CORNELL UNIVERSITY LIBRARY Cornell University Library TC 175.K97 The new formula for mean velocity of dis 3 1924 021 435 809 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924021435809 THE NEW FOKMULA MEAN VELOCITY OF DISCHARGE MYERS AND CANALS. For obtaining the value of any one of the fo representation Apply a straight-edge to the three points, indies line thns obtained will intersc The four dotted lines illustrate t 1st Case. Given n = 0-010, J - 0-0< 2nd Case. Given n = 0-017, J =0-01 » 3rd Case. Given E = 200, e = 45, 4th Case. Given E = 15, c =16, Wots. — Values corresponding to English measures are a = 1-81*.-*- VabbL&s ~df~M/,,th ~~SK © o o i Y i 1 r- 'i i'm 1 ! 'i CO *3< rt cp t> ^ ffi o T © O _i_ 8 8 CHAPTER I. L. The New Formula of D'Akcy and B/zin and Humphreys and Abbot, for determining Mean Velocities of Discharge of Rivers and Canals. In recent times two extremely valuable works on hydraulics have been published, which have thrown a /new light on one jf the most important branches of that science, the laws of motion of water in rivers and canals. They/are, the 'Recherches Hydrauliques' of DArcy and Bazin, l$g5 ; and the 'Theory of Motion of Water in Rivers and Canals,' by Captain Humphreys and Abbot, 1867, the latter of which was trans- lated into German by Grebenau. These two works far surpass all others yet written that treat on this branch of hydraulics. Both of them bring forward a very large number of results af experiment and observation that have been most carefully obtained and deduced, and are justified by the highest authority; both of them also propose new formulae, which essentially differ, not only from each other, but also from all previous formulae of Prony, Chezy, Eytelwein, St. Venant, &c. ; this difference is the more striking, as the whole of these formulas have been based on carefully conducted observation and experiment. In explanation of tins, and with reference to the two modern formulae, we would notice that the two latter are results deduced from observations made under extremely different conditions ; those of the French engineers, D'Arcy and Bazin, having been taken on small canals, and those of the American engineers, Humphreys and Abbot, on 2 THE NEW FORMULA FOR MEAN VELOCITY. [CHAP. I. • very large rivers, like the Mississippi. Both formulae are correct within certaiD limits, but neither can have any pre- tension to general application, as the former of the two is inapplicable to large rivers with low inclinations, and the latter'to small discharges with greater fall. To decide which of these two formulas is preferable and more useful generally, and to enable us to base our decision on practieal considera- tions, we have made a collection of all known observed results that bear on the subject, together with some that are of special interest from having been conducted on streams of extremely high inclination, and have compared these results with those deduced from the measurements by the formulae. 2. The previously accepted Formula. The well-known formula of ordinary use, v = c .J rs, in which ~ v is the mean velocity of discharge, VV> WXh^m" . r is the mean hydraulic radius, or the quotient of the water section by the wetted perimeter, s is the inclination of the water surface, and c is the experimental coefficient, is that of Chezy and Eytelwein ; it was assumed that it gave correct results under all cases and conditions of inclina- tion and dimension, a fallacy that vanished only after a long time, with the discovery that the coefficient c was not a con- stant but a variable quantity. In the formulae of De Prony and Weisbach the coefficients c vary with the velocity of the water, but their results differ but slightly from those afforded by the former formula with the coefficients of Eytelwein. More recent researches have however shown that the variation of the values of c depends on very varied influences, and can be more correctly determined and expressed than by simply treating it as dependent on the variation of the velocity v. ohap. i.] flow in open channels. 3 3. The New Formulae of D'Aecy and Bazin. In the ' Recherches Hydrauliques ' of D'Arcy and Bazin, 1865, the coefficients c are made to vary, not with the velocity, but with the values of r, the hydraulic mean radius, and with the conditions of the section. These con- ditions are classed in four categories, which, naturally, do not include every degree of roughness of the wetted peri- meter, but are merely averages assumed for convenience in determining the coefficients. D'Arcy and Bazin have deduced their formulae from their own new experimental observations on artificial canals, 2 metres wide, 1 metre deep, and about 600 metres long, whose beds and banks were constructed of various different materials, as well as from other observations on rivers and canals. They gave various forms to the section of their canal, and thence discovered that the semicircular form was that most favourable to a rapid discharge, while they also demonstrated that the form of section was not by any means the most important influence on the velocities and discharges of open channels. 4. The New* Foemula of Humphreys and Abbot. -T-he-Am grican engineers, Humphreys and Abbot^ propQggd. an entirely new formula, based on a vast number of frequently repeated measurements of discharge on the lower Mississippi and its affluents. At page 138 of Grebenau's translation of their work, we find that the extremely inge- nious formula deduced by them for velocity is based on the following law, established by their own experiments: That the velocities at different depths below the surface in a vertical plane vary as the abscissae of a parabola, whose axis is parallel to the water-surface, and representthe maximum Tvelocity ; and thus, the position of this axis once determined, the velocity at any depth in this vertical plane can be obtained b 2 4 THE NEW FORMULA FOE MEAN TELOCITY. [CHAP. I. from the parabolic curvature. This law is also confirmed by the experience of D'Arcy and Bazin. Since, therefore, this new formula is deduced from observations on large rivers of low inclination, and has also been proved to hold good for rivers and small streams with small inclinations, it becomes important to discover whether it is also correct for discharges of high inclination. Should that be the case, it will then have a claim to general application. 5. Practical Examination of the New Formula. The collection, given on the following page, of observed measurements of discharge on the Wildbachschalen, near Lake Thun, under conditions of very high inclination of channel, affords a ready answer to this important question, without entering into unnecessary details or lengthy discus- sion. The data and dimensions there given, the observed velocities of discharge, and the velocities calculated according to the well-known formulae of Chezy-Eytelwein, of D'Arcy and Bazin, and of Humphreys and Abbot, comprise every- thing that is required. Besides those above mentioned, we have collected another series of measurements of discharge in Switzerland, that is also applicable to this question; some of them are from streams on the Jura series by Professor Trechsel, some from well-maintained river channels in Canton Graubiindten by Oberst La Bicca, and others from the Linth-and-Escher canals by Engineer Legler. The whole are eighty-five in number. The comparison of the observed with the cal- culated results shows that for steep inclinations the American formula gives far too small velocities of discharge, and that the formulas of D'Arcy and Bazin give results which are generally much better, and in some cases very good. We hence infer that the American formula has no claim to OHAP. I. J PLOW IN OPEN CHANNELS. general application, and would be much improved by the introduction of variable coefficients. The conclusion is also forced on us, that any formula that would possess any adequate claim to universal utility must necessarily be very complicated, and hence unsuited to practical requirements, • while it appears at the same time that if a good general formula, somewhat resembling that of D'Arcy and Bazin, be adopted as a basis, and a collection of correct coefficients be applied to it, every purpose will be sufficiently served. It must, however, be noticed that any such formula must be applicable to all ordinary hydraulic conditions, and that the choice therefore lies between the old general formula,Twhich admits of adaptation to those of D'Arcy and BazinJ apd the new American formula. Table of Observations on the Wildbachschaxe. Length. r Inclination or Fall per 1000. Observed Velocity. Calculated Velocities. Dates. Chezy- Eytel- wein. D'Arcy and Bazin. Hum- phreys and Abbot. G'riinnbachschale. 3rd June, 1867 27th"june,"l867 )> » M » Gerbebachscnale. 27th June, 1867 » » 11 'I )» " Gontenbachschale. 26th June, 1867 >» » ii » 800 1200 200 800 1200 200 100 100 400 100 100 400 600 400 600 0-394 0-385 0-361 0-657 6-644 0-591 0-197 0-375 0-328 106-775 99-270 82-85 106-775 99-27 82-85 237-3 185-2 167-9 137-5 111-7 46-425 42-350 46-425 42-350 13-97 13-54 12-00 19-48 18-58 15-79 10-31 9-58 9-33 9-05 8-61 11-15 10-05 10-66 9-60 19-07 18-18 16-08 24-63 23-51 20-57 20-10 17-76 16-91 15-30 13-79 12-26 11-71 11-48 10-96 13-68 12-93 11-17 20-69 19-65 16-77 11-20 9-90 9-42 8-53 7-69 8-64 8-25 7-70 7-36 3-50 3-37 3 11 4-56 4-42 4-04 2-97 2-78 2-71 2-57 2-43 2-72 2-65 2-53 2-47 Summation of \ results . . / Ratios ... •• 181-70 1-00 252-31 1-39 173-58 0-96 46-83 0-26 6 the new fobmuxa for mean telocity. [chap, i. 6. Examination of the Old-established Formula and the New American one, with the View of apply- ing Series of Coefficients to either of them as a Basis. The 'old formula, \ = c V rs, whose terms have already been explained, may be said to assert the general law that the mean velocity of discharge at any section varies with the square root of the product of the sine of the inclination and the mean hydraulic radius. The value of the experimental coefficient e may be shown to vary greatly ; although fixed as a constant quantity 92 • 975 by Eytelwein, it has yet been proved by the experiments of D'Arcy and Bazin to vary between 5 and 100, while the results on the Mississippi give it not less than 256 as the highest limiting value. The new American formula, expressed in Swiss feet, is v = VO-008 299 6 -f [229-06 ^ V* - 0-090 716 Jb\\ where Jr+ 1-524 i> + W W U t5 U itv ' To simplify this rather complicated expression, Grebenau neglects the two smaller quantities represented by the first and third terms of the equation, and reduces it to the form v = c V r i v' 8 . which may be thus verbally expressed : The mean velocity of discharge at any section is the product of the square root of the p rime radius^r quotient of the sectional area by the 4 whole wetted perimeter and j jr eadth of su rfaced rod- the h™ 9 fourth root of the inclination, «Hri&p li c d - b y an experimental ^J- coefficient. The introduction of the breadth of surface of the water section into the quantities composing this equa- tion, and the resulting substitution for v, the mean radius, of CHAP. I.] ^» FLOW IN OPEN CHANNELS. 