lff^ \\^ \^ \ \^ ' K ^^ ^ ^ ^ ^ ^ ^^
1 8
Inches
Fig. 12.
An equation of the form
where K, a and b are constants, may be represented
readily by the use of logarithmic scales. The expo-
nents a and b really become the scale factors and one
scale is translated a distance log K.
Another method of treating an equation in two
variables
22 = /(2i)
is to make use of the ordinary cartesian graph. In
Fig. 15 let C be the graph of the above equation
referred to the axes OX and OY. For a given Zi say
OM, draw MP perpendicular to OX and from P drop
a perpendicular PN to OY. Then ON is the desired
value of 22. Coordinate or cross-section paper would
ordinarily be used for this type of diagram.
Diagrams representing equations in two variables
are used more to supplement the usefulness of more
complicated diagrams than to afford in themselves
a means of solving equations in two variables. In
later examples it will be found that many of the scales
are graduated for two quantities, such as cubic feet
per second and gallons per minute, on the same line.
While only one of these quantities may appear in the
formula for which the diagram is drawn, the addition
of the other often increases the usefulness of the
diagram.
4. Choice of Scale Factor.— The construction of
the scale of a function with a suitable scale factor /x
is an essential operation in the design of any perma-
nent diagram for numerical solutions. The length
L of the desired scale is limited by the size of the paper
and must satisfy the equation
L = 4Ab) - f{a)]
— §
8
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
where a and b are the limiting values of the variable
and n is the scale factor to be selected. There is
usually some choice of these limiting values a and b,
and as more than one function scale is involved the
relation of the various scales in the diagram must be
carefully studied in advance. The use of the scale
must be kept in view and the graduations arranged
so that interpolations by eye will, when possible,
yield one figure beyond the required accuracy. When
some portion of a non-uniform scale is to be most
frequently used that portion should be given the
advantage of the larger graduations by the methods
y
developed in Chapter III, as for example in the stadia
formula
It is always desirable to check various points on a
new scale by double calculations and by various known
characteristics of the function such as the magnitude
and uniformity of the rate of increase within a given
interval of the variable. The accuracy of the finished
diagram should also be checked by characteristics
of the given formula and by various numerical
examples.
Problem 1. — Construct a diagram showing the relation
between kilowatts and horsepower.
Problem 2. — Construct a diagram showing the relation
between circular pitch and diametral pitch of gear teeth.
Problem 3. — Construct the projective scale of the
function
1.7 log s + 6.5
2.4 - 0.84 log z
from a logarithmic scale.
Problem 4. — Construct a scale for values of ^
>^
^
^
^
^
^
I
—
n
^^
^
^
fe
^=^
"^
0.1 Q3 0.4 0.5 0.6 0.7
0.8 0.9 1.0 I.I
Head H in feet
Fiu. 17.
1.3 1.4 1.5
.8 1.9 2.0
on both axes. It is necessary to introduce such scale
factors when one of the independent variables, say
2i, varies through a greater range than the other z^.
Example 6. — In Fig. 17 there is shown a diagram for
Francis' formula for the discharge of water over a weir
without end contractions,
q = Z.ZWH^"-
head H is usually less than two feet while with B =
10 the discharge q runs up to 94.2 if H = 2. So the
values of q run through a range of numbers about 50
times as great as the corresponding values of H.
Accordingly it will be desirable to plot the scale for H
with a scale factor which is about 50 times that used
for the scale of q.
12
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
Diagram for the quadratic z^ + pz + q =
Fio. 18.
ELEMENTARY DIAGRAMS
13
Let
then
Mii? and
- = 3.3351
©"
is the equation of the system of curves for B. In
Fig. 17 Ml was taken as 5 and m as 0.1 so that
y = 0.2978a;'^5
is the equation of the curve system referred to natural
scales on the coordinate axes. Since each value of
0.2978.T** is multiplied by B it is necessary to plot
only the curve for 5 = 10 and divide each of its
ordinates into 10 equal parts to obtain the entire
system of curves for B. In Fig. 17a the curves near
the origin are shown drawn to a larger scale.
7. Simple Straight Line Diagrams. — The labor of
constructing diagrams such as were given above is
considerable unless the family of curves is easily
plotted. The curves can be made straight lines when-
ever Equation (2) has the form
z,f{z,)+z,giz,) + h{z,) =
or more briefly
S1/3 + S2g3 + /^3 = (3)
where /a, gz and h^ are any functions of S3, may or may
not be alike and frequently reduce to constants includ-
ing zero. The use of the same letter to denote func-
tions of different variables in what follows will not
necessarily mean that the functions are the same
although such may sometimes be the case. In general,
for example, /(si) or /i will not denote the same
functions as/(z3), etc.
Whenever Equation (2) has a form which may be
reduced to the above form (3) by suitable transforma-
tions, set
X = ix\Z\ and y = mZi
and Equation (3) becomes
Xti-ifs + 3'Mlg3 + MlM2/?3 =
which determines a family of straight lines marked
with corresponding values of S3. Equations in three
variables such as Equation (2) occur very frequently in
engineering practice and are of particular interest here.
Example 7. — The general quadratic equation
z'- + pz^q = Q
if .r = p and y = q becomes
s= -f xs -I- y =
This is a straight line system and the original Equation
is of the type (3) where
When as many lines of the system have been drawn
as the diagram will comfortably admit at suitable
intervals of z it is seen that for moderate values of the
coefficients p and q it is possible to solve any quad-
ratic by reading the roots written on the lines passing
through the corresponding intersection point of the
lines X = p and y = q. It will be necessary to
interpolate for all the quantities p, q and z. See
Fig. 18.
8. Anamorphosis. — It is possible in a large class of
equations which do not fall under the type of Equation
(3) to reduce the needed family of curves to straight
lines. It will first be shown how this may be done
graphically with a single curve and then the method
will be extended to apply to a family of curves.
and
= 1,
Suppose there is given in Fig. 19 a single curve C
corresponding to some particular value of the quantity
S3 in the equation
/(S,S2S3) =
This curve may be changed to a straight line L
which will serve equally well to determine either of the
corresponding quantities Si and 22 as foUows. Draw
any oblique line AB and let every point P of the
curve C be projected horizontally into a corresponding
point Q upon the line L. Now inscribe N, the foot
of the ordinate of Q, with the value of 21 which is
found at M on the X axis. After a sufficient number
of points have been treated in this way the curve C
may be erased, also the old scale of Si and then the
diagram serves to determine the corresponding values
of Si and so for the value of S3 originally used. This
process was called by Lalanne "Anamorphosis."'
What has been done changes the scale on OX from
the ordinary scale for Si to a certain function scale.
To see this it is only necessary to notice that the
length ON is always a function of the length OM.
A logical extension of the above principle to all the
curves S3 of a given family is desirable. For this
purpose it will be necessary from the given equation
/(s,s.S3) =
to select a function x of Si such that when y = ^222
the entire family of curves corresponding to values
' L. Lalanne, Annales des Ponis el Chaussies, 1846.
14
DESIGN OF DIAGRAMS FOR ENGINEERIXG FORMULAS
of 23 shall be straight lines. That is, it is necessary
to change the original equation
/(212223) =
by virtue of the relations,
X = Mi/(si) y = H1.Z2
into a linear equation in x and y. A necessary and
suflScient condition is that the original equation
f{z,z.z,) =
may be reduced to the form
/(Sl)/3 + S2g3 +h = (4)
For in this Equation (4) if Zi and zj are eliminated there
results
Xpifi + Vyuigs + MiM2/'3 =
which is the equation of a family of straight lines to
be inscribed with values of 23-
Equation (4) is of the form that will yield straight
lines when a function scale is used on the A' axis only.
If. however, the ordinates also are made to depend not
simply on 22 but on a function of 22, as 7(22), there
results a method of treating equations of greater gener-
ality. Set therefore
X = /ii/i and y = ti-ifi
then when the Equation (2),/i23 = 0, has the form
/(Sl)/3+/(22)g3+//3 = (5)
it will yield a system of straight lines for the values
of 23, by virtue of these relations.
This is the principle underlying the use of "logarith-
mic cross-section paper" for plotting an equation in
two variables. This paper is a cross-section paper
ruled with logarithmic scales on the axes instead of
with the ordinary scales. Any equation in two vari-
ables which has the form
21-S2' = A-,
for example, where a, h and K are constants, may
immediately be given the form
a log zi + & log Z2 — log A' =
by taking the logarithm of both sides. The resulting
equation has the form (5). When therefore
X = log zi y = log Z2
the above equation reduces immediately to
ax -{- by — log K =
which is a straight line equation for the ordinary cross-
section paper. Or in other words, if corresponding
values of Z\ and Z2 determined from the original equa-
tion are plotted directly on the logarithmic cross-sec-
tion paper, the resulting coordinates are proportional
to the corresponding logarithms and the graph is a
straight line.
The exponents a and h determine the slope of the
resulting straight Une; i.e. — v-
This principle when used inversely is of great value
in determining the unknown exponents for an empiri-
cal formula when a sufficient number of points are
plotted on the logarithmic cross-section paper from
actual observation and are found to determine closely
a straight line.
Equation (5) is a very general type equation and
includes a large number of formulas of engineering.
Such formulas will frequently require algebraic and
sometimes logarithmic transformations in their form
before they can be identified with the type by inspec-
tion. It will be seen that the corresponding diagrams
consist essentially of three systems of straight lines
and that two of these systems are parallel to the axes,
determined by function scales on the axes.
The foregoing Equation (5) is not the most general
equation in three variables whose diagram can be
constructed by three straight line systems provided
no restriction is placed on the nature of the systems.
Such an equation is best expressed in determinant form
but can, however, be treated by much more elegant
methods than those of the present chapter.
Example 8. — The "external" or distance from the
intersection of two tangents to the curved line, in high-
way or railroad surveying is given by the formula
Z) = r tan 7
4
where T is the length of the tangent and / the acute
angle of intersection. In the field it is often desired,
before finally determining either T ox h for a given
angle, to try several pairs of values, and the diagram
given in Fig. 20 is convenient.
The formula is in the form of Equation (4) where
tan^=/(2,), 6
22, T =Ug3= - 1, //3
so that if
M26
there results the radial line system
V X „
limit of b is taken as 18 feet the diagram of Fig. 20
can be drawn with in = 0.3 and M2 = 60.
Example Q. — The mean pressure Pm of steam
expanded from an initial pressure Pi according to
the law PV = constant, is given by the formula
ELEMENTARY DIAGRAMS
q nvfcaaxxa
15
q iVNHaxxa
16
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
if measured above a back pressure of absolute zero.
R is the ratio of expansion.
1 + log. R
" R
is taken as/i and Pm as/2 then/3 = P\, gz
Mean Pressure of fxpanded Sfeam
according fo the law pv^p, V/
Pm - Absolute Mean Pressure
P,= ^, Injfial '>
/p- Ralio of Expansion =}L
•1 and
Since all the lines Pi pass through the origin it is
necessary to locate but one point on each line to draw
the system. Such points are very simply determined
by the intersections of the radial lines with the line
a; = 1 parallel to the Y axis. In general when a
system of radial Unes y = mx is to be plotted, set
4O0-
0^^
Ratio of Expansion
Fig. 21.
^3 = showing that the above equation is in the
form of (5)
/2-/3/l =
Accordingly let
1 + log. R
x = ,. ^
and y = ti.iPm.
so that there results a family of radial straight lines
(See Fig. 21.)
a; = 1 so that y = w. In the present case the scale
determined on the line a; = 1 is an ordinary scale
whose scale factor is — • Of course beyond the limits
of the paper the radial lines cannot intersect the line
a; = 1 and if it is necessary to draw additional lines
they may be determined by their intersections with a
line parallel to the Y axis at any convenient distance.
P — D
Example 10. — The formula 5 = — „ — of Example
5 may be written
Z) = P(l - 5).
ELEMENTARY DIAGRAMS
17
If /i = (1 - 5), /o = Z?, /a = P, g3 = -1, /ra = it it becomes
is in the form of Equation (5)
hh-h = o
Accordingly let x = ^{1 — S)
y = i^iD
y
ny — X = Q Z.S shown in Fig. 23.
givmg i^ " ■^
as shown in Fig. 22.
The scales on the axes are readily plotted by the
method of Article 2 (e); i.e., for a given a or /3 look
up the value of the natural tangent, add one, and find
the resulting quantity on a logarithmic scale, inscribing
the point with the value of a or /3 used.
Pihh of holes in inches
100 %
%
85 80 75 70 65
Efficiency in percent
It will be noticed that the graduations of the 5 scale
on the X axis increase toward the origin, since the
function is (1 — S).
Example 11. — The expression (tana + 1)" = (tan
/3 + 1) is useful in plotting exponential curves of the
typePF" = constant, in thermodynamics by Brauer's
method. If written
n log (tan a + 1) - log (tan /3 + 1) =
with
X = log (tan /3 + 1)
y = log (tan a + 1)
2
Example 12. — The formula for the diameter {d)
of a shaft to transmit a given horsepower (h.p.)
at a given speed (r.p.m.) is of the form
'i,
h.p.
.p.m.
If the allowable stress for a steel shaft is taken
13,500, the constant c has the value 2.87; hence
1
d^ = (2.87) 'h.p.
r.p.
If a reciprocal scale is used on the A' axis and a cubic
18
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
scale is used on the Y axis a family of straight lines Example 13. — The approximate formula for the area
for values of h.p. results. A of the segment whose height is H of a circle of radius
Ris
Let
and
y = uid^
A =
"- l2R
4b -
7
//
^6$
/////\
Af)
/
'//y/.
/
yy///
///
'/
Z./Zi
////
/.
5222
/
''z
////-/
<^3S -
//
/:////
/.
/Zzu
t -
///>
c;///.
^/
y/////,
^
/////.
^^^./;
^
5^5^
ZvZ-t'.
/////
^/
r^^-'/r^
y
/.
5222^
^^i^C^
//
2222
o
///
22^^'
^//y//.
^
2^11
:3
/
'///
^2??^
'/////.
;^
^%^^
^
///
'/^
tz&%
y///y/
^
^^
/
^Z''/
//4
tz^t-t'^
y^///A
;^
y
/./
'//^/
'yy/
SSSS^
V/'/y,
/
Or
////.
^^///
//^
2122^
y///
//////
/y V
y/A
iSil^
V
//
v/////,
^^
^^
t^W
//^/
4%Vi
^;^:;^
^^
Pi "
^
^'^
'^//////^
y//^A
''//^.
:Z
^
^
^/^
'^'^'f
^//y//.
'y.y
^
^
^^
yyyyy.
'////
3
^
V///.
^^^^
r/
Vy
^
m.
%v'%'
'
iram for
^^
^
^
m
^^
IS
«^=^
i
'/f,
^
m
^
/r-4^
(Tanfi+I)
^^
p
^
^
''/
^CTanoa-O
¥'k'
V
/■
''^
10
-_
20 ZS 50 35 40
Volues of (3 J decrees
Fig. 23.
45
SO that
(2.87)'h.p.ci;
or 9^2 - 32/2^?' + 9.728Z/^ =
This equation is in the form (5) with/i = A"^ and/2 =
as shown in Fig. 24. A second set of underscored R so that '\{ x = A''- and >> = /? a family of straight
graduations for d and h.p. have been added, covering lines for E
a larger range of numbers. %x - ZIR^y + 9.728^^ =
ELEMENTARY DIAGRAMS
19
%
jsModapjof-i
^^ "%\ %\ o\ s\ ^\ 'g\ ^\ ^\ ^
ti-
% "§>
O O
CO V-
^
o
o\ \
\\
^
\
1
9s.
^. \
\ \
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\ \
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w
w
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w
% \
k \
, \;
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\ \
, 1
---< \
\
\^ \
w
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\ ^
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h \
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w
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s \,
N. \ \
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w \
TX
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<5
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SW^
S x::
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W^WT
1
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— x^
N\
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^
XV
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^$^^
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OSI
OOJ
002 c^
0017
005
0001
O i-O o
u-> C-" o
•^ c?|o
saLjOui_ ui |J-DL|5 J.0 J3|aaiDi(]
20
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
^. oi o
-^ ^=1= -si--
ssLioui ui snipD^ J.0 sanpy^
ELEMENTARY DIAGRAMS
21
would result. This equation is difficult to plot and
the lines are poorly located for accurate reading.
If the ratio -^ = A' is used the equation becomes
_ 4 V2A-- 0.608
Then with x = iufi
results a radial system
/ii/1- and y =
-■"Ji = i^2R* there
2K - 0.608
shown in Fig. 25.
9. Special Form of Equation. — The case where
Equation (5) has the simple form
/1+/.+/3
(6)
is of special importance. It gives rise to a system of
parallel straight lines, since if a; = mi/i and y = M2/2,
the equation becomes
IX2X + idiy + ixiix-ifa =
This system of lines may be dispensed with if their
common normal through is drawn and on it the
function scale for =3 established. The function scales
for Zi and S2 must be constructed on the X and Y
axes respectively as before. The diagram then
consists essentially of three function scales whose
supports intersect at 0. It is read by finding the
unknown value of 2 where a line through the intersec-
tion of the two perpendiculars at the given values of
s on their respective axes meets perpendicularly the
scale of the unknown 2. See Fig. 26.
Fig. 26.
Since the three lines necessarily perpendicular to the
respective scales meet at constant angles they maybe
scratched on a transparent sheet which when properly
oriented on the drawing will enable the unknown
values to be read rapidly. For ordinary work, how-
ever, a diagram having the cardinal values of all three
straight line systems drawn in is found to be the best
arrangement.
Example 14. — The formula for the weir discharge
used in Example 6
may be brought into the form (6) by taking the
logarithm of both sides. There results
log q — log 3.33 — log B — ^2 log H =
Here if log q = Ji and log H = fi'it is seen that
/3= -log 3.33 - log 5
Set X = 111 log H and y = /is log q. Then the equation
of the parallel Unes for B is
^ - ^ - - log 3.33 - log 5 =
/*2 2 All *
These lines may best be drawn if the common normal
to the system is first drawn and numbered with the
values of B at the points of intersection with the
parallel lines. To do this it is necessary to determine
the angle a of Fig. 27 (above) and the correspond-
ing function scale on the normal. The angle in the
present example is 126° 52' 12" and the lines B inter-
sect the normal at distances from the origin determined
by the function
|M.[log3.33 + log5]
The completed diagram is shown in Fig. 28.
