22358 24
.11639
.018220
.21948
25
.12321 .020227
.23116 25
.11901
.019410
.22689
26 .12589 .021507
.23875 26
.12158 .020635
.23431
27
.12854 .022825
.24634 27
.12411
.021897
.24174
28
.13114 .024181
.25394 28
.12661 .023194
.24917
29
.13372 .025578
.26156 29
.12907
.024530
.25663
Awwww wwww wwww
30 .13626
.027016
.26922 30
.13150
.025906
.264II
.13877 .028497 .27690 31
32 .14126 .030022 .28463
.13391
.027323
.27164
.14373
.031593 .29241
34 .14617
.033213 .30025 34 .14098
35
.14860
36
.15102
.034882 .30815 35 .1433I
.036604. .31613 36
37 .15342
38 .15581
39 .15819
40 .16056
.038380 .32418
.040213
~ www www
32 .13628
.028781
.27919
33 .13864
.030283
.28680
.031832
.29447
.033429
.30219
.14562
.035075 .31000
37
.14791
.036773 .31787
.042105
.044060
.33232 38 .15020
.34055 39 .15247
.34889 40 .15474
.038526 .32582
.040335 .33387
.042203
.34203
32

y = 16.0
= 17.0
yγ
:
Ф
X
Y
T
Ф
X
Y
T
0.0
.00000
.000000
.00000 0.0
0.2
.0033 I
.000006
.00340 0.2
.00000
.00330
.000000
.00000
.000006
.00339
0.4
.00632
.000021
.00664 0.4
.00629 .000021
.00662
0.6
.00909
.000045
0.8
.01166
.000077
1.0
.01406
.000114
.00975 0.6
.00902
.01274 0.8 .01155
.01564 1.0 .01391
.000045
.000076 .01268
.000113 .01555
100971
1.2
.01632
.000158
.01845 1.2
.01613
.000155
.01834
1.4
.01845
.000206
.02118 1.4
.01822
.000203
.02104
1.6
.02048 .000259
.02384 1.6 .02020
.000255
.02367
1 2
1.8
.02242 .000317
.02644 1.8
.02209
.000311 .02624
2.0 .02427 .000378
.02899 2.0
.02390
.000371 .02876
3.0
.03253 .000737 .04100 3.0
.03193
.000720
.04060
456
78 9
.03957 .001166 .05210
.05133
.04576
.001652 .06253
.002187 .07243
.05641 .002766 .08191
.06111 .003384
.06549
.004038 .09988
.09104
456
789
.03876
.001136 .05154
.04475
.001607 .06180
.05014 .002124 .07154
.05505
.002684 .08085
.05959
.003280 .08983
.06382
.003911 .09851
IO .06961
12
13
14
rottin
.08073
.004726
I I .07350 .005447
.07720 .006199
.006982
.07851
.08411 .007794 .14100 14 .08176
.10848 ΙΟ
.06778
.004575 .10695
.11686 I I .07154 .005270
.12506 12
.11518
.07510
.005995
.12323
.13310 13
.006750
.13113
.007532
.13888
15 .08737 .008635
.14879 15 .08490
.008343
.14652
17
18
678
16 .09051 .009506
.09355 .010405 .16406 17
.09649 .011334
.17158 18
678
.15647 16 .08792 .009181
.15406
.09084 .010046
.16151
.09368 .010941
.16888
19
20
.09935
.012291
.17903
19
.09644 .011862
.17620
.10214
.013278
.18643
21
.10486 .014295
.19378
2 2
20
22
23
24
.10752 .015342 .20III
.II012 .016420 .20840
.11267 .017530 .21568
25 .11518 .018672 .22295
26
27
.11765 .019848 .23021
.12007 .021058 .23749
28 .12247 .022303 .24477
.023584 .25207
30 .12716 .024904 .25940
29 .12483
.21215
25
.11167 .018001
.21928
26
.11404 .019133
.22640
27
.11637 .020296
.23354
2 2 3
.
28 11868 .021494
.24068
29
.12094 .022726
.24784
30 .12319 .023995
.25503
40
Awww wwww www.
.12947 .026263
32 .13175 .027662
33 .13401 .029103
.26677
.27417
www.
.12540 .025301
.26225
32
.12760 .026647
.26951
.28162 33 .12977 .028033
.27682
.13626
.030588
.28913
.13849 .032119 .29669
.14070 .033698
35
.30433
36
37
.14290 .035326 .31204 37
.13832 .034015
38
156
Co to co
34 .13194 .029460 .28418
.29159
.13408 .030931
.13621 .032450 .29907
.30664
.14509 .037007 .31983 38 .14042 .035630 .31428
.14727 .038742 .32771 39
.14944 .040533 .33570 40
.14252
14460
.037298
.32201
.039020 .32984
2 2 2
2 2 2
23
21
22
.10430 .014798 .19786
.10680 .015836 .20501
24 .10926 .016903
.09912 .012812 .18345
.10173 .013790 .19067
33
3

y =
Y
18.0
y = 19.0
ዎ
X
Y
T
Ф
X
Y
T
0
0.0
.00000
.000000
.00000 0.0
.00000
0.2
.00329
.000006
.00339 0.2
.00328
.000000
.000006
.00000
.00338
0.4
.00625
.000021
.00660 0.4
.00622
.000021 .00659
0.6
.00895
.000045
.00967 0.6
.00889
.000044
.00964
0.8
.01144
.000075
.01262 0.8
.01134
.000074
.01257
1.0
.01376
.000III
.01547 1.0 .01363
01 1000*
.01539
1.2
1.6
.01594 .000153
1.4 .01799 .000200
.01993
.01823 1.2
.01576 .000151
.01812
.02090 1.4
.01777 .000196
.02077
.000250
.02351 1.6
.01967
.000246
.02335
1.8
.02178
.000305
2.0 .02354
.000364
3.0
.03136
.000703
.02605 1.8 .02148
.02853 2.0 .02320 .000357 .02831
.04022 3.0 .03082
.000299
.02586
889000*
456
.03800
.001107 .05100
.04381
.001564
.06110
.04903 .002066
.07069
78
9
.05378 .002607
.05817 .003184
.06226 .003794
.07985
.08868
.09722
456
78 9
.03985
.03728 .001081 .05048
.04293 .001524 .06044
.04799 .0020II .06988
.05260 .002535
.07890
.05684 .003094 .08759
.06081
.003685
.09600
IO
II
.06609 .004436
.06972 .005107
12 .07316 .005808
.10551 ΙΟ
.06452 .004307
.10415
13
14
15
17
18
cotin 678
.07645 .006536
.07959 .007291
.08262 .008073
.08050
16 .08553 .008882
.08835 .009717
.09109 .010579
19 .09375 .011468
20 .09634 .012385
21 .09886 .013328
.08332
.11360 II .06803
.12151 12 .07135 .005634
.12927 13 .07453 .006338 .12752
.13689 14 .07757 .007068 .13502
.14439 15
.15180
16
.004956
.I121I
.11989
.007824
.14239
.008606
.14967
.15912 17
.08604
.009413
.15687
.16636 18 .08869
.17354
19
18067 20
.18776 21 .09620
.010246
.16399
.09126
.09376 .011991
.011106 .17105
.17806
.012903
.18502
2 2
22
24
.10611
.10133 .014300
23 .10374 .015300
.016329
.19481 22
.09858
.013841
.19195
.20184 23
.10091
.014807
.19886
.20885 24
.10320
.015801
.20575
22 2
56
25
.10843 .017388 .21585 25
.10544
.016824
.21263
26
.11072 .018478
.22284 26
.10765
.017877
.21950
27 .11297 .019600
.22985 27
.10982
.018960
.22638
www www.
32
28 .11519 .020754
29 .11737 .021942
30 .11954 .023164
.12167 .024423
.12379
.23686 28
.11196
.020075
.23327
.24389
29 .11408
.021221
.24019
.25095 30 .11617
.022402
.24712
.025720 .26516
33
.12588
.027055 .27233 33 .12229
.12796
.02843I .27956 34 .I2429
.13003 .029849 .28684
.13208 .031311 .29418
www wwww
.25803 31 .11823
.023618
.25409
32 .12028
.024868
.26108
.026158
.26813
.027487
.27523
35 .12629
.028857
.28239
36
.12827
.030267 .28961
78
33
37
38
.13412 .032819 .30160 37
.13025
.031723 .29689
.13614
.034375 .30910 38
.13220
.033225 .30426
39
.13816
.035981
.31669 39
.13414
40 .14017 .037640
.32437 40
.13609
.034775 .31172
.036376
.31926
34

y = 20.0
y = 21.0
8
ዎ
X
Y
T
>
ዋ
X
Y
T
0
0.0
.00000 .000000
.00000 0.0
.00000
.000000
.00000
0.2
.00327
.000006
.00338 0.2
.00326
.000006
.00337
0.4
.00619
.000021
.00657 0.4
.00615
.000021
.00655
0.6
.00882
.000044
.00960 0.6
.00876
.000043
.00957
0.8
.01125 .000073
.01251 0.8
.01115
.000072
.01246
Ι.Ο
.01349
.000108
.01531 1.0
.01336
.000107
.01523
1.2
.01559
.000149
.01801 1.2
.01542
.000147
.01791
1.4
.01756
.000193
.02064 1.4 .01735
.000191
.0205 I
1.6
.01942 .000242
.02319 1.6
.01918
.000238
.02304
123
1.8
.02119 .000294
.02567 1.8
.02091
.000290
.02549
2.0
.02287
3.0 .03031 .000673
.000350
.02810 2.0
.02255
.000344
.02790
.03950 3.0
.02982
.000659
.03916
456
78
.03660 .001056
.04999
.04209
.001487 .05981
.04700 .001960 .06911
.05560 .0030II
456
78 a
.03595
.001032
.04952
.04129
.001453 .05921
.04607 .001912 .06838
.05148
.002469 .07800
.08656
.05043 .002407 .07714
.05443 .002934 .08558
9
.05944
.003584 .09484
9
.05815 .003490
.09374
10
.06304
II
12
13
14
.004187
.06644 .004816
.06966 .005472
.07274 .006155
.07569 .006862
ΙΟ
.12587
.13325 14
15
.07852 .007594
.14051 15
cot in
.10287
.06165 .004075
.11071 II .06495
.11836 12 .06808
13
.10166
.004686
.10938
.005323
.11691
.07106 .005985 .12431
.07393 .006671 .13158
.07667 .007381
.13873
16 .08125 .00835I
.14767 16
17
.08389
.009133
.15475 17
18
.00
.08645 .009939
.16176 18
678
.07931
.008115
.14578
.08188
.008873
·
.15274
.08436
.009655
.15965
19
.08894 .010771
.16870
19
.08677
.010461
.16649
20
.09136 .011628
.17560 20
.08911
.011291
.17328
21
.09372 .012510
.18245 21 .09140
.012146
.18002
22
23
24
.09602 .013418
.09828
.10049 .015315
.18927
22 .09363
.013027
.18674
.014353
.19606 23 .09582
.013933
.19342
.20284 24 .09796
.014866
.20010
2 2 2
25
.10266 .016304
.20960
25 .10007
.015824
.20675
26
.10479 .017323
27 .10690 .018371
32
33
333
34
.21637 26 .10214
.22313 27
28 .10897 .019449 .22991 28
29 .11102 .020558 .23671 29
30 .11303 .021700
31 .11503 .022876
.11701 .024086
.11896 .025333
.12090 .026618
.016811
.21342
.10418
.017826
.22007
.10618 .018870
.22672
.10817
.019945
.23343
.24353 30
.25038 31 11205
.25726 32 11396
.26419 33 .11585
.IIOII
.021051
.24014
.022191
.24689
.023364 .25366
.024570 .26048
.27117 34
35
.12283 .027942 .27821 35
36
.12474 .029307
.28531 36
4106
.I1774
.025815 .26735
.11961
.027097
37
.12665 .030714
.29248 37
38
.12854 .032167
.29972 38
78
.27428
.12145 .028420 .28126
.12330 .029783 .28832
.12513
.031190 .29545
40
39 .13042 .033666
.13230
.035214
·30705 39
.31447 40
.12696
.032641 .30266
.12877
.034142 .30997
35

y = 22.0
y = 23.0
Ф
X
Y
T
Ф
X
Y
T
0
0,0
.00000
0.2
.00325
.000000
.000006
0
.00000 0.0
.00337 0.2
.00000
.000000
.00000
0.4
.00612
.000020
.00653 0.4
.00324 .000006 .00336
.00609 .000020 .00652
0.6
.00870
.000043
.00954 0.6
.00864 .000043
.00950
0.8 .01106
.000072
.01240 0.8
.01097
.000071
.01235
1.0
.01323
.000106
.01516 1.0
.01311
.000105
.01508
1.2
.01526
.000145
.01782 1.2
.01510
.000143
.01772
1.4
.01715
881000*
.02039
I.4
.01696
.000185
.02027
1.6
.01894
.000234
.02289 1.6
.01872
.000231
.02275
1.8
.02064
.000285
.02532 1.8
.02038 .000281
.02516
2.0
.02225 .000338
.02936 .000646
.02770 2.0
.02196 .000333
.0275I
456
.04519
789
3.0
.03533
.04054
.001867 .06769
.04943 .002349 .07633
.03884 3.0
.02891
.000634
.03853
ΟΙΟΙΟΟ"
.001420
.04907
.05863
.05332 .002861
.08465
.05694 .003402 .09269
456
789
.03475
.000989
.04864
.03983
.001388
.05808
.04436 .001824 .06702
.04848 .002294 .07555
.05228
.002792
.08376
.05580
.003319
.09169
ΙΟ
.06034 .003970 .10050 ΙΟ .05910
.003872
.09939
II
.06355 .004564
.10811 II
.06223
.004450
.10690
12
13
.06659 .005183
.06949 .005826
14 .07227 .006492
15 .07493 .007182
.11554 12
.06518
.005052
.11423
.12283 13
.12999 14 .07071
.13704
.06801
.005677
.12142
.006325
.12848
15 .07330 .006996
.13543
16
17
.07750 .007895
.07999 .008631
•
.14399 16
.15085 17
18
.08240 .009390
.15765 18
678
.07580 .007689
.14228
.07822
.008404 .14905
.08055
.009142
.15575
19
.08474 .010172
.16439 19 .08283
.009902
.16240
20
21
2 2 2
25
26
56
27
.08701.010978
.08923 .011808
22 .09140 .012663 .18434
23 .09352 .013542 .19092
24 .09560 .014447
.09765 .015377 .20406 25 .09537
.09965 .016335 .21062 26
.10163 .017320
.17108 20 .08504
.010685
.16899
.17772 21 .08720
.01 1492
.17554
2223323
.08931
.012323
.18206
.09137 .013177
.18856
.19750 24 .09339 .014056
.19504
.014960
.20151
.09732 .015891
.20797
.21718 27
.09924
.016848
.21444
28
·
10358 018333
.22375 28
.10114 .017833
.22092
29
.10550 .019376
.23034 29
.10300
.018845
.22741
30 .10739 .020449
.23695 30
.10485 .019888
.23393
31
333
23
333
456
32
33
34
35
.10927 .021554 .24359 31
.IIII2 .022692 .25027
.11296 .023863 .25698 33 .I1026 .023205
www.mo
.10667
.020961 .24047
32
.10847
.022066 .24706
.25368
.I1479 .025070 .26375 34 .11203
.11660 .026314 .27057 35 .11378
*.024377
.26034
.025585
.26706
36 .11839 .027597
.27745 36
.11553
.02683.I
.27385
37
38
.12018 .028919
.12196 .030284
.28440 37
.I1727
.028115
.28069
.29142 38
.11900
.02944 I
.28760
39
.12373
40
.031692
.12549 .033146
.29852 39 .12072
.030809
.29461
.30572 40 .12243
.032221.
.30170
36
y = 24.0
y = 25.0
Ф
X
Y
T
Ф
X
Y
T


0
0.0
.00000
.000000
.00000 0.0
.00000
.000000
.00000
0.2
.00323 .000006
.00336 0.2
.00322
0.4
.00606
.000020
.00650 0.4
.00603
.000006 .00335
.000020 .00648
0.6
.00858
.000042
.00947 0.6
.00853
0.8
.01088
.000070
.01230 0.8 .01079
.000042 .00943
.000069
.01225
Ι.Ο
.01299
.000103
.01501 1.0
.01287
.000102
.01494
I.2
.01494
.000141
.01763 1.2
.01480
.000139 .01754
1.4
.01678 .000182
.02016 1.4
1.6
.01850
.000227
.02261 1.6
+6
.01660
.000180
.02004
.01829
.000224
.02247
1.8
.02013
.000276
.02500 1.8
.01989
.000272
.02484
2.0
.02168
.000327
.02732 2.0
.02142
.000322
.02714
3.0 .02849
.000622
.03823 3.0
.02808
.00061I
.03794
456
789
.03419
.03915 .001359
04357
.05755
.001784 .06638
.04759 .002242 .07481
.05129 .002728 .08291
9 .05472 .003241
.000969 .04822
456
.03366
.000951
.04782
.03850 .001331 .05704
.04282
.001746 .06577
718
.09074
78 9
.04674
9
.002193 .07410
.05035 .002667 .08210
.05370 .003168 .08983
10
.05794
.003780
II
.06098
.004343
I I
12
.06386
.004929
.09834
.10575
.11298 I2 .06261
ΙΟ .05684
.003693
.05980 .004242 .10465
.09733
.004813
.11179
13
14
15
34 in
.0666I
.005537
.I2008 13 .06529
.005406
.11880
.06924 .006168
.07176 .006821
.12704 14
.06785
.006021
.12567
.13390 15
.07031
.006657
.13244
16
17
18
678
.07420 .007496
.14066 16
.07268 .007314
.13912
.07655 .008192 .14734 17
.07498
.007993
.14571
.07882 .008910
.15395 18
.07719
.008692
.15223
19
.08104 .009650
.16051 19
.07935
.009413
.15870
20
21
.08319
.08529 .011197
.010412
.16701 20
.08145
.010156
.16512
.17347 21 .08349
.010920
.17149
22
23
.08734 .012005
.08935 .012836
24 .09132 .013691
.17990 22
.08549 .011708 .17784
.18631 23
.08744 .012517
.18416
.19270 24
.08936 .013350
.19047
N N N
26
567
25 .09324 .014571 .19908 25
.09514
.015476 .20545
27 .09701 .016407 .21183
28 .09885 .017365 .21822
29 .10066 .018350 .22463
.10246 .019364
.09491 .015994 .20935
28 .09669 .016927
2. 2
29
.21565
.09846 .017886 .22198
.23105 30 .10021 .018874 .22831
56
2 2 2
.09124 .014206
.19676
26
27
.09309 .015088
.20305
30
www.
31
32
.10423 .020408
.10598 .021483
33 .10772 .022590
.2375I 31
.24400 32
1 2
.IO194
.019891 .23469
.10364
.020937
.24109
.25053 33 .10533 .022015
.24753
دبا دبا دیا
34
.10944
35
.III15
36
.11285
.023730
.024905
.026116
.25710 34
.26373 35
456
.10701 .023125
.25402
.10868 .024269
.26057
.27042 36 .11033 .025448
.26716
37
.11454
38
.11622
.027366
.028654
.27717 37
.28399 38
78
.11197
.11360 .027919 ,28056
.026665 .27383
39
.11789
.029985
40 .11956 .031358
.29090 39 .11523 .029214
.29789 40 .11686
.030551
.28737
.29427
37

y = 26.0
y = 27.0
Ф
X
Y
T
Ф
X
Y
T
0
0.0
.00000
.000000
.00000 0.0
.00000
.000000
.00000
0.2
.00322
.000006
.00335 0.2
.00321
.000005
.00334
0.4
.00600
.000020
.00646 0.4
.00597
.000020
.00645
0.6
.00847
.000041
.00940 0.6
.00842
.00004I .00937
0.8
.01071 .000069
.01220 0.8
.01063
.000068
.01215
1.0
.01276 .000101
.01487 1.0
.01265
.000100
.01480
1.2
.01466
.000137
.01745 1.2
.01452
.000136
.01736
1.4
.01643
.000177
.01993 1.4
.01626
.000175 .01982
1.6
.01809 .000221
.02234 1.6
.01790
.000218 .02221
1.8
.01966 .000268
2.0 .02116 .000317
.02469 1.8 .01944
.000264 .02454
.02697 2.0
.02091
.000312 .02680
3.0
.02769 .000600
.03766 3.0
.02731
456
78
.03315
.000933 .04743
.03788
.001305 .05655
.042II
.04594
.04946 .002610 .08132
9 .05273 .003099
.001711 .06518
.002147 .07341
.08896
456
789
.000590 .03739
.03266 .000916 .04706
.03729 .001280 .05608
.04142 .001677 .06461
.04517 .002104 .07275
.04861 .002556
.08057
.05181
.003034
.08812
ΙΟ
.05580 .003611
.09637
10
.05481
.003534 .09545
II
.05869 .004147
.10360 II
.05763
.004057
.10259
12
.06143 .004704
.11065 12
.06030
.004601
.10956
13
.06404.005283
.11757 13
14
.06654 .005883
.12436 14
15
.06894.006503
.13104 15
34 in
.06285 .005167 .11640
.06529
.005752 .12310
.06764
.006358 .12971
16
18
.07125 .007144
17 .07349 .007806
.07565
.13764 16
.008488
678
.06989
.006984 .13623
.14415 17
.15059 18 .07418 .008296
.07207 .007630 .14266
.14902
19
.07775 .009191
20 .07980 .009915
.15698 19 .07624
.008982
.15533
.16331 20 .07823
.009688
.16159
21
.08179 .010660
.16961 21 .08018
.010416
.16781
22
.08374 .011428
.17588 22 .08208
.011165
.17400
23
.08564 .012217
.18212 23
.08394
.011935
.18017
24
.08751 .013029
.18835 24 .08576 .012727
.18632
2 2 2
25
.08935 .013864
.19456 25
26
.09115 .014723 .20077 26
96
.08755 .013542
.19246
.08931
.014380
.19859
27 .09292 .015607
.20699 27
.09104 .015242
.20474
Awww wwww www w☺ N
29
30
28 .09466
.09639
.09809
.016516
.017451
.018414
.21321 28
.09274 .016130
.21088
.21945 29
.22571 30
.09442 .017042 .21704
.09608 .017981 .22322
.09977 .019405
.23200
.10143
.020425 .23832
.10308 .021475 .24468
34
35
36
.10634
.10795
.10472 .022557 .25109 34 .10255
.023672 .25755
.024821 .26406 36
37 .10955
.026007 .27064
38
.III14
.027229 .27729
39
40
.11273 .028492 .28401 39
.11431 .029795 .29082 40
Awww www www
.09772
.018948 .22944
.09935
.019943 .23568
.10095
.020967 .24196
.022023
.24829
35
.10413
.023111 .25467
.10570
.024232
.26110
37
.10726
.025388
.26760
38
.10882 .026581
.27417
.11037 .027812
.28081
.11190 .029084
.28754
38

Υ
y = 29.0
= 28.0
❤
X
Y
T
X
Y
T
0
0.0
.00000
.000000
.00000
0.0
.00000
.000000
.00000
0.2
.00320
.000005
.00334
0.2
.00319
.000005
.00334
0.4
.00594
.000020
.00643
0.4
.00591
.000020
.00642
0.6
.00836
.000041
.00934 0.6
.00831
.000041 .00931
0.8
.01054
.000067
.01210 0.8
.01047
.000067 .01205
1.0
.01254
.000099
.01474
1.0
.01243
.000098
.01468
1.2
.01438
.000134
.01728 I.2
.01425
.000132
.01719
1.4
.01610
.000173
.01972 1.4
.01594 .000171
.01962
1.6
.01771
.000215
.02209 1.6
.01752
.000212
.02197
1.8
.01922
.000260
.02440 1.8
.01901
.000257
.02426
2.0
.02066
.000308
.02664
2.0
.02043
.000303
.02648
3.0
.02695
.000581
.03712 3.0
.02660
.000571
.03687
4
.03219
.000900
.04670
5
.03673
.001256
.05562
6
.001645
.04077
.06406
78 9
.04444
.002062
.07211
.002505
.04780
.05093 .002972 .08731
.07984
456
78 9
.03174
.000884
.04635
.03619 .001233
.05518
.04015 .001614 .06353
.04374
.002022
.07150
.04703 .002456 .07914
.05009
.002913
.08654
10
.05386
.003461
II
.05662
.003972
12
.05923
.004504
.09456 IO
.10161
.10851 12
.05296
.003391
.09371
II
.05565 .003891
.10068
.05821
.004412
.10750
13
14
15
34in
.06172
.005057
.06411
.005629
.06640 .006221
.11527 13
.12190 14 .06298
.12843 15
.06064
.004952
.11419
.005512 .12074
.06522
.006091 .12720
16
.06860 .006832
.13487 16
.06737
.006688
.13357
17
18
78
.07073
.007463
.14123
.07279
.008114
.14752 18
78
.06945
17
.007305
.13986
.07147
.007941
.14608
19
.07480
.008784
.15375 19
.07343 .008596
.15224
20
.07675
.009474
.15994 20
.07534 .009271
.15836
21
.07865
.010185
.16609 21 .07720 .009966
.16444
22
.08050
.010916
23
.08232
.011668
24
.08410
.012442
.17221 22 .07901
.17830
.08078
23
.18438 24 .08252
.010680
.17049
.011415
.17651
.012172 .18252
25
.08585 .013238
26
.08756 .014056
.19651 26
27
.08925 .014898
.20258 27
567
.19045 25 .08423 .012950
.18852
.08590 .013750 .19452
.08755 .014572 .2005I
28
29
.09092
.09256
.015765
.20865 28 .08919
.015419 .20652
.016656
.21474 29 .09079
.016290
.21254
30
.09418 .017573
.22085 30 .09238
.017187
.21858
31
.09578
.018517
.22699 31 .09394
.018109
.22465
32
.09737
.019488
33
.09894
.020489
34
.10049
.23316 32
.23937 33
.021520 .24562 34
.019058
.09549
.23075
35
.022582
.10204
.25192 35
36
456
.09703
.09854 .021044 .24306
.020037 .23689
.10006
.022082
.24929
.10357
.023676 .25828 36 .10155 .023151 .25558
37 .10510
.024805 .26470 37
38
.10662 .025970
.27119 38
39
.10813
40
.10963
.027172
.028413
.27775 39
.28440
40
78 9 O
.10305
.024254
.26192
.10453 .025393
.26834
.10601
.026567
.27482
.10748 .027779 .28140
39