7 J. a new term X» or prime radius, which is about a half of the former, causes a great alteration in the corresponding values of the coefficient. A still more important difference be- tween the American and the old formula is the introduction of the fourth root of the sine of the inclination into the basis of the formula, instead of the square root ; the law of incre- ment of a series of fourth roots varying greatly from that of a series of square roots. Hence, before deciding which of these two formulae is more suited to our purpose as a general basis, it is first necessary to determine whether mean velocities in similar sections and under corresponding inclinations of every degree happen to vary more exactly with the square roots or with the fourth roots. In order to decide this important point, we have selected, from the five hundred observed results given by D'Arcy and Bazin in the 'Recherches Hydrauliques,' thirty-three cases having different inclinations, but similar in other respects ; and from a collec- tion of about one hundred fifty observed results, made by ourselves, and taken from the work of Humphreys and Abbot, the collection of Grebenau, the observations of Trechsel, La Eicca, and Legler, as well as our own, we have selected fifty-two cases of similar results having different inclinations. In all we have chosen eighty-five, cases that are suited to the purpose, and have compared the observed velocities with the square roots, the cube roots, and the fourth roots of their inclinations. The results are that out of the first set of thirty-three cases, twenty-seven had their velocities varying more nearly with the square roots, five with the cube roots, and one with the fourth root ; and out of the second set of fifty-two cases, thirty cases had their velocities varying more nearly with the square roots, nine with the cube roots, and thirteen with the fourth root. It may also be observed, that the whole of the fourteen cases in which the velocities vary more nearly with the fourth root are cases of extremely low inclination, being those of the Mississippi system, the streams 8 THE NEW F0EMULA FOE MEAN VELOCITY. [CHAP. I. of Grebenau, and one single case of D'Arcy and Bazin. We will hence conclude, that for most falls, with the excep- tion of those that are very low, like that of the Mississippi, the mean velocities in similar sections are more in accor- dance with the square roots of the sines of the inclinations, and that the simple and useful old-established formula v =c*/rs with variable coefficients not only gives good results, but is also in our opinion that most applicable to very varying conditions of inclination. Assuming therefore the general formula v = c*/rs as that most suitable to our purposes, the next matter is to obtain a series of coefficients that will be equally applicable to every degree of inclination that will occur in practice. We have, however, fruitlessly endeavoured to discover any law for the construction of any single set of series of co- efficients, that would apply both to the low inclinations of observation of the American, and to the high falls of the Swiss engineers. In plotting the coefficients deduced from these observed results as ordinates to abscissae representing the inclinations, we discover that the greatest values of the former correspond to the least values of the latter, and the converse, and that no mean curve could be drawn that would be applicable throughout. It is also necessary to remark that the coefficients obtained in the same way for the American formula show a persistent increase of value with the increase of inclination; a proof that that formula gives incorrect results in this respect. On plotting the former coefficients as ordinates to abscissae representing values of r, (the mean radius, jand similarly plotting the curve of the coefficients calculated according to the formulae of D'Arcy and Bazin, we find that they approximately correspond in cases having similar conditions of section ; a confirmatiou of the correctness of the formulae of these authors as far as this is concerned. CHAP. I.] FLOW IN OPEN CHANNELS. 7. The "Vabiation of the Coefficients c with the Inclination. Having thus discovered that the coefficients e of the old- established formula generally vary with the inclinations for like values of r in such a manner that their values are greatest for the lowest inclinations, and the converse, let us consider them now solely with reference to the Mississippi observations. Their extreme limits there are c = 256 for an inclination of 0' 0034 per thousand,:; ' - " - ^ and to ° c = 154 for an inclination of ■ 0200 per thousand ; ._ _• flon - _ , L ' o e and if a curve be drawn to represent them, it. becomes a reversed hyperbola, whose ordinates decrease with the in- crease of inclination. It is therefore evident, from the extreme sensitiveness of the coefficients when applied within these limits, that the old formula is in this respect inapplic- able to extremely low inclinations, while the new American formula on the contrary is very well suited to them. This relation of the inclinations to the coefficients c holds good with the highest of the falls on the large rivers of the Mississippi series, but is more fully exemplified when the co- efficients diminish with decreasing values of r; so that for cases of smaller rivers it may be accepted that with similar values of r the difference of inclination has so small an in- fluence on the coefficient c that it may be entirely neglected without error. Since the four formulae of D'Arcy and Bazin have been found to give good results, not only in accordance with the observed results mentioned in their own work, but also with those collected by ourselves, and since they also, while jossessing no exclusive claim to general application, admit of 10 THE NEW FORMULA FOB MEAN VELOCITY. [OHAP. I. the interpolation and addition of additional series of coeffi- cients beyond those of their four categories, they may most justly be considered as correct points of departure in an ex- tensive field of variation. We will therefore assume that these formulae are of practical value to us for the purpose of gradually working out a good and complete series of coefficients. 8. The Employment of the Formula of D'Arcy and Bazin in constructing a Series of Coefficients. The following are the four formulae for mean velocity of D'Arcy and Bazin, in terms suited to Swiss feet; to each of them is also attached the corresponding expression for the value of c, the coefficient in the general formula, v = cjrs, which we have taken as a basis. In each case, as before, r is the mean hydraulic radius, and s is the sine of the inclina- tion of the water surface, or fall in a length of unity. 1st Category. — Very smooth surfaces of pure cement, or care- fully planed timber : " = ^O-OOO 045 + "'OOP 0<*5 ; = V : 0-000 045 + °^°-°^ 2nd Category. — Smooth surfaces of cut stone or brickwork, of cement with sand, or of planking : " = ^0-000- 057 + 2^W0iW J V 0-000 057 + 0-000 0133 CHAP. I.J FLOW IN OPEN CHANNELS. 11 3rd. Category. — Less carefully constructed sections in rubble : ^0-000 072 + °^°°°?°° V 0-000 072 + -° 000600 ith Category. — Sections in earth : " = ^0-000 084 + ?^°i52? =v : 0-000 084 + £^008500 These four expressions indicate a great variation in "the values of the terms of the formulae corresponding to the varieties of quality of the surface. We may hence conclude that the observations of D'Arcy and Bazin prove that the degree of roughness of the wetted perimeter forms a very important influence on the value of the coefficient on small sections of discharge; the respective proportions of these four formulae also show that this influence decreases with the increase of the sectional area, and, although it never en- tirely vanishes, is inconsiderable in very large rivers like the Mississippi. We may also remark, that these four categories admit of the interpolation and addition of a large number of cases of different conditions, and can thus be made to include and produce smaller values of the coefficient e than those afforded by the fourth category ; they might then become applicable to the coefficients calculated by ourselves from the observed results on the Aar, and the streams in Canton Graubundten, which are encumbered with detritus. 12 THE NEW FORMULA FOR MEAN VELOCITY. [CHAP. I. The necessity and the mode of introducing these inter- polated and additional categories, suitable to the cases that ■ occur, will necessarily be partly dependent for exactitude on the correctness and sufficiency of knowledge of the details of the observations ; the effect of the various degrees of inclina- tion on the coefficients, previously mentioned, must also be borne in mind. With reference to the observed results on the Wildbach- schale, previously quoted, we may notice that the G'riinn- bachschale and Gerbebachschale, whose walling is much damaged, can very well come under the third category. This, however, is not applicable to the more recently con- structed G-ontenbachschale, which have a better walling than that supposed in the third category, and a worse walling than that of the second. The coefficient e calculated for one of them when r = ■ 375 according to the third formula is 65, and gives too small a mean velocity, while that accord- ing to the second formula is 100, which gives too" high a mean velocity ; the actually correct coefficient being 83, or approximately a mean between the two ; the walling of the Gontenbachschale being in point of fact a mean as regards smoothness between rubble and ashlar. We must therefore not overlook the fact that we here require a category of coefficients interpolated at about midway between Categories II. and III., under conditions of section that differ sufficiently from those of either of them to justify-its adop- tion ; we must also determine more exactly the conditions of section applicable to these three categories. With reference to our observed results on rivers and streams whose beds and banks are encumbered with de- posit, it is evident they cannot come under Category No. IV. of sections in earth, as Formula No. IV. gives values of coefficients c that are too large for them. This is very natural, as part of the living force of the water is destroyed CHAP. I.] FLOW TN OPEN CHANNELS. 13 by the deposit;, the larger the boulders, and the greater the quantity of them obstructing the section of flow, the more will the velocity of the water be reduced. In the formula 1 @ . -y = a -\ — , which expresses the effect of the roughness, and in which the factors a and /S are the divisors in the formulae of D'Arcy and Bazin, these factors will increase with the size and quantity of the deposit, and may hence vary very much for different cases in the same river : they will increase with high water and with motion of the boulders, and decrease with low water and with their deposition. For our purposes we shall not go far wrong if we calculate these velocities in channels encumbered with detritus for one single value of r only, and make them correspond to those obtained by Formula IV. for sections in earth with a radius of • 7 ; or, which is the same thing, if we calculate our coefficients for this purpose from a formula, v 7 0-000 7 0-000 120 + and consider this as the basis of a new or a fifth category of coefficients. We here attach a table of calculated coefficients resulting from the above five formulas, which are applicable to all values of r that are likely to occur in practice ; and in order to afford a trustworthy guide for their employment, we give also immediately following them a table of practically deter mined coefficients, obtained by ourselves from direct velocity measurements in a considerable number of oases, together with the calculated coefficients corresponding to them, and the differences between the two. A careful examination of these two collections, and a comparison of the similar cases occurring under similar conditions, will aid us in eventually 14 THE NEW FORMULA FOE MEAN VELOCITY. [CHAP. I. determining and adopting a final series of coefficients that will be both correct and sufficiently comprehensive for all practical purposes. 9. Table of Calculated Coefficients applicable to the General Formula v = c \/Tl, arranged as to Condition of Section according to the Four Categories of D'Arcy and Bazin, and a Fifth one of the author. ^n s ua^> ^ex~ Explanation. The quantities given in the three columns, corresponding to all values of r required in practice, are values of the following expressions : c is the variable coefficient in the formula v — c >J r s. c^r = »isa variable quantity, dependent on c, useful in obtain- ing values of v corresponding to different values of n] 8. -J— = n is a variable quantity, useful in calculating values -of », when v and r are given, as is shown by putting the v 2 formula in the form s = -, - • c 2 r The quantities are applicable to Swiss feet. OHAP. I.] FLOW IN OPEN CHANNELS. 15 CATEGOEY I. Vbbt smooth Surfaces of pubb Cement, well-planed Timber, etc. V; 0-000 045 + 0-000 0045 r c 1 (« = nttf) o-oi 44-95 4-495 • 0-0495000 0-05 86-07 19-245 27000 01 105-41 33-333 9000 0-2 121-72 54-433 3375 0-3 129-10 70-711 2000 0-4 133-33 84-327 1407 0-5 136-08 96-225 1080 0-6 138-01 106-90 875 07 139-44 116-67 735 0-8 140-54 125-71 633 0-9 141-42 134-61 555 1-0 142-13 142-13 495 11 142-72 149-69 446 1-2 143-22 156-89 406 1-3 143-65 163-79 373 1-4 144-02 170-40 344 1-5 144-34 176-78 320 1-6 , 144-62 182-93 299 1-7 144-87 188-89 280 1-8 145-10 194-67 264 1-9 145-30 200-28 249 20 145-48 205-74 236 20 148-70 665-00 23 100 149-00 1490-00 2 16 THE NEW FORMULA FOE MEAN TELOCITY. [chap. CATEGOEY II. Smooth Surfaces, Ashlar, Brickwork, Planking, etc. V; 000 057 + 0-000 013 3 r c fj r = m 1 -=- = * G £ r (s = n «2) 0-01 26-85 2-685 0-1387000 005 55-64 12-442 64600 0-1 72-55 22-042 19000 02 89-98 40-252- 6175 0-3 99-34 51-963 3378 0-4 105-26 66-574 2256 0-5 109-37 77-336 1672 0-6 112-39 87-057 1319 0-7 114-71 95-971 1086 0-8 1 116-54 104-24 920 0-9 118-03 111-98 797 1-0 119-27 119-27 703 11 120-33 126-20 628 1-2 121-20 132-76 567 1-3 121-96 139-06 517 1-4 122-65 145-12 475 1-5 123-22 150-91 439 1-6 123-74 156-52 408 1-7 124-20 161-94 381 1-8 124-62 167-20 358 1-9 125-00 172-30 337 2-0 125-34 177-26 318 20 131-69 588-92 29 100 132-30 1323-00 3 CHAP. I.J FLOW IN OPEN CHANNELS. 