In general when an equation is of the form (6) and
the resulting system of lines for 23 is given by the
Equation
HlX -\- jui V + M1M2/3 =
this last equation may be put into the normal form
X cos a -h y sin a — p =
where
and sin a
Vmi" + M2" "VViM-mT"
and where the scale on the normal is determined by
the function fi with the scale factor
Vyur + 112'
TM.86
Example 15.— The formula H = 0.38 ^y;^ gives
the friction head H in feet per 1,000 feet of water flow-
ing in a pipe of diameter d with a velocity of V feet
per second. In logarithmic form the equation is
log H + 1.25 log d - log 0.38 - 1.86 log F =
22
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
X
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^
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DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
ELEMENTARY DIAGRAMS
25
li X = ixi log H and y= ti^ log d, there results a system
of parallel lines for V
+ 1.25^
log 0.38- 1.86 log V = Q
Figure 29 shows the completed diagram with /U2 =
1.25 and mi = 1-0. The normal bisects the angle
between the axes and the scale on it is
P = -^ (log 0.38 + 1.86 log V)
If the line system for V is to be drawn, it is of course
not necessary to draw the normal since the lines of
the system cross the X axis in points determined by
the scale
p = log 0.38+ 1.86 log V
upon eliminating log V between the two logarithmic
equations there results
1.86 log Q = 1.86 log ^^^11^ + 4.97 log J + log H
and since x = log H, y = 1.25 log d, the system of
parallel straight lines for Q is
^ + iS ^ = 1-86 log Q - 1.86 log ^^
The necessary lines are added to Fig. 29 in Fig. 30.
The angle for the system Q is 75° 53'.
Example 16. — The velocity V with which a jet of
steam issues from a turbine nozzle having a friction
factor Y is
V = 223.8 V(l - Y)(H, - H~i)
where {Hi — Ho) is the "Heat drop" or number of
V=J(2t Veloci+L) in f+.pcrsec
wn
O G> O o '^ O
s
s
5
^
^
S:S
5
s
^=^
^
^:
100 no no 130 140 ISO 160 ITO 180 190 200 210 220 230 140 2S0 260 TO 280 Z90 300 320 M 3M 330 400
H,-H2= Heat Drop, B.+.U.
Diagram for Steam Jet Velocities, V = 223.8^(1 -Y) {Hi- Hi)
Fig. 31.
The discharge Q is equal to the velocity of flow
multiplied by the cross-section of the stream. For a
circular pipe of diameter d the discharge is
id'-V
It is possible to supplement the diagram of Fig. 29
by new lines which will give the discharge. The
example illustrates a general method available for use
when four variables occur in this way in two equations.
Since
log Q = log 0.7854 + 2 log J + log V
and
log H + 1.25 log d - log 0.38 - 1.86 log V =
British thermal units of energy available. Figure
31 shows a diagram for this formula with the following
analysis :
log (ffi - Hi) + log (1 - F)
h 2 log 223.8 - 2 log F =
If
« = Ml log (^1 - ^2)
y = fj.2 log (1 - F)
the parallel lines for V have as their equation
HiX + Miv - MiM-2[2 log F - 2 log 223.8] =
The normal is located from
T, and sin
V Ml" + M2"
26
DESIGN OF DIAGRAMS FOR E\GI\EERL\G FORMULAS
12 23, 24 25 2& 21 28 29 50 31 52 35 54 35 36 57 38 59 40
Fig. 32.
ELEMENTARY DIAGRAMS
27
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DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
and the scale on it is
M1M2
+ Al2
^ [2 log F - 2 log 223.8]
The origin is not shown on the diagram.
Example 17. — An empirical formula giving the
number of pounds of wind resistance R in an automo-
bile offering a square feet of wind resisting area at 5
miles per hour is
R = 0.003a52
Passing to logarithms
log a + 2 log S + log 0.003 - log i? =
If X = til log a
y = y-i log R
then
Ml M2
the scale on the normal is
+ 2 log 5 + log 0.003 =
P = , V — :, [2 log S + log 0.003]
V Ml" + M2"
The diagram is shown in Fig. 32.
Example 18. — In Fig. 33 is shown a diagram includ-
ing parallel straight lines, for the Equation
£ = 0.232 log ^^
which gives the inductive voltage E per ampere per
mile of double wire for alternating currents, where r is
the radius of the wire and d is the spacing, both in
inches. As the size of the wire is usually expressed
by the gauge the latter was used in constructing the
diagram. To correct E for various frequencies the
constant must be varied; the present diagram is
drawn for both 25 and 60 cycles. It is not necessary
to pass to logarithms in order to bring this equation
into a form similar to type Equation (6)
0;|2 = logJ-log0.78r =
" "^ = '^'0232
y = U2 log d
and
^ + log 0.78 r =
the third system is
/*1 A*2
10. Hexagonal Diagrams. — For Equation (6) above,
the resulting equation for the lines of the variable
23 may be given a special form by setting mi = M2
when the range of the values of Zi and z^ permits.
The scale factor for the Z3 scale on the normal reduces
1 1
then to ~~7^. The factor ~7^ may be dispensed with
by choosing the axes for the Zi and 22 scales at an
angle of 120° and establishing the Zg scale on the
bisector of this angle. It can be proved from Fig. 34
that if from any point P perpendiculars are drawn to
three scales there shown the following geometric
relation holds
OMi + OMi = OM3
This relation is easily seen by observing that in
Fig. 35, AMi = M^B so that 20M^ = OA-\-OB, but
OA = 2OM1 and OB = 20Mi, whence the relation
above
Fig. 35.
If now OMi = m/i
OM2 = m/2
OM3 = m/3
it follows always that
/l+/2=/3
for the values of 2 found at the corresponding points
Ml, M2, M3.
This form of diagram is called the hexagonal form
from the fact that the lines involved are the diagonals
of a hexagon.
Example 19. — The formula of Example 15
71.86
H
may be readily represented by a hexagonal diagram if
written
(log H - log 0.38) + 1.25 log d= 1.86 log V
Figure 36 shows the completed diagram. While the
scale factors of all three scales must be the same, the
coefficients 1.00, 1.25, and 1.86 determine the unit
length of the scales. The constant log 0.38 in the H
ELEMENTARY DIAGRAMS
29
30
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
function shows that the logarithmic scale for H must
be moved to the left until 0.38 is at the origin.
The hexagonal diagram may be supplied with a
suflScient number of scales to solve equations of the
form
/1+/2+/3+ . . . +f„ = (7)
Write the equivalent system
//0+/3 = //I
/r„-3+/„-
/n
On a suitably inscribed diagram enter with 21 and 22
and obtain a temporary point M on a blank scale.
Then from the point where the z^ perpendicular cuts
fitf^-,f^*f^*fs'0
the perpendicular from M drop a perpendicular to
locate N and proceed in this way until s„ is reached.
The arrangement of scales for n = 5 is shown in Fig.
37. Another treatment of Equation (7) will be found
in Article 21 of Chapter V.
Problem 1. — The illustrative examples of this chapter
(5 to 18 inclusive) may in most cases be represented by
diagrams of types other than those used. Investigate
all feasible tjqjes for each formula given.
Problem 2. — The volume V of the frustrum of a cone of
height h is
V = j^h[D' + Dd + d']
where D and d are the diameters of the bases.
Using D and d as si and 33 show how the system of curves
V
for S3 = -, may become a family of concentric circles and
construct the diagram.
Problem 3. — Boussinesq's appro.ximate formula for the
perimeter of an ellipse L with semi-axes a and b is
L = ^[%(a + 6)- y/^b\
Show that with a and b as 21 and =2 the curves for zz = L
may become circles tangent to both coordinate axes if a
suitable angle is chosen for YOX.
Problem 4. — Draw all feasible diagrams for
^ = e'^
Ti
the ratio of belt or rope tensions Ti and T^ for a coefficient
of friction / and an angle of wrap d.
Problem 5. — Determine the corresponding formula when
a set of observations of two variables result in a parabola
symmetrical to the Y axis when plotted on logarithmic
cross-section paper.
Problem 6. — Construct a diagram for the cubic equation
z' + pz -\- q = similar to that of Example 7, page 13,
with regular scales for p and q on the axes.
Problem 7. — The capacity of a silo is given by K. J. T.
Eckblaw as
d' /h^-
\20
C =
2M
256 V20
where C is the capacity in tons, h the height in feet and d
the diameter in feet.
(a) Construct a diagram using parallel straight line
systems.
(b) Construct a diagram using a radial straight line
system.
Problem 8. — F. W. Taylor gives the expression for the
pressure upon a cutting tool when cutting cast iron
p = cd'^'' F^'
where P is in pounds, D is the depth of cut in inches and F
is the feed in inches. The quantity C is taken as 45,000
for soft cast iron up to 69,000 for hard cast iron. Con-
struct a convenient diagram.
Problem 9.— The expression P„ = 3.463Pi(i?-" - 1) is
used in determining the mean effective pressure P„, when
air is compressed from an initial absolute pressure Pi
pounds per square inch and R is the ratio of the final to the
initial pressure. Devise a diagram with parallel straight
lines.
Problem 10. — In problems involving compound interest
the expression i? = (1 + >■)" is the basis of all such com-
putations. Devise a useful diagram for this expression.
Problem 11. — Devise and construct a convenient
diagram which may be used to determine the correct
revolutions per minute for pieces of work of various
diameters (in inches) when certain cutting speeds (in
feet per minute) are desired in various rotary machines.
Problem 12. — Look up the formula by Grashof for the
flow of air through orifices and construct a diagram for use
only within the limits for which the formula is applicable.
Problem 13. — Construct a diagram for the two formulas
of Example 15 using ordinary logarithmic cross-section
paper with equal scale factors on the axes. Plot H on
OX and Q on OY.
ELEMENTARY DIAGRAMS
31
d- DIamc+er o-f Sha-f-f in inches
T 6 5
20
^0 40 SO inches
/L - Overhangs Disiance befweencenierofbearinc^
and cenfer of crank pin.
-10
t-30
Diagram for equaiibn c(= iJs.f F(L -i-yL^+R ^)
^^
Diameter of Shaff for Combined Bending and Tw/siihg
Place poinh of dividers on values of L and R. Wifti L as center
swing arc to fioHzonta I axis j project verticallg to value ofFj
horiiontallu to value off^) verficallg to "d" ttie required
diameter of shaft.
FiQ. 37a.
32
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
Diagram giving safe unif stress ibr
PLATE GIRDER WEBS WITHOUT STIFFEMERS
From Coopers formula S= — ^?-
3000f
ELEMENTARY DIAGRAMS
33
34
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
Problem 15. — Construct a diagram for the expression
H = 0.0274F2 + 0.0141Z,Fi"
Problem 14. — Construct a diagram for the formula
X = 2^/(80 + 741. Hog 7)10-"^
which gives the inductive reactance x in ohms of a trans- for the friction loss H in condenser tubes L feet long when
mission line when D is the spacing of the wires in feet, r is the water velocity is V feet per second through the tubes,
the radius of the wire in inches and / the frequency. L is in feet of water head.
Take / as 60 cycles or some desired frequency and include Problem 16. — Analyze the methods of construction
a scale for wire sizes in connection with the r scale. used in Figs. 37a, d7b, 37c and 37d.
500
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T= Absolute Temperature
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\
'Os
"^-e
r-"
^
/
/
"-
^
^
\
/;
^
^
/
^
■^-■
^
^.
^
/ 1 1 '2
Amperes per cm r
-^ — liiijiifiilii J 1 — 'tttt
T*r+-
r+n-
-mT
T+T+
U-J
TTTT
Ur+
't+T
AH 1
p^%iii,i,i,
^T-TT-
i>
L
T+T+
w
1^
A'tZ'lo'
A'lUlO^
iOlO . 1.00 » 05 0.? 0.2
+0.S
Values of I
Fig. yrii.
CHAPTER III
ALIGNMENT DIAGRAMS OR COLLINEAR NOMOGRAMS*
11. General Tjrpe of Eqixation and Method of Treat-
ment.^ — There will be considered in this chapter a
great class of formulas which may be written in the
determinant form
h Si 1
h g2 1
/a g3 1
(8)
in which /i and gi are functions of Zi, f2 and g2 func-
tions of 22, etc. Such knowledge of the elementary
properties of determinants of the third order as may
be gained from the reading of Appendix A will be
assumed.
A distinguishing characteristic of the determinant
of Equation (8) is the presence of the same variable
in the elements of each row. There are many prac-
tical formulas which may be reduced to this form and
their diagrammatic representation is of much value.
Such formulas lead to a new form of diagram which
will be called the alignment diagram because its key
is the alignment of three points.
It is proved in analytic geometry that if three points
Pi, Pi, Ps with the coordinates (xiyi), (xiyi), (xzys),
respectively, lie on a straight line (are collinear) the
coordinates satisfy the relation
yi 1
yi 1
ya 1
Xiy2 + Xiys + xzyi — Xzyi
— Xtyx — ^iV3 =
which expresses the fact that the point P^ixiy-^ lies
on the line joining the points P\{x\y^ and P-iixiy-^ and
whose equation is
^ yt - yi
Xi - Xx
The problem is then to establish a relation between
the variables Zi, z-i, Zz of Equation (8) and the position
of three corresponding and inscribed variable points
in the plane such that whenever three values of z are
solutions of Equation (8) there shall correspond three
such points in a straight line. When this relation is
established, a straight edge applied through two points
' Appendix A should be read before this chapter.
marked with known values of z.- Zj must pass through
one or more points marked with the value of z* which
satisfies Equation (8).
This problem is most easily solved by using the
parametric form of the equations of plane curves
where z is the parameter. The equations
X = /(z)
g(2)
are the parametric equations of a plane curve C. For
every value of the parameter z they determine a point
P on that curve. Three such sets of equations will
likewise determine three curves and the forms of the
curves will depend on the nature of the functions /
and g.
Three sets of such parametric functions may always
be determined directly from Equation (8). If the
three pairs of equations
Xl =/l
X2 = fi
Xs = Ji
yi = gi
3'2 = g2
yz = g3
are formed, using the elements of the determinant of
Equation (8) in the order shown, they may be con-
sidered as the parametric equations of three plane
curves Ci, C2, C3. These equations will be called the
defining equations. When the curves are plotted,
points are inscribed with corresponding values of z
and thus three curved function scales are obtained.
There is then established a direct correspondence be-
tween values of z and points P on the plane curves C.
36
DESIGN OF DIAGRAMS FOR EXGIXEERIXG FORMULAS
(See Fig. 38.) It is seen therefore that if Xiyi{i = 1,
2, 3) in the equation
a:i
yi
1
X^
y-i
1
Xi
3-3
1
are the coordinates of the points of the curves defined
by the three pairs of equations above, then Equation
(8) is always satisfied by values of Z\, Zi, 23, which
determine coUinear points.
When an engineering formula or equation in three
variables is given for which a diagram is desired, the
first step is to write it in the determinant form.
Equation (8) is the general type equation in three
variables for which corresponds a collinear nomogram.
Usually, however, an equation or formula does not
present itself in a determinant form nor especially in
this rather simple determinant form. Since it is
always necessary to establish the defining equations
before constructing a diagram it is very desirable to
become familiar with the necessary determinant nota-
tion at once. Equation (8) with all the elements of
the last column unity is called the reduced determinant
form. It is almost always necessary to establish a
first determinant form for any given equation and then
transform it by the laws of determinants into the
desired form above.
There is no general method by which any equation
of the form
/(21Z223) =
may be given a first determinant form and in fact not
all equations in three variables may be written in that
determinant form.
Special cases of Equation (8) have been studied and
the necessary and sufi&cient conditions developed for
identifying a given equation with them. The work
involves partial differentiation and is not usually
needed in practice.'
12. Diagrams with Three Parallel Straight Scales.
In the expanded form of Equation (8) which is
fig2 + figs + figi ~ figi - fzgi - flg3 = (9)
should ane or more of the functions / or g reduce to
a constant and especially to zero the equation becomes
much simplified. For example, the equation
/l+/2+/3 = (6)
previously discussed in Chapter II, Article 9, results if
gi = -1, g2 = 1, gs = 0, and /a = - g
> Clark, J., TWorie Gfn^rale des Abaques d'Alignment de toute
Ordre, Rivue de Micanique, 1907, No. 39. Also d'Ocagne, Nos. 152-
153, Traits de Nomographie.
A correspond
(6) is
ng first determinant form of Equation
/: -1 1 1
h 1
1
1
=
Although this is a reduced form of the equation, in
the sense defined above, it is usual to write this equa-
tion in the form resulting from an interchange of the
first two columns thus
-1 /l 1
1 fi 1
0-§ .
= (10)
The defining equations^ of the three corresponding
scales are
^=-1 y=f.
x= \ y=fi
x= y=-^i
and the scales are consequently graduated -on three
equidistant parallel lines. This is perhaps the sim-
plest form of collinear nomogram or diagram of
alignment.
Example 20. — By the method of "end areas" the
volume of earthwork per station on railway and high-
way construction is given by the formula
KV = {p, + Pi)
where V = volume in cubic yards,
A' = a constant depending on the length of sec-
tion and scale,
pi and p2 are average planimeter readings in
square inches from the cross-section
drawings.
Comparing this formula with Equations (6) and (10)
it is seen that the necessary defining equations are
-1
y = pi
1
y = pi
KV
The diagram (for the scale of cross-sections 4 feet = 1
inch) may be constructed with the vertical unit one-
tenth of an inch and the horizontal unit 5 inches. If
desired the scales may be broken and repeated to
avoid unduly enlarging the diagram. See Fig. 39.
It usually happens that for the range of values of
the variables involved in the Equation (6) it is neces-
' Henceforth it will be sufficiently clear that three curves are under
consideration without using subscripts to distinguish the coordinates
of their respective points.