γ
y = 35
y = 30.0
Ф
X
Y
T
Ф
X
Y
T
0
0
0.0
.00000 .000000
.00000 0.0
.00000
.000000
.00000
0.2
.00318
.000005
.00333 0.2
.00314
.000005 .00331
0.4
.00588
.000019
.00640 0.4
.00575
.000019
.00633
0.6
.00826
.000040
.00928 0.6
.00802
.000039
.00914
0.8
.01039
.000066
.01201 0.8
.01003
.000063
.01179
1.0
.01233
.000097
.01461 1.0
.01186
.000092 .01432
1.2
.01412
.000131
.01711 1.2
.01353
.000124 .01673
1.4
.01579
.000169
.01952 1.4
.01508
.000159 .01906
1.6
.01734
.000209
.02186 1.6
.01653
.000197 .02131
1.8
.01881
.000253
.02412 1.8
.01789
.000237 .02349
2.0
.02020
.000299
.02633 2.0
.01917
.000280 .02561
3.0 .02627
.000562
.03662 3.0
.02476
.000522 .03548
456
.03131 .000869 .04601 4
.03567 .001211
.05476
.03955 .001584 .06303
789
76
.04306 .001984
.07091
.04629 .002409 .07848
9 .04929
.002856
.08579
+36
789
.02937
.000803
.04447
.03335
.001116
.05284
.03689
.001455
.06073
.04009
.001819 .06824
.04302 .002205
.07546
.04574 .002612 .08243
10 .05209
.003325
.09288
ΙΟ
.04829 .003037
.08919
II
.05473 .003814
.09979
I I .05068
.003481
.09576
12
.05724 .004324
.10653
12 .05295
.003943
.10219
13
14
15
34 in
.05962 .004852
.11314 13
.05512
.004422
.10848
.06190
.005400
.06409 .005966 .I2602 15
.11963 14
.05718
.004918
11466
.05917
.005431
.12074
16
17
18
678
.06620
.00655I
.13231 16
.06108
.005961
.12673
.06824 .007155
.13853 17 .06292
.006508
.13265
19
20
21
22
23
.07022 .007777
.07213 .008418
.07400 .009078
.07581 .009757
.07759 .010457
.07932 .011176
.14469 18 .06471
.007071
.13850
15079 19
.06645
.007651
.14431
15684 20
.06813
.008248
.15006
.16285 21
.06978 .008863
.15578
.16884 22 .07138
.009495
.16147
24
.08102 .011916
.17479
.18074 24 .07449
23 .07295 .010146
.16714
.010815
.17279
25
.08269 .012677
26
.08434 .013459
.19260 26
27
.08595 .014264
.19853 27
567
.18667 25 .07600 .011503
.17843
.07749 .01221I
.18407
.07894 .012939
.18971
28
30
.08754 .015092
29 .08911 .015944
.09066
.20447 28 .08038
.21042 29 .08180
.016821 .21640
Awww wwww www.
.09219
32 .09370
33 .09520
34 .09669
.017723 .22240
.018652 .22843
.019608
.020593
.23450 33
www∞∞
.013687
.19536
30
.08320
.014457 .20102
.015249
.20670
.24061
35 .09817 .021608 .24677
36
.09964
.022654
.25299
37
.10109
.023733 .25926
38
.10254
39 .10399
40 .10543
.27851
.024846 .26561 38 .09394
.025994 .27202
.027180
Awww wwww www.
.08459 .016065
.21240
32
.08596 .016904
.21814
.08731 .017768
.22391
34
.08866
.018658
.22972
35
.08999 .019575
.23557
36
.09132 .020520
.24148
37
.09263
.021495
.24745
.022500
.25347
39
.09525
.023538
.25957
40 .09655
.024609
.26574
40

y = 40
y = 45
Ф
X
Y
T
X
Y
T
0
0.0
.00000
.000000
.00000 0.0
.00000
.000000
.00000
0.2
.00309 .000005
.00328 0.2
.00306
.000005
.00326
0.4
.00562
.000018
.00625 0.4
.00551
.000018
.00619
0.6
.00779 .000037
.00901 0.6
.00759
.000036
.00888
0.8
.0097I .000061
.01159 0.8
.00942
.000058
.01141
1.0
.01143
.000088
.01405 1.0
.01105
.000084
.01380
1.2
1.4
2 4
.01301
.01446
811000*
.01639
1.2
.000151
.01864 1.4
1.6
.01582
.000186
.02082 1.6
246
.01254
.0001 12
.01608
.01392
.000144
.01827
.01519
.000177
.02038
1.8
.01709
.000224
.02293 1.8
.01639
.000212
.02242
2.0
.01828
.000264
.02497 2.0
.01751
.000250
.02440
3.0 .02347
.000488
.03449 3.0
.02236
.000460
.03361
456
78 a
.000748
.03468
.02774
.03142 .001037 .05117
.001351 .05875
.04314
456
.02635
.000703
.04196
.02977 .000972 .0497I
.03280 .001263
.05702
.03762
989100*
.06596
9
.04032
.002041 .07288
.04282 .002414 .07956
58 9
7 .03554 .001574
.06397
.03804 .001904
.07064
.04037 .002251
.07708
ΙΟ
II
.04516 .002805
.04736 .003213
.08604 ΙΟ
.04254 .002613
.08332
12
.04945
.003637
.09235 II
.09850 12
.04458
.002992
.08939
.04651
.003385
.09531
13
14
15
16
17
18
3+ in 6 7∞
.05143 .004077
.10453 13 .04835
.003793
.10112
.05333 .004532
.I1045 14 .050II
.004215
.10681
.05515
.005003
.11627 15
.05179
.004651
•
11242
.05690
.005489
.12201 16
.05342
.005101
.11794
.05859
.005990
.12768 17
.05498
.005565
.12340
.06023 .006507
.13328 18
.05650
.006044
.12879
19 .06182 .007039
.13884 19
.05797 .006536
.13414
20
21
.06337
.06488 .008150
.007586
.14435 20
.05941
.007043
.13944
.14983 21
.06080 .007565
.14471
23
222222222
.06635
.008729 .15527
.06779 .009326 .16070 23
24
.06920 .009939 .16611
25
.07058 .010569
.17151 25
26
27
.07194 .011217 .17691
.07328 .011884 .18230 27
222 222
.06216 .008102
.14995
.06350 .008654
.15517
24
.06480 .009221
.16038
26
567
.06608 .009805
.16557
.06734
.06858
.010405
.OI I022
•
17076
.17596
28
.07459 .012570 .18771 28
.06980
.011657
.18116
29
.07589 .013275
.19313 29 .07100
.012310
.18637
30
.07718
.014001 .19857 30 .07219
.012981
.19160
UJ
31
.07844 .014748
78
96
www www wwww
32
.07970
.08094
.015516 .2095I 32 .07452
.016308 .21503
34
.08217 .017123
.22059
35
36
.08339
.017963
.22620
.08461 .018828, .23185
.08581
.019720
.23756
.08701
.02064I .24333
40
.08821
.08940
.021591 .24916
.022571
.25506
دیدی در دی دی دی
در در دی
38
39
.20402 31 .07336
.013672
.19685
.014383 .20213
33 .07567
.015116 .20744
35
456
34 .07681 .015870
.07794 .016647
.21279
.21818
.08240 .020003
40 .08350 .02091I
36 .07907 .017448 .22362
37
78 9
,08018
.018273 .22911
.08129
.019125 .23465
.24027
.24595
41
y = 50
γ
= 55
1

Y
T
Ф
X
Y
T
Ф
X
0
.00000 .000000
.00000 0,0
.00000
.000000
.00000
0.2
.00302 .000005
.00324 0.2
.00298
.000005
.00322
0.4
.00540 .000017
.00613 0.4
.00530
.000017
.00606
0.6
.00740
.000035
.00877
0.6
.00723
.000034
.00866
0.8
.00915
.000056
.0I 124
0.8
.00890
.000054
.01 108
1.0
.01071
.000081
.01357
1.0
.01039
.000078
.01336
1.2
.01212
.000108
.01579 I.2
.01174
.000103
.01553
1.4
.01342
.000137
.01792 1.4
.01298
.000132
.01761
1.6
.01463
.000169
.01998 1.6
.01413
.000162
.01961
1.8
.01576 .000202
.02196 1.8
.01520
.000193 .02154
3.0
2.0 .01682 .000237
.02139 .000435
.02389 2.0
.01620
.000227
.02342
.03282
3.0 .02052
.000414
.03211
456
78 9
.02513 .000663
.04091
.02834 .000915 .04842
.03117 .001188 .05549
.03374 .001480 .06222
.03608 .001788 .06867
.03825
456
.02406
.000629
.03997
.02708 .000867
.04726
.02976
.001124
.05412
78
.03217
.001398 .06065
.03438 .001688 .06691
.002112
.07490
9
.03642
.001994 .07295
10
.04028
.002451
.08093 ΙΟ
.03833
.002313
.07880
II .04219
.002805
.08680 II
.04012
.002645
.08449
12
.04400
.003172
.09253 12
.04182
.002990 .09004
13
.04571
.003553
.09813 13
.04343
.003348
.09548
14
.04736 .003947
.10364 14
.04498
.003719
.10082
15 .04893 .004354
.10905 15
.04646
.004101
.10607
16
17
18
678
.005208
.05333 .005655
.05045 .004774 .I1439 16 .04788
.05191
.004496 .11124
.11966 17
.12488 18
78
.04926 .004904 .11635
.05059 .005323
.12141
19
.05470 .006114
.13004 19
.05188 .005756 .12641
20
.05604 .006588
.13517 20
.05314
.006200
.13138
21 .05734 .007075
.14026 21
.05436
.006658
.13632
22
.05861 .007576
.14532 22
.05556
.007128
.14122
23
.05985 .008091
.15036 23 .05673 .007612
.14611
24
.06107 .008620
25
.06227 *009165
26
.06344 .009725
27
.06460
.010300
.15539 24
.16041 25
.16542 26 .06009
.17044 27
.05787
.008110
.15098
.05899
.008621
.15585
.009147 .16071
.06118
.009688
.16557
28
.06573
.010892
.17546 28
.06225
.010244
.17043
29
.06685 .011501
.18050 29
.06330
.010816
.17531
Awww wwww wwww
.06796
.012128
.18555 30
.06434
.0I 1404
.18021
.06906
32 .07014
33 .07121
.012772
.19062
34
35
36
.013436 .19572
.014119 .20085
.07227 .014822
.07333 .015547 .21122
.07437
.20601
.016294 .21647 36 .07036
www www
.06537
.01 2009
.18512
.06638
.012632
.19006
.06739
.013273 .19503
34
.06839
.013934 .20004
35
.06938
.014614 .20508
.015315 .21017
40
37 .07542
38 .07645
39 .07748
.07851
.017064 .22177 37 .07134
.017858
.016038 .21531
.22713 38 .07231
.016784
.22050
.018678
.23255 39 .07328
.017553
.22575
.019524
.23803 40
.07424
.018348
.23107
42

y = 60
.65
γ
>
X
Y
T
Ф
X
Y
T
0
0
0.0 .00000
0.2
.00294
.000000
.000005
ဝဝဝဝ
0.0
.00000 .000000
.00000
.00320 0.2
.00291
.000005
.00318
0.4
.00520
910000*
.00601
0.4
.00511
.000016
.00595
0.6
.00706
.000033
.00856 0.6
.00691
.000032
.00846
0.8 .00867
.000052
.01093 0.8
.00846
.00005 I
.01079
1.0
.ΟΙΟΙΟ
.000075
.01316 1.0
.00984
.000072
.01298
1.2
I.4
1.6
2+0
.01139
.000099
.01528 I.2
.01107
.000096
.01506
.01257
.000126
.01731 I.4
.01221
.000121
.01704
.01367 .000155
.01927 1.6
.01325
.000149
.01896
1.8
.01469
.000185
.02116 1.8
.01423
.000178
.02080
2.0
.01565 .000217
.02299 2.0
.01514
.000208
.02259
3.0
.01975
.000395
.03145 3.0
.01906
.000378
.03086
4456
.02310
.000599 .03911
.02597
.000824 .04621
.02850
.001067 .05289
456
.02225 .000572
.03833
789
.03078
.001327 .05924
.03287 .001602 .06532
.03480
.001890
78 9
78
.02498 .000786 .04525
.02738 .001018 .05176
.02955 .001264 .05796
.03154 .001525 .06389
.03337 .001799 .06961
.07119
10
II
12
.03661 .002192
.03830.002506
.03991 .002832
.07688 ΙΟ
.08242
.08782 12
.03508 .002086 .07515
II
.03669
.03822
.002384
.002694
.08055
.08581
17
13 .04143 .003171
14 .04289 .003521
15 .04429 .003882
16 .04564 .004256
.04694 .004640
.09310 13
.09829 14
.10340 15
34in
.03967 .003015 .09096
.04105 .003347 .09601
.04238 .003690
.10098
.10843 16
.04366 .004045
.10588
.11339 17
.04489 .004410
.11072
18
.04819 .005037
.11830 18
.04608
.004786
.11551
19
.04941 .005445
.12317 19
.04724
.005173
.12025
21
22
20 .05060
.05176
.05288 .006741
.005865
.12800 20
.04837
.005571 .12495
.006297
.13279 21
.04946
.005981
.12962
.13756 22 .05053
.006403
.13426
23
.05399 .007198
.14231 23 .05158
.006836
.13889
24 .05507 .007668 .14704 24 .05260 .007282
.14350
25 .05613 .008151 .15177
26 .05717 .008647 .15649
27 .05819 .009158 .16122
28 .05920 .009683 .16595 28
2 2 2
567
25 .05361 .007740
.14811
26
.05460 .008211
.15271
27
.05557
.008695
.15731
.05653 .009193
.16191
30
.011350 .18022
32
29 .06019 .010223
.17069 29 .05747
.06118 .010778 .17544 30 .05840
31 .06215
.06311
A www.
www w
36
34 .06500
25 .06593
.06686
.19471
.013809 .19961
.014471 .20455
www wwww.
.009705 .16653
.010232 .17116
.010774 .17581
.011938 .18502 32 .06023 .011332 .18049
.06406 .012543 .18985 33
.06113 .011906 .18519
.013167
.06203 .012497 .18993
.05932
456
.06291 .013107 .19470
.06379
.013734
.19951
37
.06778
.015154 .20954 37
.06467
.014381
.20437
38
39
.06870
.06961
.015858
.21459 38
.06554
.01 5049
.20928
.016584 .21969 39
.06640
.015738
.21425
40
.07052
.017334
.22485 40 .06726
.016449
.21928
43

y = 70
80
γ
Ф
X
Y
T
op
X
Y
T
0.0 .00000
0.2 .00288
.000000
.000005
.00000 0.0
.00000
.000000
.00000
0.4
.00502
910000*
.00317 0.2
.00590 0.4 .00486
.00282
.000005
.00313
.000015
.00580
0.6
.00677
.00003 I
.00837 0.6 .00651
.000029 .00820
0.8
.00827
.000049
1.0
.00959
.000070
1.2
.01078
.000093
1.4
.01188
.0001 18
.01066 0.8 .00792 .000047
.01280 1.0 .00915
.01484 1.2 .01026
.01679 1.4 .01127
.01041
990000*
.01249
.000087
.01445
OI 1000*
.01633
1.6
.01287
.000144
.01866 1.6
.01220
.000134
.01813
1.8
.01381
.000171
.02047 1.8
.01306
.000160
.01987
2.0 .01468
.000201
.02222 2.0
.01387
.000187
.02155
3.0
.01843 .000363
.03031 3.0
.01733
.000337
.02933
456
.02148 .000548 .03761
.02408
.000753 .04438
.02638
.000974
.05074
456
.02013
.000507
.03633
.02253
.000695
.04281
.02464
.000898
.04891
78
.02845 .001209
.05678
.03034
.001458 .06258
9
.03209 .001719
.06816
689
7
.02653
.001114
.05470
8
.02827
.001342
.06025
.02987
.001582
.06560
ΙΟ
.03372
.001992
.07358
IO
.03137
.001832
.07078
II
.03525
.002276
.07884
I I .03278
.002093
.07582
12
.03671
.002571
.08398
12 .0341I
.002363
.08074
13
.03809
.002877
.08901 13
.03537 .002644
.08556
14
15
.04067
.0394I .003194
.003521
.09394 14
.09879
15
45
.03658
.002934
.09028
.03774 .003233
.09492
16
.04189 .003858
17
.04306 .004206
.10829 17
18
.04420
.004564
.11296 18
678
.10357 16 .03885 .003542
.09950
.03993 .003861 .10402
.04097
.004189
.10849
19
.04530 .004933
.11759 19
.04198
.004527
.I1291
20
.04637 .005313
.12218 20
.04296
.004874
.11730
21
.04742
.005703
.12674 21
.04392
.005232
.12167
22
.04844
23 .04943 .006517
24 .05041 .006942
.006104
.13127 22
.13578 23 .04576
.14028 24
.04666
.04485
.005599 .12600
.005977
.13032
.006366
.13463
2006
25
.05137
.007378
.14477 25
.04754 .006765
.13893
.05231 .007826
27 .05323 .008287
.14926 26 .04840 .007176
.15375 27 .04924
28 .05414 .008761 .15825 28 .05008
29 .05504 .009249 .16275 29 .05090 .008479
30 .05592 .00975I .16727
.14323
.007598
.14752
.008032
.15182
.15613
330
.05171
.008938
.16046
www www.
.05680
.010267
.17181 31
32
33
.05938
.011908 .18558
.06022
.012488 .19024
.06106
.013085 .19493
.05767 .010798 .17637
.05853 .011344 .18096
1 2 3
333
.05251 .009410 .16480
32
.05331 .009896
.16917
33
.05409 .010397
.17356
دی در دی
34
35
456
.05487
.010912
.17798
.05564
.01 1443
.18243
36 .05641 .01 1990
.18693
37
.06189 .013701 .19968
37
38
.06272
39
.06354
40
.014337 .20447
.014993
.06436 .015669
38
78
.05717
.012554
.19146
.05793 .013136 .19605
.20932 39 .05868 .013736 .20068
.21422 40 .05944 .014355 .20538
44

Y
100
= 90
Υ
Ф
X
Y
T
ዋ
X
Y
T
0
0
0.0
.00000
.000000
.00000 0.0
,00000 .000000
.00000
0.2
.00276
.000004
0.4
.00472 .000014
.00310 0.2 .00271
.00571 0.4 .00458
.000004
.00306
.000014
.00562
0.6
.00628 .000028
.00804 0.6
.00607
.000027
.00790
0.8
.00760
.000044
.01019 0.8
.00733
.000042
.00999
1.0
.00876
.000062
.01220 I.0
.00842
.000059
.01 195
1.2
.00980
,000082
.014II 1.2 .00940
.000078
.01379
1.6
I.4 .01074
.01161 .000126
.000104
.01592 1.4
.01766 1.6
+6
.01029
.000098
.01555
.OIIIO .000120
.01724
123
1.8
.01242 .000150
.01934 1.8
.01186
.000142
.01887
2.0
.01318 .000175
.02097 2.0
.01257
.000166
.02044
3.0
.01640 .000315
.02846 3.0
.01558
.000296
.02770
456
.01900
.000473 .03521
.02122 .000648
.04146
.02318
689
.000836 .04733
.02494 .001036 .05291
.02655
.001247 .05825
.02803 .001469
.06340
456
78 9
.01802
.000445
.03423
.02010
.000608
.04027
.02193
.000784
.04594
.02357
.000971
.05134
.02507
.001 168
.05650
.02646
.001375
.06147
ΙΟ
II
12
.03072
.03195
.02942 .001701
.001942
.002193
.06838 ΙΟ
.02775
.001592
.06629
.07323 II .02897
.001817
.07098
.07797 I2
.03012
.002051
.07555
13
.03312 .002452
.08260 13
14
.03424 .002721
.08714 14
15
.03531 .002998
.09161 15
17
18
678
16 .03635 .003284
.03734 .003579
.0960I
16
.10036
17
.03830 .003883
.10466 18
34 in 16 7∞
.03121 .002293
.08002
.03226 .002544
.08441
.03326
.002802
.08873
.03422 .003069 .09298
.03515 .003344
.09718
.03605 .003627
.10133
19
.03924 .004195
.10891 19 .03692 .003919
.10544
20
.04015 .004517
21 .04103 .004847
.11314 20 .03777
.004219
.10952
2 2 2
2 2 2
167
22
.04189
23
.04274
.11733 21
.005187 .12151
.12566
.005537
.03859 .004527
.11357
22
.03940 .004845
.11760
23
.04018 .005171
.12162
24
25
26
.04438 .006266 .13394
.04517 .006646 .13807
27 .04596 .007036 .14220
.04356 .005896 .12980 24
.04095 .005506 .12562
27
2 2 2
567
25 .04171 .005850
.12961
26
.04245
.006205
.13360
.04318 .006569
.13759
28
.04673 .007438
.14633 28
.04390
.006943
.14158
29 .04749
.007851
·
15048 29
.04461
.007329
.14559
333
35
36
Oux www w
.04824 .008275
.15464 30 .0453I
.007725
.14960
.04898 .008712
.15882 31 .04600
.008132
.15363
.04972
33 .05044
34 .05116
.05188
.009162
.16301 32 .04669
.008551
.15769
.009625
.16723 33 .04737
.008983
.16177
.ΟΙΟΙΟΙ
.010592
37 .05329
.05259 .011098
.011620
.17577 35
.18009 36
.18445
4560
.17149 34 .04804 .009427
.16587
.04870 .009885
1
.17001
.04937 .010357
.17418
38
39
40
.05399 .012158
.05469 .012713
.05538.013285
.18886
.19332 39
.19784
78 a
333 +
37
38
40
.05002 .010843 .17839
.05068 .011345 18265
.05133 .011862 .18696
.05197 .012396 .19132
45

(B) VALUES OF X' Y' & T'.
Ф
0
I
3 P1 + P₁³
= 0.01
Y
X'
Y'
I
3 P₁ + P₁3
= 0.001
Y
T'
Ф
Χ'
Y'
T'
0
0.2
.00000
.000000
.00000 0.2
.00000
.000000
.00000
0.4 .04040
.000208
.01187 0.4
.05447
.000278
.01376
0.6
.07243
.000486
.02244 0.6
.09336
.000615
.02541
0.8
.09943
.000815
.03214 0.8
.12455
.000994 .03584
1.0
.12301 .001184
1.2
.14407
.001588
.04121 Ι.Ο .15099
.04979 1.2
.001408
.04545
.17417
.001853
.05444
I.4
.16321
.002022
.05796 1.4
.19493
.002323
.06296
1.6
.18080
.002482
.06580 1.6
.21383
.002817
.07108
1.8
.19712 .002966
.07335 1.8
.23123
.003333
.07888
2,0
.21239 .003472
.08066 2.0 .24740
.003870
.08640
2.2
.22676 .003999
.08775 2.2
.26253
.004424
.09367
2.4
.24035
.004545
.09464
2.4
.27678 .004996
.10073
2.6
.25326
.005108
.10136
2.6 .29027
.005585
.10760
2.8
.26557 .005688
.10792 2.8 •30310
.006190
.I1429
3.0
.27734
.006284
.11434 3.0 .31533
.006809
.12084
3.2
.28863
.006896
.12063 3.2
33
3.4 .29949
.007522
.12679 3.4
3.6
.30996
.008162
.13285 3.6
246
.32703
.007443
.12724
.33826
.008090
.1335I
.34907
.008751
.13966
3.8
.32007
.008816
4.0
-32985
.009482 .14466 4.0 .36955
.13880 3.8
.35949
.009425
.14570
.ΟΙΟΙΙΟ
.15164
5.0
.37460
.012995 .17269 5.0
.41543 .013712
.18002
6
.41398 .016781
.19902
7
.44940 .020811
.22404
8
.48174
.025065
.24801
679
.45563 .017575
.20663
.49166 .021675
.23187
8 .52448
.025993
.25601
9
.51163 .029529
.27110
9 .55477
.030516
.27925
ΙΟ
.5395I .034191
.29346
IO .58297
.035234
.30175
II
.56570
.039044
.31521
II
.60945
.040138
.32361
12
13
14
15
16
234 no
.59048 .044082
.33643
12 .63446
.045225
.34493
.61403 .049301
.63651 .054698
.35719
13
.65822
.050490
.36579
.37757 14
.68089
.055931
.38624
.65807 .060272
.39760 15
.70261
.061547
.40635
.67880 .066021
.41734 16 .72349 .067336
.42616
17
.69881 .071946
.43683
17 .74363
.073300
.44572
20
21
23
24
18 .71816 .078048
19 .73694 .084329
-75519 .090791 .49412
.77297 .097437 .51293
22 .79032 .104271 .53163 22
.80728 .III297
•55026
.82389 .118519 .56882 24
.45610 18 .76310
.47519 19 .78198
.079439
.46505
.085756
.48419
20 .80033
.092253
.50317
21 .81820
.098933
.52203
.83563 .105800
.54078
23
.85267 .112858
.55944
.86936 .120III
.57805
25
.84018 .125943
.58735 25
.88571 .127566
.59662
26
.85618 .133575
.60587 26 .90178
.135227
.61517
27
.87192 .141421
.62439 27 .91757
.143102
.63372
28
.88742 .149488
.64293 28 .93312 .151198
.65230
29
.90269
.157784
.66151 29 .94845
.159521
.67091
30
.91778 .166316
.68015 30 .96358
.168081
.68958
33
31
.93268 .175094 .69887 31 .97853
.176886
.70833
33
wwww
32 .94742
.184128 .71768 32 .9933I
.185946
.72717
.96202
.193426 .73661 33 1.00795
.195270
.74612
34
.97648
.203001 .75566 34 1.02246
.204870
.76520
35 .99084
.212864
.77487 35 1.03684
.214758
.78443
46

I
3 P₁ + P₁³ =
0.0001
3 P₁ + p₁s
о
γ
Υ
Ф
X'
Y'
Τ
Ф
X'
Y'
Τ'
0
0
0.2
.00000 .000000
.00000
0.2
0.4
0.6
.05669 .000289
.09649 .000633
.01403 0.4
.02581
.00000
.05701
0.6 09691
.000000
.00000
.000290 .01407
.000635 .02586
0.8
.12819
.001019
.03633
0.8 .12867
.001021 .03639
1.0
.15497
.001438
.04600 1.0
.15549
.001442
.04608
1.2
.17839
.001887
.05504 1.2
.17894
.001891
.05511
1.4
.19934
.002362
.06359
1.4 .19991
.002366
.06366
1.6
.21838 .002860
.07175
1.6 .21897
.002865
.07182
1.8
.23589 .003379
.07957
1.8
.23650
.003385
.07965
2.0
.25216
.003919
.08711 2,0
.25277
.003924
.08719
2.2
.26738 .004476
.09440 2.2
.26800
.004482
.09448
2.4
.28170
.005052
.10148 2.4
.28233
.005058
.10156
2.6
.29525
.005643
.10836
2.6
.29589
.005649
.10845
2.8
.30813
.006250
.11507 2.8 .30877
.006257
.11516
3.0
.32041
.006872
.12163
3.0
.32106
.006879
.12172
دب دب حب حب ج
3.2
3.4
3.6
246
.33215 .007508
.12804
3.2
.34342
.008158 .13432
.35426
.008821
.14049
3.8
.36471
.009496
.14654
4.0
.37480
.010184
.15249
5.0
.42080 .013795
.18090
2 46
33
∞
345
•33281
.007515
.12813
3.4
3.6
.34408
.008165
.13441
.35493
.008828
.14058
3.8
-36538 .009504
.14663
4.0 .37547 .010192
.15258
5.0
.42148 .013803
.18100
648
.46108
.017666
.20754
76
.49717
.021773
.23280
•53005 .026098
.25696
678
.46177
.017676 .20764
.49787
.021784
.23290
.53076 .026109
.25706
10
106
9
.56038 .030627
.28022
9
.56109 .030638
.28032
.58862 .035350
.30272
ΙΟ
.58933 .035362
.30283
II .61512 .040260
.32460
II
.61584
.040273 .32470
12
13
14
15
16
23 + 16
.64016
.04535I
.68662
.66394 .050621
.056067
.34593 12
.36680
.64088
.045365
.34604
13 .66466
.050635
.36690
.38726
14
.68735
.056081
.38737
.70836
.061686 .40738 15
.72925
.067480 .42720 16
56
.70908 .061701
.40749
.72998 .067495
.4273I
17
.74940
.073448 .44675 17 .75013
.073463
.44687
18
.76889 .079591
.46609 18 .76962
.079607
.46620
19
.78778 .085911
.48524 19 .78851 .085928
.48535
20
.80614 .092412
21
.82402 .099095
22
.84146 .105965
23
.85851 .113026
.50423 20
.52309
.54184 22
.56051
.80687
.092429 .50434
21
.82475
.84219 .105983
.099113
.52320
.54195
23
.85924
.113044
.56063
24
.87520 .120283
.57912 24
.87593
.120301
.57924
2 2
25
.89156 .127740
.59769 25
.89230
.127759
.59781
26
.90763
.135405
.61625
26
.90837
•
135424
.61637
27
.92343 .143283
.63480
27
.92417
•
143302
.63492
28
.93899 .151381
.65338 28
.93973
.151401
.65350
29
.95433
.159707
30
.96946 .168270
ww
31
.98441
32
.99920
.67200 29
.69067 30
.177078 .70942 31 .98515
.186140 .72826 32 .99994
.95506
.159727 .67212
.97020
.168290
.69079
.177098
.70954
.186161
.72838
33 1.01384
34 1.02835
.195467
.74722 33 1.01458
.195488 .74734
.205069
.76630 34 1.02909
.205091
.76642
35
1.04274
.214959
.78553
35|| 1.04348
.214981
.78565
47