17 CATEGOEY III. Moderately well-constructed Sections in Bubble, etc. V; 000 072 + 0-000 060 c c ij r = m 1 c'r 0-01 12-83 1-283 0-6072000 005 30-54 6-830 214400 0-1 38-57 12-199 67200 0-2 51-85 23-187 18600 0-3 60-63 33-210 9067 04 67-12 44-448 5554 0-5 72-17 51-031 3840 06 76-25 59-063 2867 . 0-7 79-63 66-624 2253 0-8 82-48 73-771 1837 0-9 84-94 80-582 1540 1-0 87-04 87-039 1320 1-1 88-91 •93-251 1150 1-2 90-54 99-177 1017 1-3 9202 104-92 908 1-4 93-33 110-43 820 IS 94-49 115-73 747 1-6 95-56 120-88 684 1-7 96-54 125-87 631 1-8 97-45 130 74 585 1-9 98-25 135-43 545 2-0 99-01 140 03 510 20 115-47 516-40 37 100 117-36 1173-63 4 18 F THE NEW FORMULA FOE MEAN TELOCITY. 0^ -w-W" "^v'^^* 5 ? CATEGORY IV. [OHAP. I. Sections in Earth. V, 000 084 + 0-000 350 9 c C tj r — m 1 r c 2 r (i)=cV") (v = m V«) (s = n d 2 ) 0-1 16-70 5-282 0-0358400 0-2 23-25 10-443 91700 0-3 28-27 15-486 41700 0-4 32-29 20-423 23975 0-5 35-64 25-225 15745 0-6 38-71 29-985 11122 0-7 41-38 34-585 8343 0-8 43-87 39-252 6494 0-9 45-99 43-624 5254 1-0 48-00 48-002 4340 11 49-86 52-298 3656 12 51-59 56-518 3131 1-3 53-21 60-666 2717 1-4 54-72 64-743 2386 1-5 56-14 68-753 2115 , 1-6 57-47 72-697 1892 1-7 58-74 76-585 1705 1-8 59-93 80-401 1547 1-9 61-06 84-166 1412, 2-0 62-14 87-875 1295 2-1 63-16 91-529 1194 2-2 64-14 95-132 1105 23 65-07 98-685 1027 24 65-96 102-19 957 2-5 66-81 105-64 896 2-6 67-63 109-05 841 2-7 68-42 112-42 791 2-8 69 17 115-74 746 2-9 69-90 119-03 706 30 70-59 122-27 669 31 71-26 125-48 635 32 71-91 128-64 604 33 72-54 131-77 576 34 73-14 134-86 550 35 73-72 137-92 526 36 74-28 140-94 499 3-7 74-83 143 -.94 483 3-8 75-44 147-07 463 3-9 75-87 149-82 445 4-0 76-36 152-72 427 4-1 76-84 155-59 413 4-2 77-31 158-43 398 4-3 77-76 161-24 385 4-4 78-20 16403 372 4-5 78-62 166-78 359 CHAP. I.] FLOW IN OPEN CHANNELS. 19 c ,J r = m 1 — = n r c*r (« = Cv/r») (•=»Vo (t = nift) 4-6 78-94 169-32 0-0000348 4 7 79 44 172 22 337 4 8 79 83 174 90 327 4 9 80 21 177 55 317 5 80 58 180 19 308 5 1 80 94 182 80 299 5 2 81 30 185 38 291 5 3 81 64 187 95 283 5 4 81 98 190 49 276 5 5 82 30 193 01 268 5 6 82 62 . 195 51 262 5 7 82 93 198 00 255 5 8 83 33 200 73 248 5 9 83 53 202 92 243 6 83 82 205 32 237 6 1 84 10 207 72 232 6 2 84 38 210 10 226 6 3 84 65 212 47 221 6 4 84 91 214 82 217 6 5 85 17 217 15 212 6 6 85 43 219 47 208 6 7 85 67 221 76 203 6 8 85 92 224 00 199 6 9 86 15 226 31 195 7 86 39 228 56 191 7 1 86 61 230 79 188 7 2 86 84 233 01 184 7 3 87 06 235 22 181 7 4 87 27 237 40 177 7 5 87 48 239 58 174 7 6 87 69 241 74 171 7 7 87 89 243 89 168 7 8 88 09 246 02 165 7 9 88 28 248 14 162 8 88 47 250 24 160 8 1 88 66 252 34 157 8 2 88 85 254 45 154 8 3 89 03 256 48 152 8 4 89 20 258 53 150 8 5 89 38 260 58 147 8 6 89 55 262 61 145 8 7 89 72 264 63 143 8 8 89 89, 266 61 141 8 9 90 01 268 54 139 9 90 21 270 62 137 9 1 90 37 272 60 135 9 2 90 52 274 56 133 9 3 90 67 276 52 131 9 4 90 81 278 41 129 9 5 90 97 280 39 127 9 6 91 11 282 30 125 9 7 91 26 284 54 124 9 8 91 40 286 12 122 9 9 91 53 288 01 121 o 2 20 THE NEW FORMULA FOR MEAN VELOCITY. [CHAP. I. c*/r = m 1 r (v = ct/rs) (•=»//') (s=»i> 8 ) 10-0 91-67 289-89 0-0000119 10-1 91-80 291-76 117 10-2 91-94 293-62 116 103 92-06 295-47 114 10-4 92-19 297-32 113 10-5 92-32 299-15 112 10-6 92-44 300-97 110 10-7 92-56 302-79 109 10-8 92-68 304-59 108 10-9 92-80 306-39 106 11-0 92-92 308-18 105 11-1 93-04 309-97 104 11-2 93 15 311-74 103 11-3 93-26 313-51 102 11-4 93-37 315-26 101 11 -5 93-48 317-01 100 11-6 93-59 318-75 98 11-7 93-70 320-49 97 11-8 93-80 322-21 96 11-9 93-90 323-93 95 120 94-00 325-63 94 12 1 94-10 327-33 93 12-2 94-20 329-03 92 12-3 94-30 330-71 91 12-4 94-39 332-40 • 90 12-5 94-49 334-08 90 12-6 94-58 335-74 89 12-7 94-68 337-40 88 12-8 94-77 339-06 87 12-9 94-86 340-71 86 130 94-95 342-35 85 131 95-04 343-97 85 13-2 .95-12 345-59 84 133 95-21 347-21 83 13-4 95-29 348-83 82 13-5 95-38 350-44 81 136 95-46 352-04 81 13-7 95-54 353-63 80 13-8 95-62 355-23 79 139 95-70 356-81 79 14 95-78 358-39 78 141 95-87 360-00 77 142 95-94 361-52 77 143 96-01 363-07 76 144 96-09 364-63 75 14-5 96-16 366-18 75 146 96-24 367-73 74 14-7 96-31 369-26 73 14-8 96-38 370-79 73 14-9 96-45 372-31 72 15-0 96-53 373-84 72 15-1 96-59 375-35 71 15-2 96-66 376-85 70 15-3 96-73 378-35 70 CHAP. I.] FLOW IN OPEN CHANNELS. 21 e */ r = m 1 r (y = c/rs) (v = m*/s) c 2 r 15 4 96-79 379-85 0-0000069 15 5 96-86 381 35 69 15 6 96-93 382 83 68 15 7 97-00 384 33 68 15 8 06 385 81 67 15 9 12 387 28 67 16 19 388 75 66 16 1 25 390 21 66 16 2 31 391 67 65 16 3 37 393 12 65 16 4 43 394 57 64 16 5 49 396 02 64 16 6 55 397 46 63 16 7 61 398 88 63 16 8 67 400 33 62 16 9 '72 401 74 62 17 78 403 16 62 17 1 84 404 58 61 17 2 89 405 99 61 17 3 95 407 40 60 17 4 98-00 408 82 60 17 5 06 410 21 59 17 6 11 411 60 59 17 7 17 413 00 59 17 8 22 414 40 58 17 9 27 415 77 1 t 58 1 57 18 82 417 15 18 1 37 418 51 ^ 57 18 2 42 419 89 57 18 3 47 421 73 56 18 4 52 422 •62 56 18 5 57 423 97 56 18 6 62 425 •32 55 18 7 67 426 •67 55 18 8 72 428 •02 55 18 9 76 429 ■38 54 19 81 430 •71 54 19 1 86 432 05 54 19 2 90 433 •37 53 19 3 95 434 •71 53 19 4 99-00 436 ■03 53 19 5 04 437 •34 52 19 6 08 438 •36 52 19 7 13 • 439 •97 52 19 8 17 441 •28 51 19 9 21 442 •59 51 20 26 443 •90 51 30 102-24 40 103-83 50 104-83 60 105-51 70 106-00 80 106-37 100 106-90 22 THE NEW FORMULA FOE MEAN "VELOCITY. [OHAP. I. CATEGOEY V. Fob Sections covered with Detritus, corresponding to those of the Streams in Canton Graubundten. •V; 000 120 + 0-000 7 r c r c r c 01 12 3 53 10 73 0-3 20 4 58 13 76 0-5 26 5 62 16 78 1 35 7 67 20 80 2 46 10. Table of Experimental Values of Coefficients in the formula v = c \/~rs obtained from velocity Observations. Explanation. The first three columns give the actual values of r, s, and v, as obtained by measurement ; the fourth column gives the value of E, the coefficient resulting from experiment ; the columns I. II. III. IV. give values of the corresponding calcu- lated coefficients in these respective categories according to the formulae of D'Arcy and Bazin; and the last column gives the difference. ^ fyU~wtfJ> The quantities are in Swiss feet I I. Sections in Masonry, Semicircular. Geriebachschale. 0-197 237-3 10-31 58 90 52 III +6 , 185-2 9-58 55 » +o 1 167-9 9-33 51 23 ., - 1 s 137-5 9-05 50 „ -2 1 111-7 8-61 48 1) V j) ,■ -4 Rather damaged. The successive decre- ments in these coeffi- cients is due to the employment of an aver- age, instead of an exaot, value of r. CHAP. !•] FLOW IN OPEN CHANNELS. 23 Fall per 1000. Coefficients. II. HI. IT. Difference. I I I I I | | 6. G'riinnbachschale. I 0-394 0-385 106-775 13-97 68 105 67 it III + l 99-270 13-54 69 104 66 ., ,, +3 82-850 12-00 69 103 64 „ +5 106-775 19-48 73 78 40 ,, -5 99-270 18-58 73 ., 78 40 ,. -5 82-850 15-79 71 •• 76 39 „ -5 0-328 0-375 » 3-0 Gontenhachschale. 46-425 10-66 86 101 62 Ill + 24 11-15 ^ 42-350 9-60 81 101 62 .. ,, +19 ,,-20 | 46-425 11-15 84 104 65 ., „ +19 ,,-20 f 42-350 10-05 83 104 65 „ +18 ,,-21 ' d. Mill-lcnts, Diemerstein. Rather damaged. Little water, but clear. Turbid water with detritus. New and well constructed. It is evident that these are means between Cate- gories I. and II. Section in Sandstone. J-40 1-40 70 99 61 HI + 9 Fall per 1000. Coefficients. IV. Difference. Remarks. II. Sections in Eaeth. a. Brooks, fflibengraben, Hochenbach, Speyerbach, Lautercanal, Canal at Ried von Marmels, Canal in England. 0-6 1-300 1-45 52 39 IV + 13 ) 0-9 0-778 1-46 56 46 „ +io 0-9 0-797 1-49 56 46 „ +io 1-5 0-667 1-85 59 56 „ + 3 1-6 0-267 1-83 88 57 „ +31 1-8 0-664 214 61 60 „ + 1 2-35 0-500 1-92 56 65 „ - 9 2-50 0-063 1-13 91 67 „ +24 The inclinations are gene- rally low. The greatest dif- ferences occur with the least inclinations. The sections appear to be better than that allowed for by the formula, with the exception of the last but one, which is evi- dently strong. Ohio Speisecanal, River Hague, Yssel, Ohio (Point Pleasant), Rhine below the Tssel. 0-698 2-72 54 75 0-698 3-03 59 76 0-165 2-49 87 81 0-156 2-56 85 84 0-117 2-77 105 84 0-093 2-51 100 86 0-117 2-92 97 88 IV -21 „ -17 „ + 6 „ + 1 „ +21 „ +14 „ + 9 Iln the Chesapeake Ohio Speisecanal there is grass or weeds, and the inclination is high; this is expressed by the coefficients. ' The re- mainder have lower incli- nations, and hence higher coefficients. 24 THE NEW FORMULA FOB MEAN VELOCITY. [CHAP. I Fall per 1000. w Coefficients. Rbmabks. B IV. Difference. 1 c. The Tiber at Rome, the Rhine at Speyer, Waal, the Shine at Pannerden, and at Byland. 9-9 9-9 11-5 11-7 17-1 0-131 3-41 97 92 0-112 2-96 89 92 0-104 3-16 93 93 0-100 3-28 98 94 0-098 3-57 89 98 IV + 5 Jl — 3 + 4 » — 9 d.- Bayou Lafourche, Bayou Plaquemine, the Great Newka. 13-0 13-6 16-0 16-3 16-8 18-1 19-1 0-044 2-79 119 95 IV + 24 0-037 2-84 128 95 „ +33 0-144 3-96 84 97 „ -13 0-045 3-08 115 97 „ +18 0-036 2-81 129 98 ,, +31 0-015 2-05 127 98 „ + 29 0-206 5-20 .84 99 „ -15 Low inclination. High inclination. Low inclination. High inclination. e. Newa, Mississippi. 46-4 32-0 54-3 59-8 66-8 67-3 68-6 74-4 75-1 77-0 78-3 0-014 3-23 145 102 IV + 43 0-022 3-52 130 100 „ + 30 0-030 5-56 139 103 „ + 36 0-048 6-32 120 103 ,, + 17 0-064 6-95 108 104 ,. + * 0-044 6-82 128 104 „ + 24 0-068 6-96 103 104 ,. - 1 0-017 5-89 166 105 „ + 61 0-020 5-93 154 105 „ + « 0-003 4-03 253 105 „ +148 0-004 3-98 234 105 „ +129 Low inclination. High inclination. Low inclination. High inclination. Slight inclination. Very slight inclination. /. Linth Canal. 2 6 2 6 2 4 7 90 9-3 0-29 3-47 89 81 IV+ 8 0-30 3-90 92 84 ,, + 8 0-31 4-22 93 85 „ + 8 0-32 4-49 93 87 „ + 8 0-33 4-83 96 88 » + 8 0-34 5-00 95 89 » + 6 0-34 5-14 96 89 >, + 7 0-35 5-31 96 90 „ + 6 0-36 5-48 96 90 „ + 6 0-37 5-62 95 91 ,. + * The Linth canal has a ra- ther smoother section than that of the Fourth Category. Its coefficients run higher than, but yet tolerably pa- rallel to, those of D'Arcy and Bazin. CHAP. I.] PLOW IN OPEN CHANNELS. 25 FalJ per 1000. Coefficients. IV. Difference. III. Sections obstructed by Detbitus. a. Acer. 3-250 1-27 4-37 68 72 IV- 4' 4-122 1-09 6-37 95 77 „ +18 4-769 1-78 5-67 62 80 „ -18 5-597 1-09 7-38 94 83 „ +11 6-351 1-27 6-38 71 85 » -I* 6-900 1-87 5-93 52 86 „ -34 7-350 1-78 7-05 62 87 „ -25 8-819 0-14 2-04 58 90 „ -32 11-855 0-28 3-06 53 94 r » ~ 41 12-005 o-io 2-30 66 94 „ -28 15-510 0-10 3-53 90 97 ,, - 7 17-526 0-12 4-40 ■ 96 98 „ - 2 6. Escher Canal. 3-815 4-487 4-821 1-001 1-217 1-550 c. The Meuse at Misox. 11-87 393 36 48 11-87 5-63 47 52 11-87 7-71 57 57 IV -12 „ - 5 d. The Shine at Domleschgerthal. 0-255 •073 •086 •128 •335 •329 •344 1-320 1-366 2-000 1-970 2-227 2-429 2-465 2-997 2-997 2-99? 3110 3 195 3-475 Some of the measurements are doubtful. The influence of the detritus is generally very evident. 3-00 6-46 60 75 IV- 15 3-00 7-80 67 78 » -n 3-00 10-87 90 80 „ +io 5-77 1-27 33 26 IV + 7 6-43 3-70 35 48 11 -13 6-43 4-38 52 49 11 + 3 6-43 4-60 54 50 11 -+ * 6-43 5-13 55 54 11 + 1 6-43 5-24 57 54 11 + 3 7-73 4-83 45 54 11 - 9 7-96 7-00 59 53 11 + 6 7-73 3-91 38 54 It - 16 7-73 5-97 43 62 11 - 19 7-96 7-20 55 62 11 - 7 7-03 6-77 76 64 It + 12 7-73 6-07 44 66 11 -22 7-96 7-40 52 66 11 - 14 7-55 7-25 48 70 11 -22 7-75 7-40 49 70 11 -21 7-96 7-54 49 70 11 -21 7-03 8-38 57 71 1> - 14 7-96 8-83 52 72 11 -20 7-96 9-67 56 73 » -17 The detritus is large. ? Influence of detritus. Some of these results gene- rally indicate the influence of the detritus. 26 THE NEW FORMULA FOE MEAN VELOCITY. [OHAP. I. Fall per 1000. Coefficients. IV. Difference. Remarks. 1-267 2-373 3-531 3-638 3-650 4-365 0-423 0-776 1-229 I I v. The Plessur at Thur. 9-65 6-10 55 53 IV + 2 9-65 10-15 67 66 » + 1 9-65 10-36 56 73 ,, -17 9-65 13-80 74 74 9-65 14-17 75 75 9-65 13-07 68 78 „ -10 /. The Shine at Sheinwald. These results agree well generally, with the exceptiqn of two. 14-20 2-37 31 33 IV 2 14-20 4-60 44 43 + 1 14-20 6-13 46 52 » — 6 11. Kbmaeks on the Series of Observations of D'Aroy and bazin. The ' Becherches Hydrauliques ' of D'Arcy and Bazin contain fifty series, comprising three hundred and seventy measured observations of cases similar to the foregoing, which afford a large number of experimentally obtained coefficients c for the formula v = c <\/Tl. We have plotted them to scale according to their respective categories, in conjunction with the curves of the coefficients calculated from the four formulae ; they indicate the following results : Category I. — Very smooth Sections in Cement, planed Timber, etc. The coefficients afforded by the series Nos. 2, 24, 25, 28, and 29, group themselves generally close to the calculated coefficients obtained by formula No. I. ; in the semicircular sections in cement, series Nos. 24 and 25, the coefficients are higher than those of the formula, and increase very rapidly with the values of r. CHAP. I.] FLOW IN OPEN CHANNELS. 