ALIGNMENT DIAGRAMS OR COLLI NEAR NOMOGRAMS
IT- 1000
950
8 —
7.11
-100
E-iso
-II
^
37
Fig. 39.-Diagram for KV = [p, + p.) V = Volume of Earthwork, cu. yds. p, and p, = Average Planimeter Readings, sq. in
38
DESIGX OF DIAGRAMS FOR ENGINEERING FORMULAS
sary to introduce scale factors and sometimes it is
desirable to establish the scales at unequal distances.
Suppose that it is desired to introduce the scale factors
ixi and Hi on the parallel scales for Si and s-> and to
estabUsh these scales at distances 5i and 5.. from the
Y axis. It is then necessary to determine how the
third scale shall be graduated.
The new defining equations for the first two scales
would necessarily be written
X = -&i y = Hifi
X = Si y = fi-f-
and it may be assumed temporarily that the third
scale equations will have the form
X = F3 y = G3
where 7^3 and G3 are to be functions of /s and involve
the new constants.
To determine F3 and G3 so that points originally
corresponding to any set of solutions of Equation (6)
shall remain collinear in the changed diagram it is
necessary that the equation
- 5 f^ifi 1
62 M=/2 1
F3 G3 1
shall be satisfied by virtue of Equation (6). Upon
expanding this equation and substituting the value of
/i from equation (6) there results
{t^i52 - Mi^3)(/2 +/3) - {F3 + 5OM2/2 +
(6i + 52)G3 =
Since this equation must hold for any values of the
independent variables S2 and S3, then the coefficient of
/2 and the term not involving /« must vanish identi-
cally, that is
;il52 — Ml^3 "~ M2^1 ~ ^2^3 =
and G3(5i + §2) + (mi52 - miFs)^ =
_ 5i/X2 — S-y/Xi r, — — ^'^'--^^
Ml + V-1
whence
G3
+ M2
and the defining equations of the third scale are
_ hiy.\ — Sifi2 _ _ ft 11^2 f 3
Ml + AI2 All + M2
It is seen that when (§2^1 — Si/i2) = 0, the new
scale will remain on the Y axis and the constants may
usually be so chosen that this is true. It is to be
observed also that the scale factor of the third scale is
independent of 61 and 52- Frequently 5i and &2 may
be chosen equal in which case /xi and fj.2 must also be
equal if the third scale is to remain on the Y axis;
that is to say if the three scales are to be at equal
/UlM2
distances. The quantity -
+ M2
may be called the
As a check on the work the values of F3 and G3
above determined may be substituted in the last
determinant equation with the result
-Si /Xl/l 1
62 /.2^2 1
52M1 — S1JX2 —1x1112/3 J
MlMi C ^l + ^2)
Ml + M2
[/. + /2+/3]=0
Ml + ^2 Ml + M2
It is well to point out here that the effect of the
introduction of the above scale factors and the change
of moduli is to apply a projective transformation^ (see
Appendix B) to the original geometric configuration.
A projective transformation when applied to all the
variable elements of the first two columns of such a
third order (reduced) determinant has the effect of
manipulating the elements of the determinant by the
laws of determinants and the net 'result is always
merely to multiply it by a constant. In the present
case the constant is
_ MlM2(5i + S2)
Ml + M2
To understand how the above theory of the scale
factors is applied, the formula for volume by "end
areas" of Example 20 may be resumed. The use of a
horizontal unit of 5 inches and a vertical unit of one-
tenth of an inch was equivalent to the introduction
of the values
5i = So = 5, Ml = M2 = ^io
in order to change the defining equations for the dia-
gram to
-5
x = 5
x =
Pi
y=ro
P2
KV
y=^
It is to be observed strictly that in all the equations
above /s is the value appearing in Equation (6).
By using a logarithmic transformation any equation
of the form
Z{' = KZ2W (11)
(a, /3 and y = constants) maybe written in the form
of equation (6) thus
a log zi - fi log S2 - 7 log S3 - log A' =
The corresponding diagram has three parallel loga-
rithmic scales defined by the equations
X = —I y = a log Si
X = 1 y = /3 log S2
x= y = -J^(7logs3 -H log A')
' The projective transformation has the equations
(miSz 4- M2ii)» -I- {f^i&2
.«.)
scale factor /is of the third parallel scale.
(in - tit)x + (mi -t-
2MiM2y
- /X2)X + U, -I- M2)
■M2)
ALIGNMENT DIAGRAMS OR COLLINEAR NOMOGRAMS
39
The following equation
z'z-z^Zi'' = constant (12)
may be similarly treated. The logarithm of the con-
stant can of course be associated with any one of the
variables desired for convenience in constructing and
using the diagram.
Example 21. — An illustration of Equation (11) is
afforded by the formula for the volume of a torus or
ring of circular cross-section
V = 2AQ74:Dd^
Taking logarithms of both sides of this equation it
may be written
2\ogd + log D - log F -I- log 2.4674 =
A corresponding reduced determinant form is therefore
-1 • 2 1og^ 1
1 log Z? 1
log 2.4674 - log V
2 1
so that the three scales, when no scale factors are
used, are defined as follows:
x= -1 y = 2\ogd
x= 1
x=
Iog2.4674-logF
If the same limiting values are chosen for d and D it is
seen that the scale for d will be twice as long as that for
D. In order to have these scales of the same length
and covering the same range of values and so arranged
that both may be read with equal accuracy, choose
Ml = 1 ^2 = 2
For convenience let {tnh — fii^i) = so that 82 =
2S1 and the scale factor for the V scale will be
MlA'2 2
M3 = X ^ ■?
Ml + W o
The constant term log 2.4674 in the V function simply
determines the initial point of the logarithmic scale
for V, (see Fig. 40) , for which the equations are
x = 0, y = M[log V - log 2.4674]
It frequently happens that two parallel scales will
extend in opposite directions from the X axis and
whenever this is so a displacement of the scales along
their supports is desirable in order to dispose them to
better advantage on the sheet. In the following
example the K and R scales are started from a line
making an angle of 45° with the X axis at the initial
point of the A scale while the original distance between
the scales is preserved. Geometrically this is the
effect of carrying out upon the original diagram a
projective transformation whose equations are
xi = X yi = X + y + 1
and consequently alignment is preserved.
Example 22. — The area of a segment of a circle of
radius R and height H is given by the exact formula
H
H-
A = RH arc vers =- - — (R - E)
Since H appears always divided by R, write -^ =
then
A = R'[aTc vers A" - V2K - K-{1 - K)]
and passing to logarithms
log yl = 2 log R + log [arc vers A' - V2K - h
so that the reduced determinant form is
-1 log^
■2 log 7?
(1 - A)]
Yi log[arc vers A -^2K - ^=(1 - K)]
Figure -41 shows the diagram for this formula con-
structed with unit scale factors. When A' = 1 and
when A = 2 the corresponding areas are respectively
semi-circles and circles.
In most practical examples the displacement of a
scale whose graduations increase in a downward
direction from the X axis is best effected as in this
example by simply starting it from a point above the
X axis on a 45° line through the origin.
Example 23. — Figure 42 shows a diagram for the
formula
P = CF^D'^'
which is given by F. W. Taylor' for the pressure on a
tool when cutting cast iron, where
F = feed in inches,
P = pressure in pounds,
C = 45,000 for soft cast iron,
C = 69,000 for hard cast iron,
D = depth of cut in inches.
Passing to logarithms, the formula becomes
H log F + ms log D = logP- log C
and the scales are defined by the equations
X = 1 >' = ^•^ log F
a; = — 1 y = ^Hs log D
x= y = Viilog P - log C]
The constant C is associated with the P scale in order
that its extreme values given above may be used in
placing the graduations on the P scale. The diagram
thus gives the maximum and minimum values of P
for any D and F.
Example 24. — In correcting a barometer reading
at a temperature /i to a temperature / for which the
barometer is calibrated the correct reading in English
units would be
h = //,[1 - 0.000101(/i - t)]
» Trans. A. S. M. E., vol. 28.
40
10-
9-1
8
7-
64
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
r3000
2000
1000
E-900
800
100
hWO
500
-400
-300
200
150
2-
O
100
•90
80
•TO
GO
50
•40
[-10
r20
1-10
9
8
7
■G
■5
■4
E-5
Diagram for
fhe Volume of a Torus
V-2.4G74d^D '
rlO
9
8
-T
^5
-2
!-•
100 ■
ALIGNMENT DIAGRAMS OR COLLI NEAR NOMOGRAMS
41
D/'agram for fhe
ExacfArea of a Ci'rcu/ar Sepmenf
A = fi^[arc vers^~ IIMB(R.ff)]
r-I.O
1-1
G
4^
nor
o
-2.0
• 1.5
■ 1.25
-1.0
■0.9
•0.8
hoi
■CG
■0.5
■0.4
1-O.J
-0.2
-I.S
-2.0
D
-3.0
1.0-1
0.9-
0.8-
0.7-
0.&-
0.5-
0.4-
• 0.1
F-4.0
-5.0
-&.0
t-7.0
-8.0
9.0
42
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
l.(W-1.0
.90
54
-V«
.35-
-»/..
05
5 ^
-"/„
■S
-v„
S-:
-"/„
o _
5 -
-Vm
p.
o —
Q J
-"/„
JB^
-4
I
-'Vm
T
-V,»
.20-
-"/„
.19-
-v..
10.000-
9000-
8000-
7000-
-§-10.000
9000
7000
6000
p
;
r
n
o
a
u
s
2000-
- 3000
1
5
1
1
"S
1500-
M
w
-
J3
:S
1 2000
'%
&
.
1
;
•
h
i
1000-
- 1500
a
g
90O-
-
o
1
800-
-
\
Fig. 42.— Diagram for P= CF^ T^K
ALIGNMENT DIAGRAMS OR COLLI NEAR NOMOGRAMS
43
where hi is the observed height. The correction there-
fore is
E = h\ O.OOOIOIA/]
or, passing to logarithms,
log E - log 0.000101 = log hi + log M
If
X = -bi y = ixi\og hi
X — h y = fi-i log A/
it would be desirable to apply the methods of Article
4 to determine the scale factors /Ji and ;ii2 in order to
extend the scale resulting from the short range of
numbers for hi. If the hi and At scales are each to be
10 inches long and the following limits chosen:
hi from 26 to 31 inches of Mercury
A^ from 1 to 70 degrees Fahrenheit,
then
Ml = 130 and ^2 = 5.42
Taking (/ii52 — fi2&i) = as before, there results
6, _ Ml _ 130 24
6o ~ M2 ~ 5.42 ~ 1
and for m3
5.19
Ml + M2
The defining equations then are
x= -24 y = 130 log /fi
X = 1 y = 5.42 log At
x= y = 5.19 [log £- log 0.000101]
and the diagram appears in Fig. 43.
Example 25. — A modification by Grashof of Napier's
Rule for the flow of steam through an orifice is some-
times used for steam nozzles in the following form
„ P'-"Ao
^ ~ 60
where
F = flow of steam in pounds per second,
P = absolute initial pressure in pounds per square
inch,
A = area at throat in square inches.
If T/ritten in the logarithmic form, then
log F + log 60 = 0.97 log P + log ^0
In order to use the same units for the F and the ^o
scales let
^'"~ = nn7 Ml = 1
0.97
then if
^2 _ M2 _ _1_
6i " Ml ~ 0.9
the three scales are
X = 0.97
x= -I
x=
See Fig. 44.
1
1.97
0.508
y = logylo
y = log P
y = 0.508[log F + log 60]
Example 26. — The Royal Automobile Club (Eng-
land) automobile engine rating gives the rated horse-
power, HP, of N cylinders of bore D inches and
stroke 5 inches as
D'-NS
12
HP
or
m
+ log 12 = 2 log Z) +
In order to read the D and 5 scales on a diagram with
equal ease, let
X = —di y = Ml 2 log Z>
X = &2 y = M2 log 5
HP
log
N
log 12
if Ml = K, M2 = 1 then ms = M- See Fig. 45.
It is to be observed that when it is desirable to
displace one or more of the parallel scales in a diagram
it is not necessary to start the downward scales
from a Une making 45° with the X axis but any angle
a whatever may be used. The equivalent projective
transformation in the case of a diagram with scales
originally at distances Si, 5o from the Y axis would
have the equations
Xi = X yi — {x -\- 5i) tan a -\- y
Equations of four variables of the form
/l+/2+/3+/4 = (13)
may be represented by parallel scale diagrams and
will be discussed in Chapter IV together with the
more general type
/1+/2+/3+/4+ . . . +/„ =
13. Diagrams with Straight Scales and Two Only
Parallel. — It is easily seen that the equations
x = y = gi
x=\ y= go
x=fz y =
where gi, gi, fs are functions of Zi, Zi, Zi respectively,
would define a diagram in which there would be two
parallel scales and a third straight scale perpendicular
to them. What is the corresponding equation in
three variables for which such a diagram would be
useful?
Before deciding this question it is well to state that
all equations or formulas are subject to a great variety
of algebraic and other transformations: clearing of
fractions, factoring, removal of radicals, separation
or combination of constants, etc., which all tend to
change the appearance of any given equation.
An equation corresponding to the particular type ■
of coUinear nomogram or diagram of alignment to be
discussed is not diflacult to establish for it is only
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
31.0-
0.2190-1 ^10
0.?000— I I ,^
50
^40
30.0 —
^ ?9.0
2S.0-
27.0
2G.0— 1
Diagram for Correcfing
Readings of Mercury Baromehsrs
from formula h=h,[l- 0. 000 1 01 (i, - ffj
h- Correcied heighi of column
h, = Observed heic/hf of column
fj = Observed iemperafure
"f - Calihrafion iemperalure
- - ' -OTiobo
0.0800-
0.0600^
? 0.0400-
0.0200-
0.0100-
50—
0.0060—
0.0040-5
0.0050-^
0.002G—
—30
■20
-10[|
•9 1/5
■5
—4
—3
-7 ^
6-
L,
50-
4-0 4
30
20-
10-
7 —
5-=
0.5-
ALIGNMENT DIAGRAMS OR COLLI NEAR NOMOGRAMS
500-
200-
150-
100;
GO-
50-
40-
50
20-
10^
I
7
there result the
2.5 sin 2a
54
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
R ..H V
\
fi>= 70O5«L
--?7''30_
\
\
\
\.
Fig. 50.— Diagram for the Stadia Formulas F = /?-/? sin' a + cos a and K
ALIGNMENT DIAGRAMS OR COLLINEAR NOMOGRAMS
55
The values of a are graduated on a quartic curve.
The diagram is combined with the diagram for the
horizontal distance worked out above as the two
quantities H and V are always computed together.
The reader will readily see that the same transforma-
tion was necessarily applied to both parts of the
diagram to improve the arrangement of the values of
the vertical angle. Figure 50 shows the combined
diagram.
It is not of course necessary that the two straight
scales be parallel when there is but one curved scale.
Below is given an example where the two straight
scales are graduated on the axes of coordinates.
Example 33. — The formula for the mean hydrauhc
radius of trapezoidal sections of canals may be written
li(b + h cot 4>)
^ = b + 2//V1 + cot-
Where R = mean hydraulic radius
h = depth of water
b — width of canal bottom
= angle which the side slope makes with
the horizontal.
Write ^ =
equation is
and
li and a final determinant
K
1
1
tan y = — tan
The values of the angle are graduated on an
equilateral hyperbola crossing the X axis at a; = — H ■
The diagram is shown in Fig. 51.
When h is given there is of course not much disadvan-
tage in computing R from the value of -r •
If it were required to use scale factors to establish
a diagram for this equation above which has actually
the form
g, 1
/o 1=0 (22)
h g3 1
or expanded, the form
gl/3+/2g3-/2gl = (23)
then the projective transformations developed in
treating Equation (18) are available. The equations
were
(m2 — lil)x -\- Ml
>'l
(21)
(mi — Ml)^+ Ml
Then the new defining equations corresponding to
Equation (22) would be
(M2
5M2/2
- Atl)/2 +
5M2/3
(M2
l)/3 +
MlM2g3
0/3 + Ml
Another simple reduced determinant equation for
which there are two parallel straight scales and a
curved scale is
-1 «i 1 I
1 g2 1 1=0 (24)
I /a g3 1 I
The expanded form of this equation is
(gi + g2) - hig. - g.) - 2g3 = (25)
The defining equations for the curved scale will
undergo a change should scale factors be introduced
in the equations of the straight scales. Suppose that
it is desired to have the two parallel scales at equal
distances 8 from the Y axis and to use the scale
factors Ml and m2 respectively. The first defining
equations as before have the form
X = -8 y = Migi
X = 8 y = H2g2
and the third defining equations must be assumed to
have the form
X = F3 y = Gi
where Fz and d are to be functions of /s and gz alone
and will involve the constants 5, mi and m2. The
reduced determinant form of the equation will then
become
I -5 • Migi 1 !
I 5 M2g2 11 =
\ Fz Gz 1 I
and upon expanding there results
5(Migi + M2g2) - Fzijiigi - H2g'i) - 26G3 =
But from Equation (25)
g2 - 2gz + fzg2
which substituted in the equation above yields
g2 5m2+5m
I+/3
I+/3
8lJ.lg3 — Fzfllgz
fz-l ^=J = ^
and this equation must be true for every value of ga.
56
DESIGX OF DIAGRAMS FOR ENGINEERING FORMULAS
Diagram for the
'ean Ht/draulic Radius of
Trapezoidal Canals
ALIGNMENT DIAGRAMS OR COLLI NEAR NOMOGRAMS
57
Therefore the coefficient of gi and the term not
involving gi must vanish identically, that is
b
(mi
+ M2)/3 + (mi - M2)
(m:
- M2)/3 + (mi + M2)
2miM2^3
(mi — M2)/3 + (mi + M2)
and the changed form of the equations of the curved
scale of the diagram become
(ah + ix-2)fi + (mi — M2)
M2)/;
2MiM2g;
+ (mi + M2)
(26)
(mi — M2)/3 + iP\ + M2)
The above equations are very important for the
construction of diagrams discussed in the succeeding
sections of this book. They are the result of the appli-
cation of the projective transformation
^ (mi + tii)x + (mi — M2)
M2)-V +
2miM2}'
(m, + M2)
(27)
'' (mi - M2)X + (mi + M2)
to the points of the figure as originally defined.