(C) COEFFICIENTS FOR THE CUBIC LAW OF
RESISTANCE TO OGIVAL HEADED SHOT.
V
Kv
Kv
f-s
Kv
V
Ky
Kv
Κ. ΣΚ
g
60
g
f-s
Κ. ΣΚ
مع
g
g
900
64.4
64
910 64.8
129
2.001
2.014
2.00 1400
4.02 1410
920 65.3
195 2,029
6.04 1420
104.0 4914 3.231 152.65
103.4 5017 3.212 155.86
102,8 5120
3.193 159.06
930 65.9 260
2.047
8.09 1430
102.2
5222
3.175 162.23
940 66.6
327
2.069
10.16
1440 101.6
5324
3.155 165.39
950
67.4
394
2.094
12.25
1450 || 100.9
5425
3.134 168.52
960 68.4 463
2.125
14.38
1460 || 100.2
5525
3.112 171.63
970
69.6 532
2.162
16.54
1470 99.4
5624
3.089174.72
980
71.0 603
2,206
18.75 1480
98.7
5723
3.065 177.79
990
72.8'
676
2.262
21.01 1490
97.9
5821
3.041 180.83
1000 75.0
751
2.330
23.34 1500
97.2
5918 3.018 183.85
IOIO
77.5 829
2.408
25.75 1510
96.4
6015
2.994 186.84
1020 80.4
909
2.498
28.25 1520
95.5
6110
2.968 189.81
1030 83.9 993
2.606
30.85| 1530
94.7
6205
2.942 192.75
1040 88.2
1081
2.740
33.59❘ 1540
93.8
6299
2.915 195.67
1050 92.8 1174
2.883
36.47 1550
93.0
6392 2.889 198.55
1060 97.2 1271
3.019
39.49 1560
92.2
6484 || 2.864| 201.42
1070 100.8
1372
3.131
42.62 1570
91.4
6575 2.839204.26
1080 103.4
1475
3.212
45.84 1580
90.6
6666
2.814 207.07
1090 105.1
I 100 106.0
1581
3.265
49.10 1590
89.8
6756
2.790 209.86
1687 3.293
52.39 1600
89.I 6845
2.768212.63
1793 3.312
3.327 59.03 1620
1150 108.2
1160 108.5
1210 108.9 2877
1200 108.9
108.9 2768
IIIO 106.6
II20 107.1 1900
1130 107.5 2008 3.339 62.37 1630
1140 107.9 2116
3.351
2224 3.361 69.08 1650
2332 3.371 72.46 1660
1170 108.7 2441 3.377 75.83 1670
1180 108.9 2550 3.381
79.21 1680
1190 108.9 2659 3.383
||
3.383 82.60 1690
3.383 85.98 1700
82.1
3.383 89.36 1710
1220 108.9
108.9 2986|3.382
3.382 92.74
1230 108.8
3094 3.381
3.381 96.13
1240 108.8 3203 3.380 99.51
1250 108.7 3312 3.378 102.88
1260 108.6 3420 3.375 106.26| 1760
1270 108.5 3529 3.370 109.63 1770
1280 108.3 3637 3.364 112.99 1780
1290 108.1 3745 3.358 116.35 1790
1300 107.9 3853
3.352 119.70 | 1800
1310 107.7 3961
3.345 123.05 1810
1320107.4 4068
1330 107.I 4175
1340 106.8 4282
1350 106.4 4389
1360 106.0 4495
1370 105.6 4600
1380 105.1 4705
1390 104.6 4810
1400 || 104.0 4914
79.6 8105 2.473 251.76
79.0 | 8184 || 2.454 254.22
78.4 8262 2.435 256.65
77.8
8340 2.417 259.07
77.2
8417|| 2.398 261.47
76.6
8494 2.380 263.85
76.1
8570 2.364 266.21
3.337 126.38
1820
75.5
8645 2.345 268.56
3.328 129.71
1830
74.9
8720 2.327 270.88
|
3.317 133.03
1840
74.3
8794 2.308 273.19
3.305 136.33 1850
73.7
8868 2.289 275.48
1860
73.1
8941 2.271 277.75
1870
3.293 139.63
3.280 142.91
3.265 146.17
3.249 149.42 1890
1880
3.231 152.65| 1900
72.6 9014 2.255 280.01
72.0 9086 2.237 282.24
71.5 9157 2.221284.46
70.9 9228 2.202 286.67
55.71 1610
88.4
6933
2.746 215.38
87.8
7021 2.727 218.10
87.1
7108
2.706 220.81
65.72 1640
86.5 7195
2.687 223.50
85.9 7280
2.668 226.16
85.3
84.6 7450
7366
2.650 228.81
2.628 231.44
84.0 7534
2.609 | 234.05
83.3
7618
2.588 236.64
82.7 7700
2.569 239.21
7782 2.550 241.76
1720
81.5
7864 2.532 244.29
1730
80.8
7945 2.510 246.80
1740
80.2
8025 2.491 249.29
1750
48

(D)
Po
3 tan o + tan ³
ዎ
P
.0
.I
.2
•3
.4
.5
.6
.7
.8
.9
.000000
.005236
.010472
.015708
.020945
.026181
.031418
.036656
.041893
.047132
I
.052371
.057610 .062850
.068091
.073333
.078576
.083819
.089064
.094310
.099557
2
.104805 .110054
.115305
.120557
.125811
.131066
.136323
.141581
.146842
.152103
345678
.157367
.162633
.167901
.173171
.178442
.183717
.188993
.194272
.199553
.204836
.210122
.215411
.220702
.225996
.231293
.236593
.241895
.247201
.252509
.257821
.263136
.268454
.273775
.279100
.284428
.289760
.295095
.300434
.305777
.311123
.316474
.321828
.327186
.332549
.337915
.343286
.348661
.354040
.359424
.364812
.370205
.375602
.381004
.386411
.391823
.397239
.402661
.408088
.413519
.418956
.424398
.429846
.435299
.449757
.446222
.451691
.457167
.462648
.468135
.473628
9
.479126
.484631
.490143
.495660
.501184
.506714
.512251
.517794
.523344
.528900
10
.534463 .540033
.545610
.551194
.556785
.562384
.567989
.573602
.579222
.584850
II
.590485
.596128
.601779
.607438
.613104
.618778
.624461
.630151
.635850
.641558
12
.647273
.652997
.658730
.664471
.670221
.675980
.681748
.687525
.693310
.699105
13
.704910
.710724
.716547
.722380
.728222
.734074
.739936
.745808
.751690
.757581
14
.763483 .769396
.775319
.781252
.787196
.793150
.799115
.805091
.811078
.817076
15
16
17
56 7
.823086 .829106
.835138
.841181
.847236
.853302
.859381
.865471
.871573
.877687
.883813
.889952 .896103
.902266
.908442
.914631
.920832
.927047
.933275
.939515
.945769 .952036
.958317 .96461I
.970919
.977241
.983577
.989927
.996291
1.002669
18 1.009062
1.015469 1.021891
1.028327
1.034779
1.041245
1.047727
1.054224
1.060736
1.067263
19 1.073807 1.080366 1.086941
1.093532
1,100139
1.106762
1.113402
1.120058
1.126731
1.133421
20
1.140127
1.146851 1.153592
1.160350
1.167126
1.173919
1.180730
1.187559 1.194407
1.201272
49
4

(E) TABLE OF VALUES OF
[v versin q].
f-s
V
ΤΟ
20
30
4º
5º
60
70 80
90
100
110 120 130 14°
V
150 160
179
180
f-s
100 0.02
0.06
0.14
0.24
0.38 0.55
0.75
0.97
1.23
1.52
1.84 2.19 2.56 2.97 3.41 3.87 4.37
4.89 100
200 0.03 0.12
0.27
0.49
0.76 I.10
1.49
1.95
2.46
300 0.05
0.18
0.41
0.73
1.14
1.64 2.24
2.92
3.69
400 0.06
0.24
0.55
0.97
1.52
2.19 2.98
3.89
4.92
500 0.08
0.30
0.69 I.22
1.90
2.74 3.73
4.87
6.16 7.60 9.19
600 0.09
0.37
0.82
1.46
2.28
3.29 4.47
5.84
700 0.11
0.43
0.96
1.71
2.66
3.83
5.22
6.82
800 0.12
0.49
1.10
1.95
3.04
4.38
5.96
7.79
900 0.14
0.55
1.23
2.19
3.42
4.93
6.71
8.76
1000 0.15
0.61
1.37
2.44
3.81
5.48
7.45 9.73
1100 0.17 0.67
1.51
2.68
4.19
6.03
8.2010.71
1200 0.18 0.73
1.64
2.92
4.57
6.57
8.94 11.68
1300 0.20 0.79
1.78
3.17
4.95
7.12
1400 0.21 0.85
1.92
3.41 5.33
7.67
1500 0.23 0.91
2.06
3.65
5.71
8.22
1600 0.24
0.97
2.19
3.90
6.09
8.76 11.93 15.57 19.70
1700 0.26
1.04 2.33
1800 0.27
1.10 2.47
1900 | 0.29
1.16
2.60
2000 0.30
1.22
2.74
4.87
7.61 10.96
2100
0.32
1.28
2.88
5.12
7.99 11.50
2200 || 0.34
1.34
3.02 5.36
8.37 12.05
2300 0.35
1.40
3.15
5.60
2400 0.37 1.46
3.29
5.85
8.75 12.60
9.13 13.15
I 100
I 200
7.39 9.12 11.02 13.11 15.38 17.82 20.44 23.24 26.22 29.37. 600
8.62 10.63 12.86 15.30 17.94 20.79 23.85 27.12 30.59 34.26
|
9.85 12.15 14.70 17.48 20.50 23.76 27.26 30.99 34.96 39.15
11.08 13.67 16.54 19.67 23.07 26.73 30.6734.86 39.33 44.05
12.31 15.19 18.37 21.85 25.63 29.70 34.07 38.74 43.70 48.94
13.54 16.71 20.21 24.04 28.19 32.67 37.48 42.61 48.06 | 53.84
14.77 18.23 | 22.05 26.22 30.76 35.65 40.89 46.49 52.43 58.73
9.69 12.65 16.01 19.75 23.88 28.41 33.32 38.62 44.30 50.36 | 56.80 | 63.63 1300
10.4413.6217.24 21.2725.72 30.59 35.88 41.59
30.59 35.88 41.59 47.70 54.23 61.17 | 68.52 | 1400
11.18 14.60 18.47 22.79 27.56 32.78 38.44 44.56 51.11 58.11 65.54 73.42 1500
24.31 29.40 34.96 | 41.01 47.53 54.52 61.98 69.91 78.31 1600
4.14 6.47
9.31 12.67 16.54 20.93 25.8331.23 37.15 43.57
31.23 37.15 43.57 50.50 57.93 65.86 74.28 83.20 1700
4.38
6.85
9.86 13.42 17.52 22.16 27.35 33.07 39.33 46.13 53.47 61.33 69.73 78.65 88.10 1800
4.63 7.23 10.41 14.16 18.49 23.39 28.87 34.91 41.52 48.70 56.44 64.74 73.60 83.02 92.99 1900
14.91 19.46 24.62 30.38 36.75 43.7051.26 59.41 68.15 77.48 87.39 97.89
|
| 2000
15.65 20.44 25.85 31.90 38.58 45.89 53.82 62.38 71.56 81.35 91.76 102.8
|
16.40 21.41 27.09 33.42 40.42 48.08 56.39 65.35 74.96 85.22 96.13 107.7 2200
17.14 22.38 28.32 34.94 42.26 | 50.26 58.95 68.32 78.37 89.10 100.5
112.6 2300
17.89 23.36 29.55 36.46 44.09 52.45 61.51 71.29 81.78 92.97 104.9 117.5 |2400
2100
3.04 3.67 4.37 5.13 5.94 6.81 7.75 8.74 9.79 200
4.56
5.51
6.56 7.69 8.91 I0.22 11.62 13.11 14.68
6.08 7.35 8.74 10.25 11.88 13.63 15.50 17.48 19.58
10.93 12.81 14.85 17.04 19.37 21.85 24.47
300
400
500
700
800
900
1000
50

(F)
TABLE OF VALUES OF [1000 ÷ v]³
V
f-s
I
2
3
4
5
6
7
8
9
500 8.0000 *9522 *9048 *8577 *8110 *7647 *7188 *6732 *6280 *5831
510 7.5386 4944 4506 4071 3639 3211
4071 3639 3211 2787 2365 1947 1532
520 7. 1120 0711 0305 *9903 9503 *9107 *8714 *8323 *7936 *7551
530 6.7170 6791 6415 6042 | 5671 | 5304 4939 4577|| 4217 3861
540 6. 3507 3155 2806 2459 2116| 1775
1775 1436 1100 0766 0434
550 6.0105 *9779 *9454 *9132 *8813 *8495 *8180 *7867 *7557 *7249
560 5.6942 6638 6337 6037 5739
6037 5739 5444 51
5444 5151 4859 4570 4283
570 5.3998 3715 3433 3154 2877 2601| 2328 | 2056| 1786|| 1519
580 5. 1253 0988 0726 0465 0207 *9950 *9694 *9441*9189 *8939
590 4. 8690
7473 7235 6998| 6762 | 6529
600 4.6296 6066
5158 4935 4713 4493 4274
610 4.4057 3841
2991|| 2782|| 2574| 2368| 2163
8444
8199
7955 7713
7473
5836
5609 5383
3626
3413 | 3201
620 4. 1959
1757
1556
1356 1157
0960 0764
| |
0569 0376 0184
630 3.9992 9803
9614
9427 9240
9055 8871
8688 8507 8326
640 3.8147 7969 7792
7616|
7441 | 7267
7094
6922 | 6751 | 6582
5749 5586
5423
5262 5101 4942
650 3.6413 6246 6079 5914
2434 1295 0163 *9038
660 3.4783 4625 4469| 4313 4158 4004 3851 3699 | 3548 3398
670 3.3249 3100 2953 2806 2660 2515 2371 2228 2086 1944
680 3. 1803 1663 1524 1386 1249 1112 0976 0841 0707 0573
690 3.0441 0309 0177 0047 *9917 *9788 *9660 *9533 *9406 *9280
700 2.9155 9030 8906 8783 8660 8539 8418 8297 8177 8058
710 2.7 9399 8222 7051 5887 4730 3578
720 2.6 7918 6805 5698 4597 3502|| 2413 1331 0254 *9183 *8118
730 2.5 7058 6005 4957 3915 2879 1848 0823 *9803 *8789 *7780
740 2.4 6777 5779 4787 3800 2818 1842 0870 *9904 *8943 *7988
750 2.3 7037 6092 5151 4215 3285 2359 1438 0522 *9611*8704
760||2.2 7803 6906 6014 5126 4244 3365| 2492
2492 1623 0758 *9898
0758*9898
770 2.1 9042 8191 7344 6502 5664 4830 4001 3175 2354 1538
780 2.1 0725 *9917 *9112 *8312 *7516 *6724 *5936 *5152*4372 *3596
790 2.0 2824 2056 1291 0531 *9774 *9021 *8272 *7526 *6785 *6047
800 1.9 5313 4582 3855 3132 2412 1696 0983 0274 *9568 *8866
810 1.88168 7473 6781 6092 5407 4726 4047 3372 2701 2032
|
820 1.8 1367 0705 0047 *9391 *8739 *8089 *7443 *6801 *6161 *5524
830 1.7 4890 4260 3632 3008 2386| 1767|| 1152 0539 *9929 *9322
840 1.6 8718 8117 7519 6924 6331 5741 5154 4570 3988 3409
850 1.6 2833 2260 1689 1121 0556 *9993 *9433 *8876 *8321 *7769
860 1.5 7219 6672 6127 5585 5045 4508 3974 3441 2912 2385
870 1.5 1860
1337 0817 0299 *9784 *9271 *8761 *8252 *7746 *7243
880 1.4 6741 6242 5745 5251 4758 4268 3780 3294 2811
2329
890 1.4 1850 1373 0898 0425 *9955*9486 *9020 *8555 *8093 *7633
900 1.3 7174 6718 6264 5812 5361 4913 4467 4022 3580 3140
9101.3 2702 2265 1830 1398 0967 0538 0111*9686*9262 *8841
920 1.2 8421 8003 7587 7173 6761 6350 5941 5534 5128 4725
9301.2 4323 3923 3524 3127 2732 2339 1947 1557 1169 0782
940 1.2 0397 0014 *9632 *9252 *8873 *8496 *8121 *7747 *7375 *7004
950 1.1 6635| 6268 5902 5537 5174 4813 4454 4095 3738 3382
960 1.1 3028 2676 2325 1975 1627 1280 0935 0591
0249 *9908
970 1.0 9568 9230 8893 8558 8224 7891 7560 7230|6902| 6574
51

f-s
O
9
I
8
2
7
6
3
5
4
V
3374
980 1.0
9801.0
.0 6248 5924 5600 5278 4958 4638 4320 4004 3688
990||1.0 3061| 2749| 2439| 2130 1822
1822 1515 1210 0905 0602 0301
1000 1.0 0000 *9701 *9402 *9105 *8810*8515 *8221 *7929 *7638 *7348
10100.9 7059 6771
1020 0.9 4232 3956
1030 0.9 1514 1249
1040 0.8 3900
1050 0.8 6383
1060 0.8 3962
oo.
3725
1401
1070 0.8 1630
10800.7 9383 9164
1090 0.7 7218 7006|
1100 0.7 5131 4927
11100.7 3119 2922
1120 0.7 1178 0988 0798
1130 0.69305 9121 8938
1140 0.6 7497 7320 7143
5916 5632 5350 5069 4789 4510
3132 2860 2589 2319 2050 1781
0456|| 0194 *9934 *9673 *9414 *9157
7882 7630 7378
5404
3019
5161 4920
2785| 2552
7128 | 6879 6631
4679| 4439 4200
2320| 2089
1859
0721
0496 0272
0049 *9826 *9604
8291 8075
6165 | 5957
7859 7645 7431
5750 5543 5337
6485| 6199
3680 3406
0983 0719
8644 8388| 8135
6137|| 5892 | 5648
3488 3253
1174 0947
8944 8725 8508
6795| 6584 6374
4723 4520 4318 4116 3915 3715 3516 3317
2725 |
2529 2334 2140 1946 1753 1561 | 1369
0609 0421 0233 0046*9860 *9674 *9489
8756 8574 8393 8213 8033 7854 7675
6967|| 6791 | 6617| 6443| 6269| 6096 5923
11500.6 5752 5580 | 5410 | 5240 5070 4901 | 4733 4565 4398 | 4232
11600.6 4066 3900 3736 3571 3408 3244 3082 2920 2758 2597
11700.6 2437 2277 2118
1959 1801 1643 1486 1330 1173 1018
11800.6 0863 0709 0555 0401 | 0248|| 0096 *9944 *9793 *9642 *9491
1190 0.5 9342 9192 9043 8895 8747 8600 8453 8307 8161 8015
1200 0.5 7870 7726 7582 7439 7296 7153 7011 6869 | 6728 6587
12100.5 6447 6308 6168 6030 5891 5753| 5616| 5479 | 5342
12200.5 5071 4935 4801
4399 4266 4134 4001
1230 0.5 3738 3607 3477 3347
3088 2960 2831 2703
1240 0.5 2449 2322 2196 2070
|
12500.5 1200 1077 | 0955 | 0833
1260||0.4 9990 9872 9753 | 9635|| 9518|
12700.4 8819 8704 8589 8475 8361
7461
12800.4 7684 7572
1290 0.4 6583 6475
6367
1300 0.4 5517 5412
4666 4532
3218
1944
0712
5206
3870
2576
1323
ΟΙΙΟ
8934
1819 1695 1570 1447
0591 | 0470 0349 0229
9400 | 9283| 9167|| 9050
8247 8134 8021 7908
7908 7796
7350 7239 7129 7019 6910 6801 6692
6260 6153 6046 5939 5833 5727 5622
5307 5203 5099 4995 4892 4789 4687|| 4584
13100.4 4482 4380 4279 4178 4077 3977 3877 3777 3677 3578
1320 0.4 3479 3380 3282 3184 3086 2989 2891 2794 2698 2601
13300.4 2505 2409 2314 2219 2124
2124 2030 1935 1841 1748 1654
13400.4 1561| 1468 || 1375|| 1283
1283 1191 1099 1008 0916 0825 0735
13500.4 0644 0554 0464 0374 0285 0196 0107 0018 *9930 *9842
13600.3 9754 9667 9579 9492 9405 9319 9233 9147 9061 8975
8720 8636| 8551|| 8467| 8383 8300 8217 8134
7886| 7804 7722 7640 7559 7478 7397 7316
7075| 6995| 6916|| 6836 6757 6678 6600 6521
13700.3 8890 8805
13800.3 8051 7968
13900.3 7235 7155
1400 0.3 6443 6365 6287 6210
14100.3 5673 5597 5522 5447
14200.3 4925 4851 4778 4704
6133 6055 5979 5902 5825 5749
5371 | 5296 | 5222 5147 5073 4999
4631 4559 4486 4413 4341 | 4269
14300.3 4197 4126| 4054| 3983 3912 3841 3770 3700 3630 | 3560
14400.3 3490 3420 3351 3281 3212 3143 3075 3006 2938 2870
1450 0.3 2802 2734 2666 2599 2532 2465 2398 2331 2265 2198
14600.3 2132 2066
460|0.3
2001 1935 1870 1804 1739 1674 1610 1545
14700.3 1481|| 1417 1353 1289 1225 1162 1099 1035 0973 0910
14800.3 0847 0785 0722 0660 0598 0537 0475 0414 0352 0291
52

f-s
O
9
I
8
2
7
6
3
5
4
นา
V
1490 0.3 0230 0169 0109 0048 *9988*9928 *9868 *9808 |*9748 *9689
1500 0.29630 9570 9511 9452 | 9394|| 9335 9277 9219 9161 9103
15100.2 9045 8987| 8930 8872 | 8815 8758 8701 8645 8588 8532
15200.2 8475 8419 8363 8307 8252 8196 8141
1530 0.27921 7866| 7811|| 7757 7703 7649 7595
1540 0.2 7380 7327 7274 7221 7168 7115 7063
5209
15500.26854 6802| 6750 6698| 6647|| 6596| 6544
1560 0.2 6341 6290 6240 6189 6139 6089 6039
1570 0.2 5841 5791 5742 5693 5644 5595 5547
15800.2 5353 5305 | 5257
5161 5114 5066
14 5066
15900.2 4878 4831 | 4784 | 4737 4691|| 4644| 4598
1600 0.2 4414 4368 4323 4277 4232|| 4187|| 4141
16100.2 3962 3917 3873 3828 3784 3740 3696
|
1620 0.2 3521 3477 3434 3391 | 3348
3434 3391 3348 3305
16300.2 3091|| 3048 3006| 2964 | 2922
1640 0.2 2671 2629 2588 2547 2506
16500.2 2261
2180 2140 2100
16600.2 1861 1822 1782 1743 1704
2221
8086 8030 7975
7541 | 7487 7434
7010 6958 6906
6493|| 6442
5989 5940
5498 5449 5401
6391
5890
5019 | 4972 | 4925
4552 4506 4460
4096 | 4051
4007
3652 3608 3565
3262
3219 | 3176 | 3133
2880
2838
2796 2754 2712
2302
2465 2424
2465 2424 2383| 2342
2060 2020 1980
1665 1626
1665|| 1626|| 1587
1941 1901
1548|| 1510
1670 0.2 1471 1432
1680 0.2 1090 1052
1690 0.2 0718 0681
1394
1015
0644
1356|| 1317
0977 0940 0903| 0865| 0828 | 0791
|
0608 0571 0535 0499 0462 0426
1279 1241 1203 1165 1127
0754
0390
1700 0.2 0354 0318 0282
17100.1 9999 9964|| 9929||
17200.1 9652 9618|| 9584
0247
0211 0176 0140 0105 0069
0034
9894
|
9859 | 9825 | 9790
9825 9790 9756 9721
9687
9550
9516| 9482 9448
9482 9448 9414 9381 9347
1730 0.1 9314 9280
9247
9213
1740 0.1 8982 8950
8917
8885
8852
1750 0.1 8659 8627
8595
8563
8532
9180 9147 9114
9147 9114 9081 | 9048| 9015
8820 8787 8755 8723 8691
8500 8468
8500 8468 8437 8405 8374
1760 0.1 8343 8311
8280
8249 8218
8187 8156
8187 8156 8126 8095 8064
1770 0.1 8034
8003
7973
7942 7912
7882 7851
|
7882 7851
7821 7791 7761
1780 0.1 7731
7701
7672
7642 7612
7583 7553
7583 7553 7524
7494|| 7465
1790 0.1 7436 7407
7377
7348
7319
7290
7262
7290 7262 7233 7204 7175
1800 0.1 7147 7118
7090
7061
7033
7005| 6976|| 6948|| 6920|| 6892
1810 0.1 6864 6836
6808
6781
6753
6725 6698
6725| 6698|| 6670| 6642 | 6615
6560 6533 6506
1820 0.1 6588
18300.1 6317 |
6291 | 6264|6237|| 6211
1840 0.1 6053 6026 6000 5974
18500.1 5794 5768 5743 5717
1860 0.1 5540 5515 5490 5465
1870 0.1 5292 5268 5243 5219
1880 0.1 5050 5026 5002 4978
1890 0.1 4812 4789 4765 4742
19000.1 4579 4556 4533 4511|
19100.1 4352 4329 4307 4284
19200.1 4129 4106 4084 4062
1930 0.1 3910 3888
5170 5146 5122 5098 5074
4954 4930 4906 4883 4859 4836
4718|| 4695 4672 4649| 4626|| 4602
| |
4488|| 4465|| 4442 4419 4397|| 4374
4217
4239 4217| 4195|
3997
4019
4173| 4151
3997 3975 3953 | 3932
3781 | 3760 || 3738|| 3717
4262|| 4239
3867| 3845
4041|| 4019
3824 3802
1940 0.1 3696 3675
1960 0.1 3281 3261
3612|| 3591
19500.1 3486 3466 3445 3424 3404 3383
3240 3220 3200 3180
3654 3633
3570
| |
3549 3528 3507
3363
3342 3322 3301
3160
3140 3120 3100
6479
6452 6425
6452 6425 6398| 6371 6344
6184| 6158| 6131 | 6105| 6079
5948
5922 | 5897
5922 5897 5871 5845 5819
5692
5666|| 5641
5666 5641 | 5616| 5591
5565
5441
5416 5391
5416 5391 5366 | 5342 5317
5195
1970 0.1 3080 3060 3040 3020 3000 2981 2961 2941 2922 2902
1980 0.1 2883 2863 2844 2824 2805
1990 0.1 2689 2670 2651 2632 2613
2786 2766 2747 2728 2709
2594, 2575 | 2556 | 2538| 2519
1
53
(G) TABLE OF VALUES OF
t
,000
1.0
.001
Eeet.
0.00
0.01
0.02
.000 .000
.002 .002 ,002 .003
.006 .007
.008 .009
.000
h=1/2 gt²=16.0954 t² FEET.