27 Category II. — Sections in Ashlar, Brickwork, and Planking, etc. The coefficients given by fifteen series agree very well with the curve of coefficients corresponding to formula No. I. ; but the sections in plank show greater variability than those in stone, more especially those that are of a semicircular form. The results of the above very varied constructions of section show that the coefficients that correspond to recti- linear sections do not vary much. Category II. to III. — Sections rougher than Ashlar, Brick- work, and Planking, but smoother than dry Bubble. This new category adopted by ourselves, and placed as an arithmetic mean between Categories II. and III., is not men- tioned by D'Arcy and Bazin. The necessity of this new category as a special class is, however, clearly shown from the examination of series Nos. 12, 13, H, 27, 30, and 31, as well as those of the Gontenbachschale at Lake Thun. The series Nos. 12, 13, and 14 are rectangular sections in plank- ing, the planks being 09 foot wide, placed 0*033 foot apart: series No. 27 is a semicircular section of firmly punned gravel • 03 to • 07 foot thick ; the Gontenbach- schale is also semicircular, but is made of new and well- constructed large dry rubble. In both sections the derived coefficients fall in a mean curve lying midway between those of Categories II. and III. The series Nos. 30 and 31 have very small sections of plank covered with canvas, and give coefficients which fall between those of formula No. II. and those of the new class midway between Categories II. and III. ; they may hence be almost considered as belonging to Category No. II. 28 THE NEW FORMULA FOR MEAN VELOCITY. [CHAP. I. Category III. — Ordinary dry Bubble. To this category belong series Nos. 4, 32, 33, 45, as well as those of the Gr'riinnbachschale and the Gerbebachschale at Merligen on Lake Thun, which are semicircular in section and much damaged. Category III. to] IV. — Worse than ordinary Bubble and better than Earthen Sections, being an arithmetic mean between Categories III. and IV. This class is not proposed by D'Arcy and Bazin, but is a natural result of the examination of the following series : Series No. 5, rectangular, made of well punned gravel • 10 to 0*15 foot thick; series Nos. 15, 16, and 17, sections in planks, nailed on transversely, 0" 09 foot broad and 0*167 foot apart ; series No. 35, bad masonry ; series Nos. 44 and 46, rectangular, of damaged masonry, having their beds covered with stones and mud ; lastly, the Alpbachschale at Meiringen, of old and very much damaged rubble. Category IV. — Sections in Earth. To this belong series Nos. 34, 37, 38, 41, 42, 47, 48, 49, and 50. Some of these are entirely in earth, without any vegetation on the bed or banks; some of bad masonry, covered with moss and plants, or having their beds covered with stones and mud; some rocky sections, etc. ' To this category also approximately belong a large number of obser- vations on the Seine, Saone, Hayne, Canal du Jard, as well as those of the Swiss, and those of the American engineers on the Mississippi and its tributaries. CHAP. I.] FLOW IN OPEN CHANNELS. 29 Category V. — Sections obstructed by Detritus. This is not one of D'Arcy and Bazin's categories, but is the result of observations on rivers having their beds and banks obstructed by detritus, principally those of La Eicca, Legler, etc. To this class belong series Nos. 36, 40, and 43, in the sections of which occur many plants, grass, rocks, and stone strewn about. The determination of the final coefficients for all these classes will be subsequently explained. Further reference as to the observations of D'Arcy and Bazin may be made by consulting their ' Becherches Hydrauliques ; ' the re- maining observations we have already given in the table at pages 25 and 26, paragraph 10. 12. The Coefficients of DAect and Bazin foe CALCULATING MEAN FEOM MAXIMUM VELOCITIES. SLfcfcV l > ^ The numerous and accurate observations of D'Arcy and v~e ' ^ Bazin have demonstrated that the ratio of mean to maximum velocity in any section, till lately believed to be from • 80 to- 0*83, is not a constant quantity, but a variable one, a fact also noticed by others. Their formula for calculating mean from maximum velocities is as follows : — = 1 + 25-56 x/^r > ot«.-». = 25-56 jTi ^ Wv, £ &&n* r Category I. Category II. Category III. Category IV. o-i 0-80 0-74 0-62 0-2 83 0-78 0-67 : 51 0-3 83 0-79 0-70 0-54 0-4 84 0-80 0-72 0-56 0-5 84 0-81 0-74 0-58 0-6 84 0-81 0-75 0-60 0-7 o- 84 0-82 0-76 0-62 0-8 o- 85 0'82 0-76 0-63 0-9 o- 85- 0-82 0-77 0-64 1 o- 85 0-82 0-77 065 2 o- 85 ' 0-83 0-80 0-71 3 0-80 0-74 4 0-80 0-75 5 0-81 0-76 6 0-81 0-77 7 0-77 8 0-78 9 ., 0-78 10 ,. ., 0-78 11 0-78 , 12 0-79 ! 20 1 •• 0-79 14. Examples Explanatoey of the Use of the Table of Coefficients of D'Aecy and Bazin, given at pages 14 to 22. (Swiss feet are used in these examples, as well as in the table.) Example 1. A channel of trapezoidal section with side slopes of 45° and an inclination, s = • 0008, has to discharge 5 cubic feet per second at maximum, when the surface of the water will stand at 1 foot below the surface of the ground ; the soil is loam, with one-third sand : what will the bottom width be, and what the depth of excavation ? CHAP. I.J FLOW IN OPEN CHANNELS. 31 The method of approximation is best suited to this case. The formula to be used is v = c^/rs. Assume as a first approximation a bottom width of 3 feet, and a depth at high water of 1 foot. Then the cross section will be 4 square feet, and the wetted perimeter will = 3 + 2 V 2 = 5-8, and r will = =^ = 0-69 ; the coefficient c corresponding to this value of r in Category IV. is 41*11, but as the soil is loamy and tolerably smooth we may take it as 42. Applying these values in the formula we obtain v = 42^0*69 X 0-0008 = 0-987 and 2 = 4 x * 987 = 3 * 95 cubic feet per second instead of 5 cubic feet per second. In order to correct this, either the bottom width or the depth of wetted section must be increased ; the latter mode is preferable, on account of its occupying a smaller breadth of land. Assuming therefore for a second approximation a depth of 1 • 3 feet, the cross section becomes (3 + 1-3) x 1 ' 3 = 5 * 59 square feet, the wetted perimeter 3 + 2v / 2 , 6 = 6 - 2,r will = 0*9, and c in Category IV. will = 46 : hence v will = 46 V 0*9 x 0-0008 = 1*24, and q will = 1-24 x 5-59 = 6-93 cubic feet per second. As in the first approximation the discharge resulting from a depth of 1 foot was 1 cubic foot per second too little, and in the second, that from a depth of 1 ■ 3 feet was 1 ■ 93 cubic feet per second too much, we cannot be far wrong in putting the correct depth at 1 * 1 feet, the bottom width as 3 feet ; and then the depth of excavation will be 2 * 1 feet. 32 THE NEW FORMULA FOR MEAN VELOCITY. [CHAP. I. Example 2. Obtain the bottom width and depth of a planked rectangular channel, which will have maximum dis- charge of 3 '5 cubic feet per second, with an inclination of 0-001. Assume for a first approximation a bottom width of 2 feet, and a depth of 1 foot. Then the cross section = 2 square feet, the wetted peri- meter = 4 feet, hence r = • 5 foot, and c in Category II. will be 110. Therefore v = 110 VO-5 x 0-001 = 2-46 feet per second and q = 2 • 46 x 2 = 4-92 cubic feet per second. For a second approximation reduce the bottom width to ■ 7 foot ; the new quantities resulting are then, the cross section = 1 ■ 4 square feet, the wetted perimeter = 3*4 feet, r = - 41, ande = 105, hence v = 105V0-41 x 0-001 = 2-13, and q = 2-13 x 1-4 = 2-93. In the first case the discharge was 1*4 cubic feet too much, and in the second • 52 too little ; if then we assume a correct depth of ■ 8 instead of 1 ■ and ■ 7 foot, the error will be very small. The sides of the channel will then be not more than 1 foot in height. Example 3. To calculate the discharge of a channel. a. The maximum discharge obtained by repeated obser- vations with floats is 5-27 cubic feet per second; the section taken as a mean of those at the two ends and at the middle of the length of channel under observa- tion, is 210 square feet, and the mean wetted perimeter is 57 • 5 feet. Hence 210 r = WFl = 3 ' 65 - OHAP. I.] FLOW IN OPEN CHANNELS. 33 The mean velocity is obtained from the maximum by applying a coefficient of reduction, given in the last table, of 0-75. Hence v = 0-75 X 5-27 = 3-95, and q = 3-95 X 210 = 829-5 cubic feet per second, or in round numbers 830. b. If the inclination and dimensions of the channel are given, let the cross section be taken as 117 square feet, the wetted perimeter at 32 feet ; and the inclination as S = 0- 000 753; then will r = 3 -656, and e the co- efficient will in Category IV. be 74 • 6. Hence v = 74-6 V3-656x 0-000753 = 74-6 X 0-0525 = 3-92 feet per second, and q = 3-92 x H7 = 458-6 cubic feet per second, or in round numbers 460. Example 4. What is the inclination to be given to a channel, having a maximum discharge of 3 cubic feet per second, that has to be conducted down sloping ground of a soil not allow- ing of a mean velocity of water of more than 3 feet per second ? Let the section be trapezoidal with side slopes of 1 to one, its bottom width 3 feet, and its depth 1 foot. Then the cross section will be 4 square feet, the wetted perimeter 5 ■ 8 feet ; and r will = 0-69, and the coefficient n for Category IV. will be 0-000 8621, and hence S = nv* = 0-000 8621 x 9 = 0-0077. The suggestions afforded by these examples will aid in the choice of coefficients for various cases. 34 THE NEW FORMULA FOB MEAN VELOCITY. [CHAP. I. 15. The Formula and Categoeies op Gauckler. The two new formulae of G. Ph. Gauckler, Engineer of the Ponts et Chaussees and the works on the Ehine at Colmar, are given in a treatise, ' Etudes Theoriques et Pratiques sur l'Ecoulement et le Mouvement des Eaux,' in the Comptes Rendus of the Academie des Sciences. They are : 1st. For inclinations exceeding O'OOOT, Jv = affrf/s. 2nd. For inclinations less than 0-0007, % v = fi $ r tf 8. These two equations may be reduced to the forms v = a? $r jrs, v = p #7*». Mons. Gauckler, from a comparison of the observations of D'Arcy and Bazin, Dubuat, Woltmann, Briinings, Poiree, Emmery, and Leveille, determines the values of a and /3 to be as follows in different sections, according to his six Categories for Swiss feet. Categories. 1. Aahlar and cement 2. Ordinary good masonry 3. Sections with masonry side walls and\ the bottom in earth / 4. Canals entirely in earth 5. Canals in earth, with grass on the\ sides J 6. Rivers 7-7 to 8-1 7-2 „ 7-7 7-0 „ 7-2 6-3 „ 7-0 6-0 „ 6-3 5-8 „ 6-0 First as regards Gauckler's first formula : If we calculate a series of coefficients e for the general formula v = Cy/rs from those given by Gauckler, for all his six categories, and for a series of values of r, and plot them to the same scale as CHAP. I.] PLOW IN OPEN CHANNELS. 35 the corresponding coefficients of D'Arcy and Bazin, we find that the limits of the former are much greater than those of the latter ; for instance, for a value of r = 2, the coefficients of Gauckler's first formula vary between -42 and 168, and those of Bazin between 62 and 145. We also notice that for very small values of r, in the first category the coefficients of D'Arcy and Bazin are higher than those of Gauckler, while in the last category they are lower, and that in the first category the successive increments of e generally rise more steadily according to Gauckler than according to D'Arcy and Bazin, while in the last category, and especially from r = • 01 to • 02, they first decrease more rapidly, and after- wards increase more slowly than those according to D'Arcy and Bazin. We give here following the calculated coefficients of Gauckler for his six categories obtained from his first for- mula for Swiss feet. Secondly, as regards Gauckler's second formula, suited to streams having inclinations less than 0'0007, where if v = /8 II r V s - We have calculated a large number of values of the coefficient /3 from the results of observation, and find that they correspond tolerably well with the Series Nos. 41 to 50 of D'Arcy and Bazin ; while on the contrary the values of /S are from 5 ■ 3 to 5 • 4, or less than the minimum fixed by Gauckler at 5 ■ 8, for the observations on the Ehine at Ger- mersheim of Grebenau, for those on the Linth canal, Nos. 5 to 10 of Legler, and for those on the Mississippi and its affluents in cases where the inclinations are considerable : again, when the inclinations on the Mississippi are small the values of /8 increase and reach 7 ■ 8. d 2 36 THE NEW FORMULA FOR MEAN VELOCITY. [OHAP. I. 16. Table of Coefficients a for the First Formula of Gauckler, in his Six Categories, adapted to Swiss Feet. l. 2. 3. 4. 