It is seldom that the functions in an engineering
formula similar to Equation (8) are of such general
form that more than one curved scale results in the
diagram and indeed no rule can be given for the
introduction of scale factors when the defining equa-
tions are of the most general form. It is impossible
for example to introduce different pairs of values of
8 and m in the first two defining equations for if no
restriction were placed on the nature of the functions
/ii /a, ^1, §2, the first two curves originally defined
might intersect in one or more points and to use the
scale factors 5i and mi; and §2 and m2 would generally
demand that the same points of intersection of the
curves supporting the two original scales must move
in different directions at the same time and take new
positions. It is necessary, consequently, to leave to
the reader the introduction of desirable scale factors
in those cases of Equation (8) not already treated.
It will be necessary to take advantage of the particular
form of the individual equation in hand and to use the
general methods here developed. A thorough under-
standing of the use of the projective transformation
which is developed in Appendi.x B is very helpful.
15. Diagrams of Alignment with One FixedPoint. —
Equation (16) of Article 13 may be written
J\ - M'2 = (28)
and given the determinant form
/l g'2 1 I
/3 1 1=0 (29)
1
Then the three pairs of equations
= 1
=
define respectively: All the points of the plane, all
the points of the line y = 1 (a straight function scale),
and the origin. A diagram may be designed on
suitable cross-section paper with abscissas as values
of /i and ordinates as values of g^ and inscribed with
corresponding values of the variables Si and 22, and
with a scale of the function /s on the line y = 1 . Then
the values of 2 which constitute a solution of Equation
(29) are collinear. Since the index always passes
through the origin it may be scratched on a piece of
celluloid pivoted at that point.
Example 34. — The formula of Francis
q = 3.33S^^
yields a diagram of the above type and the defining
equations are conveniently:
10
3.335*
See Fig. 52.
Whenever the functions/i and g^ are linear functions
of the variables 21 and 22 respectively ordinary cross-
section paper may be used quickly to establish the
desired diagram. It is of course optional which
function /s or go is used in the first row of the
determinant.
By using logarithmic cross-section paper, equations
of the form
/. - g^' =
may readily be solved for a limited range of the vari-
ables involved to almost any desired degree of
accuracy. Passing to logarithms
log/, - 23 logg2 =
and with the defining equations from the determinant
of Equation (29) there results
log/i
23
y = log g2
y= 1
v=
There is an ordinary scale on the line y = 1 and the
logarithmic cross-section paper is inscribed with
values of Z\ and 22.
Example 35. — Frequently in thermodynamics the
equation
PV" = C
58
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
- . :,_ _r 1 '
_ n ^ 1 1 1 1 1 1 1 1 1 |g
1
1
^
1 1
T
1
—
; 1 1 : j
'
o
1 ! ! '
! 1 i !' ' 1
o'«- .Ml ' ' ^ -L .
'
" ± '
\
i o
\ 1 ' j
1
\ i 1
' 1
^- :
1 ~r !£
i 1 M
-t-^ ^ ' '
■
T
1 1 !
' \ ! 1 '
I ! \ i ' ' !
S-i I 1 : . 1 , I
S
' 1 \ f 1 1 1
4- -u _L -V ! 1 JL
_T^ I •, 1
s
1 \:
_1_ 1 ^|
1 ^ =t" - - '
_
1 X i
+ T~ " V " '
I ^
i i o
X
it o
1
X:
"^ o
V " "h -
^
-
s-i- Jl
1
\
!
X
\
1
a
\
\ X" °
T
0'1~
^ _
-
t
so 3^
-
"X
0- — ±___:: :±: : :: :"
:: ':: :::: :::x:: liSo
ALIGNMENT DIAGRAMS OR COLLINEAR NOMOGRAMS
59
^S*."?'.H(
:>l^ SS
«? r- vs IT)
O o a c.
^ ;
P>^7-/l,
^
//
///.
1/1/
t»S
05
//
//^/u
t"
OO
//A
///
/ /
'////
77 /
/
/
//
{//\ 1
"* _ .
//
/7//
/ /
--
///
/VV.
/y//
^.///.
^ /
///^/^
<^///
1
'/////
z/;^
f////
/
M////
'///
I/M////
y
my
/ 7
o
4
c./
12 .
§--
7
5 --
/
/
v*. --
^P
IT)
/^
Ym
o
///^
/, f/f/i
■* --
//////>
ri
<=>•
///////
'/l
c/>
/
/////^
//.
-^^
//^^
c-J
/
/
'///■
<^
w
S
/ /
//A
' / /
^ //>
'/
--
/
/ //
///,
//
///
V///
i^
//
///
////
c>
M-^
///
vZZ
■"
o^Z-
/ /
////
''TuL
-^^^'^
/ /
/ ' / /
'/i
-^s/.
-//
/ / '
//
u^/
'//
////
/
- F-7-
/ /
fo/sZ^
/ /
///
- /y-
//,
//
^/^iz
//,
V
"-/6
'/A
/
-.^'6
'//
"? ?7-
/
;:/Z'^
JL
•^ /v
./'/
~
;#
go
OK
>OC5C> c> <=>
/o'oiodH vi> u-J ^
>-|>^
60
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
arises, and with a given set of values of C and w it is The defining equations of a suitable diagram of the
desired to find an indefinite number of closely deter- type under discussion are then
mined points on the curve plotted with P as ordinates
and 1' as abscissas. It is usually desired to find addi-
tional pairs of values Pi and Vo to satisfy the equation
PiFi" = PiVi" and the diagram is shown in Fig. 53.
= log AP
y = log AV
= —n
y= 1
=
^ =
S5 6.0 t.5 70 IS 8. 65 90 9.5 10
.10 .11 .IZ .13 .14- .15 .16 .0 .16 .19
where Pi and Fi have been determined. This equa-
tion may be written
[log P, - log P,] + «[log Vi - log V2] =0
logAP+ wlogAF=
While the diagram consists essentially of the loga-
rithmic cross-section paper with the scale for n and the
origin clearly marked upon it, in this figure the various
positions of the index have been drawn in as straight
lines for n. This is allowable since the equation
log AP-t- «log AF =
ALIGNMENT DIAGRAMS OR COLLINEAR NOMOGRAMS
61
10.0-1
9.0-
8.0-
7.0-
G.O-
5.0-
4.0-
5.CM
l_?.o-
C5
^1.0-
^0.9-
1 0.8-
.E 0.7-
1 0.6-
n 0,5-
o
0.4-
0.3-
0.7-
Diagram for
yL8G
JUS
H= 0.38
and
o
20-1
15.0-
10.0-
9.0
8.0H
TO
G.O
5.0-
4.0
3.0-
7.0-
1.0-
0.9-
0.8-
0.1-
0.&-
0.5^
0.4-
o.s-
plO
-9
8
1-7
500--
400
300
tiooo
700-
4-500
400
300
-1-200
150
50-
40 -f
30-
70-
^ 10-
I00-i|=:8O
-60
-40
-30
.70
o
1/5
gO.5
=E 0.4
:^ 0.3
0.7
O.I
l?-[-G
GG-
Go4-5
54
48-
42
3^-3
30-
24-
77-
70-
18-
-f-
■z lP >„)]
when Xi represents distances from a center line, and y, cuts
and fills at the corresponding points; construct a diagram
with a sliding index to compute A for values of Xi and yi
varying by tenths up to 20 feet.
Problem 13. — Devise a combined diagram to handle the
following relation in thermodynamics
Ti ^ \pj " " \YJ
Problem 14. — Construct a diagram of three parallel
straight scales for the expression of Problem 14 of Chapter
II.
Problem 15. — The tractive resistance R in pounds of an
automobile of weight W pounds when moving at a speed of
V miles per hour is given by Prof. E. H. Lockwood as
i? = 15 + mbW + .075F2
Construct a diagram for this formula.
Problem 16. — Construct a diagram consisting of four
parallel straight line scales upon which the collineation
of four points will serve to solve the two equations
P. = iixy) = 2i 2ixy) = 22 (N)
as shown in Fig. 57.
Through every point P of the plane will pass a curve
of each system inscribed with its corresponding value
of z. This configuration of curves will be called the
curve net Nu for Si and Zo. A line perpendicular to
OX is seen to cut out an indefinite number of pairs
of values of Zi and Zi. Every point M
of OX may thus be regarded as supplied
with all the pairs of values of Zi and Z2
which correspond to the curves intersect-
ing on the line PM. These value pairs
cannot all be written at the point M but
are nevertheless definitely attached to it.
Furthermore, given the value of Zi there
is but one' corresponding value of S2 to
be found upon PM.
If now every point M on OX is regarded
as supplied in this way with its values of
Zi and zo, the line OX becomes a certain
kind of scale. Each length OM deter-
mines uniquely a line MP on which lies a
certain set of values Z1Z2.
Let OM = xi and consider the line
Xi = Xi and the curves
4>iixy) = zi 'i{xy) = Z2
Eliminating x and y from these three
equations yields an equation in Zi, 02 and
x which may be written
/(Z,Z2) = X,
All the values of Z1Z2 which satisfy the above equations
belong to the point M.
Conversely, given a value of Zi (or Z2) and the point
M (which is equivalent to assuming the value of Xi),
there is in general but one value of Z2 (or Zi) which will
satisfy the last equation. It is thus convenient to
define the configuration of Fig. 57 as a binary function
' If the line PM intersects the curve corresponding to Zi in n points
there will of course correspond n values of Zj, etc.
scale for the function fu on OX which is called the
support.
Similarly eliminating x from the equations (N)
leads to the result
g(ziZ2) = y
The curve net of Fig. 57 thus completely determines
also a binary function scale for gio on OY.
Frequently a pair of functions /12 and gn occur in an
equation of four variables for which a diagram is to
be constructed, and when the equation is put into
Fig. 57.
the determinant form analogous to Equation (8) it is
necessary to interpret the defining equations
X = /i2 y = gn
It is evident from the foregoing that these two equa-
tions define a curve net. It is merely necessary to
eliminate Z2 and Zi successively and there is obtained
again
iixy) = zi n{xy) = Z2
65
66
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
Since the only necessary equations for a binary scale
on the A' axis are
X = /i2 and y =
they are called the defining equations for the binary
scale. In constructing a curve net Nu for this scale
it is seen that either of the functions (fi or (pi may be
arbitrarily chosen. When, however, one function,
say 01, is chosen the other function 02 is determined by
eliminating Si from the equations
i cannot be a function of x alone. Exam-
ples below will show, however, that a suitable choice of
the arbitrary function aids in the solution.
The binary scale is really then a special case of a
curve net resulting when either of the functions fn
or gi2 defining a curve net reduces to zero or any
constant. When either Zi or Zi only is absent from/12
or from gi2 one set of curves in the corresponding net
will be a system of parallel straight lines. (For
another special case see Article 23 of Chapter VI.)
When a binary scale has been established on either
axis or upon a line parallel to either axis it is obvious
that the necessary curve net may be moved by trans-
lation parallel to the other axis provided that the
straight line support remains fixed. Obviously then
a new origin of the axes of coordinates may thus be
chosen for plotting the necessary curve net and this
method is sometimes of much advantage in improving
the plan of the diagram.
17. Collinear Diagrams with Two Parallel Scales
and One Curve Net. — Consider now the equation in
four variables Zi(i = 1, 2, 3, 4), which may be given
the determinant form
/i ^1 1
/2 g2 1 =0 (30)
/34 ^34 1
A simple case arises when this reduced determinant
equation may be written analogous to Equation (18)
of Chapter III:
1 gi 1 I
g, 1=0 (31)
/34 gZi 1 I
When this equation is expanded there results
g2+/34(gl-§2)-g34 = (32)
Assume temporarily that the scale factors are
unity and there results a set of defining equa-
tions from Equation (31):
X = 1 y = gi
X = y = gi
X = fsi y = g34
and the last two equations define a curve net.
It is thus necessary to study a collinear nomo-
gram or diagram of alignment consisting of two
parallel straight scales and a set of points
defined by an inscribed curve net. To each point
of the curve net corresponds a pair of values 2122
attached to the two curves passing through that
point. The equations of the curve net are readily
written by eliminating z^ and 23 successively from
the last two equations and they become
<^3(^>') = 23 i{xy) = 24
The resulting configuration is shown schematically
in Fig. 58.
Given three values of Zj, the diagram of Fig. 58 con-
stitutes a complete graphic solution for the unknown
value of z. Suppose that Z4 is unknown: The line
P1P2 cuts then the curve Z3 in the point P through
which passes a curve marked 24. The proof that this
value of 24 is the value sought is left to the reader.
For certain equations in four variables there is thus
realized an important type of colUnear diagram. To
be solvable by such a diagram an equation must be
reducible to the form (31). Obviously the parallel
scales may be placed at a distance 5 and the scale
factors Ml and 1x2 employed if the equations of the
curve net are determined from the third pair of
defining equations as modified by the Equations (21)
ALIGNMENT DIAGRAMS FOR FORMULAS IN MORE THAN THREE VARIABLES 67
of Chapter III.
equations
There results then for the defining
= S
=
y = Migi
y = fi2g2
(33)
^^2/3 4
A^lM2g34
M2/34 — Ml(/34 — 1) ' /J2/34 — A'l(/34 — 1)
The choice of the constants S, m and ^2 should of
course be made not only with the first two scales in
view but also with the resultant changes in the curve
net fully in mind. No plotting should be undertaken
until a thorough study of the equations has been
made in order to obtain the desired range of values
of the variables involved and at the same time to
reduce as far as possible the required computation
for plotting the curve net.
Example 36. — A very good illustrative example is
afforded by the complete cubic equation
s' + aiz^ + a.z + as =
which may be given the determinant form
2- + 2
Whence if 6 = 10, ,
X = 10
2- + 2
10^
;- + =
= ' + (73
are the defining equations for the diagram which is
shown in Fig. 59. In plotting the curve net for the
variables s and a^ the 2 lines parallel to the Y axis
are plotted first and then it is observed that the
successive as curves determine regular scales on each
2 line with a new scale factor for each. It is only
necessary to plot the curves for the values of a^ equal
to —10, 0, and 10 successively to determine com-
pletely the system of curves. The scale factor on
each z line is seen to be , • In the diagram the
2" -f- 2
dotted line shows the position of a straight edge set to
solve the equation 2' + 42^ - 4s + 0.5 = 0. The
straight edge is set from ai = +4 to ai = —4 and
gives the value of s = 0.69 at its intersection with the
curve 03 = 0.5.
Another simple case of Equation (30) which results
in the same form of diagram is
-1 g: 1
1 ^2 1 =0 (34)
/34 ^34 1
The expanded equation has the form
2g34 + /34(gl - g2) - (g. + g2) = (35)
and the defining equations with the scale factors
determined by the aid of Equations (25) and (26)
for the analogous case of three variables, are
(mi + M2)/34 + (mi
Mlgl
M2g2
(36)
M2)/34 + (mi + M2)
y
2MlM2g34
(mi — M2)/34 + (mi + M2)
The presence of the constants 5 and m in the third pair
of equations allows control to some extent of the dis-
position of the resulting curve net. Whenever the
scales for the first two variables extend in opposite
directions in the diagram it is desirable to apply a
transformation as in Section 12 of Chapter III. This
is done in the following illustrative examples.
From the last pair of defining equations in (33) and
(36) it is seen that whenever Z3 or Z4 is absent from 734
there results a system of straight lines parallel to the
Y axis and they are determined by a scale on the X
axis most conveniently. Whenever /34 or ^34 is zero
(or when 734 is constant) there result the defining
equations of a binary scale on the Y axis or on the X
axis (or on the line x = constant) respectively.
Another special case occurs which leads to a curved
binary scale and is discussed in Chapter VI.
Example 37. — As an illustrative example of Equa-
tion (34) consider Kutter's formula for the flow of
water in open channels,
1.81132 , 0.00281
41.6603
V =
1+ 41.6603 +
0.00281 ] n
S \VR
Vrs
Where V = velocity in feet per second
S = tangent of inclination of surface
R = mean hydraulic radius
n = Kutter's coefficient of channel bottom.
The above formula may be modified by setting
1,000
outside the radical.' There results
1.81132
44.4703 +
V =
1 +
44.4703 w
-VRS
hich, if 44.4703
-1
VR
a, and 1.81132 = b, reduces to
V I
-VS 1
-niVR + an) {an + b)R \
This substitution is known as Flynn's modification of Kutter's
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
ALIGNMENT DIAGRAMS FOR FORMULAS IN MORE THAN THREE VARIABLES
as a first determinant form. The reduced determi
nant form is then found to be
-1 V 1
[an + b)R
1 -VS
-n(VR+an) ^
ian + b}R + n{VR+an)
The defining equations written from Equations (36)
above are
X = -S
X = 5
2VS
X = 5-
b)R - uiu(VR + an)
i{an + b)R + fjLMVR + an) '
For convenience then in plotting there may be chosen
S = 10, Ml = 0.8 M2 = 80.0
whence the scale equations:
a; = -10 y = 0.8F _
a; = 10 y = -gOVs
(an + b)R - lOOniVR + an)
{an + b)R + 100n{VR + an) ■' "
Since ^34 is here zero there is a binary scale on the
X axis. One system of curves in the net defining the
binary scale may well be chosen as the parallel lines
y = 2VR
and there follows upon eliminating R the cubic curves
for n
_ (an + ^>)v^ - 20(yn(y + 2 an)
"" ~ ^^{an + b)y' + 200n(y + 2an)
All these cubics pass through the point x = — 10,
y = and are asymptotic to the vertical linex =10.
(See Fig. 60.)