1.002
.003 .004 .005 .006 .007 | 008.
Feet. Feet. Feet. Feet.
0.000.000
100*
100°
100*
.009
Feet. Feet. Feet.
Feet.
Feet.
.003
.009
.000 .001
.004 .004
.010 .OII
.005 .005 .006
.012
.013
.014
0.03 .014 .015 .016 .018
.019
.020 .021 .022
.023 .024
0.04
.026 .027
.028.030
.031
.033
.034
.036 .037
.039
0.05
.040 .042 .044 .045
.047
.049
.050
.052
.054
.056
0.06
.058.060.062 .064
.066
.068
.070
.072
.074 .077
0.07
.079.081.083.086
.088
.091
.093
.095
.098
.100
0.08
.103.106
106.108 III .114
·
.116 .119
.122
.125
.127
0.09
.130.133 .136.139
.142 .145
.148
.151
.155
.158
0.10
.161.164.167.171
.174 .177
.181
.184
.188
.191
0. II
.195.198 .202 .206 .209 .213
.217 .220
.224
.228
0.12
.232.236.240.244
.247
.251
.256 .260
.264
268
0.13
.272 .276 .280 .285
.289
.293
.298 .302
.307
.3II
0.14
315 320 325 .329
.334
.338
.343
.348
.353
.357
0.15
.362.367.372 .377 .382
.387
.392 .397 .402
.407
0.16
.412.417 .422 .428
.433
.438
.444 .449
.454
.460
0.17
.465.471.476 .482 .487
.493
.499 .504
.510
.516
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.25
0.927 .935 .943 .950 .958 | .966
1.006 .014 .022 .030 .038 .047
.521.527.533 .539 .545
.581.587.593.600 ,606 .612
.644.650.657.663|| .670
.710.717.723 .730 .737 .744
.779.786.793.800 .808 .815
.851.859.866 | .874 .881 .889
|
.551
.557
.563
.569
.575
.618
.625 .631
.637
.676
.683
.690
.696
.703
.751
.758 .765
.772
.822
.829 .837
.844
.896
.904 .912
.919
.974
.982 .990
.998
.055
.063
.071
.080
0.26 .088 .096.105 .113 .122
.130
.139
.147
.156 .165
0.27
0.28
0.29
0.30
0.31
0.32
.173.182 .191 .200 .208
.262 .271 .280 .289 .298
.354.363 .372 .382
.449.458.468.478 .487
.547.557.567| .577 .587
.648.658.669.679
.217
.226
.235
.244 .253
.307 .317
.326
.335 •344
.391
.410 .420
.690
0.33
0.34
.753.763 774 .785
.861.872 .883.894
.796
.905
0.35
1.972 .983.994 *.006
*.017
.401
.497 .507 .517
.597
.607
.700 .711
.806 .817 .828 .839
.850
.916 .927 .938 .949 .960
*.028 *.040 *.051.063 *.074
.429 .439
.527
.537
.617
.628 .638
.721 .732 .742
0.36 2.086.098 .109 .121
.133
.144 .156 .168 .180 .192
0.37
.203.215 .227.239
.251
.263 .276
.288 .300 .312
0.38
.324.336.349 .361
-373
.386 .398
.411
.423 •436
0.39
.448|.461 .473 .486
.499
0.40
.575.588 .601 .614
0.41
0.42
.962
.102
.202 .216 .230
.245
0.45
0.46
0.47
.259 .273 .288 .303 .318 .332
.406 .421 435 .450 .465 .480
.555.571.586 .601 .616
.347 .362 .376
.391
.495 .510 .525
.540
.632
.647 .662 .678
.693
0.48
0.50 4.024 .040.056 .072 .088
.708.724 739 755 .770
0.49 3.865.880 .896 .912
.928
•
.786
.802 .817 .833
.849
.944
.960
.976 .992 *.008
.105
.121
.137 .154 .170
•
.679
.785 .799 .812 .826
.511 .524 .537
.627 .640 .653 .666
.706 .719.732 745 .759 .772
.839 .853.866.880 .894 .907 .921
0.43 2.976.990 *.004 *.018 *.032 *.046 *.060
0.44 3.116.130.144.159 .173 .187
.935 .948
*.074 *.088
.550 .562
.692
·
54

t
.008 .009
.000
.006 .007
100*
.005
.002 .003
.004
0.63
0.64
"
Feet.
0.50 4.024.040
Feet. Feet. Feet.
.056.072
Feet.
.088
Feet. Feet. Feet.
.105 .121
Feet. Feet.
.137
.154
.170
0.51
.186.203 .219.236
.252 .269 .285
.302
0.52
.352 .369 .386 .403
.419
.436
.453
.470
•
.319 335
.487.504
0.53
0.54
.521.538 .555 | .573
.693.711.728 .746
.590
.607 .624
.641
.763
.781 .798
.816
.659.676
.833.851
0.55 4.869.887 .904 .922
.940
.958 .976
.994
*.012 *.030
0.56 5.048 .066 .084.102
.120
.138
.156
.174
.193 .2II
0.57
0.58
.229.248.266 .285
.414 433 .452.471
0.59 .603 .622 .641 .660
.641.660
0.60 .794 .814 .833 .852
0.61 5.989 *.009 *.028 *.048
.527
.698 .717 .737
872 .891 .911 .930
*.068 *.088.107.127
|
0.62 6.187 .207 .227 .247 .267 .287 .307 .328
.388.409 .429 .449 .470 .490 .511
.593.613.634 .655 .675
.303
.322
.340 •359
.377 .396
.489
.508
·546
.565.584
679
.756.775
.950
.969
*
.147 *.167
.531
.696
.717
.738
.348.368
-759-779
-352 .372
0.65 6.800.821 | .842.863| .884
.905
.926
.948
.969.990
0.66 7.011 .032 .054.075
.096
.118
.139 .161
.182 .204
0.67
.225.247 .268 .290
.312
•333
-355 -377
.399 .421
0.68
.443 .464 .486 .508
.530
.552
.574
.597
.619 .641
0.70
0.69 .663 .685 .708 730
7.887.909 .932.954
.752 .774
.797
.819 .842.864
.977
.000
*.023.045.068.091
0.71 8.114.137.159.182 .205
0.72 .344 367.390 | .414 .437
.228
.251 .274 .298 .321
.460
.484
.507 .530 .354
0.73 .577 .601 .624
.648||
.671
.695
.719
.743
.766 .790
0.74 8.814 838 .862
.885
.909
.933 .957
.981 *.005 *.030
0.75 9.054 .078.102
.126
.150
.175 .199
.223 .248
.272
0.76 .297 .321 .346 .370
.395
.419
.444
.469
.493 .518
0.77
.543.568.593 .617
.642
.667 .692
.717
.742.767
0.78 9.792 .818 .843.868
.893
.918 .944
.969
*
.994 *.020
0.79 10.045 .071 .096 .122
0.80 .301 327 353 378
0.81 .560 .586 .612 .639 .665
0.82 10.823.849.875 .902 .928
0.83 11.088.115.142 .168 .195
0.84 .357 .384.411 .438.465
0.85 .629.656.684 .711 .739
0.86 11.904 .932 .960 .987
0.87 12.183 .211 .239 .267
0.88 .464.493.521 .549
0.89 12.749 .778 .807 .835
0.90 13.037 .066 .095 .124
0.91 .329 .358 .387.417
*.015.043
.147
.173 .198
.224
.250 .275
.404 .430
.456
.482
.691
.717 .744
.955
.982.008
.508.534
.770 .796
*.035*.061
.222
.249 .276
.303 .330
.493
.766
.520 .547 .574 .602
.794 .821
.849 .876
.071*.099 *.127 *.155
.295 .323
.35I .379
.408.436
.578
.606
.635 .663
.692 .721
.864
.893
.951
.153
.183
.922
.212 .241 .270 .299
.979 *.008
.446
.475
.505
•534 .564 .594
0.92 .623.653.682 .712
0.93 13.921.951.981 *.011
0.94 14.222 .252.282 .313
0.95 .526.557.587.618 .649
0.96 14.834.864 .895 .926 .957
0.97 15.144.175 .207 .238 .269
0.98 .458 .490.521 .553
.553.584 .616
0.99 15.775 .807.839 .871 .903
1.00 16.095 .128.160 .192 .224
.742
.772
.801
.831 .861 .891
*
.041
*.071
*.101
*
.343
.374
.404
.131.161 * *.192
.435 .465.496
.679 .710 .741 .772 .803
.988
*.020
*.051.082 *.
*.113
.301 .332 .364 .395 .426
.648 .680 .711 .743
.935
.967
.999 *.031 *.063
.257
.289
.322 354 .386
55

t
.000 .001 .002 .003
.004
.005
.006
.007
.008 009.
Feet. Feet.
Feet. Feet.
Feet.
Feet.
Feet.
Feet. Feet. Feet.
1.00 16.095 .128 .160 .192
.224
.257
.289
.322
1.01 16.419 .451
.484.517
.549
.582
.615
1.02 16.746 .779 .811 .844
877
.910
.943
-976*
354 386
.009.042
.647 .680 .713
1.03 17.076 .109.142.175 .208
.242
.275
.308 .342 .375
1.04 17.409 .442 .476 .509
||
.543
-577 .610
.644 .678 .711
1.05 17.745.779 .813 .847
.881
.915
.949
.983 *.017.051
1.06 18.085 .118.153 .187
1.07 18.428 .462 .497.531
1.08 18.774.808 .843 .878
.222
.256
.290
.566
.600
.324
.635 .670 .704 .739
.359 -393
.843.878.913
.948
.983
*.018 *.053 *.088
1.09 19.123 .158.193 .228 .264
.299
.334 .369
.405 .440
1.10 19.475 .511| .546
.582
.617
.653 .688
.724
||
1.11 19.831 .867 .903
.938
.974
*.010.046 *.082
1.12 20.190 .226.262
.298
-335
.371 .407 .443
1.13 20.552.589 .625 .661 .698 .734 .771 .808
1.14 20.918.954 .991 *.028 *.065*.101.138 *.175
1.15 21.286 .323 .360 .397 .435 .472 .509 •546
1.16 21.658 .695 .733 .770 .808
.108.146| .184
1.17 22.033 .071
.487.425.563
1.18 22.411 .449 .487 .425 .563
1.19 22.793.831 .869.908
.869.908.946
.760 .795
*.118.154
.480.516
.844 .881
.212.249
.583
.621
.845
.222
.883 .920
.260
.958.995
.297
.335 .373
.602
.640
.678
.716 | .754
.985 *.023
*.062
*.100 *.139
1.20 23.177.216.255.293 •332 .371 .410 .449
.487 .526
1.21 23.565 .604 .643 .682 .721 .760 .Soo .839 .878.917
1.22 23.956.996*.035 *.074 *.114.153.193.232 *.272 *.311
1.23|| 24.351 .390 .430| .470 .509 .549
.589 .629 .669 .708
1.24 24.748.788 .828 .868
.908 .948
.988.028.069*.109
1.25 25.149.189 .230 .270
1.26 25.553 .594 .634 .675
1.27 25.960 *.001 *.042 *.083 *.124 *.165
1.28 26.371 .412 .453.494
.536 -577
1.29 26.784 .826 .867 .909 .95I
.310
.35I
.391 .432 .472 .513
.716
.756
.797 .838
.879 .919
*.206 *.247
*.288 *.330
.619
.660
.701 .743
.992
*.034 *.076
*.118 *.159
||
1.30 27.201 .243 .285 .327
.285.327
.369
.411 .453
.495 .537 .579
1.31| |27.621.664.706 .748
.790
.833
.875 .917
.960 *.002
1.32 28.045 .087| .130 .172
.215
.257
.300
.343
.386.428
1.33 28.471.514 .557 .600
.643
.686
.729
.772
.815 .858
*.117
*.160
*
.204
.552 .595
*.247 *.290
.639 .683.726
.945
.989 *.033
*.077 *.121*.165
.386
.430 .475
.519 .563.608
.830
.875 .919
.964 *.008 *.053
1.34 28.901.944 .987 *.030*.074
1.35 29.334 .377 .421 .464 .508
.277
1.36 29.770 .714 .858 .902
1.37 30.209.254 .298 .342
1.38 30.652.697 .741 | .785
1.39 31.098.143.187 .232
1.4031.547.592 .637 .682 .728
1.41 31.999*.045 *.090 *.136 | *.181
1.42 32.455 .500 .546 .592 .638
1.43 32.913 .960 *.006 *.052*.098
1.44 33.375 .422 .468 .515 | .561
1.45 33.841.887 .934 .981 *.028
1.46 34.309 .356 .403 .450 .497 .544
1.47 34.781.828.875 .923 .970
1.48 35.255.303 .351 .398 .446
1.49 35.733 781 .829 .877 .926
1.50|| 36.215 .263 .311 .360| .408
.322 .367 .412 .457 .502
.773
.818 .863 .909 .954
*.227 *.272*.318 *.363 *.409
.684 .730 .776 .821 .867
*.144 *.190.237.283 *.329
.608 .654 .701 .747 .794
*.074 *.121
*.168.215
.215 *.262
.592 .639 .686 .733
*.018 *.065*.113.160 *.208
.494
.974
.456
*
.542 .590 .638 .685
*.022 *.070 *.118.166
.505 .553 .602 .651
56

(H) COEFFICIENTS FOR THE CUBIC LAW OF Resistance.
PROJECTILES WITH OGIVAL-HEADS.
V
Ky
Kv
f-s
Kv
Log
Ky V
Σ
f-s K₁
g
100
IIO
120
130
140
مع
g
مه
578.1 17.959 1.2543
525.5 16.325| 1.2129
481.7 14.964 1.1750
444.7 13.815 1.1404
412.9 12.827 1.1081
150 385.4 11.972 1.0782
100
Ky Kv Kv
Log
g
مه
مه
O 520 117.3 3.644 | .5616| 298.53
17.10 530115.0 3.573 .5530302.14
32.72 540 112.8 3.504 .5446305.68
180
160 361.3 11.224 1.0501
170
340.1 10.565 | 1.0239
321.2 9.978 .9990
47.08 550 110.7 3.439 .5364 309.15
60.39 560 108.7 3.377 .5285 312.56
72.77 570 106.7 3.315 .5205 315.90
84.36 580 104.6 3.249 .5118 319.19
95.24 590 102.5 3.184 .5030 | 322.40
105.51
600 100.5 3.122 .4944 325.56
190
200
210
289.0
275.3
304.3 9.453 .9756
8.978 .9532
8.552 .9321
220 262.8
8.164
230|| 251.3
7.807
240 240.9
7.484
115.21 610 98.6 3.063.4862 328.65
I24.42 620 96.8 3.007 .4781 331.68
133.18|| 630 95.1 2.954.4704 334.66
.9119 141.54 640
.8925 149.52 650
.8741 157.16 660
93.5 2.905.4632 | 337.59
91.9 2.855.4556 340.47
90.5 2.811 .4489 343.30
250 || 231.3
7.182
6.909
6.651
260 222.4
270 214.2
280 206.5 6.415 .8072 184.84 700
290 199.3 6.191 .7918 191.14 710
300 192.7 5.986 .7771 197.23 720
310 186.5
5.794 .7630 203.12 730
320 180.8
5.617 •7495 208.82
740
330 175.5 5.452 .7366 214.35 750
.8563 164.49
.8394 171.53 680
.8229 178.31 690
670
89.1 2.768.4422 346.09
87.7 2.724 .4352 348.84
86.3 2.681 .4283 351.54
340 170.6 5.299
350 166.0 5.158
360 161.9 5.029
370 158.0 4.907
380 154.4 4.795
390 151.1 4.693
400 148.0 4.599
410 145.2 4.512
420 142.5 4.427
430 139.8 4.343
440 137.2 4.262
450 134.6 4.181
.7242 219.73 760
.7125|| 224.96 | 770
.7015 230.05 | 780
.6908 235.01 790
.6808 239.86 800
.6715 244.61 810
.6627 249.25 820
.6544 253.81 830
.6461 258.28 840
.6378 | 262.66 | 850
.6296 266.96 860
.6213 271.19 870
.6129 275.33 880
.6042 279.39 890
-5957 283-37❘ 900
460 || 132.0 4.101
470 I29.4 4.020
480 126.9 3.942
490 124.4 3.865 .5872 287.28 910
500 121.9 3.787 .5783 291.10 920
510 119.6 3.715 .5700 294.85 | 930
84.9 2.637.4211 354.20
83.7 2.600 .4150 356.82
82.6 2.566.4093 359.40
81.6 2.535 .4040 361.95
80.6 2.504 .3986 364.47
79.6 2.473.3932 366.96
|
78.7 2.445.3883 369.42
78.0 2.423.3844 | 371.85
77.4 2.404 .3809 374.26
76.8 2.386 .3777 | 376.66
76.2 2.367 3742 379.03
75.6 2.348.3707 381.39
75.2 2.336.3685 383.73
75.1 2.333 .3679 386.07
75.0 2.330.3674 388.40
75.0 2.330.3674 | 390.73
75.0 2.330.3674 | 393.06
75.0 2.330.3674 395.39
75.0 2.330.3674 397.72
75.0 2.330.3674 400.05
75.0 2.330.3674 | 402.38
75.0 2.330.3674 404.71
75.0 2.330 .3674 407.04
75.0 2.330.3674 | 409.37
57
(H) COEFFICIENTS FOR THE CUBIC LAW OF RESISTANCE.

411.70 1420|| 103.5
414.03 1430 102.9
|
75.0 2.330 .3674 | 416.36 | 1440 || 102.3
|
V
Ky
f-s
Ky
مع
Ky
Log-
g
Kv
Σ
V
V
f-s Kv
go
Ky
g
Log
KvKv
g g
940
75.0 2.330 .3674
950
75.0 2.330 .3674
960
970
75.0 2.330.3674
980
75.0 2.330.3674
990
75.0 2.330.3674
1000
75.0 2.330.3674
ΙΟΙΟ 75.I 2.333 .3679
1020 75.3 2.339 .3690
1030 76.7 2.383
.3771
432.69 1510
435.13 1520
437.73 1530-
1040 80.8 2.510
.3997
1050 87.3 2.712 .4333
1060 94.0 2.920 .4654
1070 98.7 3.066| .4866
1080 102.2 3.175 .5017
1090 104.9 3.259 .5131
1100 106.9 | 3.321 .5213
IIIO 108.4 3.367.5272
418.69 1450|| 101.6
421.02 1460|| 100.9
423.35 1470 || 100.1
425.68 1480 99.4
428.01 1490
430.34 1500
440.55 1540
443.55 1550
446.68 1560
3.215.5072 560.81
3.197 .5047 564.01
3.178.5022 | 567.20
3.156 .4991 570.37
3.134.4961 | 573.51
3.110 .4928 | 576.63
3.088 .4897 579.73
98.6 3.063.4862 582.80
97.9 3.041 .4830 585.86
97.1 3.016 .4794 588.89
96.2 2.988 .4754 591.89
95.3 2.960 .4713 594.86
94.4 2.933 .4673 597.81
93.6 2.908 .4636 600.73
92.8 2.883 .4598 603.62
449.90 1570
453.19 1580
456.54 1590
92.0 2.858 .4561 606.49
91.2 2.833 .4523 609.34
90.4 2.808.4484
612.16
459.92 1600 89.7 2.787 .4451 614.95
89.0 2.765 .4417 617.73
88.3 2.743.4382 620.48
463.32 1610
466.73 1620
470.13 1630
473.54 1640
476.94 1650
480.35 1660
483.75 1670
487.16 1680
490.56 1690
493.97 1700
497.37 1710
500.77 1720
504.18 1739
507.57 1740
I 120 109.2 3.392.5305
1130|| 109.6 3.405 | .5321
1140|| 109.6 3.405 .5321
1150 109.6 3.405 .5321
1160|| 109.6|| 3.405 | .5321
1170 109.6 3.405 .5321
1180|| 109.6 3.405 | .5321
1190 109.6 3.405 .5321
1200 109.6 3.405 | .5321
1210 109.6 3.405 .5321
1220 109.6 3.405 .5321
1230 109.5 3.402.5317
1240 109.53.402 .5317
1250 109.4 3.398 | .5312
1260 109.3 3.395 .5308
1270|| 109.2| 3.392 | .5305
1280 109.0 3.386.5297
1290|| 108.8 3.380.5289
1300 108.6 3.374 .5282
1310 108.4 3.367.5272
1320 108.1 3.358 | .5261
1330 107.8 3-349 .5249
1340 107.5 3.339 | .5236
1350 107.I 3.327 | .5221
1360|| 106.7 3.315 5205
1370|| 106.3 3.302 .5188
1380|| 105.8 3.287.5168
1390 || 105.3 | 3.271 .5147 551.08 1870
1400 104.73.252 .5122 554-34 1880
1410 104.1 3.234 .5097 557.58 1890
510.97 1750
514.36 1760
517.74 1770
521.12 1780
524.49 1790
527.85 1800
87.6 | 2.721 .4347 623.21
86.9 2.700 4314 625.92
86.2 2.678.4278 | 628.61
85.5 2.656.4242 631.27
84.8 2.634.4206 633.92
84.2 2.616 .4176 636.54
83.6 2.597 .4145 639.15
83.0 2.578.4113 641.73
82.4 2.560.4082 644.30
81.8 2.541 .4050 | 646.85
81.2 2.522 .4018 649.39
80.6 2.504 .3986 651.90
80.0 2.485.3953 654.39
79.5 2.470 3927 656.87
78.9 2.451.3893 659.33
78.4 2.435.3865 661.78
77.8 2.417.3833664.20
77.3 2.401.3804 666.61
76.8 2.386 .3777 669.01
76.2 2.367.3742 671.39
75.7 2.352 .3714 673.75
531.20 1810
534.54 1820
537.88 1830
|
541.20 1840 75.2 2.336.3685 | 676.09
544.51 1850 74.7 2.321 .3657 678.42
547.80 1860 74.2 2.305 .3627 680.74
73.6 2.286 .3591 683.03
73.1 2.271 .3562 685.31
72.6 2.255
3532 687.58
58
(H) COEFFICIENTS FOR THE CUBIC LAW OF RESISTANCE.

V
Κν
Κν
f-s
K,
Log
g
| 00
M
مه
Kv Kv
Ky
g
V
Ky
f-s
Ky
Log
g
g
go
مة
1900 72.1
2.240
•3503
1910|| 71.6
2.224
.347I
1920 71.2
2.212
•3448
689.83 2350 62.6
692.06 2360 62.0
694.28 2370|| 61.4
1.945 .2889 784.36
1.926 .2847 786.29
1.907 .2804
788.21
1930 70.8
2.199
.3422
696.48 2380 60.8
1940 70.4
2.187
-3399
1950 70.0
2.175
-3375
698.68 2390 60.2
700.86 2400|| 59.6
1.889 .2762
1.870 .2718 791.98
790.11
1.851
.2674 793.85
1960 69.7
1970 69.4
1980 69.2
1990 69.0
2000
2010
68.8
68.6
2.165-3355
2.156 .3337
2.150 .3324
2.143 .3310
2.137 .3298
2.131.3286
703.03 2410|| 59.0
705.18 2420 58.4
707.34 2430 57.8
709.48 2440 57.2
711.62 2450 56.7
713.76 2460 56.2
1.833
.2632795.69
1.746
1.814 .2586|797.51
1.796.2543 | 799.32
1.777 .2497 801.10
1.761.2458 802.87
.2420 804.63
2020
2040
2050 68.1
68.4
2030 68.3
68.2
2.125 .3274
715.89 2470 55.7
1.730 .2381 806.37
2.122
.3268
718.01 2480 55.2
1.715
.2343 808.09
2.119 .3261
720.13 2490 || 54.8
1.702
.2310 809.80
2.116
.3255
2060|| 68.0
2070 67.9
2.112
2.109
-3247
.324I
722.25 2500 54.4
724.37 2510 54.0
726.48 2520 53.7
1.690
.2279 811.50
1.678 | .2248 | 813.18
1.668
.2222 814.85
2080 67.9
2.109 .3241
2.106
.3235
2090 67.8
2100 67.8 2.106 •3235
728.59 2530 53.4
730.70 2540 || 53.1
732.80 2550 52.9
1.650
1.659 .2199 816.51
.2175 818.17
1.643.2156 | 819.81
2130 67.6
2110 67.7 2.103 .3228
2120 67.6 2.100 .3222
2.100 .3222
734.90 2560 52.7
737.01 2570|| 52.6
739.11 2580 52.5
1.637 .2141
821.45
1.634
1.631
.2133 823.09
.2125 824.72
2140 67.5
2150 67.4
2.094
2.097.3216
.3210
2160 67.3
2.091 .3204
741.20 2590|| 52.5
743.30 2600 52.4
745.39 2610 52.4
1.631 .2125 826.36
1.628
.2117 827.99
1.628
.2117 829.61
2170 67.2
2.088 .3197
747.48 2620|| 52.4
1.628
.2117831.24
2220
2180 67.2
2190 67.1
2200 67.0
2210 66.9
66.8
2230 66.8
2240 66.7
2250 66.6
2260 66.5
2270|| 66.4
2.088 .3197
749.57 2630 52.3
1.625
.2109 | 832.87
2.084.3189
751.65 2640 52.3
1.625
.2109 834.50
2.081 .3183
753.73 2650 52.3
1.625
.2109 836.12
2.078 .3177
2.075 .3170
755.81 2660 52.2
757.88 2670 52.2
1.622 .2101 837.74
1.622 .2101 839.36
2.075 .3170
2.072.3164
2.069 .3158
759.96 2680 52.2
762.03 2690|| 52.1
764.10 2700|| 52.1
2.066 .315I
766.17 2710|| 52.1
2.063 .3145 768.23 2720 || 52.0
2280|| 66.2 2.056 .3130 770.29 2730 52.0
1.622 .2101 840.99
1.618
1.618
.2090 842.60
.2090 844.22
2290 65.9 2.047 .3111
2300 65.5 2.035 .3086
2310|| 65.0 2.019 .3051
2320 64.4 2.001 .3013
2330 63.8
63.8 1.982 .2971
|
2340 63.2
1.963 .2929
772.35 2740 | 52.0
774.40 2750 | 52.0
776.43 2760 52.0
778.44 2770 52.0
780.43 2780 52.0
782.40 2790 || 52.0
1.618 .2090 845.84
1.615 .2082 847.46
1.615.2082849.07
1.615.2082850.69
.2082 852.30
1.615
1.615 .2082 853.92
1.615 .2082 855.53
1.615 .2082 | 857.15
1.615.2082 858.76
59
(J) TABLE OF Values of Rv.
PROJECTILES WITH OGIVAL-HEADS.