5 and 6. r a = a = a = a = a = 10-389 to 12-222 9-289 to 10'389 8-311 to 9-289 6-966 to 8-311 6-111 to 6-966 0-05 66 to 91 52 to 66 42 to 52 29 to 42 23 to 29 01 74 „ 102 59 „ 74 47 „ 59 33 „ 47 25 „ 33 0-2 83 „ 114 66 „ 83 53 „ 66 37 „ 53 29 „ 37 0-3 88 „ 122 71 „ 88 57 „ 71 40 „ 57 31 „ 40 0-4 93 „ 128 74 „ 93 59 „ 74 42 „ 59 32 „ 42 0-5 96 „ 133 77 „ 96 61 „ 77 43 „ 61 33 „ 43 0-6 99 „ 137 79 „ 99 63 „ 79 45 „ 63 34 „ 45 0-7 102 „ 141 81 „ 102 65 „ 81 46 „ 65 35 „ 46 0-8 104 „ 144 83 „ 104 67 „ 83 47 „ 67 36 „ 47 0-9 106 „ 147 85 „ 106 68 „ 85 48 „ 68 37 „ 48 1-0 108 „ 149 86 „ 108 69 „ 86 49 „ 69 37 „ 49 1-25 112 „ 155 90 „ 112 72 „ 90 50 „ 72 39 „ 50 1-50 115 „ 160 92 „ 115 74 „ 92 52 „ 74 40 „ 52 1-75 118 „ 164 95 „ 118 76 „ 95 53 „ 76 41 „ 53 20 121 „ 168 97 „ 121 78 „ 97 54 „ 78 - 42 „ 54 25 126 „ 174 101 „ 126 80 „ 101 57 „ 80 44 „ 57 3 130 „ 179 104 „ 130 83 „ 104 58 „ 83 45 „ 58 4 136 „ 188 109 „ 136 87 „ 109 61 „ 87 47 „ 61 5 141 „ 195 113 „ 141 90 „ 113 63 „ 90 49 „ 63 7 149 „ 207 119 „ 149 96 „ 119 67 „ 96 52 „ 67 10 158 „ 219 127 „ 158 101 „ 127 71 „ 101 55 „ 71 15 170 „ 235 135 „ 170 108 „ 135 76 „ 108 59 „ 76 20 179 „ 246 142 „ 179 114 „ 142 80 „ 114 62 „ 80 17. The Formation of a New and Final Set of Twelve Classes, instead of the previous Categories. The fifty series of observations mentioned in Bazin's work comprise only a very small number of values of r, to which a moderate number of curves or equations are applicable. The same is the case, but in a higher degree, with the observa- tions of Dubuat, Woltmann, Briinings, Poiree, Emmery, etc. Hence we may observe that the formulae of Gauckler may with an extension of the values of a and /8 give quite as good results as those of D'Arcy and Bazin, and perhaps even better, as they are more comprehensive and include the CHAP. I.J PLOW IN OPEN CHANNELS. 37 extreme values of r. A series of coefficients however that are obtained directly from observed results of all degrees and conditions are far more useful and comprehensive ; they are of more value to the practical engineer, as they possess an exactitude dependent entirely on the correctness of the ob- servations, and at the same time offer to the scientific an opportunity for deriving theoretical deductions that may be quite as correct as any hitherto made. Such a series of working coefficients e for the formula v = c y/Vs adapted to Swiss feet, as are all the foregoing tables, are given in the following table. They are separated into twelve new classes, in accordance with the various conditions under which the observations were made, and are dependent on the observations given in Series Nos. 1 to 50 of D'Arcy and Bazin, those of Dubuat, Poiree, Emmery, Leveille, Funk, Briinings, Woltmann, and Bonati; also given in the 'Eecherches Hydrauliques,' as well as others taken from the collection of Grebenau, and on the observations of engineers in Switzerland. These observa- tions are referred to their respective classes in the following list. From the evident incompleteness and deficiency for our purposes of this collection of observed results, it would be highly desirable to increase it by many more ; more espe- cially for the case of rivers and channels impeded by detritus. 18. The New Classes of Coefficients. The series referred to are those of D'Arcy and Bazin. Class I. Well-planed timber planks \ foot wide; rect- angular. Section, Series Nos. 28 and 29. Pure cement, semi- circular. Section, Series No. 24. 38 THE NEW F0EMULA FOB MEAN VELOCITY. [CHAP. I. Class II. Pure cement, rectangular section, Series No. 2. Cement with one-third fine sand from the Saone, semi- circular section. Series No. 25. Class III. Planking, semicircular section, Series No. 26. Class IV. Planking, mill-leats, rectangular, trapezoidal and triangular in section. Series Nos. 6, 7, 8, 9, 10, 11, 18, 10, 20, 21, 22, and 23. In these the coefficients o increase with the inclinations, which vary from • 001 487 to ■ 008 433. Class V. Small channels in ashlar and brickwork, rect- angular sections. Series Nos. 1 (Baumgarfcen), 3, and 39. Class VI. Planks covered with canvas, £ foot wide, rect- angular sections. Series Nos. 30 and 31. Planking of laths • 09 foot wide, nailed at distances apart of • 033 foot, rect- angular sections. Series Nos. 12, 13, and 14. In these the coefficients c increase with the decrease of inclination. Well-punned gravel, £ to f inch thick, semi- pircular section. Series No. 27. Good dry rubble, semicircular section. G-ontenbachschale at Lake Thun. Class VII. Well-punned gravel, £ to inch thick, rect- angular section. Series No. 4. Eubble in cement, with the bed damaged and covered with mud, rectangular section. Series Nos. 32 and 33. Good masonry in a well-constructed section, rectangular. Series No. 45. Dry rubble of dressed stone, damaged, semicircular section. G'runnbachschale and Gontenbachschale, at Lake Thun. Class VIII. Well rammed gravel, 1 to 1£ inches thick, rectangular section. Series No. 5. Dry rubble, in bad condition, trapezoidal section, Series No. 35. Masonry, damaged, with the bottom covered with stones and silt, rectangular section. Series Nos. 44 and 46. CHAP. I.] PLOW IN OPEN CHANNELS. 39 Planking, with boards • 09 foot broad, nailed at distances of If inches apart ; rectangular section. Series Nos. 15, 16, and 17. Here the coefficients e increase with the decrease of in- clination. Dry rubble, old and much damaged, semicircular section. Alpbachschale at Meiringen. Class IX. Small channels in earth, partly stony soil with a few plants, and partly muddy and covered with grass. Series Nos. 37, 38, 41, 47, 48, 49, and 50. Masonry, in bad condition, with moss and weeds. Series Nos. 34 and 42. Glass X. Small channels in' earth, with plants and grass, and strewn with stones. Series Nos. 36, 40, and 43. Class XI. Streams and rivers. Baumgarten's observations forming Series Nos. 1, and 41 to 50. Those of Poiree and Emmery on the Seine, of LeVeille on the Saone, of Dubuat on the Jard and Hayne, of Funk on the Weser, of Briinings on the branches of the Rhine, of Woltmann (3 ?), of Bonati, etc., on the Po and Tiber, of Legler on the Linth canal, of Grebenau on streams and on the Rhine in Bavaria, of Humphreys and Abbot on the Mississippi and its affluents, of Destrem on the Great Newka and Neva, etc. In these cases the coefficients e increase with the decrease of the inclination. Class XII. Channels of rivers and canals impeded by detritus. Observations of La Ricca on the Rhine at Dom- leschgerthal and Rheinwald, on the Meuse at Misox, on the Plessur at Thur, and those of Legler on the Escher canal. 40 THE NEW FORMULA FOR MEAN VELOCITY. [CHAP. I. 19. Table showing the Eange of r I. II. III. IV. V. VI. 0-02 0-02 76 ,, .. 30 0-045 0-050 90 .* » ... 46 0-06 it , , ,. 0-075 100 m , ,. 55 0-08 t ( t , ., 0-10 106 , , , , ,. 61 0-12 ,, ,. .. 0-14 76 to 95 ., 016 126 ., , , 79 „ 98 ., 68 0-18 t , 81 „ 100 ., 0-20 130 117 t1 83 „ 103 87 72 0-22 ., , , 84 „ 105 ,, 0-24 ,, 86 „ 107 ,, 0-26 136 121 t , 88 „ 109 76 0-28 89 „ 110 ., 0-30 124 90 „ 111 94 79 0-32 , , 92 „ 112 .. 0-34 m , , , 93 ,, 114 0-36 f t ,, ,, 94 „ 115 tm 82 0-38 , , , , 95 „ 116 , , .. -0-40 136 129 109 96 „ 116 99 85 0-42 g , , , 98 „ 117 ., .. 0-44 t t 99 „ 118 , , 0-46 , , , , tt 100 „ 118 . , 87 0-48 100 „ 119 0-50 140 133 113 101 „ 120 103 89 0-55 103 „ 121 91 0-60 144 136 117 106 „ 122 107 93 0-65 , , 107 „ 123 ,, 95 0-70 148 139 120 108 „ 124 111 96 0-75 110 „ 126 ,, 98 0-80 152 142 123 111 „ 127 114 99 0-85 tt ,112 „ 128 100 0-90 156 145 126 113 „ 128 117 101 0-95 114 „ 129 102 1-00 159 148 128 114 „ 130 121 103 1-10 162 150 130 124 105 1-20 165 152 132 127 107 1-30 , , , , . 130 ,. 1-40 ,, , , t , t 133 ,. 1-50 t t # , , 136 ,, 1-60 ,, 139 „ 1-70 , , t , ,, , 142 1-80 , , t , t 145 ,, 1-90 , , ft 148 ., 2-00 " '• " ' 151 •■ CHAP. I.J FLOW IN OPEN CHANNELS. 41 observed Coefficients. (For Swiss feet.) VII. VIII. IX. X. r XI. xn. o- 25 25 to 33 ,, ,, , , ,, o- 50 30 „ 42 ,, .. 0- 75 33 „ 49 ,, , , 1- 00 42 to 58 35 „ 54 ,, , , f , 1- 5 39 „ 61 ,, ,, 2- 54 to 70 42 „ 66 ., ,, , , 2- 5 44 „ 69 ,, , , , , 3- 63 to 78 47 „ 72 #1 , , 3- 5 49 „ 74 .. .. , , . , 4- 69 to 84 51 „ 77 ,, , , # , 4- 5 53 „ 79 ,, , , ,, 5 73 to 88 54 „ 81 57 38 to 52 •• 6 7 8 76 „ 92 78 „ 95 81 „ 97 •• 6i •■ 9 10 82 „ 99 , 84 „ 101 65 42 to 58 •• •• 11 12 13 85 „ 102 86 „ 103 87 „ 104 ■• 68 •• •• 14 15 88 „ 106 89 „ 107 •• 71 46 to 63 •• 16 17 18 90 „ 108 91 „ 108 91 „ 109 *■ 73 •• 19 20 92 „ 110 92 „ 111 75 49 to 66 21 93 „ 111 77 ., 22 93 „ 112 78 52 to 70 23 94 „ 113 79 24 94 „ 113 • .. 80 54 to 72 . , ,, ., 81 ,, , 82 56 to 74 35 to 51 ,, . . 84 ,. . 85 59 to 77 37 to 53 .. .. 86 , , . ». 87 61 to 79 39 to 55 28 to 41 .. 88 64 „ 81 41 „ 57 29 „ 43 .. 90 66 „ 83 43 „ 58 30 „ 44 , .. 91 67 „ 84 45 „ 60 31 „ 46 .. 92 69 „ 85 47 „ 62 33 „ 47 . .. 93 71 „ 87 49 „ 64 34 „ 48 .. 94 72 „ 88 50 „ 65 35 „ 50 .. 95 73 „ 90 52 „ 67 36 „ 51 96 75 „ 91 53 „ 69 37 „ 52 .. 97 76 „ 92 55 „ 70 38 „ 53 .. 98 77 „ 93 56 „ 72 38 „ 54 • *• •* 42 the new formula fob mean velocity. [chap. i. 20. Determination of the Final Coefficients foe the Twelve New Classes in Metrical Measures. The four formulae of D'Arcy and Bazin have the form : r 8 v =^/- r while the general formula we have adopted as a basis is v = c ijrs, in which the coefficient e would be, according to D'Arcy and Bazin, in which the values of a and /3 for Swiss feet are In Category I. a = 0-000 045, ft = 0-000 004 5 ; „ II. a = 0-000 057, p = 0-000 013 3; „ IH. u, = 0-000 072, /? = 0-000 060 0; „ IV. a =0-000 084, /2 = 0-000 350 0; and in our new Category V. a = 0-000 120, /3 = 0-000 700 0. These quantities (a and yS) being in all cases small and incon- venient, the formula may be improved by putting it into another form. Eeducing the expression - , it becomes 1 a r 1 or r r i. I a la 2 u. ar-|-/3 a /3 r+ a CHAP. I.J FLOW IN OPEN CHANNELS. 43 and putting - = a, and -^ = I, it becomes a P ab = a — r + 6' and / ab The values of e in each of the above categories for Swiss feet then become as follows, both in exact and in simplified round numbers : In Category I. c = 4/22 222 • 2222 -r+0-1 ° r >o»o-J^. In Category II. e = ^/n 544 ■ 4093 r+ 0-2333 ->/>"»-£** In Category III. c = fi/lB 899 • 11574 r+ 0-8333 ■«>•»-, IT. In Category IV. c = ^/ll 905 • 49 603 r+ 4-1666 -VUM._«~. In Category V. c = f^J 8333"- 48 611 r + 5-8833 .^8000-t 8 -. The following is also a corresponding reduction and sim- plification of the same coefficients for metrical measures : Category I. " \A , 0-000 004 5 == V 6667- 20 ° 0-000 15 + —"-^1 r +°-° 3 44 THE NEW F0BMTJLA FOE MEAN VELOCITY. [OHAP. I. Category II. c = ^/- 0-00019 0-000 013 3 -V 5286 - 370 r+0-07 Category III. -V- 0-000 24 + 0-000 060 '= v/4160-- 1040 + 0-25 Category IV. c = *y 0-000 28 0-000 350 -sj 3568 - 4460 r+1-25 Category V. -«/ 0-000 40- 0-000 070 -*/ 2500 4375 J- + 1-75 The values of these expressions corresponding to different values of r, for metrical measures, are given in the following table : r I. II. III. IV. V. r I. II. III. IV. V. o-oi 40-8 25-7 12-6 5-3 3-8 0-8 80-2 69-6 56-3 37-3 28-0 0-03 57-7 39-7 21-1 9-2 6-5 0-9 80-3 69-9 57-1 38-7 29-1 0'05 64-6 46-8 26-4 11-7 8-3 1 80-4 70-1 57-7 39-8 30-1 0-07 68-3 51-3 30-2 13-8 9-8 2 46-9 36-5 o-io 71-6 55-6 34-5 16-3 11-6 3 50-2 39-7 0-15 74-5 59-9 39'5 19-6 14-0 4 ■■ * 52-2 41-7 0-2 76-1 62-4 43-0 22-2 16-0 5 53-5 43-0 0-3 77-9 65-3 47-7 26-3 19-1 6 . 54-4 44-0 0-4 78-8 66-9 50-6 29-4 21-6 7 55-0 44-7 0-5 79-3 67-9 52-7 31-9 23-6 8 t . 55-5 45-3 0-6 79-7 68-7 54-2 34-0 25-3 9 56-0 45-7 0-7 80-0 69-2 55-4 35-8 26-7 OO 81-6 72-5 64-5 59-8 50-0 In the last-mentioned formulae Bazin has adopted a mean value of the coefficients a and /3 for each category. These formulas are wanting in mutual dependence, and have the CHAP. I. J FLOW IN OPEN CHANNELS. 