The V and 5 scales would naturally lie in opposite
directions from the A' axis but to secure a better
disposition of these scales and thus reduce the size
of the sheet, they have been moved by using the
projective transformation
Xi = X yi = X + y + 10
which moves all points along their ordinates a distance
equal to the abscissa plus 10. Thus the line y =
becomes the line y = x -\- 10 which is the line MN
in the diagram. From the nature of the binary scale,
however, there is no need of transforming the curve
net for the variables n and R and this has not been
done in the figure. The points on the binary scale
are simply transferred by the parallel vertical straight
lines from the X axis to the diagonal which thus
becomes the new support.
Example 38. — Another example of Equation (34) is
afforded by Bazin's formula for the flow of water in
open channels which is
87
-Vrs
0.552 +
VR
where V, R, and 5 have the same meaning as above
and m is Bazin's coefficient of bottom condition.
The first determinant form of the equation may be
written
1 V
-V5 1 =
(0.552i? + m) S7R
and the reduced form of the equation is then
1
S7R - 0.552^yR - m
V
-VS
87 R + 0.552VR + m
The corresponding scale equations are
x= -b
M.F
-M-2V5
^ Mi87jg - M-2 (0.552 Vjg + m)
Mi87i? + M2(0.552\/^ + m) ^ " ^
There is again a binary scale on the X axis which is
determined by setting
S = 10, Ml = 0.8 M2 = 80 as above and also y = 2-\/R
whence
x= 10,
87y= - 200(0.552y + m)
87y= + 200(0.552y + m)
The m curves of the corresponding net are six cubics
for Bazin's six values of m. These cubics have a
singular point at x = — 10, y = and are asymptotic
to the line x = 10. Figure 61 shows the finished
diagram originally plotted with a modulus of one
inch. The 5 scale and the binary scale have been
transformed as in the preceding example.
Equation (31) is no simpler than the essentially
equivalent form
fa gzi 1
/i 1 1=0
h 1
which has the expanded form :
/34+g34(/2-/l)-/2 =
and for which the corresponding diagram will consist
of two horizontal (instead of vertical) parallel scales
and a curve net. To introduce scale factors into
the corresponding defining equations there are avail-
able Equations (21) of Chapter III. With the obvious
changes in the role of the respective coordinates x and
y there results from these equations
IJ-\IJ-iX 5/J2V
A'l)^
(m2
i)y +
70
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
ALIGNMENT DIAGRAMS FOR FORMULAS IN MORE THAN THREE VARIABLES 71
Fig. 61. — Diagram for
72
DESIGX OF DIAGRAMS FOR ENGINEERING FORMULAS
Example 39. — One form of the fundamental formula
for bond calculations is
A
C
the quantity
tV
-K4^)
where A --=
g =
C =
« =
purchase price
nominal interest rate
effective or yield rate of interest
redemption price
term of bond in years
•^ - 1 + i
This bond formula has the reduced determinant form
R 1
g' 1 1
=
1-^;;] 1-^^
in which for convenience the symbol A~\ is used for
\ — V" , S —A
— -. — and in which g = ^ and R = j;- This is
an excellent example of the form of Equation (31)
above. By letting /i2 = 1 and tx\ = ft. and substituting
successively in the modified Equations (21) above the
respective pairs of elements from the determinant
there result the defining equations
x = R y =
X = ixg' y = 5
ixv" 8 A ^1
X = 7- 1 y = 7-
1^ - An -' 1^ - A-i
The curve net for i and n defined by the last pair of
equations may best be plotted as follows:
The ratio
from which
Sx
d.i-v"
5x — ifiy
and when this value of v" is substituted in the second
equation
bA-, -5(1-1'")
//i - 1 + v"
there results
hix - 1)
which defines a pencil of lines through x = 1, y =
as the i-lines. The equation of the w-curves may be
shown to be
' y
/ x + 2y - 1 Y_ 1-
\ y ) X
if both n and 8 are taken unity but it is not necessary
to attempt to plot from this equation. Instead
resume the equation
X /./f"
is tabulated in standard works
and
on bonds, life insurance, etc. and is designated 5
rewritten
5- = (1 + 0" - 1
"' i
From these double entry tables when / is constant the
values of Sn[ vary for n only, thus the equation
X M '
will determine a pencil of lines through the origin
varying for values of n. These lines intersect the
corresponding j-line in points necessarily on the
respective w-curves. Thus the n-curves may easily
be plotted.
The useful range of values of the ratio ^ is
from say "^^oo to ^^^loo and to be effective in
actual bond calculations this ratio must be readable
to the nearest thousandth or tenth of a per cent, conse-
quently if the scale should show one per cent as one-
half inch the effective portion would be 22 ^ inches
long and unity would be represented by 50 inches.
The choice of a» and 5 must then be made and it is
obvious that if d is greater than unity the line y = S
on which is to be shown the g' scale will be not only
off any drawing of dimension less than 50 inches verti-
cally but also (since g' will never be much greater than
Ho) unless n is large g' will be too close to the Y axis
to appear on any reasonably sized drawing where
unity is 50 inches. The remedy for both these
troubles is to choose oblique axes with a very acute
angle. When this is done and with 8 = %o and n =2
there results the completed drawing shown in Fig. 62.
It is observed that the nominal interest scale is
inscribed g and not g'. This is because the normal case
is redemption at par and then g' reduces to g the nor-
mal rate. With ;tt = 2, one per cent on the g scale is
represented by one inch. The auxiliary net of lines
for the segregation from the binary scale of the ratio
pr into the purchase price A and the redemption
price C is effected by the equations
A
The choice of m = 2 is dictated by the behavior of
the n curves for a reasonable range of useful terms
n and was determined only after several trials.
The examples here worked out are special cases of
Equations (31) and (34) which are both special cases
of the more general Equation (30) which equation
would in general require two curved scales and a curve
mA
m = constant
ALIGNMENT DIAGRAMS FOR FORMULAS IN MORE THAN THREE VARIABLES 73
LEGEND
A = Purchase Price or Present Value.
C = Redemption Price.
§ = Ratio of Ditiidend to Redemption. or Diiiidend Rate with Redemption at Par.
i = Effective Interest Rate.
n • Term of Years.
Purchase Price A
ift o >5 o
DIAGRAM
FOR THE
FUNDAMENTAL BOND FORMULA
1
c " (i+ir i
FOR AiNY VARIABLE IN THE FORMULA
74
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
net for its diagram. Examples encountered in prac-
tice seldom require such a type of diagram but treat-
ment of the scales by some projective transformation
would doubtless be needed for any such example.
It is to be observed that whenever one set of curves
in a curve net becomes a system of straight lines, then
the plotting of the second set of curves can often be
simplified by finding indirectly their intersections with
this plotted line system. Such was essentially the
method used in Examples 36 and 39.
More generally, when one set of curves of a curve
net has been plotted, the second set can be plotted by
determining indirectly points of intersection with
individual curves of the plotted first set. To do this
hold constant in either of the defining equations the
value of the variable parameter corresponding to
given curve, while the second variable corresponding
to the desired set is allowed to vary and draw the
resulting lines parallel to one of the axes. There will
thus be determined on the plotted curve a series of
points of intersection corresponding to successive
values of the second variable. These points for
constant values of the second variable on successive
curves will lie on a curve of the second set. In
particular if, as above explained, the first set of
curves is a system of straight lines, then the curves
of the second system can always be found by plotting
corresponding points of intersection of this first
system of lines with the system of lines parallel to
either one of the axes. In Example 39 an auxiliary
set of lines through the origin was used to advantage
instead of the parallel lines determined by a defining
equation.
18. Collinear Diagrams with Three Curve Nets.—
These diagrams and indeed diagrams with two curve
nets are largely of theoretic interest but there are
special cases of practical value.
Consider first an equation of six variables in the
determinant form
M
gl2
1
/34
^34
1
Aa
g66
1
(37)
By setting
/./
1,3,5
2,4,6
there are obtained three curve nets constituting a
collinear diagram for this equation. The key to
the solution of the diagram is obvious from the
schematic Fig. 63. Should the active range of the
variables involved determine curve nets which unduly
overlap or confuse the diagram, some device such as
different colors will be needed to make the drawing
of practical value. In most cases that occur in
practice the curve nets reduce to binary scales and
seldom are there more than two.
Example 40. — As an illustrative example consider
the equation for the angular distance 2 of a celestial
body east or west of the meridian from the north
point.
where L = the latitude of observer
p = the polar distance of the object
// = the altitude of the object
5 = H(h +L + p)
This equation may be solved by a diagram with two
binary scales, but since z must usually be determined
cos
/cos 5 cos (S — p)
cos L cos //
Fiu. 63.
at least to the nearest 30 seconds no diagram of any
practical value can be drawn small enough to repro-
duce here successfully. The variables are z, S,
(S — p), L, and h. After squaring both sides of the
equation it may be written in the reduced determinant
form
cos 5 cos {S - p) ]
=
cos L cos //
1 -|- cos L cos h
There may be written in a manner analogous to Equa-
tion (14) of Chapter III, the defining equations
.V- = y = Ml cos 5 cos (5 - p}
cos L cos //
fin cos L cos // + n-i '
The first two equations define a binary scale on the Y
axis and the variables 5 and (S — p) may be separated
with the curve net
X = cos S y = fiix cos (5 — p)
which gives two systems of straight lines. The second
equation pair defines the scale of length M2 measured
downward on the line x = b. The third equation
ALIGNMENT DIAGRAMS FOR FORMULAS IN MORE THAN THREE VARIABLES 75
pair determines a binary scale on the X axis and is
constructed with the simple curve net
— All cos Ly
Ly
■ cos h
The L curves are then equilateral hyperbolas passing
through the origin and with asymptotes parallel to
the coordinate axes.
Problem 1. — Consider the cubic equation
s^ + a,c2 + consisting respectively of lines
parallel to the Y axis and hyperbolas passing through the
origin and tangent to the X axis at that point.
Problem 5. — Professor C. H. Forsyth has given' a
formula for the premium or discount per unit on a bond
if the "amortization factor" accumulates at a rate of
interest r which is different from the effective or yield
rate of the bond i. If k denotes this premium or discount
then with the notation of Example 39 and redemption at
face value or par the formula is
9 - i
where the changed symbol — denotes that this tabulated
^Bulletin, km. Math. Soc, vol. XXVII, p. 451.
quantity ^ is to be here taken at the rate r. Show that
this formula is a special case of Equation (37) with five
variables and with a corresponding diagram which consists
of a straight line (cross-section) net for g and k, an ordinary
scale for ; on the Y axis and a binary scale on the line
-T = — 1 for — and that the segregation of the n and r
Hues in the binary scale net can be obtained by setting
X = r — \ and plotting the n curves by determining
points on the r line corresponding to changes in n for
constant r.
Problem 6. — In the above problem show that the
equations for the curve net for g and k can also have the
equations
-bu^k
(*12 - Ml)^
y =
(M2
.)fe+A
if scale factors 6, y.i, ^2 are introduced by Equation (21) of
Chapter III and that consequently an ordinary cross-
section net for g and k results when jui = ix^.
Problem 7. — The so-called premium formula for bond
valuation is with the usual notation
k=(g- i) /l'„-i
where A % indicates that ^ ^ is to be evaluated at the rate i.
When the bond is bought at a discount k is negative.
Compare this equation with that of Example 39 and
discuss the advantages, Lf any, for design of the correspond-
ing diagram.
Problem 8. — Show that the equation for the «-curves
in the curve net of the diagram of the above equation are
_i+_y_ .
y / I -\- y - X
given by the equation
(^)"
Problem 9. — If all interests" are payable m times a year
and the amortization factor accumulates at a nominal rate
r then the premium formula of Professor Forsyth becomes
where 5;;^ is to be evaluated for mn periods at rate —
and where ; is the nominal rate to be realized. Show how
this equation in the five variables k, g, j, r, n, can also be
diagrammatically represented for values of m from w = 1
to m = 4.
Problem 10. — Show in Example 36 how the cubic curves
for as could have been plotted by first plotting a system of
lines parallel to the X axis which would intersect a given
2-line parallel to the Y axis in points corresponding to
values of 03.
CIL\PTER V
DIAGRAMS OF ALIGNMENT WITH TWO OR MORE INDICES
19. Diagrams of Double Alignment. — Sometimes
a formula or an equation may be given the form
/l2 = /34 (38)
and may be replaced by a pair of equivalent equations
/l2 = /2 34 = fi
where h is an auxiliary variable. Assume now that
each of these equations can be represented by a
diagram of alignment. By determining the value of h
from one diagram the value of either one of the
remaining variables, say Zz or Z4, could be found from
the second diagram. If, however, both equations can
be represented with the same scale for h, a single
Fig. 63a.
figure with four z scales and one h scale would consti-
tute a complete diagram for the original equation.
Such a diagram is called a diagram of double align-
ment or a diagram of double collinealion. The scale for
// is called the hinge or pivot scale and need not be
graduated unless this is desirable for convenience in
locating the temporary point about which the index
4 sin }i cos ^
A- 1
is turned for its second position. The type of diagram
and the way to the solution is shown in the schematic
figure. Fig. 63a.
The diagram is often more conveniently arranged
when the part including the Zi and z^ scales is super-
imposed in the other part of the figure, but in many
cases where the scales are on parallel supports greater
accuracy and ease of use will result when the pivot
scale is chosen between the scales for each part of the
diagram and the indices are placed in the form of a
letter X as shown in Fig. 69, page 8.
Some thought should be given to the way in which
the variables are grouped on either side of the equality
sign in Equation (38) so that those variables which
are perhaps more closely related or those which have
about the same range of numbers may be used in the
same half of the diagram. It is usually necessary to
use different schemes of scale factors for each auxiliary
equation and the only restriction on the equations
is that they be of type (8) of Chapter III. It is always
required, however, that the hinge scale have the same
defining equations in every respect in order that the
same value of h shall be determined in both diagrams
by corresponding values of the variables.
Example 41. — The formula of Example 33 for the
mean hydraulic radius of trapezoidal sections of
canals will be arranged as an example of the diagram
of double coUineation. The formula is
1 + K cot
1 +2K cosec 4> b
where R is the mean hydraulic radius, H the depth of
water, b the breadth of canal bottom, and (t> the angle
of slope with the horizontal. The formula should
first be r&written in the form of Equation 38.
R _ J _ sin -f Jv" cos
H~ ^ ~ sm4> + 2K
and then in the reduced determinant equation forms
h 1
= =
1 -R
ih '
Since K will never be greater than unity it is evident
that the unit of the drawing for the first determinant
must be fairly large and it may be chosen as 4
inches. The scale for the angle will then be a circle
with a radius of one-half unit or two inches, which
is ample, as the angles need not be measured
closer than degrees. No scale factors are needed.
DIAGRAMS OF ALIGNMENT WITH TWO OR MORE INDICES
77
In the second equation it is seen that the R scale may
be inconveniently long with the unit of the drawing
as 4 inches and yet since the value of the H function
is never greater than unity the horizontal scale must
not be contracted. What is needed then is to extend
the horizontal scale and contract the vertical scale
and at the same time leave the // scale unchanged.
choosing 5 = 2.5.
defining equations
rt = 2.5
x=
5
There results for the second set of
H
1.0 it
0.5-
Diagram for
R.-H'-ihtl-
Jcosec^
The result is accomplished with a little study by first
writing the second equation in the form
1
1_
l + H
This equation is of the type (17) of Chapter III and
Equations (21) are applicable to the defining equations.
Sufficient contraction \^^ll result if ni is taken }4, then
M2 must be unity in order to preserve the h scale
intact. The scale for H may be extended and at the
same time the position of the R scale improved by
The completed diagram is shown in Fig. 64 which has
been rotated 90° to improve its position on the sheet.
The limiting values of the variables chosen here are
general and the graduations could be greatly refined
for special work, for small drainage ditches for example.
Example 42. — Unwin's formula for the flow of
steam in pipes
II' = 87.5
■I-
Ppd"
11 +
3.6
where W = number of pounds flowing per minute
D = density in pounds per cubic foot
p = loss of pressure due to friction, in pounds
per square inch
78
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
d = nominal inside diameter of pipe in inches
L = length of pipe in feet,
may be written
11'
8'Wf
= //
VD
^
3.6
If L is taken as 100 feet and p expressed as the loss in
pressure per 100 feet of length, the formula is similar
to Equation (38)
Tl'
^ = /, = ^/D
S.75VP
or log ir — log 8.75
3.6
\/l+ _
■4 log p = \ogh
MlogD + Mlog j^3J5
log //
Both of these auxiliary equations are similar in form
to Equation (10) and yield three parallel straight scales.
The diagram is shown in Fig. 65 with the defining
equations
= PI log ir
= -M2[log 8.75 + H log/']
X = -8i
X = bi
x=
N.+^^
64
The proportion
«i
was used
log \'D
^ = — =l = ^=ii2
&2 Hi 4 54 m
to give a sjonmetrical diagram with convenient scale
lengths. Since D, the density, depends upon the
steam pressure the corresponding pressures were
plotted in place of the various densities. To the scale
for W was added a scale for the approximate boiler
horsepower. The indices show the setting to deter-
mine the flow in a 6-inch pipe carrying steam at 140
pounds gauge pressure allowing a drop of 3 pounds
for each 100 feet.
Any equation or formula of the form
(m, », r, s, = constants)
may be replaced by the equivalent system of equations
p' 6"
and two corresponding first determinant equations are
1 //
-a"" 1
/>' 1
1 h
-q' 1
b- 1
(40)
The choice of the reduced determinant forms of these
equations may be made by first adding either the first
or the second columns to the third to form a new third
column in each determinant. The choice will of course
be made with a view to the economy of calculation
for the resulting scales on the A' or Y axis. Equa-
tions (40) are of the type (14) of Chapter III.
The above equation (39) may be written
m log a — r log p = log h = s log q — n log b
and the two auxiliary equations will be similar to Equa-
tion (10) and require simply four logarithmic scales
on as many parallel straight lines. The determinant
equations are
- 1 m log a 1
1 -rlogp 1=0
logh 1 I
1—1 s log q 1 !
1 -n]ogb 1 { =
1 logh 1
and the defining equations of Section 12 including 8
and M apply to each.