V
O
I
2
f-8
3
4
5
6
7
8
9
Diff.
+
1025.8 27.4
29.0 30.6 32.1
1140.3
41.5 42.7 43.9 45.0
1251.27 52.23 53.16 54.07 54.96
33.5
135.0 36.3
37.7 39.0
1.5
46.1
47.2
48.3
49.3 50.3
I.I
55.83
56.67 57.50 58.30 59.09
87
New Auro
69
55
13 59.86 60.61 61.34 62.06 62.76 63.45 64.12 64.78 65.42 66.05
14 66.67 67.27 67.86 68.44 69.01 69.56 70.10 70.63 71.16 71.67
15 72.17 72.66 73.14 73.61 74.07 74.52 74.97 75.40 75.83 76.25 45
16 76.66 77.07 77.46 77.85 78.23 78.61
1780.39 80.72 81.05 81.38 81.70 82.02
1883.51 83.79 84.07 84.35 84.62 84.88
1986.15 86.39 86.62 86.86 87.09 87.32 87.54 87.76 87.98
20 88.40 88.61 88.81 89.01 89.21 89.41 89.60 89.79 89.97
21 90.34 90.52 90.70 90.87 91.04 91.21 91.38 91.54 91.70
2292.02 92.18 92.33 92.48 92.63
2393.49 93.63 93.76 93.89 94.02
24 94.78 94.90 95.02 95.14 95.25
2595.917 6.023 | 6.128 | 6.232
26 96.924 7.019 7.112
7.112 7.205
2797.823 7.907 | 7.991 | 8.073
78.98 79.34 79.69 80.04
38
82.32 82.63 82.93 83.22 32
85.15 85.40 85.66 85.90 26
88.19
90.16
23
20
91.86
17
92.78 92.93 93.07 93.21 93.35
94.15 94.28 94.41 94.53 94.66 13
95.37 95.48 95.59 95.70 95.81 II
6.334 6.436 6.536 6.635| 6.732 | 6.829 101
7.296| 7.386| 7.475
8.155 8.236 8.315 8.394 8.473 8.550|| 81
15
7.564 7.651 7.737 90
|
41
28 98.627 8.702 8.777 8.852 8.925 8.998| 9.069 | 9.141
9.141 9.211 9.281|| 73
2999.349 9.418| 9.485 | 9.552| 9.618| 9.683 | 9.748| 9.812| 9.875| 9.938|| 65
30||
30 0.000 0.062 0.122 0.183 0.242 0.301 0.360 0.417 0.475 0.532 59
31 0.588 0.643 0.699 0.753 0.807 0.861 0.914 0.967 1.019 1.070 54
32 1.122 1.172 I.222 1.272 1.321 1.370 1.418 1.466| 1.513| 1.560 || 49
33 1.606 1.652 1.698 1.743 1.788 1.833 1.877
1.743 1.788 1.833 1.877 1.920 1.964 2.007 44
34 2.049 2.091 2.133 2.174 2.215 2.256| 2.296| 2.336| 2.376| 2.415
35 2.454 2.493 2.531 2.569 2.606 2.643 2.680 2.717 2.753 2.789|| 37
36 2.825 2.860 2.895 2.930 2.965 2.999 3.032 3.066 3.099 3.132 34
37 3.165 3.197 3.229 3.261 3.293 3.324 3.355
38 3.477 3.507 3.536 3.566 3.595 3.624 3.653
39 3.765 3.792 3.820 3.847 3.874 3.900 3.927
40 4.031 4.056 | 4.081 | 4.106
41|| 4.275 4.299 | 4.322 | 4.345
42 4.502 4.523 4.545 4.566
3.386 3.417 3.447|| 31
3.681 3.709 3.737 29
| |
3.953 3.979 4.005 27
4.131 | 4.156 4.180 4.204 4.228 4.252|| 25
4.368 4.390 4.413 4.435 4.458 4.480 23
4.588 4.609 4.630 4.651 4.671 4.692
| 21
43 4.712 4.732 4.752 4.771 4.791 4.811 4.830 4.850 4.869 4.888 20
44 4.906 4.925 4.943 4.961 4.980 4.998 5.016 5.034 5.051 5.069 18
45 5.087 5.104 5.122 5.139 5.156 5.173 5.189 5.206 5.223 5.239 17
46 5.256 5.272 5.288 5.304 5.320 5.336 5.352 5.368 5.383 5.399 16
47 5.414 5.429 5.444 5.459 5.474 5.489 5.504 5.519 5.533 5.548 15
48 5.562 5.576 5.590 5.605 5.619 5.633 5.646 5.660 5.674 5.687 14
49 5.701 5.714 5.728 5.741 5.754 5.767 5.780 5.793 5.806 5.819 13
50 5.831 5.844 5.856 5.869 5.881 | 5.894 5.906 5.918 5.930 5.942 12
51 5.954 5.966 5.978 5.990 6.001 6.013 6.024 6.036 6.047 6.059 II
52 6.070 6.081 6.092 6.103 6.114 6.125 6.136 6.147 6.158 6.168 II
53 6.179 6.190 6.200 6.211 6.221 6.232 6.242 6.252 6.262|| 6.272 IO
54 6.283 6.293 6.303 6.313 6.322 6.332 6.342 6.352 6.361 6.371 ΙΟ
60
(J) TABLE OF VALUES OF Rv.

V
f-s
о
I
2
3
4
5
6
7
8
9
Diff.
+
55 6.3806.3901.3995.4089 | .4183.4276 .4368 .4460 | .4551|
.4733 .4823.4912.5001.5090.5178 .5266| .5353| .5439
.5612.5697.5782.5866.5950 .6034 .6117 .6200 .6282
irerin
56
57
58 6.6446.6527.6608 .6689.6769.6849 .6928 .7007
.7242.7319.7396.7473.7549.7625 .7700 .7775
.7999.8072.8146.8219.8292 .8364.8436 .8508
59
60
62
.4642 || 94
.5526|| 88
.6364 84
.7086
.7850
|
.7164|| 80
.7925 || 76
.8579
.8650|| 72
61 6.8721 .8792.8862.8932.9001 .9070 .9139 .9208 .9276 .9344 69
.9411.9478.9545.9612.9678.9744| .9810 .9875 .9940 *.0005 66
.9875.9940
63 7.0069 .0133 .0197 .0260.0324.0387 .0450 .0512 .0574 .0636 63
647.0698.0759.0820.0881.0942.1002 .1062 .1122 .1181 .1240 60
.1299.1357.1416.1474.1531.1589|| .1646 .1704
.1874.1930.1986.2041.2097.2152 .2207 .2261
65
66
69
567
♡♡♡♡ ☹☹☹ dino 50 o OHN MED ON DOT NM +
∞ ∞ ∞
67|| 7.2424.2478.2531.2584 .2637.2690 .2742 .2795
68 .2950 .3001.3052.3103.3154.3205 .3255 .3305
|
3454.3503.3552 .3601.3650 .3698 3747 .3795
|
70 7.3939.3986.4033 .4080.4127.4173 .4219 .4265
.4402.4448.4493.4538.4582.4627|| .4671 .4715
72 .4846.4890.4933 .4976 .5019.5061 .5104 .5146
7.5272.5313.5354.5396 .5437 .5477 .5518 .5559
.5396.5437.5477
.5679.5719.5759 -5799 .5838 .5877 5916 .5955
.6071.6109.6147 .6185 .6223.6261
71
73
74
75
77
78
79
80
81
со
.1760 .1817 58
.2316
.2370 55
.2847
.2898 53
3355
.3404 || 51
.3843
.3891 49
.4311
.4357 46
.4759
.4803 45
.5188
.5230 43
||
.5599
.5639|| 41
5994
.6033 39
.6298.6335
.6373
.6410 38
.6736
.6771 36
.7083
.7117 34
.7415
7448|| 33
.7734
.7765 32
.8040
.8070 30
.8333
.8361 || 29
76 7.6447 .6483 .6520 .6556 .6592.6628 .6664 .6700
.6806.6841.6876.6911.6946|.6980
.6806.6841.6876 .6911.6946.6980 .7015 .7049
.7150 .7184.7217.7251.7284.7317 .7350| .7383
7.7480.7512.7544 .7576.7608|.7640| .7671 .7703
.7796.7827.7858 .7889 .7919.7950 .7980 .8010
.8100 8129.8159 .8188 .8217.8246 .8275 .8304
82 7.8390 .8418.8446.8474.8502.8530 .8558 .8585.8613
.8667.8694 .8721.8748 .8774 .8801 .8827 .8853 .8880
.8931 .8957.8983.9008 .9034 9059 .9084 .9109 .9134
85 7.9184 .9208 .9233 .9257 .9281.9305 .9329 .9353 9377
.9233.9257.9281.9305
86 .9424.9448.9471.9495 .9518.9541.9564 .9587 .9609
87 .9655.9677.9699 .9722 .9744 .9766 .9788 .9810 .9831
83
∞ ∞
84
.8640 28
.8906|| 27
.9159 25
.9401 || 24
.9632|| 23
.9853 22
88 7.9875 .9896.9917.9939.9960.9981 *.0002 *.0023 *.0043 *.0064 21
887.9875.9896
89 8.0085 .0105 .0125 .0146 .0166.0186 .0206.0226 .0246 .0265|| 20
.0285.0305.0324.0344.0363|.0382| .0401 .0420.0439
90
.0458 19
91 8.0477.0496 .0514 .0533 .0552.0570 .0588 .0607 .0625 .0643 18
.0661.0679.0697 .0714 .0732.0750 .0767 .0785 .0802 .0820 18
.0661.0679.0697.0714.0732.0750
.0837.0854.0871 .0888.0905.0922 .0939 .0955 .0972 .0989 17
.0837.0854.0871.0888.0905.0922
92
93
94 8.1005.1022 .1038.1054 .1071.1087 .1103 .1119
III .1135 .1151 16
.1167.1182.1198 .1214.1229.1245 .1260 .1276
1321.1336.1351.1366.1381.1396 .1411 .1426
95
96
·
98
99
.1291|| .1306|| 16
.1440 .1455 15
·
97 8.1470.1484.1499.1513.1527.1542 .1556 .1570 .1584| .1598|| 14
.1612.1626.1640.1654 .1667.1681.1695 .1708 .1722 1735 14
.1749.1762.1776.1789.1802.1815.1828 .1841 .1854 .1867|| 13
100 8.1880.1893.1906.1918.1931|.1944 .1956 .1969 .1982 .1994 13
.2006.2019.2031.2043.2055.2068 .2080 .2092 .2104 2115 12
.2127.2139.2151.2163 .2174.2186 .2197 .2209 .2220 .2231 12
ΙΟΙ
102
61
(J) TABLE OF VALUES OF Rv.

V
f-s
O
I
2
3
4
5
6
7
8
9 Diff.
+
105
1038.2 2427 2238 2649 2759 2867|| 2975
104 3500 3602 3703 3803 3902 4000
4471 4562 4652 4740 4828 4915
3082 3188 3293
4096| 4192| 4286
5001 5086 5170
3397 ||108
4379|| 98
5253 87
106
8.2 5336 5417 5498 5578
5658
5736
5814
58915968
5891 5968
6043 79
107
6118 6193 6267 6340
6413
6485
6557
6628 6699
6769|| 72
108
6839 6908 6977 7045
7113
7181
7248
7315 7381
7447 68
109 8.2 7512 7577 7642 7706 7770
7834
7897
7960 | 8022
8085 64
I10
III
116
1410
117
8146 8208 8269 8330 8390
8747 8806 8864 8922 8980
112 8.2 9322 9378| 9434 9489 9545
113 9873 9927 9981 *0034 *0088
114 8.3 0403 0455 0507 0559 0610 0662
115 8.3 0915 0966| 1016 1065 1115 1165
1459 1508 1556| 1604|| 1652
1890 1937 1984 2030 2077 2123
8450
8510
8570 8630
8689 60
9038
9095 | 9152 | 9209
9266|| 58
*
9600 9655 9710 9764
0141 *0194 *0246 *0299
0713 0764 0814
9819 55
*0351 53
0865|| 51
1214 1263 1313
1362 50
1700 1748 | 1795
1843 48
2169| 2215| 2261
2307 46
119
I 20
118 8.3 2352 2397 2443 2488 2533
2799 2843 2887 2930 2974 3017
3232 3274 3316 3359 3401 3442
1218.3 3649 3690 3731 3772
2577
2622 2666 2711
2755 || 45
I22
123
3061 | 3104| 3146
3484 3526 3567|| 3608|| 42
3812 3852 3893
4052 4092 4131 4171 4210 4249
4443 4482 4520 4559 4597
3189 43
3933 | 3973
4012 40
4288
4327 4366
4405 39
4635
4673
4710 4748
4786|| 38
124 8.3 4823 4860 4898 4935 4972
5008
5045 5082 | 5118
5155 37
125
5191 5227 5263| 5299
5335
5371
5406 | 5442 | 5477
5512 36
126
5548 5583 5618 | 5653 |
5687
5722
5756 | 5791 | 5825
5859|| 35
127 8.3 5894 5928 5962
5996 6029 6063
6096|| 6129| 6163
6096 6129 6163
6196 34
128
129
6229 6262 | 6294 6327
6554 6586 6618 | 6650| 6681 | 6713
6360| 6392 |
6425 6457 6490
6425 6457 6490 6522 33
6745 6776 6808
6839 32
131
132
1 2
130 8.3 6870 6901 6932 6963
7177 7207 7238 7268
7476 7505 7535 7564
6994 7025 7055 7086
7116
7147 31
7298 7328 7357
7387
7417
7446 30
7594| 7623| 7652
7681
7710
7739 29
133
8.3 7768 7797 7825 7854
7882
7911 7939 7967 7996 8024|| 28
134
8052 8080 8108 8135
|
8163
8163 8191
8191
135
8327 8355 8382 8409
8435
8462
8674
8700
8218| 8246 | 8273 8300 28
8489 8516 8542 8569 27
8726 8752 8778 8804 8829|| 26
140
141
136 8.3 8595 8621 8648
137 8855 8881 8907 8932 8958 8983 9009 9034 9060 9085 26
138 9110 9136 9161 9186 9211 9236 9260 9285 9310 9334 25
139 8.3 9359 9383 9407 9432 9456 9480 9504 9529
|
9456 9480 9504 9529 9553 | 9577 || 24
9601 9625 9649 9673 9696 9720 9744 9768 9791 9815 24
9838 9862 9885 9909 9932 9955 9978 *0002 *0025 *0048 23
||
142 || 8.4 0071
143
0299
0321 0344 0366|| 0389
0094 0117
0094 0117 0140 0163|| 0185
|
0411
144
0522
145 8.4 0740
0544 0566| 0588 | 0610
0761
146
0953 0974
147
1161 1182
0208 0231 0254
0433 0455
0631 0653 0675 0697
0783 0804 0826 0847 0868 0889 0911
0995 1016 1037 1058 1079 1099
1203 1223 1243 1264 1284
0276|| 23
0478
0500|| 22
0718|| 22
0932 21
I 120
1141 21
148 8.4 1365
149
1566
150
1764
1304 1325
1385 1406| 1426|| 1446|| 1466|| 1486|| 1506||
1586 1606 1626 1645 1665 1685 1705
1783 1803 1823 1842 1862 1881 1900 1920 1939 20
1345 20
1526
1546|| 20
1724
1744 20
62
(J) TABLE OF VALUES OF R、.

155
156
158
159
78 9
161
162
2591 2609 2626
2769 2787 2804
2875 2893 2910 | 2928 | 2945| 2963|| 2980
2963 2980
157 8.4 3050 3067 3085 3102 3119 3137 3154
3222 3239 3256 3273 3290 3307 3324
3391 3408 3424 3441 | 3458 | 3474 | 3491
160 8.4 3556 3573 3589 | 3605| 3622|| 3638 3654
|
3719 3735 3751 3767 3783 3799 3815
387938953911 3927 3942 | 3958 | 3974
4021
| |
2591
V
о
I
2
f-s
3
4
5
6
7
8
9 Diff.
+
2016
2092 2111 2130 19
152
153
2279
96
151 8.4 1958 | 1978 | 1997
2035 2054 2073
2149 2167 2186 | 2205 2224 2242 | 2261
2298 2317 19
2335 2354 2372 2390 2409 2427 2445 2464 2482| 2500
| | |
154 8.4 2518 2536 2554 2573
2698 2716 2733 2751
50505
456
18
2822
2644 2662 2680 18
2840 2857
18
2998
3015 3033 18
3171
|
3341
3188 3205 17
3358 3374 17
3507 3524
3540 17
3671 3687
3831 | 3847
3703 || 16
3863 16
3990 4005
|
16
4237
4389
| |
163 8.4 4037 4052 4068 4083
164 4191 4207|| 4222
165 4343 4359 4374
166 8.4 4493 4508 4523 4538
167 4641 4656| 4670 | 4685 | 4699|| 4714 | 4728 4743 4758 | 4772 || 15
168
4786 4801 4815 4830 4844 | 4858 | 4872 | 4887 | 4901
491514
4099 4114 4130
4114 4130
4145
4161
4176 || 15
4252 4268 4283 4298
4313
4328|| 15
4404 4419 4434 4449
|
4553 4567 | 4582 | 4597 | 4612
4464
4478|| 15
|
4626 || 15
169 8.4 4929 4943 4958
4972
170 5070 5084 5098
5112
171 5208 5222 5235
5249
|
4986 5000 5014
5125
5125 |
5139 5153
5263 | 5276 | 5290
5276 5290
5398 5412 5425
5028 5042 5056|| 14
5167 |
5304
5181 | 5194 14
| ||
5317 5331
|
14
5438
|
5452 5465|| 13
5571
5584 5597 13
5701
5714 5727 13
5829
5842 | 5855 || 13
5956 | 5968 | 5981 || 13
6080 | 6092 | 6105
12
179
180
6203 6215 6227
6323 | 6335| 6347 12
12
6442 6453 6465
| |
6418 6430
|
12
6558 | 6570
6582 || 12
II
I I
172 8.4 5344 | 5358 | 5371 | 5385
1728.4
173 5478 5492 5505 5518 5531 5544 5558
174 5610 5623 5636 | 5649 | 5662 | 5675| 5688
5675 5688
175 8.4 5740 5753 5766 | 5778 5791 5804 5817
5804 5817
176 5868 5880 5893 5905 5918| 5931 5943
| |
177 5993 6006 6018 | 6031 | 6043 6055 6068
6055 6068
178 8.4 6117 6129 6142 6154 6166 | 6178 6190
| | |
6239 | 6251 | 6263 | 6275 | 6287 | 6299 | 6311
6299 6311
6359 6371 6383 6395 6406
181 84. 6477 6489 | 6500 | 6512 | 6524 6535 6547
| |
182 6593 6605 6616 6628 6639 | 6651 | 6662 | 6673 | 6685 | 6696
183 6708 6719 6730 6742 6753 6764 6775 6787 6798 6809
184 8.4 6820 6832 6843 6854 6865| 6876 6887
6876 6887 6898 | 6909 | 6920 || 11
185 6931 6942 6953 | 6964 | 6975| 6986 | 6997
6986 6997 7008 7019 7030
186 7041 7052 7063 7073 7084 7095 7106
187 8.4 7149 7160 7170| 7181 7192
188 7255 7266| 7276 7287
189 7360 7371 7381 7391
190 || 8.4 7464 7474 7484 7494
191 7566 7576 7586 7596 7606
| |
192 7666| 7676 | 7686 | 7696 | 7706|
193 8.4 7765 7775 7785 7795 7804
7863 7872 7882 7892
7959 7968 | 7978
II
ΙΙ
7117 7127 7138
|
II
7202 7213
7224
7234 | 7245
I I
7297 | 7308
7402
7308
7318
7318
7329
7339 7350 ΙΟ
7412 7422
7433
7443
7505
7453 IO
7515 7525
7515 7525 7535 7545 7556 || 10
7616 7626 | 7636 | 7646 | 7656|| 10
7716 | 7726
7716 7726 7736 7746 7755 ΙΟ
194
195
7987
7997
7814 7824 7834 7843 7853 10
7901 7911 7920 7930 7940 7949
|
8006 8016
8006 8016 8025 8034 8044
IO
196 8.4 8053 8063 8072 8081
8091
8100 8109
S100 8109 8119 8128 8137
197
8146 8156 8165 8174
8183
8192 8202 8211
S220 8229
198
8238 8247 8256 8265
8274
8283 8292 8301
8310 8319
9
63

(J) TABLE OF VALUES OF Ry.
V
O
I
2
f-s
3
4
บา
5
6
7
8
9
Dift.
+
200
201
203
204
199 8.48 3280 3369 3458 3547
4164 4251 4338
5032 5118 5204 5289
202 8.48 5886 5971 6055 6139
6725 6808 6891 6974 7056 7139 7221
7549 7631 7713 7794 7876 7957 8038
2058.48 8361 8441 8521 8601 8681
4426
3635 3724 3812 3900 3988 407689
4513 4600 4686 4773 4859 4946 87
5375 | 5460 | 4546 | 5631 | 5716|| 5801 || 86
6224 6308 6391
6475
6559
6642 84
7304 7386
7468 || 83
8119
8200
8280 81
8761 | 8840
8920
8999
9078 || 80
206
207
9157 9236 9315 9394 9472 9551| 9629
9941 *0018 *0096 *0173 *0250 *0327 *0404
208 8.49 0710 0786 0861 0937 1013 1088 1163
209
9707 9785 | 9863|| 78
*0481 *0557 *0633 77
1239
1239 1314 1389 75
1463 1538 1613
1687
210
2203 2276 2349
2422
1761 1835 1909
2495|| 2568 | 2640
1983 2057 2130 74
2713
2785| 2857|| 73
211 8.49 2930 3002 3073
212
213
3643 3714 3784 3855
4344 4413 4482
215
216
3217 3288 3359 3431
3925| 3995 4065 4135
4620 4689 4758 4827
2148.49 5031 5100 5167 5235 5303 5370 5438 5505
5706 5773 5840 5907 5973 6040 6106 6173
6371 6437 6503| 6568 | 6634| 6699| 6764
3145
3502
3572 71
4205
4274 70
4551
4895
4963 69
5572
5639|| 68
6239
6305 67
6829
6894
6959 65
218
219
221
224
225
230
231
217 8.49 7024 7089 7153 7218 7282
7218 7282
7666 7730 7794 7857 7921
8299 8361 8424 8486 8548
220 8.49 8920 8981 | 9042 9104 9165
9531 | 9591 | 9651 | 9712| 9772
222 8.50 0130 0189 0249 0308 0367
2238.50 0719| 0777 | 0835| 0893| 0951 1009 1067 1125
1298 1355 1413 1470 1527 1584| 1641| 1698
1868 || 1925| 1981 | 2037 2093|| 2150 2206| 2262
2268.50 2429 2485 2540 2596|| 2651| 2706|| 2761| 2817 2872
227 2981 3036 3091 3145 3200 3254 3308 3363 | 3417
228 3525 3579 3633| 3687| 3741
3687 3741 3794 3848 3901 3955
|
229 8.50 4062 | 4115 4168| 4221
4221 4274
4591 | 4644| 4696| 4749|| 4801
5115 5167 5219 | 5271
5271 5323
7346 7411
7346 7411
7475
7539
7603 64
7984 8047 8110 8173
8236 63
8611 8673 8734 8796
8858 62
9226 9287 9348 9409
9832 | 9892| 9952 *0011 *0071 60
0426 0484 0543 0602 0660|| 59
9470 61
1183
1240 58
1755
1811 57
2318
2373 56
2926 55
3471 54
400854
4327 4380
4433 4486
4538 53
4854
4906
4958 5011
5063 52
5375
5427
5479 5531
5582 52
233
232 8.50 5634 5686 5737 5789
6149 6200 6252 6303 6354|| 6405| 6456| 6507| 6558
5841 5892
5944
5995 | 6046 |
6098 52
6609|| 51
234
6660 6711 6762 6813
6864 6914 6965
7016| 7066
7117 51
235 8.50 7167 7218
7268 7319 7369
236
237
76717721
8171 8221
7771 | 7821 | 7872
8271 | 8321
8321 8370
7419 7470 7520
7922 7972 8022 8071
8420 | 8470 8519 8569
7570
7621|| 50
8121 50
8618 50
239
240
238 8.50 8668 8717 8767 8816 8866 8915 8964 9014 9063 9112 49
| |
9161 9210 9260 9309 9358 9407 9456 9505 9553 9602 || 49
9651 | 9700 | 9749 9797 9846 | 9895 9944 9992 *0041*0089 49
||
241 8.51 0138 0186 0235 0283 0332 0380 0428 0477 0525 0573 48
242
0621 0670 0718 0766 0814 0862 0910 0958
1054 48
243
I 102 1150 1198 1245 1293 1341 1389 1436 1484|| 1532 48
244 8.51 1579 | 1627|. 1674|| 1722 1769|| 1817 1864 1912 1959 2006 47
245
2053 2101 2148 2195 2242 2289| 2336 2383 2430 2477 47
2524 2571 2618 2665
2711 2758 2805 2852 2898 2945 47
246
1006
64
(J) TABLE OF VALUES OF R、.