45 disadvantage of having two variable coefficients, while that proposed by us has only one. It will also be observed, from an inspection of the formulae and from the preceding table of Bazin's coefficients, that when r = o, e = o, and that when r is of infinite value, the values of c become 81-65, 72'55> 64 '55, and 59*76, in their respective categories, results which would lead one to the almost inadmissible conclusion, that in rivers of unlimited dimensions the influences of various conditions of roughness of the surfaces of their channels would still be appreciable to an important degree in the discharge. Although the calculation of results based on infinite dimensions may be considered impossible, we can- not neglect the indications afforded by them, which in this case lead us to believe that, if in the case of a very large river, like the Mississippi, the channel were lined for certain distances with various materials, such as smooth cement, plank, rubble, ashlar, or coated with vegetation, then the resistance or friction resulting from these various degrees of roughness of surface would be so appreciable that its in- fluence would be felt throughout the whole of such an enor- mous section of water, and the quantity of water discharged would be affected in the same way as is known to be the case in small canals — a very doubtful conclusion. We know that the amount of resistance must be far less on the whole in very large rivers than in small channels, if we take it in proportion to the whole cross section of the water in each case. For example, if we take two cross sections, one of 10 and the other of 20,000 square metres, the resulting resistances taken in proportion to the sections are as 0- 000 01 to 0-000 000 02. We therefore conclude that in a river of unlimited dimensions of section, the resis- tance would be infinitely small. We can also hence assume without error, that in the case of infinite dimensions the differences of influence of various degrees of roughness of 46 THE NEW FOBMTJLA FOE MEAN VELOCITY. [CHAP. I. the wetted perimeter are not constant quantities, and in this respect we would prefer the formula of G-auckler as more correct ; it is, however, in itself unimportant which value in that case should be given to e, in the formula v = c\/rs, for under either assumption v will be infinite. To return to the formula c = \J a —7 , already de- duced from that of D'Arcy and Bazin ; this may be much simplified by modifying it so as to include only one variable coefficient throughout all the categories; and if, in accor- dance with the results of previous examination, we put a = 100 in all categories, and obtain corresponding new values for b, the relation between the two coefficients, as well as the corresponding results, may be made to remain unaltered, whatever may be the values of r. A further simplification of the above formula may be effected by reducing it to the form ab c = a — Jr-\-b This simple formula has been found on trial to give at least as good results as those of D'Arcy and Bazin in ob- taining values of the variable coefficient e. As it appears that -the four categories of D'Arcy and Bazin are both too few in number, and are placed at intervals apart that are far too large, we have effected a further improve- ment by departing from their system of categories, and adopting a system of classification of twelve classes suitable for practical employment in obtaining coefficients applicable to any observed dimensions and conditions. We give here following a table of the values of these coefficients, calculated on our principles, and arranged ac- cording to our twelve new classes, for metrical measures ; as well as a table of observed results, giving the differences in OHAP. I.] FLOW IN OPEN CHANNELS. 47 each case between the coefficient as practically and as theo- retically obtained. It will be noticed that these coefficients have not been modified so as to make any allowance for the influence of the inclination of the water surface, which we have previously shown to be important, in cases of high inclination combined with small values of r. This matter will be taken into con- sideration subsequently. At present we have confined our- selves to the more usual cases of ordinary inclination, and have contented ourselves with deducing one practical for- mula, that takes into consideration all other influences, that is supported both by the observed results of Bazin and those on the Mississippi, and is hence suited to general application. 21. Values op the Calculated Coefficients e fob the Formula v = c \/7~$, arranged in Twelve Classes. (For Metrical Measures.) r I. IL III. IV. V. VI. VII. VIIL IX. X. XI. XII. 0-01 45-5 40-0 33-3 27-0 22 •2 18-2 15-2 122 9-7 7-6 5-6 3-9 003 59-0 53'6 46 '4 39-0 33 1 27-8 23-6 19-4 15-7 12-4 9-4 6-6 0-05 65-1 59-9 52-9 45-3 39 •0 33-2 28-6 23-7 19-4 15"5 11-8 8-4 0-07 68-8 63-9 57-0 49-5 43 •1 37-1 32-1 26-9 22-2 17-8 13-7 9-8 o-io 72-5 67-8 61-2 53-9 47 •5 41-3 36-1 30-5 25-4 20-6 15-9 11-5 0-15 76-3 72-0 65-9 58-9 52 5 46-2 40-9 35-0 29-4 24-1 18-9 13-7 0-2 78-8 74-9 69-0 62-3 56 1 49-8 44-4 38-3 32-4 26-8 21-1 15-5 0-3 82-0 78-5 73-2 67-0 61 54-9 49-5 43-2 37-1 31-0 24-7 18-4 0-4 84-0 80-8 76-0 70-1 64 4 58-4 53-0 46-7 40-4 34-1 27-5 20-6 0-5 85-5 82-5 77-9 72-4 66 9 611 55-8 49-5 43-2 36-7 29-7 22-5 0-6 86-6 83-8 79-5 74-2 68 9 63-3 58-1 51-8 45-5 38-9 31-7 24-1 0-7 87-5 84-8 80-7 75-6 70 5 65-1 59-9 53-8 47-4 40-7 33-4 25-5 0-8 88-2 85-6 81-7 76-8 71 9 66-5 61-5 55-4 49-0 42-3 34-9 26-8 0-9 88-8 86-4 82-6 77-9 73 67-8 62-9 569 50-5 43-8 36-2 28-0 10 89-3 87-0 83-3 78-7 74 69-0 641 58-2 51-8 450 37-5 29-1 2 ,, . .. 60-3 53-7 459 36-7 3 ., .. ., 65-0 58-7 50-9 41-5 4 ,. .. .. 68-3 62-1 54-5 45-0 5 .. .. .. 70-6 64-8 57-3 47-8 6 . ., .. 725 66-8 59-5 50-1 7 ., .. ,. .. 74-0 68-5 61-3 52-0 8 ,, . .. .. 75-2 69-9 62-9 53-7 9 , , ,, ., ,. 76-3 71-1 64-2 551 00 100 100 100 100 100 100 100 100 100 100 100 100 48 THE NEW FORMULA FOR MEAN VELOCITY. [OHAP. 22. Table of observed Eesults, with their corre- sponding Coefficients. Series of Materials and Form Mean Dimensions. Class JD'Arcy of and Bazin. of Section. r s Surface Ereadtb. Depth. Coefficient. No. 28 /Carefully planed timber — \ 0-022 0-00489 0-10 0-042 II 29 /Carefully planed timber — \ Pure cement — semicircular . . 0-016 0-01524 0-10 0-024 1 + 2 24 0-250 0-00142 1-00 0-45 1 + 2 2 „ rectangular .. 0-150 0-00506 1-81 0-18 H + l 25 (Cement with one-third sand — 1 0-260 0-00138 1-00 0-49 II 26 Unplaned plank — semicircular 0-280 0-00152 1-10 0-49 III -2 21 „ trapezoidal 0-250 0-00152 1-40 0-38 IV 22 WMWJWVW^ 0-200 0-00488 1-30 0-30 III -3 23 „ triangular 45° 0-200 00465 0-57 III - 2 6 „ rectangular 0-200 0-00221 1-99 0-26 IV- 2 7 >» » 0-160 0-00489 1-99 0-19 III -3 8 » d 0-140 0-00816 1-99 0-16 III-l 9 >» » 0-220 0-00147 1-99 0-28 IV- 1 10 n u 0-140 0-00587 1-99 0-17 III - 1 11 v »» 0-130 0-00838 1-99 0-15 III 18 >! ») 0-200 0-00460 1-20 0-28 III -2 19 » >» 0-150 0-00427 0-80 0-25 IV + 2 20 Eammed gravel — o-ioo 0-00598 0-48 0-19 IV + 1 27 ( 0-01-- to 0-02»- thick— \ 0-230 0-00136 1-00 0-41 IV 4 ( 0-01™' to 0-02 m - thick— \ 0-200 0-00497 1-83 0-26 VII 5 ( 0-03 m ' to 0-04 m - thick— \ Laths nailed on — 0-220 0-00497 1-80 0-30 VIII - 3 12 • 01 m - apart — rectangular 0-230 0-00147 1-96 0-31 VI 13 o-oi»- „ 0-170 0-00597 1-96 0-20 VI + 2 14 o-oi»- „ 0-150 0-00886 1-96 0-18 VI + 2 15 0-05-- „ 0-290 0-00147 1-96 0-40 IX + 1 16 0-05-- „ 0-210 0-00600 1-96 0-27 IX + 1 17 0-05-- „ 0-190 0-00886 1-96 0-24 IX + 1 12 Ashlar— rectangular 0-540 0-00084 2-59 0-93 III + l 39 » » .... 0-180 0-00810 1-20 0-26 IV-1 3 Brickwork „ .... 0-170 0-00502 1-91 0-20 IV -1 CHAP. I.] FLOW IN OPEN CHANNELS. 49 ■ 1 Series of u^ :.i j 1? Mean Dimensions. Class D'Arcy iuowruue uuu curui and j of Section. Bazin. r s Surface Breadth. Depth. of Coefficient. No. 32 f Bubble, damaged and covered\ \ with silt— rectangular . . J 0-160 0-10076 1-80 0-19 vn+ i 33 Ditto ditto— „ 0-200 0-03686 1-80 0-27 vn+ § 1-4 Bough rubble „ 0-190 0-06000 1-00 0-29 VIII -2£ 1-3 j» ,j 0-220 0-02900 1-00 0-36 VIII + 4 1-6 » » 0-250 0- 01400 1-00 0-47|VIII+l£ 1-5 » M O-270 0-01220 1-00 0-49 VIII - 1 [Bough rubble, the bed 44 :| covered with Btones and ( silt — rectangular 0-450 0-00032 2-00 080 IX + 3 45 Ditto ditto — ditto 0-400 0-00032 2-00 0-70 IX 35 ( Ditto ditto, damaged — trape-\ 0-370 0-01422 1-50 0-70 IX -1J Goutenbachschale, at Lake Thun, | dry rubble, new and in good| 0-100 0-04400 1-70 018 V-2 G'runnbachschale, dry rubble,1 damaged — Bemicircular . . . . J 0-140 0-09927 2-80 0-25 VII -1 Gerbebachechale, ditto ditto 0-059 0-16800 114 0-00 VII -2 Alpbachschale at Meiringen, much) 0-220 0-02740 2-50 0-36 IX -2 Canals, Streams, and Rivers, in Earth. Marseilles Canal — rounded 0-87E 0-000430 6-00 1-35 X-3J Jard Canal „ 0-600 0-000400 6-00 1-35 XI + 2 Chesapeake-Ohio Canal — rounded 1-122 0-000698 6-90 2-40 XII + 1 Canal in England „ 0-740 0-000063 5-40 1-20 IX + 2J Lanter Canal near Neuberg „ 0-554 0-000664 9-00 0-55 XI + 2i Pannerden Canal „ 3-120 0-000224 170-0 3-00 XI -1£ Linth Canal — trapezoidal 2-400 0-000340 37-5 3-30 XI + 4 Canal at Marmels „ 0-705 0-000500 8-00 0-78 XI-3 Hiibengraben „ .... 0-179 0-001300 1-48 0-24 X + 2 0-266 0-000787 3-40 0-35 X + l 0-446 0-000667 5-00 0-60 X-3 50 THE NEW FORMULA FOE MEAN VELOCITY. [CHAP. I. Mississippi Bayou Plaquemine Bayou La Fourche" Ohio ' Tiber Newka Newa Weser (Schwartz) . . . , Elbe Bheinarme in Holland (Brihiings) Seine at Paris Seine at Poissy, Triel, and Meulan Saone at Boconay Haine Bhine at Speyer Rhine at Germersheim — pebbles Bhine at Basle — pebbles Lech — pebbles Saalach — pebbles Salzach — pebbles Ysaar Plessur — pebbles Bhine at Bheinwald Mosa at Misox Bhine at Domleschgerthal Esoher Canal Simme at Lcuk 20-000 5-130 Mean Dimensions. 4-000 4-048 2-883 5-309 10-796 2-900 3-325 3-800 3-700 4-100 3-600 1-600 2-964 3-308 2-100 0-963 0-422 1-260 1-200 070 240 380 600 240 0-500 Surface Breadth. ■000170 •000040 •000093 ■000130 ■000015 ■000014 •000200 •000310 •000150 ■000137 •000070 ■000040 ■000100 ■000112 •000247 •001218 •001150 ■001100 •001200 •002500 •009650 •014200 •011875 •007500 •003000 •010500 760 84' 67 325 73 270 370 120 •96 400 439 228 201 48 20 115 50 13 4 4 5 22 35-0 7-8 7-2 2-4 4-5 6-4 16-0 3-0 3-3 4-5 Depth 2-96 2 V 78 113 0-65 60 35 40 30 40 0-75 1-50 Glass of Coefficient. X XII -2 IX X + l XI + 3 IX -1 IX + 5 XI XII XI XI XI -2 XI -3 XI XI -2 XI + 2 XII + 1 XI + 3 XII + 2 xi + H XI + 2 XI XI XI -6 XII + 4 XII + 2 ( 51 ) CHAPTER II. FLOW IN OPEN CHANNELS IN EARTH. 23. The Application of the various Formula of Eytelwein, Patzig, Hagen, Bornemann, Brunings, Bazin, Hagen (new), Humphreys and Abbot, for Determining Discharges of Canals and Rivers in Earthen Channels. It is of the utmost importance to the hydraulic engineer, that the velocity formulae he may employ in his calculations of discharge and velocity for projected canals should be such as will yield trustworthy results ; it is also of the greatest advantage to him that such tables as he uses for shortening the labour of calculation should not only be based on ac- curate formulae, but should include velocities and discharges for all cases that occur in practice, of canals in channels in earth. We have undertaken the laborious and lengthy task of calculating such tables, with the object of supplanting those now existing that are based on erroneous or defective principles, and of affording undoubtedly accurate results even for channels of extremely large dimensions. Vincent, in his ' Der Wiesenbau dessen Theorie und Praxis,' makes use of the well-known formula v = c V R J with the coefficient of Eytelwein, 92 • 9 for Prussian feet, and 50 • 9 for metrical measures. This in modern times has been shown to give results undoubtedly too large, the velocities in small canals and drains in earth being actually and invari- e 2 52 THE NEW FOBMULA FOB MEAN VELOCITY. [CHAP. II. ably less than those calculated with that coefficient; this conclusion is also supported by our own evidence. At page 71, of the edition of 1858, Vincent gives an ex- ample taken from Patzig's 'Praktische Eieselwirth,' in which the latter gives a discharge of 30 cubic feet per second for a case which, according to Eytelwein, would be 98 cubic feet per second ; according to Bazin iu Category IV., would be 66 ; and according to the new general formula of Ganguillet and Kutter, already mentioned in the 'Zeitschrift des Oester- reichischen Ingenieur und Architekten-vereins ' for the year 1869, would be 64 cubic feet per second, for a coefficient of ; roughness n = • 03 ; this last result is an exact arith- metical mean between those of Vincent and Patzig. In order to compare the results obtained in extreme cases by the various formulae, we give the following small table containing three examples taken from page 266 of Vincent's, work ; the two inclinations adopted throughout the three cases are the highest and lowest, and the sectional areas are the minimum, mean, and maximum. As to these results, we would observe that the results of Vincent and Eytelwein are entirely, and those of Hagen mostly, worthless. An example for the calculation of discharges is given at page 35 of an article in the second number of the ' Cultur- Ingenieur,' by Wasserbau-Inspector Hess. The smallest dis- charge calculated for this example, from among the results of the formulas of Eytelwein, Prony, Hagen (old), and Lahmeyer, is that of the last named, and is 45 • 89 cubic feet per second. The following comparison of this result with those obtained by the newer formulas of Bazin, Bornemann (Gauckler's system), Hagen (1868), and Ganguillet and Kutter, show that the whole of these last give results still smaller. CHAP. H.] FLOW IN OPEN CHANNELS IN EARTH. 