It is to be observed that no plotting on the // scale
is necessary but the same value of h must determine
the same point on the h scale so that the reduced deter-
minant forms of the equations must both result in the
same defining equations for the // scale even though
that scale is not graduated. It must therefore be
borne in mind that the choice of the scale factor for
the h scale must be the same for both equations.
Example 43. — As an illustrative example of the
above Equation (39) Chezy's formula for the flow of
water in open channels may be studied. The formula
is
V = fv'^
where V, R, and S have the designations of Example
37 and c is Chezy's coefiicient. The form for the
reduced determinant equations may be taken
1 h 1
-V
c
c + 1
1
h
_rH
5-w
1+s-^
It will be necessary to graduate the scales for R and
V and for S and c on the same axes, and this will
always be necessary for equations of the tj-pe (40).
Since the c and S scales will not extend beyond unit
distance from the origin it will be well to make 6 as
DIAGRAMS OF ALIGNMENT WITH TWO OR MORE INDICES
79
-0.1
20,000
Diagram for -the
FLOW OF STEAM IN PIPES
Un win's Formula
w-87.sy£EiL
W= Flow - lb. per mjn.
D = Densi'iy - lb. per cu. FF.
p - Pressure Drop - lb. per sq. in.
d - Inside Diamefer Pipe - in.
L^LengihoFPipe-FI:
-250
-2Z5
-200
-180
^
-m
--
-140
-120
-100
-90
D
-80
O
-TO
.C
-60
d-
Kf)
-50
ty
-40
-30
oT"
u
3
-20
1?,
-15
d:
-10
E
-^
-5
^
To
-2
-0
■£
IG.GOO-- 6,000
12,000
8,000
6,000
G.OOO
4.000
3,000
4,ooa|- 1,000
3.,000-- 1,500
2,000- -1,000
I.GOO-
1,200-
1,000-
800
GOO-
GO
c^30
o "20
^ 1.2
0.8
0.&
10,000
800
-MO
50(K^
400 ^
300 ^
50O::2'5a'
^4^-200
300- -150
200-1-100
IGO-
%
100
80-H-o
30
-1-10
15
"18 --9 ^
4-2
3-1-1.5
1.0
l.G-fo.8
0.&
04
0.3
0.4-L0.2
Fig. 65.
-12
■10
-8
-T
-&
-5
~^4
-5
u
2 -E^
\% J
I
-0.15
-0.2
-0.S
-0.4
-0.5
-O.G
-0.7
-0.8
-0.9
-1.0
-1.2
-1.4
-1.8
-2.0
-2.25
-2.5
-2.15
-3.0
■3.5
■4.0
-4.5
-5.0
1-6.0
-7.0
-8.0
-9.0
-10.0
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
large as the sheet will permit. Also since the // scale
and the V and R scales are in opposite directions the
X axis may well be chosen at an acute angle with the
Y axis at the outset.
If the formula is written
log F - log c
log R + M log 5
the two determinant equations are
■ 1 log F 1
1 -logc 1
log It 1
-1 ^2\ogR 1
1 i2log5 1 =0
log h 1
and the general arrangement of the figure would be
similar to Fig. 65 of E.xample 42. This latter form
of similar formulas usually results in simpler and easier
plotting.
Some formulas in four or more variables can best be
handled by combinations of simple diagrams composed
of straight line systems as described in Chapter II.
The use of such diagrams is very common and obvi-
ously all that has been said regarding the use of a
hinge scale h for combinations of collinear diagrams
applies equally well in such cases. (See Problem 16 of
Chapter III.) Often a simple diagram may be
combined with a collinear diagram so that the scale
on one of the axes of the simple diagram serves
also as the h scale of the collinear diagram. When two
simple diagrams are placed "back to back" or super-
imposed, of course no hinge scale is used.
20. Diagrams with Parallel or Perpendicular
Indices. — Such diagrams consist of four scales
arranged in pairs corresponding to the four variables
of an equation or formula. The scales are so disposed
that a straight line drawn through a known point
on a third scale parallel or perpendicular to a line
joining two known points on two other scales will cut
the fourth scale in the point inscribed with the value
of the unknown variable. The geometric theory
involved is as follows: The equation
yo - y. _ b
Xi — .Vi a
expresses the equaUty of the slopes of the two lines
PiPi and OP respectively as shown in Fig. 66.
This equation above may be written in the determi-
nant form
and another equation
a
b
Xi
?'i
1
Xi
}'-
1
a
b
Xs
>'3
1
Xi
A'4
1
=
regarded as a simultaneous equation would then
express the fact that the lines PiPo and P^Pi were
parallel. If a and b are eliminated from the above
two determinant equations there results
^2 — yi yu — ys
Xi — Xi Xi — Xs
Consider now an equation in four variables which has
the form
^2
^4 - ^3
(41)
/2 - /, fi- U
This equation may be regarded as the result of elim-
inating h from the two determinant equations
1 A I
.A g. 1
h
g2 1
0(42)
\i
Consequently the straight line index of a diagram with
the defining equations
= /2
will be parallel to the index of a diagram plotted with
the same coordinate axes and with the defining
equations
X = fi y = g3
x = h y = gi
since both indices will have the same slope whenever
Equation (41) or the equivalent system (42) is satis-
fied by a set of values of the variables Zi- . . .Zi.
Example 44. — Lame's formula for thick cylinders may
be arranged to afford an example of the use of diagrams
with parallel indices. As usually given the formula is
= .J
S + P
IS - P
where the letters have the meanings given on the dia-
DIAGRAMS OF ALIGNMENT WITH TWO OR MORE INDICES
81
gram for the formula shown in Fig. 67.
The formula may be written
S + P _ D'-
S - P ~ 0+ d""
Diagram for
Lame 's Formula for Thick Ojlihders
D= External Diamef-er
d- Internal "
5- Stress in Inner Surface
P^ Infernal Pressure
Dand d m I he iame uni'fs
Sand Pin f he same iinlhs
A line from dlo D Ji parallel lo ahne from PfaS
Values of d
may at once be written without reference to the first
rows involving the auxiliary variable h but usually
scale factors will be needed and since the parallelism
of all lines must be preserved the scale factors for
\0
and may be regarded as the result of eliminating //
from the two simultaneous equations
1 1
1
h
p
-p
1
s
s
1
1
h
d-
1
Z>2
1
=
A set of defining equations for both these equations
defining equations of the second set must be propor-
tional to those of the first set thus
where
The method used in this example is general. The
underlying principle is the use of a projective trans-
formation that preserves parallelism. (See Appendix
B.)
hp
y =
-y^p
.r =
-M^
y=0
8,S
y =
t^xS
Ml
X =
M-2
y = i^iD-
DESIGiX OF DIAGRAMS FOR EXGIX BERING FORMULAS
Diagram for
Lams' ^6 Formula for Thick Qj finders
D= Exhrnal Dia meter
ct= Infernal Diamefer
S'Sfress In Inner Surface
P' Internal Pre&sure
D and d in the same anih
Sand Pin the same unlls
A line fromdlo D is perpendicular toa line from PfoS
DIAGRAMS OF ALIGNMENT WITH TWO OR MORE INDICES
In the present example the scales are all straight
and readily plotted. The same units must be used
for 5 and P such as tons or thousands of pounds:
also in using the D and d scales the same units must be
employed, as inches or centimeters. The indices are
shown set for P = 4,000, 5 = 10,000, d^ Q required
D. Reading of the diagram may sometimes be
made more convenient by providing a transparent
piece of celluloid on which parallel lines are
scratched.
It is now quite evident that Equation (39) of Article
18 may be represented also by a diagram with parallel
indices. In fact Equations (40) constitute the necessary
reduced determinant equations. This Equation (39)
serves also to show that where the scales of Equation
(42) reduce to straight scales supported respectively
in pairs on the same straight lines, the necessary theory
of the parallel alignment of the indices is merely the
geometry of similar triangles. In case the supports
of the scales are parallel the segments intercepted by
the two indices are proportional. Those who are
familiar with the use of homogeneous coordinates in
geometry will recognize that the presence of zero in
place of unity in the third element position of the
determinants of Equations (42) merely indicates that
the resulting diagram with parallel indices is a special
case of the diagrams of double collineation in which
the hinge scale has been removed to infinity.
It is possible to give equations of the form (42),
which includes Equation (40) as a special case, another
simple diagrammatic representation. In this repre-
sentation the key to the solution is by perpendicular
indices instead of by parallel indices and it has some
advantage because of the fact that two perpendicular
lines scratched on a piece of transparent celluloid
wUl serve as the two perpendicular indices and both
pairs of scales may be read at one setting. Bearing
in mind that It in both determinants of Equation (42)
represents the variable slope of the two indices it is
only necessary to replace unity in the first element
position of the second (or first) determinant by —h,
and h in the second element position of the first row
by unity in order that the slopes of the two indices
shall be no longer equal but one the negative reciprocal
of the other whenever they are to determine cor-
responding values of the four variables Zi . . . Z4.
The corresponding change in Equation (41) requires
that equation to be written
gi
h-h
gi- gi
(41a)
but the defining equations are selected from this
equation exactly as they were for Equation (41). In
other words the original Equation (41) may be repre-
sented by a diagram with perpendicular indices by
writing first the two determinant equations
=
1
h
gl
g-i
1
1
1
1
h
gi
-h
gi
-u
(42a)
as a check and constructing the diagram from the
defining equations
/l
y = gi
x = gz
y = -/s
72
y = g2
X= gi
y = -U
Example 45. — From the formula of the preceding
example another set of defining equations may be
written which will yield a diagram with perpendicular
indices as follows
byP
biS
biD^
y =
y = md^
and the diagram is shown in Fig. 68.
21. Diagrams for the Equation /i -f /o . • . +
/„ = 0. — Sometimes the Equation (38) in Article
19 has the simple form
/>+/.= /a +/4
(43)
and a diagram of double collineation with parallel
straight scales may be constructed by using as before
the auxiliary variable h and the two equivalent
equations
h^h-h
h + U
The corresponding determining equations with suitable
scale factors will then be (Article 12, Chapter III)
x= -5i
y = Mi/i
X = -
63
y = /is/a
X= b2
y = M2/2
X =
«4
.V = M4/4
X =
y - - , ■ ■
m -f- iJi
X =
Ma + /i4
where it
is necessary in
addition to the conditions
imposed
upon the constants m and b
in
Article 12 to
require a
so that
MiM2
M. + M2
M3M4
Ala + M4
(44)
in order that the same value of the auxiliary variable
h shall always determine the same point on the hinge
scale. The scheme of the resulting diagram is shown
in Fig. 69 and Example 42 illustrated its application.
84
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
If it is found more convenient the determinant
equations used for the Equation (43) may be written
2
-/i
/2
h
2
/4
Fig. 69.
The
h
V =
-M.A
X =
-82
y =
-M-./3
Si
>' =
M./2
x =
&2
y =
/X2/.
and a diagram with parallel indices designed,
defining equations with scale factors are
where
If 5i = 62 then of necessity mi = ^2, and the supports
of the scales coincide in pairs and the resulting diagram
is shown in Fig. 70.
f,\
/3
JL^'-""".
5
'"'"
■
-
~-
New types of diagrams may now be constructed
for the equation (7) of n variables
Introduce the auxiliary variables h\, hi, . .
and write the equivalent system of equations
/l+/2-/^l =
hx + /3 - //2 =
hi + /4 - //3 =
^„-3 +/„-!+/„=
In each of these equations except the first one and the
last one two auxiliary variables h enter and one value
of h must always have a support for each application
of the index in the diagram. For example h-i may be
determined from the first two equations written in the
form
1
h,
2
1
-h
1
1
h
1
1
h,
2
1
u
1
1
hi
1
and represented by a corresponding diagram with
parallel indices in which no support appears for h\
but in which one does appear for hi. Figure 71 shows
the scheme of such a diagram.
f,
f
h '■-
/
;
5- /
i h h -V fe
- ~ 7''"'
/1 + /2+/3 +
+/»
(7)
It is not necessary of course to use the principle of
parallel indices to construct the diagram for Equation
(7) as hinge scales can be used throughout, but is
frequently convenient to do so. The spacing of the
scales and the use of the scale factors are controlled at
each step by principles already laid down in this chap-
ter and in Chapter III.
Example 46. — The formula
JFD
for determining the actual time for turning a piece of
work in a lathe is shown in Fig. 72 and the symbols of
the formula are described on the figure. If written
log r + log 5 - log 0.2618 - log /^ - log L
- log Z? =
DIAGRAMS OF ALIGNMENT WITH TWO OR MORE INDICES
the formula is in a form similar to Equation (7). If between the scales for D and L. Rearranging it in
P, the product LD is introduced as an auxiliary variable the form
the equations are log P — log Z, = log D
log r + log 5 = log k = \og F + log P + log 0.2618 results in placing the D scale between the other two
log P = log L -\- log D scales thus permitting the auxiliary diagram for DL =
I-
W 60-
3
3
- - g
UiS
The first equation yields the following defining
equations
X = —26 y = fjilogF
x= 25 >> = M log P + M log 0.2618
C 50^ <7}
;§
Is
T -8
E'5
— 6 S
-53
Fig. 72.— Diagram for T = 0.2618 ^ t!.
P to be joined conveniently at the side of the diagram
for the first equation instead of superimposing it
upon the latter.
The defining equations for the second equation
referred to an origin on the Z> scale are
log 5
logr
The equation P = LD if plotted from its form as given
above, would require that the scale for P be located
x= -Si
x=
X = 5i
y = fi log P
>- = glogZ?
y = i^logL
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
Zl\
"m^ W.
lYtl
///
a.o- -i>»
1.8^ -IK
■g ^ -s
So.sL -2
■5 r 15
^ ■■ a
5- ^
VulbMofFactorfl/)
y
y
llih
L>».lul.
C."il
^
■'To""
^:e
IJ
078
0.007
052
37
111
100
OOOl
0.058
30
0102
oes
u
IS
0.076
38
0.122
067
<3
>'
0.080
0.130
110
070
10
0.050
0.103
0.000
100
0.1«2
0.113
IS
0.108
0.087
0.003
~' 1 0.'5«
0.12.
.,.W6
The Lewis formula for the strength of gear teeth is
ty^SPby; or, if h =KP; W=SPtKy.
A line from S \o ^V crosses the center line at the
point as a line from A" to P.
EXAMPLE.
Given: Cast Iron gear, velocity of teeth =150 ft. per
W = 3SOO lbs. "load carried by teeth.
^=0.11.
A- =4.
RequiredJ P
As shown by dotted lines, P >-l.(Me
DIAGRAMS OF ALIGNMENT WITH TWO OR MORE INDICES
87
Example 47. — In the Lewis formula for the strength
of gear teeth
W = SPb{y)
it is often desired to solve for P if & is taken as KP,
where
W = load in pounds carried by the teeth
S = stress in pounds per square inch, chosen with
reference to the velocity and material of the
teeth
P = circular pitch in inches
b = width of face in inches
K = a, constant, usually 2 to 6
(y) = a factor depending upon the number and
shape of the teeth.
In logarithmic form the formtila becomes
log IF - log 5-2 log P - log K- log (y) =
but if the auxiliary variable hi is chosen so that
it becomes
log hy - log 5 = log /? = 2 log P + log K
Figure 73 was then plotted from the defining equations
W
X = -2h y = li log hi = n log-
x= 23 y = -^logS
■iy)
-5
8
'logh
y= 2i, log P
y = filogK
The first two equations define a binary scale on the
line X = —25. The curve net drawn for this scale
may consist of the lines
X = -iy), y = Mlog|^3^J
plotted with the line x = —28 as a new Y axis.
(See Article 16, Chapter IV.)
From a table given by Mr. Lewis the values of the
velocity were added in proper correspondence with
the scale for S. Since the product of the diametral
pitch and the circular pitch is always t a scale for the
diametral pitch was added to the P scale.
The equation hi
could naturally be written
in the logarithmic form and two auxiliary (collinear)
logarithmic scales for IF and (y) added to the present
figure just as was done for L and D in Fig. 72 for the
preceding example. The range of numbers for (y) , how-
ever, is very small and it was found more convenient
to use the system of curves and establish a binary
scale on the line x = —25 as shown in the figure.
Problem 1. — Discuss the equation of Example 47 as an
equation of type (7) and construct the diagram resulting
when the binary scale is replaced by the required parallel
scales for IF and (y)-
Problem 2. — Reduce Bazin's equation for the velocity of
water to type (7) and construct a corresponding diagram.
Problem 3. — Construct a practical diagram for the
formula of E.xample 12 of Chapter II. Write useful
values of c and use four parallel lines.
Problem 4. — Gordon's formula for columns is
1 +
5a
600^2
where W = safe unit load, 2,725 to 14,450
a = coefficient, 2,000 to 3,200
I = unsupported length in inches
d = least dimension in inches
I
d
If the determinant equation for I and d is
1 h
1 - 600d^
8 to 40.
600 d^
1 + P - 1
P P
find the corresponding determinant equations for a and
PF and design a diagram of parallel alignment with suit-
able scale factors for practical use in steel design.
Problem 5. — In Section 15 of Chapter III were described
diagrams of alignment with a fixed point; show that
Equation (41) can be represented by a diagram with the
fixed point x = I, y = and two binary scales on the Y
axis.
Problem 6. — A reduction formula used in automobile
radiation tests is
62.4^1
H2
1 +
0.24^
where A
Di
Hi
OASAJ),
W "^ H\
(6,000 to 15,000) air, pounds per hour.
(80° to 115°F) mean temperature difference
(40,000 to 90,000) heat transfer observed,
B.t.u.
IF = (1,000 to 3,500) water, pounds per hour.
Show how to design a diagram of double alignment with
parallel scales for this formula and with the quantities
grouped in the pairs H,A, and W, K where K = ^'
CHAPTER VI
ALIGNMENT DIAGRAMS WITH ADJUSTMENT
Introduction. — There is introduced in this chapter
a new class of diagrams based upon fundamental
principles already developed. These diagrams
enlarge the number of types of equations to which the
principle of collineation is immediately applicable
and furnish also a general alternative method espe-
cially for those equations which cannot readily be iden-
tified with preceding types. It will moreover be
found that the types of equations already treated
may be regarded as special cases of the more general
types now to be discussed.