V
о
I
2
f-s
3
4
5
6
7
8
9
Diff.
+
247 8.51 2992 3038 | 3085| 3131
3178
3224 3271
3317
3363 3410 47
248
249
3456 3502 3548 3594
3916 3962 4008 4054
3640
3687|| 3733
3779
3825| 3870|| 46
4100
4145 4191
4237 4282
4328 46
250 8.51 4374 4419 | 4465 | 4510
4555
4601 |
4646| 4691 | 4736| 4782|| 45
251
252
4827 4872 4917 4962 | 5007
5276 5320 5365 | 5409
5052
5097 | 5142 | 5186|| 5231|| 45
5454
5498
5543 5587 5631
5676 44
253 8.51 5720 5764 5808 | 5852 | 5896 5940
254 6159 6203 6246 6290 6333 6377 6420 6464
255 6594 6637 6680 6723 6766 6809
5984 6028
6072
6115 44
6507
6550 44
6852 | 6894
6937
6980|| 43
256 8.51 7023 7065 7108
7150
257
7447 7489 7531
7573
258
7865 7907 7948
7989
8400
8806
9206 | 9245
259 8.51 8278 8319 8359
260
8685 8725 8765
261 9086 9126 9166
262 8.51 9482 | 9521 | 9560 |
263
9872 9910 | 9949
264 8.52 0255 0293 0331
266
9599 9638| 9677 9716 9755 9794 9833 39
9987*0026 *0064 *0102
9987 *0026 *0064 *0102 *0141 *0179 *0217 38
0369 0407 0445 0483 0520 | 0558| 0595|| 38
2658.52 0633 0670 0708 0745
| |
1005 1042 1079 1115
1371 1408 1444 1480
0782 0819 0857
1152 1189
0894
|
8441
8846 8886
9285
8482
7193 7235 7278 7320
7615 7656 7698 7740
8031 8072 8113 8154
8522 8563
8926 8966 9006 | 9046|| 40
7362|| 7404|| 42
7782
7823 42
8196
8237|| 41
8604
8644 41
9324|| 9364|| 9403| 9443|| 40
0931 0968 37
267
1516 1553
1226 1262
1589 1625
1299 1335 37
1661 1697 36
268 8.52 1732 1768
1804
1840
1875 1911
1947|| 1982 |
2018 | 2053|| 36
269
2088 2124 2159 2194
2229 2265
2300 2335
2369| 2404|| 35
270
2439 2474 2508 | 2543
2543
2578 2612
4647 2681
2715
2750 35
271 8.52 2784 2818 2852 2887 2921
2955
2989 3023
3057
3091 || 34
272
273
3124 3158 3192 3226|| 3259
3461 3494 3538 3561
| | |
3293
3327 3360
3394
3427 34
3594
3627
3661 3694
| ||
3727
3760 33
274 8.52 3793 | 3826 | 3859 | 3891
3891
3924 3957
3990 4022 | 4055
4087 33
275
276
4120 4152 4185
|
4185 4217
4442 4474 4506 | 4538
4538
4249
4281 4314
4346 4378
4410 32
4570 4602
4634
4665 4697
4729 32
277 8.52 4760 4792 4823 4855 4886
| |
4917
4949
4980 5011
|
5042 || 31
278
5229 5259
5290 5321
5352 31
279
5566
5597 5627
|
5657 31
27
5073 5105 5136 5167|| 5198
5383 5413 5444 | 5475 5505 5536
50.81
28 45.55
41.00
29
www www
37.04
33.57
30.52
33
27.83
34
(K) TABLE OF VALUES OF [1000÷v]³
50.24 49.69 49.15 48.61 | 48.08 | 47.56 | 47.05 | 46.54 | 46.05 || 53
| | |
45.07 44.59 44.12 | 43.66 | 43.20 42.75 42.30 41.86 41.43 46
40.58 40.17 39.76 39.35 38.95 38.56 38.17 37.79 37.41 40
36.67 36.31 35.95 | 35.59 | 35.25 34.90 34.56 34.23 33.90 || 35
33.24 32.93 32.61 | 32.30 | 31.99 31.69 31.39 31.10 30.81 31
30.23 29.95 29.68 29.40 29.13 28.86 28.60 28.34 28.08 27
27.58 27.33 27.08 26.84 26.60 26.36| 26.13 25.90 25.67 24
25.44 25.22 25.00 24.78 24.57 24.35 24.14 23.93 23.73 23.53 21
23.32 23.13 22.93 22.73 22.54 22.35 22.16 21.98 21.80 21.61 19
65
જ્ઞ
5
(K) TABLE OF VALUES OF [1000÷v]3
³
V
О
I
2
f-s
3
4
5
6
7
8
9
Diff
Aw www i
+
36
37
38
21.43 21.26 21.08 20.91 20.74 20.57
19.74 19.58 19.43 19.27 19.12
18.22 18.08 17.94 17.80 17.66
20.40 20.23 20.07
19.90 17
18.96
18.81
18.66
18.52
18.37|| 15
17.52
17.39 17.25
17.12
16.99 14
39
40
16.23
16.10 15.98
15.86
15.74 12
15.05 14.94 14.83
14.72
14.62 || 11
13.99|| 13.89| 13.79|
13.69
13.59 10
3.027
2.935
2.845
2.755
2.666|| 92
43
2.149
2.065
1.983
1.901
1.820 84
1.348
1.272
1.197
1.122
1.048 77
41
42
44
16.86 16.73 16.60 16.48 16.35
15.63 15.51 15.39 15.28 15.17
| | |
14.51 | 14.40 14.30 14.20 14.09
I 3.497 3.401 3.306 3.212 3.119
2.578 2.491 2.404 2.318 2.233
1.739 1.660 1.581 1.503 1.425
45 1 0.974 0.901 0.829 0.757 0.686 0.616 0.547 0.478 0.409 0.341 70
46 0.274 0.207 0.141 0.075 0.010 *9.946 *9.882 *9.819 *9.756 *9.694 64
470 9.632 9.571 9.510 9.450 9.390 9.331 9.272 9.214 9.156 9.099 59
48
9.042 8.986 8.930 8.875 | 8.820 8.766 8.711 8.658 8.605 8.552 54
8.500 | 8.448 | 8.397 | 8.346 | 8.295|8.245|8.195| 8.146| 8.097| 8.048|| 50
FOR VELOCITIES 500 TO 1999 f-s SEE TABLE (F).
49
200
201
1 2500 | 2481 | 2463 | 2444 | 2426 | 2407 | 2388 | 2370 2351 2333 19
2314 2296 2277 2259 2240
202 2132 2114 2096 2078 2060
2222 2204 2186
2042 2025 2007
2168 2150 18
1989 1972 18
203.1 1954 1936
|
1919|| 1901
1901
1884
1884
1866 1849 1831
1814
1796 17
204
1779 1762 1744
205 1608 1591 1574
1523
|
1727
1710
1693 1676
| | 1659
1659
1642
1625 17
1557 1540
|
1506
1489
1473
1456 17
206.1 1439 1423| 1406
1423 1406
1389
1389 1373
1373
1356
1340
1323 1307 1290 16
207
1274 1258| 1241
1225 1209
1193
II77
208
1113 1097 1081
1065| 1049
1033
1161 1145 1129 16
1017 1002 0986
0970 16
209.1 | |
I 0954 0939 0923
0907
0892 0876
0860 0845 0829
0829
081416
210
0798 0783 0767
0752
0736 || 0721
211 0645 0630 0615
0600
0585 0570
07060690 | 0675
0555 0540 0525
оббо 15
0510 | 15
212.1 0495 0480 | 0465
0480 0465
0451
0436
0421
0407 0392 0377
0363 15
213
0348 0333 0319 0304
214 0204 0190 0176 0161
||
0290
0276
0147
0147 0133
0261 0247 0233 0218 14
0119 0104 0090 0076|| 14
|
217 9786
9772 9759
215.1 0062 0048 0034 0020
216.0 9923 9909 | 9895
9909 | 9895 9882 9868 | 9854
9746 | 9732 | 9719
0006*9992
*9978 *9964 *9951
*9978*9964 *993714
9841 9827 | 9813
9813
9800 14
||
9706 | 9692 | 9679
9679
218.0 9652 9639 9625
9639 9625|| 9612 | 9599 | 9586 | 9573 | 9560 | 9547
219 9521 9508 9495 9482 | 9469 | 9456 | 9443
220 9391 9378 | 9365 | 9352 9340 | 9327 9315
221.0 9265 9252 | 9240 9227 9215
| |
9140 9128 9116 9103 9091
223 9018 90068994 8982 8970
222
9665 13
9534 13
9430 9417
9302
9404
13
9290
9277 13
9202 9190
9079 9067 9055 9042
8958 | 8946 | 8934 | 8922
9177
9165
9152 12
9030 12
8922
8910
8910 | 12
224.0 8898 8886 | 8874
8886 8874 8862 | 8850 |
225
8779
8768 8756 8744 8733 8721
8744 8733 8721
226 8663 8652 8640 8629 8617
8838 8826 8814
8803
8791 12
8709
8698
8686
8686 8675 12
8606
8595
8583
8583
8572
8572
8560 II
227.0 8549 8538 8526 8515 8504
228 8437 8426 8415 8404 8393 8382
229 8327 8316 8305 | 8295 | 8284 |
230.0 8219 8208 8198 8187
231 8113 8102 8092 8082
232 8008 7998 7988 7977
|
8493
8482
8470
8459
8459 8448 II
8371
8360
8360
8349
8349
8338 II
8273 8262
8251
8241 8230
II

8176 8166
8155 8144
8134 8123 II
8071 8062
8050 | 8039
7967|| 7957
7947 7936
8029 8019 IO
7926
7926
7916
IO
66

(K) TABLE OF VALUES OF [1000÷v]3
³
V
O
I
2
f-s
3
4
5
6
7
8
9
Diff.
+
233.0
234
235
N N
237
238
333
678
7906 7895 7885 7875
7805 7795 7785 7775
7705 7696 7686 | 7676 |
236.07 6079 5982 5886 5789
7666 ||
|
7865 7855 7845 7835 7825 7815 10
7765 7755 7745 7735
7755 7745 7735 7725 7715 IO
7647 7637 7627|| 7618|| 10
7656|
5693 5597
5501 | 5406
5501 5406 5310 5215 96
5120 5025 4930 4835
4177 4084 3991 | 3898
239.07 3250 3158 3066| 2974
4741||
4646
4646|
4552 4458
4552 4458 4364 4271 95
427195
3805
3712
3619 3527 3434 3342 93
| ||
2883|| 2792
2701 2610
2701 2610
2519|| 2429|| 91
240
2338 2248 2158 2068
1978
1888
1798 1709
1619|| 1530|| 90
241
1441 1352 1263| 1175
1086
0998
0910 0822
0910 0822
0734 0647 88
242 .07 0559 0472 0385 0298
243.06 9692 | 9606 | 9520 | 9434
244 8838 8754 8669 | 8585 | 8501
245.06 7999 7916 | 7833 7750 7667
246
173 7091 7010 | 6928 | 6847 |
247 6361 6280 6200 6119 6039
0211
0124
0038*9951 *9865 *9778 87
9349 |
9263
9178 9093 9008| 8923|| 86
8417
8333 8250 8166 8083 84
7584 7502
7419
7337
7255 83
6765 | 6684
6603
6522
6442|| 82
5959 5879
5800
5720
5641 || 80
248.06 5561 5482
249
5403 5324 5246
4775 4697 4619 | 4541 | 4464 | 4386 | 4309 | 4231
4000 3923 3847 3770 3694 | 3618 | 3542 | 3466 | 3390
5167 5088 5010
4931
4853|| 79
4154
4077 78
3315 76
259
250
251.06 3239 3164 3088 3013 2937 2862 2787 2712
2488 2414 2340 2266| 2192 2118 2045 1971
1751 1678 1605 | 1532 | 1459 | 1386| 1314
252
0737
0027
I
2637| 2563|| 75
1898 1824 74
253
1314 1241 1169 1096 73
254.06 1024 0952 0880 0809
| |
06660594 0523 0452 0381 71
255 0310 0239 01680097 *9956 *9886 *9815 *9745 *9675
256.05 9605 9535 9465 9395 9326 9256 9187 9117 9048 8979 70
257.05 8910 8841 8773 8704 8636 8568
258
8228 8160 8093 8025 7958|| 7891
7557 7490 7424 7357 | 7291 7225
261
262
264
4163| 4102
3553 3493
8500 8432 8364 8296 68
7824 7757
7824 7757 7690 7624 67
7159 7093 7027 6962 66
260.05 6896 6830 6765 | 6699 6634|| 6568 6503 6438 6373 6308 66
6243 6179 6114 6050 | 5985 | 5921 | 5857 | 5793
5857 5793 5729 5666|| 64
5602 5539 5475 5412 5349 5286 5223 5160 5097 5035 63
263.05 4972 4909 4847 4784 4722 4659 4597 4534
4348 | 4287 4225
4040 3979 3918
3735 3675 3614
4472
4410 63
3857 3796 62
3432 3372 3312
3252 | 3192|| 61
2953 2894 2834 2775 2715
2361| 2302 2243 2185 2126
1777 1719 1661 1604 1546
2656 2597 60
2068
2068 2009 | 59
1489|| 1431|| 58
265
266.05 3132 3073 3013
267 2538 2479 2420
268 1951 1893 1835
269.05 1374 1317 1259 I202 1145 1088 1032 0975 0918 0862 57
270 0805 0749 0692 0636 0580 | 0524 0468 0412 0356 0300 56
271 0244 0189 0133 0077 00229966 *9911 *9856 *9801 *9746|| 56
272.04 9691 | 9637 9582 9528 9473
273 9149 9095 9041 89878934
274 8613 8560 8507 8454 8401
275 .04 8084 8032 7979 7927 7874
276 7562 7511 7459 7408 7356
277 7050 6999 6948 6897 6847
278.04 6544 6494 6444 6394
| |
279 6045 5996 5946 5897
| |
280 5554 5506 5457 5408
|
8880 8826 8773
9419 9365 9311 9257 9203 54
8880 8826 8773 8719 8666|| 54
8348 8295 8242 8189
8137 53
7822 7770 7718 7666 7614 52
7305 7254 7203 7152 7101 51
6796 6746 6695 6645 6594 51
6294 6244 6194 6144 6095 50
5848 5799 5750 5701 | 5652 | 5603|| 49
5360 5311 5262 5214 5165 5117 49
|
6344
67
5*
(L) TABLE OF VALUES OF Tv.
PROJECTILES WITH OGIVAL HEADS.

V
O
I
2
3
4
5
6
7
8
9
Diff.
f-s
secs.
secs.
secs.
secs.
secs.
secs.
secs.
secs.
secs.
secs. +
ΙΟ
75.4 77.1
78.8
80.4
82.1 83.6 85.2
|
86.7
88.2
89.7 1.6
II
91.1 92.5
93.9
95.3
96.6 98.0 99.3
|
00.5
01.8
03.01.3
12
I
04.2 05.4
06.6
07.7 08.9 10.0 II.I 12.2
13.2
14.3 I.I
13
I
15.3 16.3
17.3
18.3 19.3
20.3
14
24.8 25.7 26.6
27.4 28.3
29.I
21.2 22.I
29.9
23.0
23.9 1.0
30.7 31.5
32.3
.8
15
33.I
33.8 | 34.6
35.3 | 36.1
36.8 37.5
38.2 38.9
39.6 .7
16
14 0.28
0.96 1.62
2.27 2.92
3.56 4.19
4.81 5.43 6.04
17
18
73
6.64 7.24
7.82
8.40 8.98
9.55 0.II
0.66
1.21 1.75 .57
15 2.29
2.82 3.35
3.87
4.38
4.89 | 5.39
5.89 6.38
6.38 | 6.87
.5I
55555
.64
19
15 7.35 | 7.83 | 8.30 | 8.77
9.23
9.69 0.14
0.59
1.03 1.47
.46
20
16 1.90
2.33
2.76 3.18
3.60
4.01
4.42
4.83
5.23 5.63
.4I
21
6.02 6.41
6.80 7.18
7.56
7.94
8.31 8.68
9.04 9.40
·38
222222223
16 9.76
0.12
0.47 0.82
1.17
1.51
1.85 | 2.19
2.52 2.85
.34
17 3.18
3.51 3.83
24
6.31 6.61
4.15 4.47 4.78
6.91 7.20 7.49 7.78
5.09 5.40 5.71 6.01
.32
8.07 8.35
8.64
8.92 .29
www wwN N N N
25
17 9.19 | 9.47
9.74 0.01 0.28 0.55
0.82
1.08
1.34
1.60
26
18 1.86 2.II
2.36
2.61 2.86 3.11
3.36
3.60
3.84
3.84 4.08
2222223
.27
.25
27
4.32 4.55
4.79
5.02
5.25
5.48
5.71
5.94
6.16 6.38
.23
28
18 6.60
6.82
7.04
7.26
7.47
7.69
7.90 8.11
8.32 8.53
.21
29
8.73
8.94
9.14
9.35
9.55 9.75
9.94
0.14
0.34 0.53
.20
30
19 0.72 0.91
I.10 1.29
1.48
1.66
1.85
2.03
2.22 2.40 .19
19 2.58 | 2.76
2.93
3.11 3.29
3.46 3.63 3.81
|
4.32 4.48
5.95
4.65
4.82 | 4.98
5.15 5.31
5.47
3.97 4.15 .17
5.63 5.79
.16
6.11 6.27
6.42 6.58
6.73
6.88 7.04
7.19 7.34 .15
333
456
33
78
34
35
36
37
38
39
19 7.49 7.64 | 7.79
8.94 9.08 9.21
20 0.30 0.43 0.56
7.93
9.35 9.49 9.63
8.08
8.22
8.37
8.37
8.51
8.65 8.79
.15
9.76
9.90 0.03
0.17 .14
0.69
0.82 0.95 1.08
1.21 1.33
1.46
.13
20 1.59 1.71
1.83 1.96
2.80 2.92
3.95 4.06
3.04 3.15
4.17 4.28
2.08 2.20 2.32
3.27
4.39
2.44 2.56 2.68
3.38 3.50 3.61 3.72 3.84 .12
4.50 4.60
.12
4.71 4.82 4.92
.II
40
41
42
20 5.03 5.13 5.24 5.34 5.45
6.06 6.16 6.256.35 | 6.45
7.03 7.12 7.22 7.31 7.40
5.55 5.65
6.55 6.65
6.74
7.49 7.59 7.68
5.75 5.85 5.96 .10
6.84
6.93
7.77 7.86
.10
.09
.09
48
50
51
uurt ☆ ☆ ☆ ☆
43
44
45
46
47
49
8.39 8.48 8.57 8.66 8.74
9.17 9.25 9.33 9.42 9.50 9.58 .08
9.99 0.07 0.14 0.22 0.30 0.38
20 7.95 8.04 | 8.13 8.22 8.31
8.83 8.91 9.00 9.08
9.66 9.74 9.82 9.91
21 0.46 0.54 0.61 0.69 0.77
1.22 1.29 1.37 1.44 1.52
1.95 2.02 2.09 2.16 2.23
21 2.65 | 2.71 2.78 2.85
3.32 3.38 3.45
3.96 4.02 4.08
0.84 0.92 1.00 1.07 1.15
1.59 1.66 1.73 1.81
1.88
2.30 2.37 2.44 2.5I 2.58
.08
882 208 888
.07
.07
2.92
3.51 3.58 3.64
4.15 4.21 4.27
2.98 3.05 3.12 3.18 3.25
.07
3.71 3.77 3.83
4.33 4.39 4.46
3.90
.06
4.52
.06
68

(M) TABLE OF VALUES OF Sv.
PROJECTILES WITH OGIVAL HEADS.
V
о
I
2
3
4
5
6
7
8
9
Diff.
f-s
feet.
feet. feet.
IO
II
12
1066 1238 1409
2715 2871 3026
4220 4363 4506
feet. feet. feet. feet. feet. feet.
1578 1745
1578 1745|| 1910 2074
3180 3333 3484 3633
4647 4787 | 4926 |
feet. +
2236 | 2397 | 2557 || 166
3782 | 3929 4075|| 151
5064 |
5200 5336 | 5471|| 139
13
5604 5737 5866 5999 6129 6257
6385| 6511|| 6637
6762 129
14
6886
7009
15
8079
8194
7132 | 7253 | 7373
8309 8422 8535
7493 7612 7730 7847
7964 120
8647 8758 8868
8978
9087|| 112
16
9196
17
18
70
9304 9411 9517 || 9623 | 9728 | 9833 | 9937
I 0244 0346 0447 | 0546 | 0645 0743 0841 0939
1230 1326 1421 1516 1610 1704 1797
0040
0142 || 105
1037
1134 98
1890
1982
2074 94
19
2 2
20
I 2165 2256| 2346 2436
3052 3139 3224 3310
21 3896 3979 4060 4142
2525 2614 2703 2791
3395| 3480 | 3564 | 3648
4223
2878 2966
3731 | 3814|| 85
4303 4384 | 4463 | 4543 | 4622
СО СО СО
89
81
2 2
22
I 4701
4779 4857 4935
5013
5090 | 5167
5244 | 5319 | 5395
77
23
5470
55455620 5694
5768
24
6206 6278 6350 6421
6492
5842 5916 | 5989 | 6061 6134
6563 6633
74
6703 6773 6843
71
2 2
25
26
27
I 6912 6981 | 7050 | 7119
7255 7322
7591 7657 7723 7789 7855 7920
8244 8308 83718435 | 8498 | 8561
84988561
7187
7390| 7457
7524
68
7985
8624
8050 8115 8179
8686 8749 8811
65
63
28
I 8873
8935
29
9480 9540
30
2 0067 0124
8996 9057 | 9118| 9179 |
95999658 | 9717 9776 9834 | 9893 | 9951 0009
0181 || 0239
0239 | 0296 | 0352 | 0409 | 0465 | 0522 | 0578
9240 | 9300 | 9360 | 9420
61
59
57
www.
31
2 0634
0689 0745 0800 0855
32
33
0910 0965 1020 1074 1128
1182 1236 1290 1344 | 1397 | 1450 | 1503
1503
1714 1766 1818 1870 1922 1973 2025
|
55
1556
1609 1661
53
2076
2127 2178
|
52
532
34
35
36
2 2229 2280 2330 2380 | 2431 | 2481
2728 2778 2826 2875 2924 2972
3213 3260
3260 3308 3355 3402 3449
2531
2580
2630 2679
|
50
3021
3069
3117 3165 49
3496
3543
3589 | 3636
47
37
38
78
39
2 3682 3728 3774 3820 3866
4137 4182 4227 4271 4316
|
4579 4623 4666 4709 4752
| |
3911 3957 4002 4047 4093
4360 | 4404 4448 4492 | 4536
| |
4795 4838 | 4881 | 4923 | 4966
46
44
43
40
41
42
43
44
45
46
47
48
50
uunit fo
49
2
|
2 5008 5050 5092 5134 5176 52185259 | 5301 | 5342 5383
5424 54655506 | 5546 | 5587 5627 5668 5708 5748 5788
|
5828 5867 5907 5946 5986 6025 6064 6103 6142 6181
2 6220 6258 6297 6335
|
6601 6639 6676 6714
6973 7009 7046 7082
27335 7371 7407 7442
7689 7724 7759 7794
8035 8069 8103 8137
40
6374 6412 6450 6488 6526 6564||| 38
| | |
6751 6788 6825| 6862 | 6899 | 6936
7119 7155 | 7191
|
7478 7513 7548
7828 7863 7897
|
8171 8205
|
8373 8407
8407 8440 8473 8506 8540 8573
8704 8737
8737 8770 8802 8835 8868 8900
9029 9061 9093 9125 | 9157 | 9189 | 9221
|
42
| |
| |
| |
39
37
7227 7263 7299
36
7584 | 7619 | 7654
35
8239
7932 7966 8001
8272 8306 | 8340
|
35
34
8606 | 8639 | 8672
33
8932 | 8965 8997
| |
33
9253 9284 | 9316
32
www www
69
(L) TABLE OF VALUES OF Ty.

secs.
V
O
I
2
3
4
5
6
7
8
9 Diff.
f-s
secs.
secs.
secs.
secs.
secs.
secs.
secs.
secs.
secs.
+
52
53
54
55
56
57
58
59
60
61
62
64
65
66
67
68
69
70
71
72
73
345
74
75
21 4.577 4.638 4.698 4.758 4.818 4.877 4.937 4.996 5.055 5.113 60
5.172 5.230 5.288 5.346 5.404 5.461 5.518 5.575 5.632 5.689|| 57
5.745 5.801 5.857 5.913 5.969 6.024 6.079 6.134 6.189 6.244 55
21 6.298 6.352 6.406 | 6.460 | 6.514 6.567 6.620 6.673 6.726 6.779|| 53
6.8316.883 6.936 6.987 | 7.039 7.091 7.142 7.193 7.244 7.295|| 52
7.346 7.397 7.447 7.497 7.547 7.597 7.647 7.696 7.746 7.795|| 50
21 7.844 7.893 7.942 7.990 8.039 8.087 8.136 8.184 8.232 8.279 48
8.327 8.375 8.422 8.469 8.516 8.563 8.610 8.657 8.703 8.749 47
8.796 8.842 8.888 8.933 8.979 9.025 9.070 9.115 9.160 9.205|| 45
21 9.250 9.295 9.339 9.384 9.428 9.472 9.516 9.560 9.604 9.647 44
9.691 9.734 9.777 9.820 9.863 9.906 9.949 9.991 0.034 0.076 43
63 22 0.118 0.160 0.202 0.244 0.286 0.327 0.369 0.410 0.451 0.492|| 42
22 0.533 0.574 0.615 0.655 | 0.696 | 0.736 | 0.776 | 0.816 | 0.856 0.896|| 40
0.936 0.976 1.015 1.054 1.094 1.133 1.172 1.211 1.250 1.288 39
1.327 1.365 1.403| 1.442 1.480 | 1.518 | 1.556 1.593 1.631| 1.668|| 38
22 1.706 1.743 1.780 1.818 1.855 1.891 1.928 1.965 2.001 2.038 37
2.074 2.111 2.147 2.183 2.219 2.255 2.290 2.326 2.361 2.397 36
2.432 2.468 2.503 2.538 2.573 2.608 2.642 2.677 2.712 2.746 35
22 2.781 2.815 2.849 2.883 2.917 2.951 2.985 3.019 3.053 3.086|| 34
3.120 3.153 3.186 3.220 3.253 3.286 3.319 3.351 3.384 3.417 33
3.449 3.482 3.514 3.546 3.578 3.611 3.642 3.674 3.706 3.738 32
22 3.769 3.801 3.832 3.864 3.895 3.926 3.957 3.988 4.019 4.050 31
4.080 4.111 4.142 4.172 4.202 4.233 4.263 4.293 4.323 4.353 30
4.383 4.413 4.442 4.472 4.501 4.531 4.560 4.590 4.619 4.648 29
22 4.677 4.706 4.735 4.764 4.792 4.821 | 4.849 4.878 4.906 4.934 29
4.962 4.991 5.019 5.046 5.074 5.102 5.130 5.157 5.185 5.212 28
5.239 5.267 5.294 5.321 5.348 5.375 5.402 5.428 5.455 5.481 | 27
22 5.508 5.534 5.561 5.587 5.613 5.639 5.665 5.691 5.717 5.743 26
5.769 5.794 5.820 5.845 5.871 5.896 5.921 5.946 5.971 5.997 25
6.021 6.046 6.071 6.096 6.121 6.145 6.170 6.194 6.218 6:243 25
22 6.267 6.291 6.315 6.339 6.363|6.387 6.410 6.434 6.458 6.481|| 24
6.504 6.528 6.551 6.574 6.597 6.620 | 6.643 6.666 | 6.689 6.711|| 23
6.734 6.756 6.779 6.801 6.823 6.845 6.868 6.890 6.911 6.933 22
22 6.955 6.977 6.998 7.020 7.042 7.063 7.084 7.106 7.127 7.148|| 21
7.169 7.190 7.211 7.232 7.252 7.273 7.294 7.314 7.335 7.355
7.375 7.395 7.416 7.436 7.456| 7.476 7.496 7.516 7.535 7.555
22 7.575 7.594 7.614 7.633 7.653 7.672 7.691 7.710 7.730 7.749 19
7.768 7.787 7.806 7.824 7.843 7.862 7.881 7.899 7.918 7.936 19
7.954 7.973 7.991 8.009 8.027 8.045 8.063 8.081 8.099 8.117 18
22 8.135 8.152 8.170 8.188 8.205 8.223 8.240 8.257 8.275 8.292 17
8.309 8.326 8.343 8.360 8.377 8.394 8.411 8.428 8.445 8.461|| 17
8.478 8.494 8.511 8.527 8.544 8.560 8.576 8.593 8.609 8.625 16
22 8.641 8.657 8.673 8.689 8.705 8.721 8.737 8.753 8.768 8.784 16
8.799 8.815 8.831 8.846 8.861 8.877 8.892 8.907 8.923 8.938|| 15
8.953 8.968 8.983 8.998 9.013 9.0289.043 9.057 9.072 9.087 15
22 9.101 9.116 9.131 9.145 9.160 9.174 9.188 9.203 9.217 9.231
14
9.245 9.260 9.274 9.288 9.302 9.316 | 9.330 9.344 | 9.358 9.371 14
9.385 9.399 9.413 9.426 9.440 | 9.453 9.467 9.481 9.494 9.507 14
76
77
78
79
80
81
82
83
Со со со
234
84
86
567
85
87
88
89
90
91
со со
92
93
94
95
96
97
98
99
no não
21
20
70