53 Authors. Vincent (Eytelwein) Hagen(1868) .. .. Bazin, Category IV. . . Ganguillet and Kutter\ » = 0-030 .. ../ a = 2 Square feet. J = 0-000 069 44 J = 0-000 416 66 a = 22 Square feet. J = 0-000 069 44 J = 0-000 41666 Discharges in cubic feet per second. 1-07 1-26 0-43 0-40 2-62 1-70 1-06 1-03 20-19 24-15 12-63 10-56 49-44 32-58 30-91 27-52 a = 80 Square feet. 7 = 0-000 069 44 7=0-000 416 66 Vincent (Eytelwein) Hagen(1868) .. .. Bazin, Category TV. Ganguillet and Kutterl n=0-030 .. ../ Discharges in cubic feet per second. 102-45 115-76 75-84 62-64 250-89 156-08 185-76 156-16 Cubic Feet per Second. Lahmeyer 45-89 Bazin, Category IV 35-61 Bornemann (Gauckler) 39-80 Ganguillet and Kutter a. For channels in good order » = • 025 .. 35- 70 6. In moderately good order n = 0-030 .. 31-06 c. For channels obstructed with detritus, and strewn with stones, &c n = 0-035 .. 26-80 25. The Formula of Bornemann and Hagen. Besides the'tables based on the above-mentioned formulae, there are some of a Prussian hydraulician based on a formula v = 83 V it J ; it is perhaps almost needless to remark tha* this gives too high discharges for small canals in channels in earth, in the same way, though not to so great a degree, as 54 THE NEW F0EMU1A FOB MEAN VELOCITY. [CHAP. II. the formula of Eytelwein. We may hence conclude that the results of the most modern experimental observations, which are those of Bazin, are not yet generally known and employed. We have already in the ' Zeitschrift des Oesterreichischen Ingenieur und Architekten-vereins,' for 1869, commented on the inapplicability of any of the old formulae that have single constant coefficients to all the various degrees of roughness of wetted perimeter; we have also mentioned that we have based our conclusions principally on the careful and valuable observations of D'Arcy and Bazin, recorded in the ' Kecherches Hydrauliques,' Paris, 1865 ; we have besides proved that any formula must assume a bino- mial form in order to give correct variable values of C, the coefficient in the general formula v = c*/ R.J. This is the case in the new formula of Bornemann, E J -= 7 x v (see ' Civil-Ingenieur '), which we consider the best of the older formulas. We have not, however, enough space here to enable us to support our opinion on this subject by bringing forward results of observation, and comparing them fully with the results of these various formulas, and we therefore refer to our previously mentioned article for further information about this formula, as well as for fuller detail as to the derivation of the formula which we have adopted. For a stronger recommendation of the new formula of Hagen we must refer the reader to the articles contained in the 'Koniglich Akademieder Wissen-Schaften,' Berlin, 1868, . and the 'Mittheilungen des Hannoverschen Gerverbevereins,' 1868; and confine ourselves at present to the following remarks on it. This formula v = 2-425 V KVTfor me- trical measures is deduced from the results of the observations of Von Brunings, made with his own tachometer, on the CHAP. II.] FLOW IN OPEN CHANNELS IN EARTH. 55 lower Ehine, from 1790 to 1792, on the Waal, the Leek, and the Yssel, on those seventy-five years afterwards, the results of the observations of the Mississippi Commission given in Humphreys and Abbot's work, on those on the Seine at Paris, by Poiree, and on those on the Rigoles de Chazilly et de Grosbois by Bazin, or altogether on sixty-six cases. While leaving the term V R unaltered, Mr. Hagen introduces the sixth root of the sine of the inclination, instead of its square root, into his formula, with the object of combining the re- sults of the experience gained on the Mississippi with that on the European rivers : the introduction of this sixth root also leads Mr. Hagen to the conclusion that the coefficient of Eytelwein, 50 * 9 fpr metrical measures, gives velocities that are nearly three times too high. A conclusion that can only be correct in some cases. In making the trials necessary for determining the ex- ponents most appropriate for the inclination, there is no objection to leaving the term J B, in the formula unchanged as the resulting errors introduced are approximately the same, when the exponents of J are taken at either £ and £, or £ and £. The American results (see Hagen's article) require an exponent of £ or J'> those of the Netherlands require J^> those of the Seine at Paris J» or Ji, and those of the Eigoles, J*. Hence the question arises whether it would not be more advisable to give the term R any other exponent instead of £, which could be suitably applied to both E and J in the velocity formula. In the article referred to the maximum and minimum values of R occurring in large rivers and small canals have very properly been taken into consideration, while however it is remarkable that the extreme values of J have been neglected, although the essential distinction between the American and the European 56 THE NEW FORMULA FOE MEAN VELOCITY, [CHAP. n. formulae lies in the difference of the exponent assigned to the inclination. All the rivers as well as all the small canals compared in his article have low inclinations, in no case exceeding 0*001 : if rivers of high as well as those of low inclination had been included, as is absolutely essential in attempting to deduce a general formula, there is no doubt that some other exponent for J would have been adopted instead of £. As also in addition to this the influence of the degree of roughness of the wetted perimeter on the velocity of discharge has been entirely neglected, in spite of the evidence afforded by the observations of D'Arcy and Bazin, the new formula of Hagen thus becomes entirely useless in calculations of discharge of the small canals and drains of the agriculturist, where this influence has most effect. This formula also appears to be not suited to artificial channels of any description, but merely to rivers ; while even in these the various grades of roughness of the wetted perimeter are doubtless productive of effect, and the results due to weeds and detritus in their channels cannot be justly neglected. The formula of Humphreys and Abbot has been pre- viously demonstrated to be useful only under special con- ditions, and to be perfectly useless for high inclinations ; since, then, the exponent in their formula is merely raised from £ to \, the same defect will show itself to a greater degree in that of Hagen, where the exponent is ^. For e'xample, in a case of welUconstructed channels in masonry in good order, having an inclination of - l, the formula of Humphreys and Abbot gives only one quarter, and that of Hagen only one-eighth, of the actually observed velocity of discharge. In cases of lower inclination the differences are not so great. We have compared several hundred results of observations on rivers of various hydraulic inclinations having the same CHAP. II.] PLOW IN OPEN CHANNELS IN EAETH. 57 degree of roughness of surface of channel, as well as similar values of E, and tried them in the expression m-- but we have never found x to be £ ; on the Mississippi alone it was found to be \, while in most cases it was approximately from | to b or averaged £. If we plot a series of values of c, for the formula e =■ , VKJ that have been obtained from observed results, and put them as ordinates to a series of abscissae representing the corre- sponding values of E, they will be seen to show a steady increase corresponding to the increase of the values of E : these increments being greatest among the smaller values of E, and less among the greater, the resulting curve falling off very much indeed among the least values of E, showing that at last when E is infinitely small, e = 0. When, however, we plot in the same way the coefficients of the Eytelwein formula, they give us a horizontal straight line, having an ordinate of 50 • 9 ; and when we plot those of 2 • 425 the formula of Hagen, in which C = ,, T ; we find them to vary not with E, but with J. These widely opposed deduc- tions show how it is that both' the formula of Eytelwein and Hagen often give results that are positively impossible ; — a fact that is also true of the formula of Humphreys and Abbot. 26. Safe Bottom Velocities. Before going on to our own formula and our tables of velo- cities and discharges, we will take the opportunity of mention- ing the maximum velocities determined by Dubuat as suitable 58 THE NEW FORMULA FOR MfeAN frELOCTTt. [CHAP. II. to channels in various descriptions /of soil, whiah are taken from Morin's ' Aide Meinbire de Mecanique Pratique/ p. 63, 1864. The first column'in the foMowiig table'gives the safe b ottom velocity, and the second' the 'mean velocity of the cross section ; the formula by which these are calculated is %• = »„ + 6 vEJ , for metrical measures. ?\ We are unable, for want of observations, to judge how far these figures are trustworthy. The inclinations certainly have no influence in this case, as the corresponding velocities are mutually interdependent, but the variation of the depth of water is most probably of consequence, and in shallower depths the soil of the bottom is possibly less easily and rapidly damaged than in greater depths, under similar conditions of soil and of inclination. Yet this effect is not very large, while that of the actual velocity of the water is of the highest importance. Hence it appears that these figures may be assumed to be rather disproportionately small than too large, and we therefore recommend them more confidently. 1. Soft brown earth 0-076 0-100 2. Softloam .. .. 0-152 0-200 3. Sand 0-305 0-400 4. Gravel 0-609 0-800 5. Pebbles 0-914 1-200 6. Broken stone, flint 1-220 1-700 7. Conglomerate, soft slate . . . . 1 • 520 2-000 8. Stratified rock 1-830 2-500 9. Hard rook 3-050 4-000 chap. ii.] flow in open channels in bakth. 59 27. The Derivation of the New Formula for Coeffi- cients of Mean Velocity. The derivation of this formula is entirely omitted in the articles of the ' Cultiir-Ingenieur,' the reader being referred to the 'Zeitschrift des Oesterreichischen Ingenieur u'nd Architekten-vereins/ 1869, where it is given at full length with explanatory diagrams. The following brief mention of the mode in which the formula is derived, is therefore extracted from that work with the view of supplying in a small degree the information that Mr. Kutter was from want of space compelled to omit in his article in the ' Cultur- Ingenieur.' The formulae of Bazin have the general form / EJ f v = /v/ -Q where c = \/ - putting it becomes ■+fi v -+fi - = a and — = b a p / a.R.J / a » =.V b where c = V b (1) or by adopting other coefficients, a', V, or a", b", it maybe put into either of the forms c= a ' v (2) «« = — y , (3) 60 THE NEW FOBMULA FOB MEAN VELOCITY. [CHAP. U. A tabulation of these coefficients, together with those based on observed results, is necessary to determine which of these three coefficients is most correct; we therefore attach the following tabulated results for the series Nos. 24, 2, 26, 6, 9, 32, 33, and 17 of D'Arcy and Bazin, which comprise values of the coefficients c, as calculated according to the three formulae already mentioned, and their differ- ences from the actual values of c, as obtained by observation in those series. Values of the Coefficients e— {Metrical). Observed. 00 Formula 1. CO Differences. Formula 2. Differences. Formula 3. Series No. 24. 73-0 73-0 0-0 73-0 o-o 73-0 00 76-8 77-6 +0-8 77-2 +0-4 77-8 +10 78-2 80-0 +0'8 79-7 + 1-5 80-1 + 1-9 81-4 81-4 o-o 81-2 -0-2 81-5 +0-1 82-2 82-5 +0-3 82-4 +0-2 82-6 + 0-4 83-3 83-3 0-0 83-3 00 83-3 o-o 83-1 84-0 +0-9 84-1 +1-0 83-9 +0-8 84-3 84-6 + 0-3 84-7 +0-4 84-4 +0-1 86-4 84-9 -1-5 85-2 -1-2 84-7 -1-7 86-9 85-2 -1-7 ' 85-7 -1-2 85-1 -1-8 87-4 85-6 -1-8 86-1 -1-3 85-4 -2-0 87-9 85-7 _2'2 86-2 -1-7 85-5 -2-4 Totals of differences 10-3 •• 91 •• 12-2 Series No. 2. 63-3 63-3 0-0 63-3 0-0 63-3 0-0 68-0 67-7 -03 67-7 -0-9 68-0 o-o 69-0 70-0 +1-0 69-2 . +0-2 70-3 +0-3 71-9 71-2 -0-7 70-5 -1-4 71-5 -0-4 71-9 72-2 +0-3 71-6 -0-3 72-4 +0-5 73-4 72-9 -0-5 72-4 -1-0 73-1 -0-3 73-6 73-5 -o-i 73-0 -0-6 73-6 0-0 74-0 73-9 -01 73-5 -0-5 74-0 o-o 74-5 74-3 -0-2 74-0 -0-5 74-3 -02 74-5 74-6 +0-1 74-4 -0-1 74'6 +0-1 74-9 74-8 -o-i 74-8 -0-1 74-9 0-0 75-1 75-1 o-o 75-1 0-0 75-1 0-0 Totals of differences - 34 •• 5-6 •• 1-8 CHAP. H.] FLOW IN OPEN CHANNELS IN EAETH. 61 Observed. (c). Formula 1. CO- Formula 2. CO- Differences. Formula 3. 0=„). Differences. Series iVo. 26. 59-4 59-4 0-0 59-4 0-0 59-4 o-o 62-9 64;2 + 1-3 63-7 +0-8 64-5 + 1-6 66-5 66-4 -0-1 65-7 -0-8 66-8 +0-3 67-9 68-1 +0-2 67-6 -0-3 68-5 +0-6 68-0 69-4 + 1-4 68-9 + 0-9 69-7 +1-7 69-5 70-3 +0-8 69-9 +0-4 70-6 + 1-1 68-8 71-1 + 2-3 70-7 + 1-9 71-3 +2-5 70-7 71-6 +0-9 71-3 +0-6 71-8 + 1-1 70-7 72'2 +1-5 71-9 +1-2 72-3 + 1-6 72-0 72-6 + 0-6 78-4 +0-4 72-7 +0-7 72-0 73-0 +i-o 72-9 + 0-9 73'0 +1-0 73-1 73-3 +0-2 73-2 +0-1 73-3 +0-2 73-5 73-5 0-0 73-5 0-0 73-5 o-o Totals of c ifferences 10-3. •■ 8-3 •• 12-4 Series No. 6. 