22. Equations in Three Variables. — It was shown
in Chapter III that an equation in three variables
/i
may be represented by a collinear diagram with three
scales when and only when it can be written in the
reduced determinant form
/3 g. 1
=
(8)
Since there is no immediate and satisfactory test for
this desired property of an equation or formula it is
usually necessary to resort to the principle of compari-
son with certain type forms and to the tentative rule
of Article 14. It is therefore desirable to free the
determinant form from restrictions as far as possible
and at the same time to preserve the principle of
collineation, or the use of the straight line index in
designing the diagram.
The fundamental property of Equation (8) is the
segregation of the variables /i, /o, /a, into rows of the
determinant.
Consider now the determinant equation of the type
/l2 gl2 1
/•23 g-23 1
fu ^31 1
=
(45)
in which the elements of each row are allowed to con-
tain at most two variables, which variables may appear
in more than one row.
Then the equations analogous to the previously
defined and much used defining equations will be of
the type
x=fi2 y = gi2
X =f23 y = g23 (46)
X = fii y = g3i
and each pair of equations will determine a curve
net, except in the case explained below in Article 23.
The three curve nets are shown schematically in Fig.
74 and it is seen that there appear two families of
curves for each variable z.
Call any set of three values of z which simultane-
ously satisfy Equation (45) corresponding values of z.
Such values of z necessarily determine values of the
functions / and g and hence by Equations (46) there
result three pairs of coordinates x and y which must
satisfy Equation (45) when substituted for/ and g.
But Equation (45) would then express the geometric
fact that three points in the three respective curve
nets are collinear.
In general, however, it would not be true that
any three points taken at random in the three
plotted curve nets and on the same straight line
would yield corresponding values of z attached to
the curves intersecting in pairs at the respective
points. It is here that the present theory departs
from the theory previously developed. What happens
in general is that there appear six values of z consisting
of three pairs of dissimilar values.
When corresponding values of z are used to select
three points in the three curve nets it is seen at once
that the same value of z is used to select a curve from
two different nets. If now one variable Zo is unknown
ALIGNMENT DIAGRAMS WITH ADJUSTMENT
=
(47)
it is evident that the index must be rotated about the
point always determined in one net by the known
values ZiZz until the same value of the unknown z-i
appears in the two remaining nets at the points of
intersection of the index with the known curves in
each net. In the iigure the line PR is rotated about
R until the points of intersection P and Q determine
the same value of the variable z^ when it is assumed
that Zi and Zs are known.
This then is the principle of collinear diagrams with
adjustment. There are many special cases and in not
a few no adjustment of the index is required because
the unknown variable appears but once. The method
is of great practical advantage especially if a given
equation is not adapted to the preceding treatment.
23. Special Forms of Equations.— Equation (45) is
a general form and is less frequent than the simpler
special cases. For example /31 and gzi may often
reduce to fz and gz respectively by skillful choice of
the elements of the determinant. The corresponding
determinant equation is then
/12 gl2 1
fiZ g2Z 1
A simpler form of diagram results from this equation.
Without scale factors the defining equations are
X = fi2 y = gi2
X= fiZ y = g23
x=Jz y = g3
The first pair of equations lead to the curve net,
F,{xy) = Zi Fiixy) = Zi
Similarly from the second pair is obtained the curve
net
Giixy) = Z2 Gz{xy) = Zz
and the third pair of equations determine a curved
scale for z with the support
S(xy) =
Still more simple is the equation
/2 g2 1
/23 g23 1
In the resulting diagram there wil
a curve net defined as follows
x=f,
x = f,
X = hz
Frequently a redundancy of variables in an equa-
tion may be reduced by the introduction of a param-
eter which is a simple function of two or more of the
variables whose values are always given, as was done
in Chapter III in the case of Example 33 for the mean
=
(48)
ill be two scales and
y =
gi
y =
g-2
y =
g2 3
hydraulic radius of trapezoids. This device will
be of advantage in several of the examples which
follow.
Example 48.- — As a first illustrative example of the
Equation (48) the quadratic equation
z^ + aiZ -{- a-i = Q
may be written in the reduced determinant form
-a, 1
2 1=0
z k 1
where k = is a parameter. The three variables
are ai, 2, and k. The defining equations are
x = —ai y =
X = y = z
X = Z y = k
To solve a quadratic equation by this diagram the
figure is entered on the X axis with the value of ai at
P and the index is then turned about this point until
the value of 2 read on the Y axis is the same as the
value read on the vertical line intersecting the index
aj
ai
is set for the two roots of the equation
z^ - 6.2z - 18.6 =
Another simple case of equation (45) is
/12 gu 1
fz gz 1 =0
fi' gi 1
Example 49. — As an illustration of this form of an
equation, the equation
21Z2 - 23 + Vl + Z2-VI + sr =
which can not by algebraic transformation be given
the form of Equation (8), may now be considered. It
may at once be written in the first determinant form
1
where it is crossed by y
In Fig. 75 the index
(49)
=
1 Vl+Zi
Zi Vl + 22- Z3
from which by adding columns one and two for a new
second column and then interchanging columns two
and three and dividing by the elements of column
three there results the reduced determinant equation
1 Zy 1
Vl+Zl
Z3
=:-=„ 1
S2 + Vl + Z2- Z2 -1- VI + 22 =
For a good arrangement of the drawing the vertical
90
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
unit may be taken one-tenth of the horizontal unit and equations leads to the equation of a family of parabolas
there follow the defining equations 2 _ ^^ n _ ^
2j for segregating the values of Z3. The finished diagram
y ~ u\ is shown in Fig. 76.
In constructing diagrams of adjustment it is desir-
able if possible to avoid binary scales as the use of the
10 diagram is complicated by their presence.
10
10 '
8
. 2
14^56
-i:.: 3: -
|I|M||||H]|I||
Jl
I
: ^
r -1^1-7
q+ ,
qq::;::::
::::::: it:::::::::::::::::::
1 III
"F::t:::x::::
-r +rTT — r--i
. \m
::::::::::!j:3::::::::::::±::::
1 1 I'M
-^'r4^--p^---L-
:.c
.1 _^ - --
- 4
i '
1 J
h^-t T 4-i
-
' ''i^'
i±::::::
, 1
i:i=|::::::::i:::g::ii:::::
ij
--4?-
::::::::::ii^^^::
HiS::::::-:T;:-:::::::::x::::::
1^4:;;-
: 2
fEMMttm™
::^r::::::::::
: '
■;;;;|-Sife;a..
H--%f:^ = = -5s44^--i^M
a
:::::l::ii;p.::::l.;
::::::::::lSS:::::::l-it:::
-t-i- -4.1 - . . i
'i
'" ! ■, r-; ''" ' ■ I ": ; :
3;;^x;::I&:::S::::;!|H
"1 —
=;r^^"^"~^^7rT
T^^^^'^sM:^^
^t
0, ::::::
--^Xi^px ' ' ' ''MI-
|f:|:::::::,,^_:::±^:^::::::::
|^;:::;:::::!::;::|::ili
::::X:-i:':^::::
1 1 1 -j-.-
i'
m
rt'::::::::::::: :::::::::::::::
tt :::::::::::::::: ::::::::::::::
is
: :::i
mm
- - +f 44+h +f++ +rf+ 4-W- -l-l-i- - -1-
rflifiMfflfflMIIIIIIIIIIIII
:
9
8 1
6 5^
2 J ^
1 8 S
Example 50. — The quadratic equation
z'^ -f aiZ -f a2 =
22 + VI + z^ 10 (z2 + Vl + 22')
The two scales for Zi are easily drawn. For the may be written in the form
curve net for Z2, Z3, since the right side of the first
equation involves only Zi, there result straight lines
parallel to the Y axis for the curves of that variable.
The elimination of the variable Zi between the last two
-1
aiZ
1
-z'
2
1
1
02
1
ALIGNMENT DIAGRAMS WITH ADJUSTMENT
91
which is another special case of Equation (45) . With
a horizontal unit twice the vertical the defining equa-
tions become
X = —2 y = Ci2
— 2^
a;= 3'= ^
x = 2 y = ai
appear at Q and R and the operation is somewhat
difficult to manage. In the figure the index is set
twice for the roots of the equation
z^ - 0.8z - 6.6 =
It will be seen that whenever any variable z appears
in but one row of the determinant equation of the
2.-S Z^.J0S_Z^6_
A binary scale is required on the line x = — 2 and
the variables ai and z may be segregated by setting
y = zx X = ai
the resulting diagram is shown in Fig. 77.
The roots are determined by turning the index about
the point P on the aj scale until the same values of z
form (45) , no adjustment of the index is necessary in
determining its value from the corresponding diagram.
It is to be noticed that the successive elimination
of two variables Zi and Z2 from two equations
X =/i2, y = gu,
will fail if the two functions fu and gu are not
DESIGN OF DIAGRAMS FOR EXGIXEERIXG FORMULAS
independent functions; that is to say in case one
is a function of the other. Sometimes this con-
dition plainly arises because both functions are
functions of the same combination of the two vari-
ables z.
results always the equation of a curve which is the
curve support for a binary scale.
In the above example it is seen that to every point
of the parabola there corresponds an indefinite number
of pairs of values z^z-i and to segregate them either
For example, suppose that
/i2 = ZiZo and gn = VziZq
then it is evident that both variables are elimi-
nated simultaneously from /12 and gn and that there
results the parabola y^ = x. Whenever both variables
are eliminated simultaneously in this way there
one of the defining equations may be used. Choosing
X = ZxZi, any simple family of cur^'es, except the
lines parallel to the Y axis, may be selected to define
one of the variables, say y = Zi ; substituting this
value of Zx in the last equation yields
X = yzi
ALIGNMENT DIAGRAMS WITH ADJUSTMENT
and all lines of the two systems which intersect on
the same ordinate {x = ZiZ->^ determine pairs of values
of Z\ and z g^i 1 I
fki gk, 1 I = (51)
An g^n 1
where the subscripts are allowed to take on any
two different values in pairs from the numbers 0, 1, 2,
3, 4, and 5, exhausts all possible cases of the equation
in five variables.
Problem 1.-
Discuss the equation
2^ 1
1 z
a I -02 1
100
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
T7m
wm^^^mw^^^^¥mi
I
'Kli
^^ ^ >'At
m
^^
^
^
^
ALIGNMENT DIAGRAMS WITH ADJUSTMENT
101
Problem 2. — Criticise the diagram for the quadratic
equation which results from the determinant form
2+Ci2 -1 1
a., 1 1
1
Problem 3. — Discuss the case of equation (51) where
Zi is unknown and when the curves for Zi are the same in
two curve nets.
Problem 4. — Discuss the possible diagram for the cubic
equation written in the form
z^ 1
1 -a^z z
1 fl3 -ai
Problem 5. — The law of cosines in trigonometry may be
written in the form
— a-
b^+c^
:(1 - 2b)
-cos A
Construct a diagram.
Problem 6. — Construct a diagram for a formula of
trigonometry which falls under the special form (Problem
3) of Equation (51).
Problem 7. — The formula
r, - r.
M
log.
of Fig. 56 is written
T,-T,
log.
for use in connection with exhaust steam feed water
heaters, where
Ti = temperature of the exhaust steam
T2 = temperature of the water leaving the heater
To = temperature of the water entering the heater
At = average temperature difference.
Show that it can be written in the reduced determinant
form
T,) 1
At
log. (T,
(Ti - Ti)
log . {T, - To)
{Ti - To)
{Ti - T,)
(T, - To)
and design a diagram with one curved binary scale which
has two systems of segregating curves that serve for the
three variables Ti, T2, To.
Problem 8. — If the formula of the preceding problem
is written
At
1 (r, - T2)
loge (7-:
1
Ts) log. (Ti-Ti)
(r, - To)
log. (Ti - To) log. (T, - To)
design the corresponding diagram.
Problem 9. — The annual sinking fund which will accrue
to 1 at the end of n years is given by the formula
1 = '
5^-(l + 0"-l
This equation may be given the determinant form
1
s-„\
i
1
i
1
1
(1 + i)"
a + 0"
1
Identify this with the last special form discussed in this
chapter and construct a diagram with suitable scales for
practical use for values of n between from 5 to 20 intervals.
Problem 10. — The accumulation of an annuity of 1 per
annum at the end of n years is given as the formula 5-;| =
; This equation has a determinant form
similar to the one of Problem 9. Construct a useful
diagram for va ues of i from 3 to 12 per cent.
APPENDIX A
DETERMINANTS OF THE THIRD ORDER
Definition. — The square array of nine numbers with
two vertical bars
rtl
bi
Cl
aa
b2
C2
03
63
Ci
is a convenient symbol for the expression,
a\bnCi + bic^az + C\a-ibz — C1M3 — aiCobz — biaiCz (1)
and is called a determinant of the third order. The
separate letters are called elements. The elements
in a vertical line form a column and those in a hori-
zontal line a row. The expression (1) is called the
expansion of the determinant. The elements aibic^
form the principal diagonal of the determinant and
the elements Cib-ia^ the secondary diagonal.
Expansion or Development of Determinants. —
When the determinant A is given, the expansion (1)
may be obtained as follows: Rewrite the first and
second columns to the right of the determinant.
The diagonals running down from left to right give
the positive terms. The diagonals running down from
right to left give the negative terms. Whenever
negative elements are present care must be taken in
determining the sign of each term in the expansion.
SIMPLE PROPERTIES OF DETERMINANTS
I. When all the elements of one row or of one column
are zero the value of the determinant is zero. This is
proved by observing that each term in the expansion
contains as factors one and only one element from
each row and each column.
II. // all the terms in a row or in a column are multi-
plied {or divided) by the same number K, the value of the
determinant is multiplied {or divided) by K. The
reasoning is the same as for I. In particular if
A" = — 1 the sign of the determinant is changed.
III. // the rows of a determinant are changed into
corresponding columns the determinant is unchanged.
Thus
IV. // two rows or columns of a determinant are
interchanged the sign of the determinant is changed.
This property may be proved for adjacent rows by
determining the change in the expansion due to inter-
change of corresponding subscripts. Repetition of
this process will extend the result to any two rows.
By virtue of III the result is true for columns.
V. If a determinant has two rows or columns identical,
its value is zero. If we interchange two rows we
obtain by IV —A, but since the interchange of identical
rows does not alter the determinant we have
ai
61
Cl
ai
ao
as
ai
62
Cl
=
bi
b.
63
az
^-3
C3
Cl
C2
cz
that is
A =
2a =
VI. // one row or column of a determinant A has as
elements the sums of two or more numbers, A can be
written as the sum of two or more determinants.
Thus
1 ai + a/ + a/
bi
Cl
Ol
bi
Cl
Oi
bi
02 + a-2 + ai'
b2
C2
a.
62
C2
+
a/
b2
\ 03 + 03 + Os"
b3
C3
03
^-3
C3
10.
5
03
b3
104
DESIGN DIAGRAMS FOR ENGINEERING FORMULAS
VII. The value of any determinant A ^5 not changed
if each element of any row or column multiplied by any
given number K be added to the corresponding element
of any other row or column.
By II and VI
b, + A'63 ^'2
c\ + Kci Co
Special Properties.
from II that
02 fls
62 ^-3 +
Ci Ci
Oi
as
62
b, =
Ci
C3
0-2 03
bo is
Co C3
+
-It results, by V immediately
b, fi I
bo Co =0
bz C3
=
provided that Ci, C2, C3 are all different from zero. A
column of unit elements may then always be intro-
duced into the equation A = 0. For even should a
zero appear in every column, by using VII a column of
elements all different from zero may be obtained and
by using IV, this column may be given the third
position. Finally the determinant may be divided
by the elements of the third column.
In the construction of engineering diagrams one of
the fundamental operations is to write certain given
formulas of three variables in the determinant
equation form.
C3
MxUtiplication of Determinants of the Third Order.
The product of the two determinants of the third
order A and Ai, is a determinant of the third order as
follows :
OiWi + bini + Cil
a^mi + btHi + f2l
flsWi + 63M1 + C3I
61
fl
nil
Wi
1
62
Co
W2
«2
2
63
Ci
mz
«3
3
O1W2 + bini + Ci2
atnio + bojto -\- co2
a^mo -\- bsno -f C32
aiW3 + bins + Ci3
02^3 + boHs + CoS
a^mi + 63M3 + CsS
To prove this result it will be sufficient to actually
carry out the expansions and multiplications. A
further proof is given by L. G. Weld "Theory of
Determinants," Chapter VI and in any work on
determinants.
A working rule for multiplication may then be
stated thus : Connect by plus signs the elements of each
row in both determinants. Place the first row of the
first determinant upon each row of the second in turn
allowing each two elements as they touch to become
products. This is the first row of the product. Perform
the same operation on the second determinant with the
second row of the first to form the second row of the
product, and again with the third row of the first
determinant to obtain the third row of the product.
Note that the product (by virtue of III) may also
be obtained by using columns instead of rows.
APPENDIX B
THE PROJECTIVE TRANSFORMATION
Definition. — A geometric transformation in the
plane is an operation which replaces one geometric
configuration by another. A one to one point trans-
formation replaces a given point by another uniquely
determined point. Under the operation of such a
transformation the locus of a given variable point
P{xy) is transformed into, or replaced by, another
definite locus traced by the corresponding point
Equations of a Transfonnation. — Usually a relation
may be written between the coordinates of a given
point {xy) and those of the transformed point (xiyi).
Such equations are called the equations of transforma-
tion. Thus for example, if every point P of the plane
is pushed outward by an impulse from the origin so
that the distance OP is doubled, there results obviously
^1 = 2x
y, = 2y
for the relations connecting the coordinates of the old
and the new points. Such a transformation is called
a dilatation. By it, circles about the origin are
transformed into circles with radii twice as great.
Straight lines remain straight, etc. A more general
dilatation is given by the equations
xi = nx yi = ixy
where ;u is any constant whatever.