(M) TABLE OF VALUES OF S
V
O
I
2
3
4
5
6
7
8
9
Diff.
52
53
54
55
56
57
58
60
61
62
63
66
67
68
69
g。°an ;
f-s
feet.
feet. feet. feet. feet. feet. feet.
feet.
feet. fect.
+
2 9347
9379
9379 9410
9660 | 9691 | 9721
9442
9752
9966 9997
99970027
0057
9473 | 9504 9535
9783 9814 9844 9875 9905 9936
00880118 0148 0178
0118 0148 0178 0208 0238
9567 | 9598 | 9629
31
კი
3 0268 | 0298
0298
0327
0357 0387
0416 | 0446
0475 0505 0534
პი
0564 | | |
0593 0622
0651 0681
0710 0739
0768 | 0797 | 0826
29
3 1141
0855 0883
1169 1198 1226
0883 0912
0941
59
1423
1701
1451| 1479|| 1507
1730 1757 1785
0970
1254| 1283
1535 1563 1591|| 1619
| |
1812 1840 1867
0998 1027 | 1055 | 1084
[112
29
1311
1339 1367 | 1395
28
|
1619 | 1646 | 1674
1895 1922
28
1949 28
64
3 1977 2004 2031 2058
2247 2274 2301 2328 2355
2514 | 2541
2541 2567 2594 2620
3 2778 2804 2830 2856
3037 3063
3063 3089 3114
3293 3318
3318 3344 3369
2085
2112
2139 2166 2193 2220
2381
2646
2408 | 2435 | 2461 | 2488
2673 2699 2725 2751
N N N
27
27
26
2882 2908 2934
2960 2985 3011
26
3140 3165 | 3191
3394 | 3419 3445
3217 3242 3267
3470 | 3495 | 3520
26
25
3 3545 | 3570
3793
3570 3595
3620
3818
3645|| 3670 | 3694 | 3719 | 3744 3769
3818 3843 3867 3892 3916 | 3941 | 3965 3990 4014
4038 4063
4063 4087 41114136 | 4160 | 4184 | 4208 | 4232 | 4256
25
25
24
70
3 4280 4305
4305 4329 4352
4376 4400 4424
4448 | 4472 | 4496
24
71
73
81
IN MED ON 285 ☹mt mom å ää☹ *ino no a
∞
72
4755 4778
45194543
4543 4567 4590
4778 4801 4825
4614 4637
4848 4871
|
4661
4684 4708 4731
24
4894
4917 4941 4964
|
23
3 4987
5010
5010 5033
5056 |
5078 | 5101
5124
5147
5170 | 5192
23
74
5215 5238
5238 52605283 5306 5328 5351
5373
| | 5373
5395 5418
23
75
5440
5463
5463 5485 5507 5529 5552 5574
5596
5596 56185640
22
76
3 5662 5684 57065728
77
78
5750 5772 5794 5815 5837 5859
5880 5902 5924 5945 5967 | 5988 | 6010 | 6031 | 6052 6074
6095 6116 6137 6158 6180 6201 6222
6222 6243 6264 | 6285
| |
22
22
21
79
80
82
3
83
84
3 6306 6326
6326 6347 | 6368 6389 6410
6451
| |
6513 6533
6533 6554 6574 6594 6615
67166736
6736 | 6756 | 6777 6797 6817
6916 | 6936
6936 6956 6975 6995 7014 7034 7053
| |
7112 7131 7150 7170 7189 7208 7227 7246
|
7303 7322 7341 7360 7378 7397 | 7416 |
6430
6451 64716492
21
6635
6655
6655 | 6676 | 6696
20
6837
|
6856 6876 | 6896
20
7053 7073
7073 7092
20
7246 | 7265 | 7284
19
7434 | 7453 | 7472
18
85
3 7490
7509 7527 7545
7564
7582 7600
7618 | 7636 | 7654
IS
86
87
7672
7691
7851 | 7868
7691 7708 7726
78687886 7903
7921
7938 | 7956 |
88
3 8025 8042
8042 8059 8076
8093
89 8195
8212 8229 8245
8262
90
8362 8378
8378 8394 8411
8427
|
7744 7762 7780
|
7973 7990 8008
8110 8127 8144
8279 | 8295 8312
8443 8460
7798
7798 7815|| 7833 18
со со
7973
17
8161 8178
17
8329 | 8345
17
8476 8492 8508
16
91
92
93
94
95
96
97
98
3 8524 | 8540 8556 8572 8588 8604 8620 | 8636 | 8652 | 8668
8684 8699
8699 8715 8731 8746 | 8762 | 8778 8793 8809 | 8824
8839 8855
8855 8870 8886 | 8901 | 8916 | 8931 | 8947 | 8962 | 8977
3 8992 9007
9007 9022 9037 9052 9067 9082 9097 | 9112 | 9126
| |
9141 9156
9156 | 9171 | 9185 | 9200 | 9215 | 9229 | 9244 | 9258 | 9273
92879302
9302 9316 9331 9345 9359 | 9374 9388 | 9402 | 9416
3 9431 9445 9459 9473 | 9487 | 9501 | 9515 | 9529 | 9543 | 9557
9571 9585
95999612 | 9626 | 9640 | 9654 | 9667 | 9681 | 9695
99 9722 97899803
9708 9722 9735 | 9749 | 9763 | 9776 | 9789 | 9803 | 9816 | 9830
16
16
15
15
15
14
14
14
14
71

(L) TABLE OF VALUES OF Ty.
V
I
2
3
4
5
6
7
8
9 Diff.
secs.
secs.
secs.
secs. secs. +
f-s
secs.
secs.
secs.
secs.
secs.
100 229. 5207 5340 5473| 5606 | 5738 | 5869 | 6001 | 6132
ΙΟΙ
102
| |
7418
6522 | 6651 | 6780 | 6908 | 7036 | 7164 | 7291
7796 792180468170 | 8294| 8417 8540 8662
103 229. 9024
104 230. 0177
105
9144 9262 9380 9496
| | |
0287 0396 0504 0610
1226 1325 1423 1520 1615
1710
1710
2609
6262|6392|| 132
7544 | 7670 || 127
8783 8904|| 123
9612 9727 9841 9954 0066|| 115
| | | |
0716 0820 0923 1025 1126 105
1804 1897 | 1988 | 2079
106 230. 2170| 2259 2347 2435 2522 2609 2694 2780 2864 2948
107 3031 3114 3196 | 3278 | 3359 3439 3520 3599 3678 3757
108 3835 3913 3990 | 4067 | 4143 4219 4295 4370 4445 4519
| | |
109 230. 4593| 4667 | 4740 | 4813 4885 4958 | 5030 | 5101
5314 5384 5454 | 5524 5593 5662 5731 5800
6004 6071 6139 | 6206 | 6272 | 6339 | 6405| 6471
IIO
III
| |
| 5172 | 5243
5868 5936
|
6537 6603
112 230. 6668 | 6733 | 6798 6863 6928 | 6992 | 7056 | 71207184 7248
| | |
7311 7374 7437 7500
7936 7997 8059 8120 8181
113
114
|
7563
94
86
80
76
72
69
66
64
62
7625 7688 7750 7812 | 7874
| | | |
8242 8303 8364 8424 8484||| 61
115 230. 8545 8605 8665 8726 8787 8847 8906 | 8965
9142 9200 | 9259 9317 9375 9433 9490 9548
| |
I
116
117
118 231. 0283 | 0338 | 0394 0449
0832 0886 0940 | 0994
1367|| 1420 | 1473 1525
231. 1889 1941 1992 2043 2095
2399 2449 2499 2549 2599
|
|
2896| 2945| 2994 3043 3091
|
119
120
121
122
123
|
|
9024 9083 59
9605 9663|| 58
삼성 ​넑읽
​56
55
9720 9777 9833 | 9890 | 994700030059 0115 0171 0227
05040559 | 0614 | 0669 | 07230778
1048 ΙΙΟΙ 1154 1208 1261 1314 54
1578 | 1630 | 1682 | 1734 | 1786 | 1838|| 52
2146 2196| 2247 2298 2348
| |
2649 2698 | 2748 | 2797 2847
|
3140 3188 | 3237 | 3285 | 3333
124 231. 3381 3429 3477 3524 3572 3619 3667 3714 3761 3808
3855 3902 3948 3995 4041 | 4088 | 4134 4180 4226 | 4272
|
4318 4364 4410 4455 4501 4546 4591 4636| 4681 4726
| | | |
|
125
126
128
129
127||231. 4771 | 4816 | 4860 | 4905 | 4949 | 4993 | 5038 | 5082 5126 5170
| |
51
50
49
47
46
45
|
44
| |
5214 5257 5301 5345 5388 5431 5475 5518 5561 5604
|
5647 5690 5732 5775 5818 | 5860 | 5902 5945 5987 | 6029
| |
43
42
130 231. 6071 6113 6155 | 6196 6238 | 6280 | 6321 | 6362 | 6404 6445
| |
42
131
132
6486 6527 6568 6609 6650|| 6690 6731 6772 6812 6852
| | |
6893|6933 | 6973 | 7013 | 7053 7093 7133 | 7173 | 7212 | 7252
4I
|
40
133 231. 7291 | 7331 | 7370| 7410 | 7449
7370 7410 7449
7488
7527
|
7566 7605 | 7644
134
135
137
138
7682 7721 77607798 | 7837
7760 7798 7837
8066 8104 8142 8179 8217
| |
136 231. 8442 8479 8517 8554 8591
8812 8848 8885 8921 8958
| |
9175 9211 | 9247 | 9282 | 9318
9247 9282 9318
139 231.9532 9567 9602|| 9638 9673
140
8665 8702 8738 8775
8994 9030 | 9067 | 9103 9139
9354 9390 | 9425 | 9461 |9496
|
9708| 9743 9778 9813 9848
98839918 | 9952 | 9987 | 0022 00560091 0125 0160 0194
| |
0297 0331 0365 | 0399 | 0433 0467 05010535
| | | |
141||232. 0228 | 0263 | 0297 | 0331
143
144
1234
34
33
33
142|| 232. 0569 | 0602 0636 | 0670 | 0703|| 0737 0770 0804 0837 0870|| 34
0904 0937 | 0970 | 1003
1003 1036| 1069
1069 | 1102 1135 1168 1201
1267|| 1299
1299 1332 1364 1397 1430 1462 1494 1527
|
145 232. 1559 1591 | 1624 | 1656 | 1688
1624 | 1656 | 1688
146 1880 1912 | 1944 1975 2007
2197 2228 2260 2291 2322 2354 2385 2416 2447 2478
147
1720 1752 1784 | 1816 | 1848
2039
32
7875
7913
7952 7990 8028
8255
8292
|
8330 8367 8405
8405 38
8628
8775
36
9496 36
9848 35
2071 2102 2134 2165
| |
www www www wwww www
39
38
37
72

(M) TABLE OF VALUES OF Sv.
V
O
I
2
3
4
5
6
7
8
9 Diff.
f-s
feet.
103 40 230.1
104 349.4
107
108
IIO
III
I
|
623.6 | 632.6 | 641.6
711.4 719.9 728.4
794.8 802.9 811.0
9.2
8.6
8.2
7.2
7.1
7.0
6.8
6.7
6.6
6.5
6.4
6.3
6.2
6.1
6.0
feet. feet. feet. feet. feet. feet. feet. feet. feet. +
100 39 842.9 856.3 869.6 882.9 896.1| 909.3 922.5 935.7 948.9 961.9 13.2
ΙΟΙ 975.0 988.1 | 001.1 014.1 027.1 040.0 052.9 065.8 078.7 091.5 12.9
102 40 104.3 117.1 129.8 142.5 155.2167.8 180.4| 192.9 205.4 217.8 12.6
242.4 254.6|| 266.8 278.8 290.8 302.7 314.5 326.2 | 337.8 11.9
360.8 372.2 383.4 394.5 405.6 416.5 427.3 438.1 448.7 11.0
105 459.2 469.6| 479.9 | 490.0 | 5C0.1 510.1 520.0 529.8 | 539.5 549.2 9.9
106 40 558.7 568.2 577.6 586.9 596.2 605.4 614.5
650.5 659.3 668.1 | 676.9 685.6 704.2 702.8
736.8 745.2 753.6 761.9 770.2 778.4 786.6
109 40 819.0 827.1 835.0 843.0 850.9 858.9 866.7 874.6 882.4 890.2 7.9
897.9 905.7 913.4 921.1 | 928.7|936.4 944.0 951.5 959.1 966.6|| 7.6
974.2 981.6 989.1 996.6 004.0 11.4 18.8 026.2 033.5 040.9 7.4
112 41 048.2 055.5 062.8 070.0 077.3 084.5 091.7 099.0 | 106.1 | 113.3
113 120.5 127.6 134.8 | 141.9 | 149.0 156.1| 163.2 170.2 177.3 184.4
114
191.4 198.4 205.4 212.4 219.4 226.4 233.3 240.3 247.2 254.1
115 41 261.0 267.9 274.8 281.7 288.6 295.4 302.3 309.1 315.9 322.7
116 329.5 336.3 343.1 | 349.8 | 356.6 363.3 370.0 376.7 383.4 390.1
117
396.8 403.5 410.1 416.8 423-4 430.0 436.6 443.2 449.8 456.4
118 41 462.9 469.5 476.0 482.6 489.1 495.6 502.1 508.6 515.1 521.5
|
| |
119 528.0 534.4 540.9 547.3 553-7 560.1 566.5 572.9 579.2 585.6
I 20 591.9 598.3 604.6 610.9 617.2 623.5 629.8 636.1 642.3 648.6
121 41 654.8 661.1 667.3 673.5 679.7 685.9 | 692.1 | 698.2 704.4 710.5
122 716.7 722.8 728.9 735.0 741.1 747.2 753.3 759.4 765.4 771.5
123 777.5 783.6 789.6 795.6 801.6 807.6 813.6 819.6 825.6 831.5
124 41 837.5 843.4 849.4 855.3 861.2 867.1 873.0 878.9 884.8 890.6|| 5.9
125 896.5 902.3 908.2 | 914.0 | 919.8 925.6 931.5 937.3 943.0 948.8
126 954.6 960.4 966.1 971.9 977.6 983.3 989.0 994.8000.5006.2
127 42 011.8 017.5 023.2 028.9 | 034.5 | 040.2 | 045.8 | 051.4 057.0 062.7
128 068.3 073.9 079.5 085.0 090.6 | 096.2 || 101.8 107.3 112.9 | 118.4
5.6
129 123.9 129.4 135.0 140.5 146.0 151.5 157.0 162.4 167.9 173.4 5.5
130 42 178.8 184.3 189.7 195.1 200.6 206.0 211.4 216.8 222.2 227.6 5.4
131 233.0 238.4 243.7 249.1 254.5 259.8 265.1 270.5 275.8 281.1
132 286.4 291.8 297.1 302.4 307.6 312.9 318.2 323.5 328.7 334.0
133 42 339.2 344.5 349.7 355.0 360.2 365.4 370.6 375.8 381.0 386.2
134 391.4 396.6 401.8 406.9 412.1 417.3 422.4 427.6 432.7 437.8
135 443.0 448.1 453.2 458.3 463.4 468.5 473.6 478.7 483.8 488.9
136 42 493.9 499.0 504.1 509.1 514.2 519.2 524.3 529.3 534.3 539.4
137 544.4 549.4 554.4 559.4 564.4 569.4 574.4 579.4 584.3 589.3
138 594.3 599.2 604.2 609.1 614.1 619.0 624.0 628.9 633.8 638.8 4.9
139 42 643.7 648.6 653.5 658.4 663.3 668.2 673.1 678.0 682.9 687.8 4.9
692.6 697.5 702.4 707.2 712.1 717.0 721.8 726.7 731.5 736.3 4.9
741.2 746.0 750.8 755.7 760.5 765.3 770.1 774.9 779.7 784.5 4.8
142 42 789.3 794.1 798.9 803.7 808.5 813.2 818.0 822.8 827.5 832.3
143
837.1 841.8 846.6 851.3 856.0 860.8 865.5 870.2 875.0 879.7
144 884.4 889.1 893.8 898.6 903.3 908.0 912.7 917.4 922.1926.7
145 42 931.4 936.1940.8945.5 950.1 | 954.8 959.5 | 964.1 | 968.8 | 973.5
146 978.1 982.8 987.4 992.1 996.7 001.3 006.0 010.6 015.2 019.9
147 43 024.5 029.1 033.7 038.4 043.0 047.6 052.2 056.8 061.4 066.0
140
141
5.8
5.7
in in
5.6
5.3
5.3
5.2
5.2
5.I
5.0
5.0
4.8
4.7
4.7
4.7
4.6
4.6
6
73

(L) TABLE OF VALUES OF Tv.
V
O
I
2
3
4
5
6
7
8
9
Diff.
secs.
secs.
secs.
secs.
secs.
secs.
secs.
secs. secs.
+
f-s
149
150
secs.
148 232. 2509 2540 | 2571 | 2602
|
2633 2664 2695 | 2726 | 2757 2787|| 31
2818 2849 2879 | 2910 | 2940 | 2971 | 3001 | 3032 3062 | 3093|| 30
3123 3153 3183 3214 3244 3274 3304 3334 3364 3394 30
151232. 3424 3454 3484 3514 3543 3573 3603 | 3633 3662
3722 3751 3781 | 3810 | 3840 | 3869 | 3899 3928
4016| 4046 | 4075 | 4104 | 4133 | 4162 | 4192 | 4221
152
153
155
3692|| 30
3958 3987 29
4192 4221 4250 4279|| 29
154 232. 4308 4337 4366 4395 4424 4453 4481 4510 4539 4568 29
4597 4625|| 4654 | 4683| 4711 4740 4768 4797 4825 4854 29
156
4882 4911 4939 4967 4996 5024 5052
157 232. 5165 5193 5221
|
5080
5108 5137 28
5249 5277
5305 5333
5361
5389
5416|| 28
158
159
5444 5472 5500 5528 5555
5721 5748 5776 5803 5831
5583 5611 5638
5666
5693|| 28
5858 5885
5913
5940
5967|| 27
160 || 232. 5994 6022 6049 6076 6103
6130 6157 6184
6211
6238|| 27
161
6265 6292 6319 6346 6373 6400 6426 6453
6480
6506|| 27
162
6533 6560 6586 6613 6640
6666 6693 6719
6745
6772|| 26
163||232. 6798 6825 6851 6877 6903 6930
6956
6982
7008
7034 26
164
165
7061 7087 7113 | 7139 || 7165 | 7191
7320 7346 7372 7398 7423 7449
7217
7243
7268 | 7294 || 26
7475
7500
7526 7552|| 26
166|| 232. 7577 7603 | 7628 | 7654
7679|| 7705
7730
7756 || 7781 | 7806 || 25
167
7933 7958 7983
8008 8034 8059 || 25
168
7832 7857 7882 7908
8084 8109 8134 | 8159 | 8184 | 8209 | 8234 | 8258 | 8283 | 8308 || 25
169 232. 8333 8358 | 8383 | 8407 | 8432 8457 8481 8506
170
171
173
174
8580 8604 8629 | 8653 | 8678
8824 8848 8872 8896 8921
8945
8702 8726 8751
8969 8993
9209 9233 9257
9447 | 9470 | 9494
9682 9705 9729
9017
8531 8555 25
8775 | 8799 || 24
9041 || 24
9281 || 24
9518|| 24
9752|| 23
9916 9938 | 9961 | 9985 || 23
0146|| 0169 0192 0215 23
0374 0397 0420 0442|| 23
172232. 9065 9089 9113 9137 9161 | 9185
9304 93289352 | 9376 | 9399 9423
9541 95659588 9612 9635 9659
175 232. 9776 9799 9822 | 9845 | 9869 | 9892
176233. 0008 0031 0054 0077 ΟΙΟΟ 0123
0237 0260 0283 | 0306 | 0329 0351
177
178 233.0465 0488 0510 0533 0555 0578 0600 0623 0645 0668 23
0690 0713 0735 0757 0780 0802 0824 0847 0869 0891 22
0913 0935 0958 0980 1002 1024 1046 1068 1090 1112 22
179
180
181 233. 1134| 1156 | 1178
182
1353 1375 1396
I 200 I222 I244 1266
1287
1309 1331 22
1418 1440 || 1461|| 1483
1505
1526 1548 22
183
1569 | 1591 | 1613
| 1634 | 1656
1634 1656 1677
1698
1720
1741 1763 21
184 233. 1784 1805 1827
1848 1869
1848 1869 1891
1912
1933 1954 1975
| |
21
185
1997 2018 2039
2060 2081 2102
2123 2144 2165 2186
21
186
2207 2228 2249
2270
2291
2312
2333 | 2353 | 2374 | 2395 || 21
188
189
187233. 2416 2437 2457 2478 2499 2520
2623 2643 | 2664 | 2685 | 2705 2726
2828 2848 2869 | 2889 | 2909 | 2930| 2950
2540 2561 2582 2602 21
2746| 2767
2787
2808
2808|| 21
2970 | 2991
3011 20
190 || 233. 3031 | 3051 | 3072 3092 | 3112
191
192
3092 3112 3132 3152 3172 3192 | 3212 || 20
|
3233 3253 3273 | 3293 | 3313 | 3333 | 3353 | 3372 | 3392 | 3412 20
3432 3452 3472 3492 3511 | 3531 | 3551
3571
3590 | 3610|| 20
194
195
193 233. 3630 3649 3669 3689 | 3708 | 3728 3747
3825 3845 3864 3884 3903 3922 3942
| |
4019 4038 4057 4077 4096 4115 4134
3767
3961
3786 3806|| 20
3980 4000 19
| ||
4153
|
4172 4192 19
74
(M) TABLE OF VALUES OF Sv.

V
f-s
150
151
152
153
O
I
2
3
4
5
6
7
8
9 Diff.
feet. feet
feet. feet. feet. +
feet.
102.7 107.3 111.8 4.6
148.3 | 152.9|| 157.4|| 4.6
193.7 198.2 202.7|| 4.5
feet. feet. feet. feet.
148 43 070.6 075.2 079.8 084.4 089.0 093.5 098.1
149 116.4 121.0 125.6| 130.1| 134.7 139.2 143.8
162.0 166.5 171.0 175.6 180.1 | 184.6| 189.2
43 207.2 211.8 216.3 220.8 225.3 229.8 234.3 238.8 243.3 247.8 4.5
252.3 256.8 261.3 265.8 270.3 274.8 279.3 283.8 288.3 292.8 4.5
297.2 301.7 306.2 310.6 315.1 | 319.6 324.1 328.5 333.0 337.5 4.5
154 43 342.0346.4 350.9 355-3359.8364.3 368.7 373.2 377.6 382.1 || 4.5
386.5 391.0 395.4 399.9 404.3 408.7 413.2 417.6 422.0 || 426.5 || 4.4
430.9 435.3 439.8 444.2 448.6 453.0 457.4 461.9 466.3 470.7 4.4
157 43 475.1 479.5 483.9 488.3 492.7 497.1 501.5 505.9 510.3
|
519.1 523.5 527.9 532.3 536.7 541.1 545.4 549.8 554.2
159
563.0 567.3 571.7 576.1 580.4 584.8 589.1 593.5 597.9
| |
160 43 606.6 610.9 615.3 619.6 624.0 628.3 632.6 637.0 641.3
155
156
158
161
162
514.7 4.4
558.6|| 4.4
602.2 4.4
645.7 4.3
||
650.0 654.3 658.7 663.0 667.3 671.6 676.0 680.3 684.6 688.9 4.3
693.3 697.6 701.9 706.2 710.5 714.8 719.1 723.4 727.7 732.0 4.3
||
163 43 736.3 740.6 744.9 749.2 753.5 757.8 762.1 766.4 770.6 774.9 4.3
|
164 779.2 783.5 787.8 792.0 796.3 800.6 804.9 809.1813.4 817.6 4.3
165 821.9 826.2 830.4 834.7 838.9 843.2 847.4 851.7 855.9 860.2 4.3
166 43 864.4 868.7 872.9 877.2 881.4 885.6 889.9 894.1 898.3 902.5 4.2
167 906.8 911.0915.2 919.5 923.7 927.9 932.1 936.3 940.5 944.7 4.2
|
940.5944.7 ||
168 949.0 953.2 957.4 961.6 965.8 970.0 974.2 978.4 982.6 986.7 4.2
169 43 990.9 995.1 999.3 003.5 007.7 0 12.0 16.0 020.2 024.4 028.6|| 4.2
170 44 032.7 036.9 041.1045.2 049.4 053.6 057.7 061.9 066.0 070.2 4.2
171 074.3 078.5 082.6 086.8 | 090.9 095.1 099.2 103.3 107.5 111.6|| 4.1
172 44 115.7 119.9 124.0 128.1 132.3 136.4 140.5 144.6 148.7 152.9 4.1
173 157.0 161.1 165.2 169.3 173.4 177.5 181.6 185.7 189.8 | 193.9 || 4.1
174 198.0 202.1 206.2 210.3 214.4 218.5 222.6 | 226.7 | 230.8 234.8 4.1
175 44 238.9 243.0 247.1 251.2 255.3 259.3 263.4 267.5 271.5 275.6 4.1
176
279.6 283.7 287.8 291.8 295.9 300.0 304.0 308.0 312.1 | 316.1 || 4.1
177 320.2 324.2 328.3 332.3 336.4 340.4 344.4 348.5 352.5 356.5 4.0
178 44 360.5 | 364.6 368.6 372.6 376.6 380.7 384.7 388.7 392.7 396.7|| 4.0
179 400.7 404.7 408.8 412.8 416.8 420.8 424.8 428.8 432.8 436.8 4.0
440.8 444.7 448.7 452.7 456.7 460.7 464.7 468.7 472.6 476.6 4.0
181 44 480.6 484.6 488.5 492.5 496.5 500.5 504.4 508.4 512.4 516.3 4.0
182 520.3 524.2 528.2 532.2 536.1 540.1 544.0 548.0 551.9 555.9 4.0
183
559.8 563.7 567.7 571.6 575.6 579.5 583.4 587.4 591.3 595.2|| 3.9
184 44 599.2 603.1 607.0 610.9 614.9 618.8 622.7 626.6 630.5 634.4 3.9
185
638.4 642.3 646.2 650.1 654.0 657.9 661.8 665.7 | 669.6 673.5 3.9
186 677.4 681.3 685.2 689.1 693.0 696.9 700.8 704.6| 708.5 712.4 3.9
187 44 716.3 720.2 724.1 | 727.9 731.8 735.7 739.6 743.4 747.3 751.2 3.9
188
755.0758.9 762.8 766.7 770.5 774.4 778.2 782.1 786.0 789.8 3.9
189 793.7 797.5 801.4 805.2 809.1 812.9 816.8 820.6 824.5 828.3 3.8
190 44 832.2 836.0 839.8 843.7 847.5 851.4 855.2 859.0 862.8 866.7 3.8
870.5 874.3 878.1 882.0 885.8 889.6 893.4 897.3 901.1 904.9 3.8
908.7 912.5 916.3 920.1 923.9 927.7 931.5 935.3 939.1 942.9 || 3.8
193 44 946.7 950.5 954.3 958.1 961.9 965.7 969.4 973.2 977.0 980.7 3.8
194
984.5 988.3 992.1 995.8 999.6 003.4 007.1 010.9 014.7 018.4 3.8
195 45 022.2 025.9 029.7 033.4 037.2 040.9 044.7 048.4 052.1 055.9 3.7
180
191
192
75