Kinds of Point Transformations.— Obviously if a
pair of equations
xi = ixy) yx = rPixy) (1)
are written at will, they may in general be regarded as
establishing geometrically a relation between the
points (xy) and the (transformed) points (xiVi) which
may be computed whenever values are assigned to
X and y; i.e., whenever any point P is selected. Now
the properties of the resulting geometric transforma-
tion will depend upon the nature of the functions
{xy) and 4/ {xy) of Equations
(1) are Hnear fractional functions with the same
denominators.
the relation
The constant coefficients must satisfy
oi bi Ci I
a, b, c. I ^ Qi (6)
03 ^3 C3
otherwise the coefficients are not restricted. The
determinant of the inequality (6) is called the determi-
nant of the transformation. It is to be observed that
the transformation Equations (2) above result from
Equations (5) if 03 and bs are chosen equal to zero and
C3 equal to unity
Properties of the Projective Transformation.—
There are two principal properties enjoyed by this
transformation which are important for the work
needed in this book. First, the transformation pre-
serves straight Hnes. Solving Equations (5) for x
and y it is found that
_ ^1^1 + Biyi + Ci ^ A2X1 + Bjyi + Ci
^ ~ Azx^ + B3yi + C3 ^ A3X1 + B3yi + C3
where A, B, C, are expressions involving a, b, c, only
and consequently a straight line
ax+by + c =
becomes
a{AiX, + B,y, + d) + b{A.x, + B.yx + C2) +
c{A3X, + B3y + C3) =
and collecting terms this equation has the final form
o'x + b'y + c = ^
where c' = aAx + bAi. + CA3, etc., and is consequently
the equation of a new or transformed straight line.
Second, the transformation may always be so
selected that any four points (no three of which are
coUinear) may be made to take any four (similarly
restricted) positions. This result is accomplished by
' Read 9^ "is different from."
selecting the suitable coefficients for the equations of
the transformation (5). To prove this property of
the projective transformation whose equations are
written in the form (5) above, assume that the four
points given are Pi, Pi, P3, Pi, with the coordinates
(wi«i), (nhni), (OT3W3), (w4«4), or more briefly, P,- with
coordinates WiH,, where i = 1 . . .4.
Let it be assumed now that these four points are to be
tiansformed into the four new positions whose coor-
dinates are respectively /),g, (j = 1 . . . 4) . There
will result immediately from Equations (5) eight
necessary relations of the form
_ fliWi + biUi + ci _ ajnii + b^nj + Ci . .
^' ~ flaw. + bini + C3 ^' " OsWi -f is".- + Cs
which the nine coefficients a, 6, c of Equations (5) must
satisfy. All these equations will be homogeneous in
the quantities a, b, c which are to be found. There is
required one more relation or equation to completely
determine the nine constants and that relation may
be selected at will and of course will be so chosen as
to reduce the labor of solving the equations as much as
possible.
Example. — Suppose that the four given points are
those with the coordinates (0 0), (0 -1), (-1 0),
(—1 1) and that it is desired to develop a projective
transformation which will transform those four
pointsinto the four points (0 0), (0 1), (1 0), (1 1),
respectively. Choosing for convenience C3 = 1 the
eight equations resulting from Equations (7) upon
substitution of these coordinate sets are :
0=0
=
1 =
1 =
C2=0
-b, + c,
-b3+ 1
-b..+ci
^- -b3 + l
-ai + ci
-03+ 1
^- -03+1
— oi — bi
-03 — b3
+
+
Ci
1
-02 - bi + C2
^- -a3-b3+ \
This set of equations reduces at once to the set of four
linear equations
— fll + 03 = I
-b, + 63 = 1
-Oi + 03 + bz = 1
-b2 + a3 + b3= 1
The solutions are Oi = 1, 03 = 0, b-i = — 1, 63 = 0.
Consequently the equations of the transformation (5)
become
x = -X y = -y
The important application of the above principle
in the present volume arises in connection with the
selection of scale factors in the design of the necessary
diagrams. Suppose in connection with a nomogram for
an equation of three variables Zi, Z2, Z3 it is desirable to
APPENDIX B
107
move the 2i and the Zn scales from the two parallel
straight lines x = —I and x = \ to the two lines
X = — 5i and x = &2 respectively and at the same time
to introduce the scale factors mi and ix-i so that the two
parallel scales will then have the defining equations
X = -by. y = Migi
X =^ bn y = //2g2
respectively. In order to determine once for all what
will be the nature of the change in the defining equa-
tions for the third scale it is only necessary to observe
that the change determined by the choice of the two
transformed scales above is sufficient to determine a
projective transformation. The four points (1 0),
(1 1), (—1 0), (—1 1), have been transformed respec-
tively into the four points {bi^) Xhni) ,{— 5iO),(— ^-ly-i)-
Following the procedure above there result the eight
equations
ai + ci
' az + f3
a. + c.
az + cz
ai-\-hi-\-Ci
' az + hs + c
02 + &2 + C2
"' 03 + 63 + Cz
-ai + ci
-az + Cz
-ai + bi + ci
'' ~ -az + bz + cz
-at -\- bi + Ci
"' ' -az + bz + cz
Selecting for convenience the
arbitrarily chosen relation
az + bz +
cz= 1
the solution of the nine equations yields
fiibi + M25i j^
^1^2 — M25l
U Cl = „
2ai2
02 = b,=
M2 C2 =
Ml ~ M2 ,
az- 2^^ bz-
(. Ml + M2
and by substituting these values in the Equations (5)
above there results for the necessary projective
transformation
(mi52 + ti2bi)x + (/ii62 — ^l■ibl)
y =
- fii)X + (mi + M2)
2MlM2y
(mi — M2)a; + im + M2)
There is a convenient modification of these equations if
(nibn — Ilibi) =
and also another convenient simplification if 5i =
62 = 5 and (/xi^i — m-jSo) 5^ from which results
_ ( mi + Ma)^ + (mi — M2)
(mi — M2):«; + (mi + M2)
_ 2miM2>'
(mi - M2):*; + (mi + M2)
These are the equations of Chapter III numbered
(26)-.
The equations developed for the introduction of
scale factors into the equations numbered (10) and
(13) in Chapter III may be obtained by the method
here used.
In the text of the present volume the supplementary
transformations that have been introduced to better
the design of diagrams are all very simple and similar
transformations can usually be selected by inspection ;
it is desirable to point out, however, that in the design
of important nomograms for permanent service the
use of the four point method here described may be
the only way that the necessary transformation can
be determined.
It is obvious from Equations (5) that if a point P
with the coordinates {m n) is to be transformed to
infinity it is only necessary to choose azbzCz so that
aztn + bznix + Cz = 0, since then the values of both
Xi and yi will be infinite. The equations of trans-
formation numbered (2) above are the most general
equations for the projective transformation which
preserves parallelism of straight lines. Such projective
transformations are called affine transformations.
The Projective Transformation and Determinants.
The condition that three points {x'y'), {x"y"),
{x"'y"') shall lie on a straight line is conveniently
expressed in the form
:' V 1
■" y" 1
:'" y'" 1
If a general projective transformation is applied to all
the points in the plane the three points in question go
over into three new points which are collinear also.
Substituting for x' and y' , etc., in the above determi-
nant the corresponding values obtained above from
Equations (5) in terms of x\ and y'l, etc., there results
A 3X1' + Bzyi' + Cz AzXi + Bzji + Cz
AxXx' ■\- Bxyx' ■\- Cx A2X1" + B2yx" + C2
Axx' + Bxv' + Cx Aox' + B.y'
Axx" + Bxy" -f- Cl A2X" + B-iy"
Axx'" -{■ Biy'" -f Cl Aix'" -\- B^y"
AzXi" + Bzyi" + C3 AzXx" + Bzyi" + C3
^1X1'" + Bxyx"'+ Cx A2Xx"' + B^yi'" + C2
AzXxx"'+ Bzyx'" + Cz AzXx" + Bzyx" + C3
and multiplying this equation by the three denomi-
nators of the elements of the first column, there
results
-f C2 Azx' + Bzy' + C:
-t- Co Azx" + Bzy" + C:
'+C2 Az^'" ^- Bzy"' + C.
108
DESIGN OF DIAGRAMS FOR ENGINEERING FORMULAS
which by the multiplication law of determinants is
1
Bi
Ci
Xi'
y\
1
Bi
c.
X
Xi"
y"
1
3
B^
C^
Xi'"
y\"
1
Since now the first determinant factor does not vanish^
the second must and hence the condition that the
transformed points lie also upon a straight line
appears at once as a result of their original coUinear
position.
If now it is desired to write the above equa-
tion in terms of the original coordinates there
follows:
Xx
y\
1
Cl
&1
Cl
X\"
yi"
1
=
^2
^2
Cl
Xx"
y'i
1
^3
bz
cz
Which may be proved by the laws of multiplication of
determinants and Equations (5).
There results then the Working Rule:
To apply a projective transformation to the variable
elements of a determinant multiply the determinant by
the determinant of the transformation. This rule may
be used as a check in the practice involved in this
voLume. The important principle, however, which the
above rule brings out is in connection with the manip-
ulation of first determinant equations to reduce them :
Every manipulation of a determinant equation by the
laws of determinants is equivalent to applying to
its elements a projective transformation. In other
words every change in the first determinant form has
corresponding to it a geometric change in the plane.
y' 1
y" 1 =0
y'" 1 I
be shown to be true from the condition VI.
INDEX
Accuracy, choice of units for, 72
of a scale, 8
Adjustment diagrams, alignment, 88
AfBne transformations, 107
Air compression, intercooler pressures, 64
horsepower, 33
Alignment diagrams, definition, 35
Amortization factor, bonds, 75
Anamorphosis, 13
Angular distance of celestial body from meridian, 74
Annuity formula, 101
Automobile engine rating, 43
radiation reduction formula, 87
tractive resistance, 64
Auxiliary variable, 76
variables, hexagonal diagrams, 30
parallel indices, 84
Barometer readings, corrected, 39
Bazin's formula, 69
Belt tensions, 30
Binary scale, curve support for, 92
defining equations for, 66
definition of, 65
methods of plotting curve nets, 74
Binary scales, segregation of variables, examples of, 97, 99
use of in diagrams with adjustment, 90
Biquadratic equation, 99
Bond formula, 72, 75
Boussinesq's formula, 30
Brauer's method, 17
Canals, mean hydraulic radius of trapezoidal sections, 55,
64, 75, 76
Celluloid, use of transparent with indices, 21, 83
Change of scale factor, 2
Chezy's formula for flow of water in open channels, 78
Chimney formula, 49
Choice of Scale Factor, 7
Circles and straight lines, diagrams of, 30
Circular segment, approximate area, 18, 64
exact area, 39
mean hydraulic radius of, 64
Coefficients as scale factors, 28
Collinear nomogram definition, 35
points, 36
CoUineation of three points, 35
Colors used to simplify diagrams, 74
Column formula, Gordon's, 87
in a determinant, 103
Combinations of simple and collinear diagrams, 80
of simple diagrams for four variables, 80
Common normal, system of parallel lines, 21
Complete cubic equation, 67', 75
Compound interest, 30, 64
Compressed air, mean effective pressure, 30
horsepower, 33
Condenser tubes, 34
Cone pulleys, open belt, 97
Cooper's formula, 32
Coordinates of ray center, 3
Corresponding values of the variable, in diagrams of
adjustment, 88
Cosines, law of, trigonometry, 101
Cubic equation, 51
complete equation, 67, 75
general equation, 94
Curve nets, 65
method of plotting one set of curves by intersections, 74
when one set of curves become straight lines, 74, 75
with adjustment, 88
Curve support for a binary scale, 92
Curved binary scale, 67
scales, 49
Curves of translation, 97
transformed to straight lines, 13
Defining equations, definition, 35
for the binary scale, 66
Derivation of new scales, 1
Deriving a scale factor, method of, 47
Development of determinants, 103
Determinant of the transformation, 106
Determinants, effect of manipulating, 38
example of how to set up, 49
of the third order, 103
properties of, 103
Determining unknown exponents from empirical formula,
14
Diagonals, of determinants, 103
110
INDEX
Diagrams, alignment, with two or more indices, 76
collinear, with two parallel scales and one curve net, 66
of adjustment, when no adjustment is necessary, 91
of ahgnment with one fixed point, 57
with adjustment, 88
with three parallel straight scales, 36
Dilatation, projective transformation, 105
Displacement of parallel scales, 43
Double alignment diagrams, 76
coUineation, 76
graduation, points of, 65
Hexagonal diagrams, 28
for n variables, 30
Hinge scale, 76, 83
Horizontal formula, stadia distance, 53
Inductive reactance, 34
voltage, 28
Intercooler pressures, in air compression, 64
Earthwork computations, 36, 64
Eckblaw's silo formula, 30
Elementary diagrams, 9
Elements, of a determinant, 103
Ellipse, perimeter of, 30
End areas in earthwork, 36, 64
Expansion of determinants, 103
External of two tangents, 14
Equal ordinary scales on both axes, 11
Equation in three variables, eight general forms, 94
Equations in more than three variables, diagrams with
adjustment, 94
in two variables, 5
Equilateral hyperbolas, 75
First determinant form, definition, 36
rule for obtaining, 51
Five variables, equations in, 99
Flow of water {see also Francis, reclamation service), 25
Bazin's formula, 69
Chezy's formula, 78
Kutter's formula, 67
Flynn's modification of Kutter's formula, 67
Four parallel straight scales for four variables, 61, 64
Four-point method in projective transformations, 106
systems of parallel straight lines, 25
variables, diagrams for, with parallel or perpendicular
indices, 80
with two parallel scales and a net, 66
Francis' weir formula, 11, 21, 47, 57, 62
Friction head, flow of water, 21
factor for steam turbine nozzles, 25
Function scale, 1
Kutter's formula, 67
Lame's formula for thick cylinders, 80
Law of cosines, trigonometry, 101
Length of a scale between limits, 7
Length of open belt, stepped pulleys, 94
Lewis formula for gear teeth, 87
Limiting values of the variable, 8
Locus, of transformed points, 104
Logarithmic cross-section paper, 14, 57, 60
transformation, 38
Log tan z from log z scale, 5
M
Machining time, 84
Manipulating determinants, 103
Manipulation of determinants equivalent to projective
transformation, 108
Mean efifective pressure of expanding steam, 14
hydraulic radius, circular segment, 64
trapezoidal section, 55, 64, 75, 76
temperature difference by Greene's formula, 97
log formula, 63, 101
Modulus, 1
Multiplication diagram, 62
of determinants, 104
N
New scales, methods of deriving from given scales, 1
Nomogram, 9
Non-parallel scales, 49
Normal form, 21
Gear teeth, Lewis formula for, 87
Gordon's column formula, 87
Grashof formula, 30, 43
Greene's formula, mean temperature difference, 97
H
Heat drop, adiabatic, 25
transfer apparatus, mean temperature difference,
Oblique axes, use of, 39
scales, 49
One fixed point in diagrams of alignment, 57
Open belt formula, 94
Ordinary scale, 1
Orifice, flow of steam through, 43
rectangular, flow of water under low head, 62
INDEX
111
Parallel indices, 80
straight lines, 21
Parameter, 9
Parametric equations, 35
Partial differentiation, 36
Perpendicular indices, 80
Pitch, gear teeth, circular and diametral, 87
Pivot scale, use of, 76
Plotting curves by dividing ordinates, 9, 13
function scale from graph of curve, 5
Point transformations, 105
Points of double graduation, 65
Principal diagonal of a determinant, 103
Projective scale, 2
transformation, 104
effect of, 38
Properties of determinants, 103
the projective transformation, 106
Purchase price of bonds, 72
P7" = constant, 17, 60, 64
Segment of circle, approximate area, 18
exact area, 39
Selection of curves in net of binary scale, 69, 74
Shaft diameter, to transmit given h.p. and r.p.m., 17
for combined bending and twisting, 31
Simple Cartesian diagrams, 9
diagrams, combined for four variables, 80
straight line diagrams, 13
Singular point, 69
Sinking fund formula, 101
Special forms of equations, for diagrams of adjustment,
properties of determinants, 104
Stadia formulas, 53
Steam, Unwin's formula for flow of in pipes, 77
Stepped pulleys, open belt, 94
Straight lines and circles, diagrams of, 30
scales, three parallel, diagrams for, 36
two parallel one oblique, diagrams for, 43
Superimposed diagram, 76
Supplementary transformations, 106
Support, of binary function scale, 65
Quadratic equation, 12, 49
diagram with adjustment
Quartic curve, 55
90
Radial lines, how to plot, 16
line systems, 14
Ratio of expansion, 16
Ray center, 3
Reciprocal scales, methods of avoiding use of, 50, 97
Reclamation service formula for flow of water, 21, 28, 61,
62
Rectangular orifice, flow of water, low head, 62,
Redemption price of bonds, 72, 75
Reduced determinant form, 36
Richardson's equation, 34
Riveted joints, 10
Row, in a determinant, 103
Rules concerning determinants, 103
Taylor's formula for tool pressure, 30, 39
Thermionic current, 34
Thermodynamics equations, 64
Thick cylinders, Lame's formula for, 80
Third order determinants, 103
scale, scale factor, 38
Three curve nets, collinear diagrams for, 74
parallel straight scales, 36
at unequal distances, 38
straight scales, no two parallel, 49
variables, most general form of equation, 93
Tool pressure, Taylor's formula, 30, 39
Torus, volume of, 39
Transformation, projective, 105
Transformations, affine, 107
Trapezoidal canals, m.h.r., 55, 64, 75, 76
Turbine nozzle, 25
Two variable equations, 5
U
Unwin's formula, flow of steam in pipes, 77
Scale factor, definition, 1
choice of, 7
development of, for three parallel scales at unequal
distances, 38
for two straight scales at right angles and one curve
scale, 55
for Cartesian diagrams, 9
Scale of log z, 1
of Vz, 1
Scales determined by curves on parallel straight lines, 67
Secondary diagonal of a determinant, 103
Vertical distance, stadia, formula 53,
Volume of frustum of cone, 30
W
Water, flow of, Kutter's formula {see also Francis, reclama-
tion service), 67
Bazin's formula, 69
Wind resistance of automobiles, 28