(L) TABLE OF VALUES OF Ty
V
O
I
2
3
4
เก
5
6
7
8
9
Diff.
f-s
secs.
secs.
secs.
secs.
secs.
secs.
secs.
secs.
secs.
secs.
+
197
198
196|| 233. 4211 4230 4249 4268 4287 4306 4325 4344 | 4362
4400 4419 4438 4457 4476 4494 4513 | 4532 4550
4588 4606| 4625 | 4644 | 4662 | 4681 | 4699 4718 4736
199 233. 4773 4791 4810
4381 | 19
4569|| 19
4755 19
200
201
203
204
206
207
209
210
212
213
7177
215
216
218
219
8059
8200
221
8338
222
224
227
228
230
231
4828|| 4846 | 4865
4956| 4974 | 4992 5010 5028 5047
5137 5155 5172 5190 5208 5226 5244
202233. 5315 | 5333 | 5351 | 5368|| 5386 | 5404 5421 5439 | 5456
5492 5509 5527 5544 | 5561 | 5579 | 5596 | 5614 | 5631
5666 5683| 5700 5717 5735 | 5752 | 5769 | 5786 | 5803
5769 5786 5803
205 233. 5837 5854 | 5871 | 5888 | 5905 | 5922
6007 6024 6040 6057 6074 6091
|
6074 6091
6174 6191 6207| 6224 | 6240 6257
208 233. 6339 6355 6372 6388 6404 6420 6437
6502 6518|| 6534 6550 6566 | 6582 | 6598
6662 6678 6694 6710 6726 6741 2757 6773 6789
211|| | |
211 233. 6820 6836 6852 6867 6883 6899
6883 6899 6914 | 6930|| 6946
6977 6992 7008 7023 7039 7054
7039 7054 7070 | 7085 7100
7131 7146 7162
792 7207
| ||
7:92 7207 7223 7238 7253||
214 233. 7283 7298 | 7313 7329 7344 7359 7374 7389 | 7404
7434 7448 7463 | 7478 || 7493 | 7508 | 7523 7538 | 7552
7582 7597 7612 | 7626 || 7641 | 7656
7641|| 7656| 7670| 7685|| 7700
217 233. 7729 7743 7758 7772 7787 7801
| | | 7787 7801 7816 | 7830 | 7845
7874 7888 7902 7917 7931 7945 7960 7974 7988
8016 8031 8045
8073 8087
8073 8087 8101 8115 8129
220 233. 8158 8172 8186
8214 8227
8214 8227 8241 | 8255
8297 8311 8325
8435 8448 8462 8476
223|| 233. 8571 8584 | 8598 | 8611
8705 8718 8732 8745
225 8838 8851 8864 8877
226 233. 8969 8982 8995 9008
9047 9059 | 9072 | 9085 13
||
9098 9111 9124 9137
9162 9175 9188 9201
|
9213 13
9226|| 9239 | 9252 9264
9290 9303 9315 9328 9341
|
| | |
13
229 233. 9353 9366 9378 9391 9404 9416 9429 9441 | 9454 9467|| 13
| |
9479 9492 9504 9517 | 9529 9542 95549567 | 9579 | 9592
9604 9617 9629 | 9642 | 9654 | 9667| 9679|| 9692 | 9704 9716
| | |
|
4883
4901 | 4920
4938 || 18
5065
5083 5101
5119 18
5262 5280
5297
18
|
5474
5474 18
5648|| 17
5820 17
5939 5956 | 5973
6107 | 6124 | 6141
5990 || 17
6157
17
6273
6290 | 6306
|
6323 || 16
6453 6469
6485 || 16
6614 | 6630
6646|| 16
6805 16
|
6961 || 16
7116 15
7268 || 15
6 in in
7419|| 15
7567|| 15
7714|| 15
7859|| 14
| 8002 14
8144 14
8269 | 8283 || 14
8625 |
8638
8758 8772
8758 8772
8890 8903
8352 | 8366 | 8380 8394 8407 8421 14
8489 8503
8489 8503 85178530 | 8544 8557|| 14
8651 | 8665 | 8678 | 8692 || 13
8785 | 8798 | 8811 | 8824 || 13
8916 8930 8943 | 8956|| 13
|
9021 9034
9150
9277
|
13
12
234
236
337
239
240
232 233. 9729 9741 9754 9766 | 9779 | 9791 | 9803 | 9816 | 9828 | 9841 || 12
|
233 9853 98669878 9890 9903 9915 9927 9940 9952 9965|| 12
|
| | | | |
9977 9989 0002 0014 0026 0039 | 0051 0063 0076 0088 12
235 234. 0100 0113 0125 0137 0150 0162 0174 0186 0199 0211 12
0223 0236 | 0248 | 0260 | 0272 0284 0297 | 0309 | 0321 | 0334 12
0346 0358 0370 0383 0395
0419 0431 0444 0456
|
238 234. 0468 0480 0492 0505
0590 0602 0614 0626
0711 0724 0736 0748
0407
12
0517 0529
0639 0651
0760|| 0772
0541
0663
0553 0566 0578 12
| |
0675 0687 0699|| 12
0784 0796 0809 0821 12
242
243
| | |
241 234. 0833 0845 0857 0869 0881 0893 0905 0917 0930 0942
||
0954 0966 0978 0990 1002 1014 1026
1074 1087 1099 IIII 1123 1135 1147
12
1159 1171 1183
1038|| 1050 || 1062
1159
12
12
76

(M) TABLE OF VALUES OF Sv.
V
O
I
2
3
4
5
6
7
со
8
9 Diff.
f-s
196
197
198
199
200
201
202
203
204
feet.
264.9
feet. feet. feet.
+
085.7 089.4 093.1 3.7
122.8 126.5 130.2 3.7
159.6 163.3 166.9 3.7
| |
196.2 | 199.8 | 203.4|| 3.6
232.5 236.1 239.7 3.6
268.5 272.1 275.7 3.6
feet. feet. feet. feet. feet. feet.
45 059.6 063.4 067.1 070.8 074.6 078.3 082.0
096.9 100.6 104.3 108.0 111.7 115.4 119.1
133.9 137.5 141.2 144.9 148.6| 152.3 156.0
| |
45 170.6 174.3 177.9 181.6 185.2 188.9 192.5
207.1 210.7 214.3 218.0 221.6 225.2 228.8
243.3 246.9 250.5 254.1 257.7 261.3
45 279.2 282.8 286.4 290.0 293.6| 297.2 300.7 304.3 307.8 311.4 3.6
314.9 318.5 322.0 325.6 329.1 | 332.7 336.2 339.7 343-3 346.8 3.5
350.3 353.8 357.3 360.9 364.4 367.9 371.4 374.9 378.4 381.9 3.5
205 45 385.4 388.9 392.4 395.9 399.4 402.9 406.3 409.8 413.3 416.7 3.5
420.2 423.7 427.1 430.6 434.1 437.5 441.0 444.4 447.8 451.3 3.5
454.7 458.1 461.6 465.0 468.4 471.9 475.3 478.7 482.1 | 485.5 || 3.4
208 45 488.9 492.3 495.7 499.1 502.5 595.9 509.3 512.7 516.1 519.4 3.4
209 522.8 526.2 529.6 532.9 536.3 539.7 543.0 546.4 549.7 553.1 3.4
556.4 559.8563.1 566.4 569.8 | 573.1 | 576.5 579.8 583.1 586.4 3.3
45 589.7 593.0 596.4 599.7 603.0 606.3 609.6 612.9 616.2 619.5|| 3.3
622.8 626.1 629.3 632.6 635.9 639.2 642.5 645.7 649.0 652.3 3.3
| |
|
655.5 658.8 662.0 665.3 668.6|671.8 675.1 678.3 681.5
206
207
210
211
212
213
215
216
219
220
221
222
684.8|| 3.2
214 45 688.0 691.2 694.5 697.7 700.9 704.2 707.4 710.6 713.8 717.03.2
720.2 723.4 726.6 729.9 733.1 736.3 739.5 742.6 745.8 749.03.2
752.2 755.4758.6 761.8 764.9 768.1 771.3 774.4 777.6 780.8|| 3.2
217 45 783.9 787.1 790.2 793.4 796.6 799.7 802.9 806.0 809.1 812.2 3.1
218 815.4 818.5 821.6 824.8 827.9 831.0 834.1 837.3 840.4 843.5 3.1
846.6 849.7 852.8 855.9 859.0 862.1 865.2 868.3 871.4 874.4 3.1
45 877.5 880.6 883.7 886.8 889.9 893.0 | 896.0 899.1 902.1 905.23.1
908.3 911.3 914.4 917.4 920.5 923.6926.6 929.6 932.7 935.7 3.0
938.7 941.8 944.8 947.8 950.9 953.9 956.9 959.9 963.0 966.0|| 3.0
223 45 969.0 972.0 975.0 978.0 981.0 984.0 | 987.0 990.0 993.0 996.0 3.0
224 999.0002.0004.9 007.9 10.9 013.9 016.9 019.8 022.8 025.8 3.0
225 46 028.7 031.7 034.6 037.6 040.6 043.6 | 046.5 049.5 052.4 055.3 3.0
226 46 058.3 061.2 064.1 067.1 070.0 | 072.9 075.9 078.8 081.7 084.7 2.9
227 087.6 090.5 093.4 096.3099.3 102.2 105.1 108.0 110.9 113.8 2.9
228 116.7 119.6 122.5 125.4 128.3 131.2 134.1 137.0 139.9 142.8 2.9
229 46 145.7 148.6 151.5 154.4 157.3 160.2 163.1 166.0 168.8 171.7 2.9
230 174.6 177.5 180.4 183.3 186.2 189.1 191.9 194.8 197.7 200.6 2.9
231 203.5 206.4 209.3 212.1 215.0 217.9 220.8 223.7 226.6 | 229.5 || 2.9
232 46 232.3 235.2 238.1 241.0 243.9 246.8 249.7 252.6 255.4 258.3 2.9
233
261.2 264.1 267.0 269.9 272.8 275.7 278.6 281.5 284.3 287.2 2.9
234 290.1 293.0 295.9 298.8 301.7 304.6 307.5 310.4 313.3 316.2 2.9
235 46 319.0 322.0 324.9 327.7 330.6 333.5 336.4 339.3 342.2 345.1 2.9
236 348.0 350.9 353.8 356.7 359.6 362.5 365.4 368.3 371.2 374.1 2.9
237 377.0 379.9 382.8 385.7 388.6|391.5| 394.4 397.3 400.2 403.1 2.9
238 46 406.0 408.9 411.8 414.8 417.7 420.6 423.5 426.4 429.3 432.2 2.9
239 435.1 438.0 440.9 443.8 446.8 449.7 452.6 455.5 458.4 461.3 2.9
464.2 467.1 470.1 473.0 475.9 478.8 481.7 484.6 487.6 490.5 2.9
241 46 493.4 496.3 499.2 | 502.2 | 505.1 508.0 510.9 513.8 516.8 519.7 2.9
242 522.6 525.6 528.5 531.4 534-3 537.3 540.2 543.1 546.1 549.0 2.9
551.9 554.9 557.8 560.7 563.7 566.6 569.5 572.5 575.4 578.3 2.9
240
243
77
(L) TABLE OF VALUES OF Ty.

V
O
I
2
3
4
5
6
7
8
9
Diff.
f-s
secs.
secs.
secs.
secs.
secs.
secs.
secs.
secs.
secs.
secs.
+
244 234. 1195
245
246
1471 | 1483|
|
1590
1602
1207 1219 1231 1243 1255 | 1267|| 1279|| 1291 | 1303
1315 1327 1339 | 1351 | 1363
1351 1363
1387 1399 1411 1423 12
1435 1447 | 1459 |
1506 | 1518 | 1530 | 1542
|
247 234. 1554 1566 1578
12
12
1375
1495
1614
1626
||
1638 1649|| 1661
12
248
1673 1685 1697
1709
1721 1733
1744
1756 1768 1780 12
249
1792
1804 1815
1827
1839
1851 1863
1874
1886 1898 12
|
251
2027
|| | | |
250 234. 1910
2062
1922 1933 1945
1957 |
1969 | 1980
1969 | 1980 | 1992
| 2004 | 2015
12
2039 | 2051
2074
252
253 234. 2260 2272
257
258
254
255
256 234. 2603 2615
2716 2727 2738
2827 2838 2849
2375 2387
2490 2501
2283 2295| 2306|
2398 | 2410 | 2421
2513 | 2524 | 2535
2626 2637 2648
| |
2749 | 2760
|
|
2860 2871 ||
2121 2132 12
2144 2156 2167 2179 2190 2202 2214 2225| 2237 | 2248 12
2318 | 2329 | 2341 | 2352 | 2364 || 12
2433 2444 2455 2467 | 2478
2547 2558 2569 | 2581 2592
2547
2558
2660 2671| 2682| 2693 | 2705
2794 2805| 2816
29042915| 2926
2086 | 2097 | 2109
II
II
2660 2671
II
2772 2783
2772 | 2783
2882 || 2893
2882 2893
I I
II
260
261
259 234. 2937 2948 | 2959 2970 2981
3046 3057 3068 | 3079 | 3090
3154 3165 3176 | 3187 | 3197
2992 3003
3014 | 3025 | 3036 || 11
3101
3208
3111
3208 3219
3219
3122 3133
3133 3144 II
3229 | 3240 | 3250 II
263
262 234. 3261 3272 3282 3293 3303
|
|
3314 3325
3314 3325
3335 | 3346 | 3356 || 11
3419 3429
3440 3450 3461 IO
272
273
274 234. 4457
275
276
277 234. 4734
278
279
280 234. 5001
281
282
4743
4824 4833
4913 4922
5010
5088 5097
5174 5183
264
266
267
3367 3377 3388 | 3398 | 3409
3523 3533 3544 3554 | 3564|| 10
3471 3482 3492 3502 3513 3523 3533
34823492 | |
4066
IO
ΙΟ
4066|| 10
265 334 3574 3585 3595 3605 | 3615 | 3626 | 3636 | 3646 | 3656 | 3667 || 10
3677 3687 3697 3707 3717 3728 3738 3748 3758 | 3768 || 10
3778 3788 3798 3808 3818 3828 3838
3828 3838 3848 3858 3868
268 234. 3878 3888 3898 3908 3918 3928 3938 | 3948 | 3958 | 3968
|
3977 3987 3997 4007 4017 4027 4036 4046 4056
4075 4085 4095 | 4105 | 4114|| 4124 | 4134 4143 | 4153 | 4163
271|| 234. 4172 | 4182 | 4192 | 4201 4211 4220 4230 4240 4249 4259 ΙΟ
4268 4278 4287 4297 4307 4316 4326 4335 4344 4354
| | |
| |
4363 4373 4382 | 4392 | 4401 | 4411 | 4420 4429 4439 | 4448
269
270
4467|| 4476 | 4485 | 4495 | 4504 | 4513 | 4523 | 4532 | 4541
4551 4560 4569 | 4578 | 4587| 4597|| 4606 | 4615| 4624 | 4633
|
4643 4652 4661 4670 4679 4688 4697 4706 4715 4725
| | |
4752 4761 4770 4779 4788 4797 4806 4815
| |
|
4842 | 4850 | 4859 | 4868 | 4877 | 4886 | 4895 4904
| |
|
4930 | 4939 | 4948 | 4957 4966 | 4975 | 4983 | 4992
| |
5018 | 5027 5036 5045 5053 5062 5071 5080
|
|
5105 5114 5123 | 5131 | 5140 5148 5157 | 5166
5191 | 5200 | 5208 | 5217 | 5226 | 5234 | 5243 5251
ΙΟ
ΙΟ
9
9
66 a
283 234. 5260 5268
284
285
5277|| 5285 | 5293 | 5302
5344 5352 5361 5369 5378 5386
| |
5427 5436 | 5444 | 5452 | 5461 | 5469
5310
|
5319 5327 5336
5394
|
5403 5411 5419
5477
|
5485 5494 5502
286|| 234. 5510 5518 5527 5535 5543 5551
5559
5567 5576 5584
287
288
5592 5600 5608 | 5616 | 5624
5673 5681 5689 5697 5705
| | |
5632
5641
|
5648 5657 5665
5713
5721
5729 5737 5745
∞ ∞ ∞
∞ ∞ ∞
8
8
290
291
289 234. 5753 5761 5769 5777 5785 5793
5832 5840 5848 | 5856 | 5864 5871 5879
5910 5918 5926 5934 5942 5949 5957
5800
5808 5816 | 5824
5887 5895 5903
5965 5973 5980.
8
8
8
78
(M) TABLE OF VALUES OF Sv.

V
f-s
feet.
I
2
3
4
5
6
7
8
9
Diff.
feet.
feet. feet. feet. feet. feet.
feet.
feet.
feet.
+
245
246
244 46 581.3 584.2 587.2 590.1 593.0 596.0 598.9 601.8 604.8 607.7 2.9
610.6 613.6 616.5 619.5 622.4 625.3 628.3631.2 634.2 | 637.1 || 2.9
640.1 643.0 645.9 648.9 651.8 654.8 657.7 660.6 663.6 | 666.5 || 2.9
247 46 669.5 672.4 675.4 678.3 681.3 684.2 687.2 690.1 693.0 696.0 2.9
248 698.9 701.9 704.8 707.8 710.7 713.7 716.6 719.6 722.5 725.5 2.9
| | | ||
249 728.4 731.3 734.3 737.2 740.2 743.1746.1 749.0 752.0 754.9 2.9
250 46 757.8 760.7 763.7 766.7 769.6 772.6 775.5 778.4 781.4 784.3 2.9
251 787.3790.2793.1 796.1 799.0 802.0804.9 807.8 810.8 813.7 2.9
252 816.6 819.6 822.5 825.4 828.4 831.3 834.2 837.1 840.1 843.0 2.9
253 46 845.9 848.8 851.8 854.7 857.6860.5 863.5 | 866.4 869.3 872.2 2.9
254 875.1 878.1 881.0 883.9 886.8 889.7 892.6 895.6 898.5 901.4|| 2.9
255 904.3 907.2910.1 913.0 915.9 918.8 921.7 924.6 927.5 930.4 2.9
256 46 933.3 | 936.2 | 939.1 | 942.0944.9 947.8 950.6 953.5 956.4 959.3 2.9
257 962.2 965.0 967.9 970.8 973.7 976.5 979.4 982.3 985.1| 988.0||2.9
258 990.9 993.7 996.6 999.4 002.3 005.1 008.0 10.8 013.7 016.5 2.9
259 47 019.4 022.2 025.0 027.9 030.7 033.5 036.4 039.2 042.0 044.8 2.8
260 047.7 050.5053.3 056.1| 058.9 061.7 064.5 067.4070.2 073.0 2.8
261 075.8078.6 081.4 084.2 087.0 089.7 092.5 095.3098.1 100.9 2.8
262 47 103.7 106.5 109.2 112.0 114.8 117.6 120.3 123.1| 125.9
263
264
272
273
274
275
276
128.6 2.8
131.4 134.2 136.9 139.7 142.4 145.2 147.9 150.7 153.4 156.2 2.8
158.9 | 161.7 | 164.4| 167.1 | 169.9|| 172.6| 175.4 | 178.1 | 180.8 | 183.5 || 2.7
265 47 186.3 189.0 191.7 194.4 197.1 199.9 202.6 205.3 208.0 210.7 2.7
266 213.4 216.1 218.8 221.5 224.2 | 226.9 | 229.6 | 232.3 235.0 237.7 2.7
267 240.4 243.1 245.8 | 248.5 | 251.2 253.8 256.5 | 259.2 | 261.9 264.5 2.7
268 47 267.2 269.9 272.5 275.2 277.9 280.5 283.2 285.9 288.5 291.2 2.7
269 293.8296.5 299.1 | 301.8 304.4 307.1 309.7 312.3 315.0 317.6 2.6
270 320.2 322.9 325.5 328.1 | 330.8 333.4 336.0 338.6 341.2 343.9 2.6
271 47 346.5 349.1 351.7 354.3 356.9 359.5 362.1 364.7 367.3 369.9||2.6
372.5 375.1 377.7 380.3 382.9 385.5 388.1 390.7 393.3 395.8 2.6
398.4 401.0403.6 406.2 408.7 411.3 413.9 416.4 419.0 421.6|| 2.6
47 424.1 426.7 429.3 431.8 434.4 436.9 | 439.5 | 442.0 | 444.6 | 447.1 || 2.6
449.7 452.2 454.8 457.3 459.8 462.4 464.9 467.4 470.0 472.5 2.5
475.0 477.5 480.1 482.6 485.1487.6| 490.1 492.7 495.2 497.7 2.5
277 47 500.2 502.7 505.2 507.7 510.2 512.7 515.2 517.7 520.2 522.7 2.5
278 525.2 527.7 530.1 532.6 535.1 537.6540.1 542.6 545.0 547.5 2.5
279 550.0 552.4 554.9 557.4 559.9 562.3 564.8 567.2 569.7 572.2 2.5
280 47 574.6 577.1 579.5 582.0 584.4 586.8 589.3 591.7 594.2 596.6 || 2.4
599.0 601.5 603.9 606.4 608.8 611.2 613.6 616.1 618.5 620.9 2.4
623.3 625.7 628.2 630.6 633.0 635.4 637.8 640.2 642.6 | 645.0 || 2.4
283 47 647.4 649.8 652.2 654.6 657.0 659.4 661.8 664.2 666.6 669.0 2.4
284 671.3 673.7 676.1 678.5 680.9 683.3 685.6 688.0 690.4 692.7 2.4
285 695.1 697.5 699.8 702.2 704.6| 706.9 709.3 711.6 714.0 716.4 2.4
286 47 718.7 721.1 723.4 725.8 728.1 730.4 732.8 735.1 737.5 739.8 2.3
287 742.1 744.5 746.8 749.1 751.5 753.8 756.1 758.4 760.8 763.1 2.3
288 765.4 767.7 770.0 772.4 774-7 777.0 779.3 781.6 783.9 786.2 2.3
289 47 788.5 790.8 793.1 795.4 797.7 800.0 802.3 804.6 806.9 809.2 2.3
281
282
290
291
811.5 813.7 816.0 818.3 820.6 822.9 825.2 827.4 829.7 832.0||2.3
834.3 836.5 838.8 841.1 843.3 845.6 847.9 850.1 852.4 854.6 2.3
79

(N)
Log
Log
Log
Log
Log
Log
Ф
www www.
78
33
37
38
8. 720 67 9. 021 17
8. 722 26
8. 724 39
8. 727 04
4 (Pop-Py-1)(Pq-Pq-2) (Pq-Pq-4)
123
456
8.719 09
8. 719 61
9. 020 38
31
32
9. 022 50
9. 322 47
34
9. 024 36
9. 026 75
9. 323 80
35
9. 325 66
36
789
8. 730 22
8. 733 94
98. 738 21
9. 029 66
9. 328 05
9. 033 12
9. 330 97
9. 037 II
9.334 43
39
IO
8. 743 02
9. 041 65
9. 338 44
40
II
8. 748 369. 046 73
9. 342 98
4I
I 2
8. 754 26 9. 052 35
9. 348 06
42
13
8. 760 70
9. 058 52
9. 353 69
43
14
8. 767 70
9. 065 24
9. 359 87
44
15 8. 775 27
9. 072 53
9. 366 61
45
17
18
678
16 8. 783 38
9. 080 37
9. 373 91
46
8. 792 08
9. 088 78 |
9. 381 76
47
8. 801 36
9. 097 77
9. 390 19
48
19
8. 811 21
9. 107 34
20
8. 821 64
9. 117 49
9. 399 19
9. 408 77
49
21
8. 832 69
9. 128 23
9. 418 94
2 2 2
22
8. 844 33
9.
139 58
9. 429 70
23
8. 856 58
9. 151 53
9. 441 06
24
50
que am not
ir ir ir
(Pq—Pq-1) (Pq—P-2)|(Pø—Pq-4)
8. 977 78 9. 270 14 9. 554 68
9. 288 01 9. 571 88
8. 996 00
9. 014 959. 306 619. 589 78
9. 034 64
9.055 10
9. 076 33
9. 098 37
9. 121 22
9. 144 91
9. 169 47
9. 194 92
9. 221 28
9. 248 59
9. 276 87
9. 306 16
9. 336 49
9. 367 90
9. 400 42
9. 434 10
9. 469 00
9. 505 14
9. 542 60
9. 581 42
9. 621 67
9. 663 42
9. 706 74
9.751 71
8. 897 09 9. 191 II
8. 911 88
25
26
2 2 2
27
8. 869 45 9. 164 09 9. 453 04
8. 882 95 9. 177 28| 9. 465 62
55
9. 478 84
56
9. 205 58
9. 492 69
57
28
8. 927 33
9. 220 71
9. 507 19
58
29
8. 943 46
9. 236 50 9. 522 34
59
30
8. 960 27
9. 252 98 | 9. 538 17
60
9. 798 43
9. 846 97
9. 897 46
9. 325 949. 608 42
9. 346 029. 627 80
9. 366 879. 647 92
9. 388 52 9. 668 81
9. 410 97 9. 690 52
9. 434 269. 713 02
9. 458 40 9. 736 37
9. 483 419. 760 56
9. 509 33 9. 785 64
9. 536 18 9. 811 63
9. 563 999. 838 55
9. 592 79 9. 866 44
9. 622 62 9. 895 32
9. 653 51 | 9. 925 24
9. 685 49 9. 956 22
9. 718 62 9. 988 31
9. 752 930. 021 55
9. 788 470. 055 98
9. 825 30 0. 091 65
9. 863 47 0. 128 61
9. 903 04 0. 166 94
9. 944 08 0. 206 67
9. 986 65 0. 247 89
0. 030 84 0. 290 65
0. 076 730. 335 05
0. 124 41 0. 381 17
0. 173 98 0. 429 10
(0) TABLE OF VALUES OF COS³.
Ф
Ф
Փ
Ф
Փ
O
I 9. 999 80
2
9. 999 21
14
3 9. 998 21
15
9. 960 71 26
9. 954 83
27
| 49
13 9. 966 17 25 9. 871 83 37 9. 707 05
| |
9. 860 98 38 9. 689 60
| | |
9. 849 64 39 9. 671 51 51
9. 450 83
50
9. 424 20
9. 396 62
456
49.996 82
16
9. 948 52
28
5 9. 995 03
17 9. 941 79
29
9. 992 84
9. 837 80 | 40 | 9. 652 76 52
| |
9. 825 46 | 41 | 9. 633 34
18 | 9. 934 62 | 30 | 9. 812 59 | 42 | 9. 613 22
9. 368 03
53
9. 338 39
54 9. 307 66
79. 990 25
8 9. 987 26
99.983 86
19 9. 927 OI
31 9. 799 20
439. 592 38
55
9. 275 77
20
9. 918 96 | 32
21
0. 910 46 33
9. 785 26
9. 770 77
44
9. 570 80
56 9. 242 69
45
9. 548 46
579. 208 33
HH
ΙΟ
9. 980 05
I I 9. 975 84
22
9. 901 50 34 9. 755 72
| |
46
9. 525 31
23
9. 892 08 35 9. 740 09
| 47
47
9. 501 35
I2 9. 971 21
24
9. 882 19 | 36 | 9. 723 87
48
9. 476 53
58 9. 172 63
59 9. 135 52
| 60 | 9. 096 91
80

UNIVERSITY OF MICHIGAN
3 9015 06440 2251