49-8 49-8 0-0 49-8 o-o 49-8 0-0 52'3 54'8 + 2-5 53-8 + 1-5 54-7 +2-4 55'0 57-3 +2-3 56-6 + 1-6 57-7 +2-7 57-0 58-9- +1-9 58-2 +1-2 59-3 +2-3 57-2 60-0 +2-8 59-5 +2-3 60-4 +3-2 60-2 60-8 +0-6 60-3 +0-1 61-1 + 0-9 60-7 61-9 + 1-2 61-5 +0-8 62-1 +1-4 60'7 62-2 + 1-5 61-7 +1-0 62-3 +1-6 61-9 62-6 +0-7 62-3 +0-4 62-6 +0-7 62-2 63-0 +0-8 62-8 +0-6 62-8 +0-6 63-7 63-2 -0-5 63'2 -0-5 63-0 -0-7 63-6 63-6 o-o 63'6 0-0 63-6 00 Totals of ( 1 ifferences 14-8 •• 10-0 ■• 16-5 Series No. 9. 49-3 47'2 -2-1 47-9 -1-4 1 46-2 -31 53 7 53-7 0-0 53-7 0-0 53-7 0-0 58-2 59-9 +1-7 59-5 + 1-3 60-2 +2-0 61-6 63-0 +1-4 62-7 + 11 633 +1-7 64-2 65-0 +0-8 64-9 +0-7 65'2 + 1-0 665 66-5 0-0 66-5 0-0 66-5 o-o 67-2 67-8 +0-6 67-9 +0-7 67-6 + 0-4 Totals of t ifferences 6-6 •• 5-2 8-2 62 THE NEW FORMULA FOE MEAN VELOCITY. [CHAP. II. (Observed. Formula 1. Differences. Formula 2. Difference?. Formula 3. Series No. 32. 37-5 41-2 42-7 451 37-5 41-5 43-8 45-1 0-0 +0-3 +1-1 00 37-5 41-4 43-7 45-1 0-0 + 0-2 +1-0 0-0 37-5 41-7 43-9 45-1 0-0 +0-5 + 1-2 0<0 Totals of d inferences 1-4 •• 1-2 1-7 -Series No. 33. 39-9 44-9 45-1 47-0 39-9 43-9 45-8 47-0 o-o +2-0 +0-7 0-0 39-9 43-8 45-6 47-0 0-0 +1-9 +0-5 00 39-9 44-1 45-9 47-0 00 + 2-2 +08 o-o Totals of d inferences 2-7 2-4 * 30 Series No. 17. 26-9 26-9 0-0 26-9 0-0 26-9 0-0 28-3 29-8 +1-5 29-4 +r-i 29-9 + 1-6 308 32-0 + 1-2 31-6 +0-8 32-1 +13 32-3 33-1 +0-8 32-8 +0-5 33-2 +0-9 33-4 33-8 +0-4 33-6 +0-2 33-9 +0-5 34-0 34-3 +0-3 34-2 +0-2 34-3 +0-3 34-7 34-7 0-0 34-7 0-9 34-7 o-o Totals of c inferences 4-2 2-8 4-6 Collection of Totals of Differences. Series 24 10-3 9-1 12-2 „ 2 3-4 5-6 1-8 „ 26 10-3 8-3 12-4 ,. 6 14-8 10-0 ... 16-5 ' ,. 9 6-6 5-2 , 8-2 „ 32 1 ; 4 1-2 .. 1-7 » 33 2-7 2-4 3-0 » 17 -4-2- 2-8 • 4-6 Totals 53-7 , » 44-6 •■ 60-4 CHAP. H.J FLOW IN OPEN CHANNELS IN EARTH. 63 The above is conclusive in demonstrating that formula No. 2 is the best of the three, and that it yields results at least as good as the established formula of Bazin; assuming therefore this form '+£ and inverting it, it becomes V 1 + 1 ' VB 1_,V_ 1 o a' «' + a' X VB ; and this is the equation to a straight line, whose abscissa = —pos and whose ordinates are - : the distance of its inter- section with the axis of the ordinates from the origin of the co-ordinates is — , and the tangent of its inclination with the "" -LI axis of the abscissae is —r. a A practical examination and comparison of these plotted coefficients with the results of observation on the Seine, Saone, Weser, a branch of the Rhine in Holland, and the Linth canal, show that this equation to the straight line, does not hold entirely good, and that the observed results on the contrary indicate a curvature ; it also shows that a' is not a constant quantity, but is dependent on the value of i '; so that V may either be taken as = no! or = n 2 a', where n represents the coefficient of roughness of the natural surface of the wetted perimeter. 64 THE NEW FORMULA FOR MEAN TELOCITY. [CHAP. H. Putting therefore the equation into the form z z may = — = in which case x = nz = a,Jn, or z may = — in which case x = n 2 z = a n. J n After much examination, and further comparison, the fol- lowing form is finally established as preferable : z = a A — .and hence a; = nz — I = an; n and by introducing these quantities, the equation becomes I We have, however, already shown that in very large rivers the coefficients c, obtained from observation, decrease with the increase of the inclination of the water-surface ; and that the formula, in order to be rendered applicable to all cases what- ever, must therefore be modified by introducing a term to suit the extremes of inclination, as well as the extreme limits of sectional area. When E = infinity, c will = z, and the coefficients z will have their values represented by a hyper- bolic curve ; the terms of the equation to which curve can then be practically determined. Hence, putting the coefficients of the formula become I m z = a-\ ^~ n J % = n z — I = (a -\- y) n > CHAP. H.J FLOW IN OPEN CHANNELS IN EABTH. 65 and the formula itself takes the final form, " 1 + (« + ")^' The effect of the introduction of these quantities into the equation is shown by comparing its values with those of the observed results on the Mississippi and other large rivers, after plotting their curves. They are found to be not only in ac- cordance with them, but also with the following series of Bazin, Nos. 6, 8, 9, 11, 12, 14, 15, 17, 32, and 33. The form of the new general formula is hence perfectly established. The values of its various terms are deduced for metrical measures from a geometrical consideration of the hyperbolic curve plotted from it, and its coincidence with that obtained from the Mississippi observations at ten points in its length. Giving to R and J successively their ultimate values, and taking again the first general form of the equation x 1 + 7^ fib in which the new value of z will be A + -=- after the intro- duction of the new term ; in the extreme case, when J is of infinite value, A will =a + -, and this is found to be = 60 ' n for metrical measures, and = I, which is found = 1, and hence - = n = 0-027 for the Mississippi; c * _i =37 . n ~ 0-027 66 THE NEW FORMULA FOB MEAN TELOCITY. [CHAP. II. therefore a = A - - = 60 - 37 = 23. n Taking again the equation z=A+ -y;m will be the tangent of the inclination of the asymptote with the axis of abscissae ; this straight line having as abscissas the values of y and as ordinates the values of z ; for the extreme case of J = • 000 003 63 and z = 487 as determined from the curve, m we obtain from the equation a = A 4- -=- where A = 60 m = 0-00155. The values of n are in the same way obtained by plotting observed results; and are found to vary between 0-009 and * 040 ; their values as thus obtained are given in the follow- ing tables, as are also those of a -( — for various values of n, n and those of -=- for various values of J. The values of us and z in the formula I+ 7k are besides given for six successive values of n, namely n = 0-010, 0-012, 0-013, 0-017, 0*025, and 0-030, in the table immediately following them. Substituting the values of the coefficients deduced in this manner in the formula n J i i / i m \ n 1 + ( a + J-)7E CHAP. H.] FLOW IN OPEN CHANNELS IN EARTH. it becomes for metrical measures 67 a8 + ± + °^» n J 0-00155^ n the formula for mean Telocity of discharge thus becoming v = 2B + 1 - + -^ ' n ' J l4-(23 + °l°f 155 ) • Vej 28. Table giving the Observed Values of the Co- efficient n, COEEESFONDING TO THEIE DATA OF OBSERVATION, IN METEICAL MEASUEES. •urtni. Breadth j The Series of T!;izin. R J at water Burface. Depth. | n No. i 28 Carefully planed plank 0-022 0-0048922 0-100-0420-0096 29 0-016 0- 0152370 0100-024000870 24 In cement — semicircular , . . 0-250 0-0014243 1-000-45 0-01005 2 „ rectangular 0-150I0- 0050600 1-81J0-18 ,0-01040 25 ( „ with one \ sand— semicircular . third\ 0-260 0-0013802 1-000-49 0-01113 26 Flank — semicircular 280 0-0015227 1-100-49 0-01195 21 „ trapezoidal . .. 0-250 0-0015213 1-400-38 0- 01255 22 0-200 0-0048751 1-300-30 0-01190 23 Plank — triangular 45° . , 0-200 0- 0046550 1-300-57 011900 6 „ rectangular 0-200 0-0022136 1-990-26 0- 13000 7 0-160 0-0048889 1-990-19 0- 01190 8 0-140 0-0081629 1-990-16 0-01150 9 0-220 0- 0014678 1-99028 0-012H0 10 0-140 0-0058744 1 990-17 0-01170 11 0-130 0-0083805 1-990-15 0-01140 18 0-200 0-0045988 1-200-28 19 0-150 0- 0042731 0-800-25 20 v » 0-100 0059829 48 019 Rammed gravel — 27 r 0-01 to ■02"' thick— ( 0-01 to 0-02»thick- semi-\ 0-230 0-0013639 1-000-41 0-0163 4 -rect-1 1 0-200 0-0049736 1-830-26 0-0170 F 2 68 THE NEW FORMULA FOB MEAN TELOCITY. [CHAP. II. Breadth The Series of Bazin. R J at water surface. Depth. n No. Battens placed — 12 • 01 m apart — rectangular 0-230 0-0014678 1-96 0-31 0-0149 13 0-01 m „ „ 0- 170,0 -0059664 1-96 0-20 0-0147 14 01 m „ „ 0-150 0-0088618 1-96 0-18 0-0149 15 0-05 m „ „ 0-290 0-0014678 1-96 0-40 0-0208 16 0-05 m „ „ 0-210 0-0059976 1-96 0-27 0-0211 17 0-05" „ „ • 0-190 0-0088618 1-96 0-24 0-.0215 12 Ashlar — rectangular 0-540 0-0008400 2-59 0-93 0-0133 3 Brickwork ,, .... 0-170 0-0050250 1-91 0-20 0-0129 39 Ashlar — rectangular 0-180 0-0081000 1-20 0-26 0-0129 Bubble— 32 / Bather damaged — rectan-l 0-160 0-1007600 1-80 0-19 0-0167 33 » » 0-200 0-0368560 1-800-27 0-0170 1-4 i. » new 0-190 0-0600000 1-000-29 0-0180 1-3 >» » >» 0-220 0-0290000 1' 00 0-36 0-0184 1-6 » » >) 0-250 0- 0140000 1-000-47 0182 1-b 0-270 0-0122000 1-000-49 0-0192 44 f With deposits on the bed\ \ — rectangular .. ../ 0-450 0-0003200 2-00 0-80 0204 46 0-400 0-0003200 2-00 0-70 00210 35 Damaged rubble — trapezoidal 0-370 0-0142210 1-50 0-70 0-0220 Other Observations. Gontenbachschale, new rubble- semicircular G'runnbachschale — semicircular — damaged G erbebachschale — semicircular — Alpbarhschale ■ — semicircular much damaged }| 0-100 ] 0-140 \l 0-059 •} 0-220 Marseilles Canal Jard Canal Chesapeake Ohio Canal Canal in England Lanter. Canal, at Newbury .. Pannerden Canal, in Holland Canal of Marmels Lintb. Canal Hiibengraben Hockenhach Speyerbach Mississippi Bayou Plaquemine Bayou Latorische Ohio, Point Pleasant Tiber, at Borne Newka 0-875 0-600 1-122 0-740 0-554 3-120 0-705 2-400 0-179 0-266 0-446 20-000 5-130 4-000 2-048 o- 0' 883 309 044000 099270 168000 027400 000430 000400 000698 000063 000664 000224 000500 000340 001300 000787 000667 000667 0001700 0000400 0000930 0001300 0000150 0-0145 0-0175 0-0185 0-0230 •35 0-0244 ■35 0-0255 •40 0-0330 •20,0-0184 •550-0262 •00,0-0254 •78,0-0301 ;• 30 0-0222 •24 0-0237 •35;0 0243 •600 '0260 ■00 0-0270 •80,0-0294 ■20 0-0200 •40 0210 0-0228 0-0252 CHAP. II.] FLOW IN OPEN CHANNELS IN EARTH. The Series of Bazin. Hewa Weaer , Elbe Bhine, in Holland . . Seine, at Paris Seine, at Poissy, &o. Saone, at Raconnay Haine 10-796 900 325 800 700 100 3-600 1-600 0-0000140 0-000200 0' 000310 0-000150 0-000137 0-000070 0-000040 0-000100 Breadth at waterjDepth. surface, 370 120 96 400 00 6-00 00 ( 3-00 0013-30 00 4-50 0-0262 0-0232 0-0285 0-0243 0-0250 0-0260 0-0280 0-0260 Channels obstructed by Detritus. The Bhine, at Speyer .. Bhine, at Grertnersheim Rhine, at Basle Lech Saalach Salzach Isaar Eschcr Canal Plessur Rhine, at Bhiuewald Mosa, at Misox Bliine, at Domlesohgerthal Simme, at Lenk 2-964 3-308 2-100 0-963 0-422 260 200 240 070 240 0-380 0-600 0-500 000112 000247 001218 001150 001100 001200 002500 003000 009650 01420 01187 00750 01050 439-00 228-17 201-27 48-00 20-70 115-00 50-00 22-00 13-00 4-30 4-00 5-00 2-96 2 ; 78 1-13 0-65 3-60 0260 0227 0300 0220 0270 0280 0305 0300 0270 0310 0310 •0350 •0345 29. Table giving the Values of the Expressions a + - and ^r- for Metrical Measures, correspond- ING TO VARIOUS VALUES OF W AND OF J RESPECTIVELY. n I a+ - n 71 l a + - n n l a+ - n 0-0090 134 0-0170 82 0-0250 63 0-0095 128 0-0175 80 0-0260 61 o-oioo 123 0-0180 79 0-0270 60 0-0105 118 0-0185 77 0-0280 59 0-0110 114 0-0190 76 0-0290 57 0-0115 110 0-0195 74 0-0300 56 0-0120 106 0-0200 73 0-0310 55 0-0125 103 0-0205 72 0-0320 54 0-0130 100 0-0210 71 0-0330 53 0135 97 0-0215 70 0-0340 52 0-0140 94 0-0220 68 0-0350 52 0-0145 92 0-0225 67 0-0360 51 0-0150 90 0-0230 66 0-0370 50 0-0155 88 0235 66 0380 49 0160 86 0-0240 65 0-0390 48 0-0165 84 0-0245 64 0-0400 48 70 THE NEW FOKMTTLA TOE MEAN VELOCITY. [CHAP. II. m m nt J J J 7 J J 0-000000 00 0-000050 31 o-oooio 15-5 1 1550 51 30 11 14 2 775 52 30 12 13 3 517 53 29 13 12 4 387 54 29 14 11 5 310 55 28 15 10 6 258 56 28 16 10 7 221 57 27 17 9 8 194 58 27 18 9 9 172 59 26 19 8 o- 000010 155 0-000060 26 0-00020 8 11 141 61 25 21 7 12 129 62 25 22 7 13 119 63 25 23 7 14 111 64 24 24 6 15 103 65 24 25 6 16 97 66 23 26 6 17 91 67 23 27 6 18 86 68 23 28 6 19 82 69 22 29 5 0-000020 77 0-000070 22 0-00030 5 21 84 71 22 31 5 22 70 72 22 32 5 . 23 67 73 21 33 5 24 65 74 21 34 5 25 62 75 21 35 4 26 60 76 20 36 4 27 57 77 20 37 4 28 55 78 20 38 4 29 53 79 20 39 4 0-000030 52 0- 000080 19 0-00040 4 31 50 81 19 0-00050 3 32 48 82 19 0-00060 3 33 47 S3 19 0-00070 2 34 46 84 18 0-00080 2 35 44 85 18 0-00090 2 36 43 86 18 0-001 1-55 37 42 87 18 2 0-8 38 41 88 18 3 0-5 39 40 89 17 4 0-4 0-000040 39 0-000090 17 5 0-3 41 38 91 17 6 0-3 42 37 92 17 7 0-2 43 36 93 17 8 0-2 44 35 94 16 9 0-2 45 34 95 16 0-010 015 46 34 96 16 100 0-02 47 33 97 16 00 o-oo 48 32 98 16 49 32 99 16 OHAP. n.] FLOW IN OPEN CHANNELS IN EAETH. 71 30. Table op the "Values op the Expressions z and x, fob metrical measures corresponding to Different Values op n and J in the Formula l + 7S , I , m . ( m\ , = a-\ h t an d » = l«+-;l» =nz — I n 3 \ J/ m = 0-010 B = 0-012 »= 013 « = o-on Inclination J z X Z X z X Z X o-oooo CO 00 00 00 00 00 OO 00 o-oooi 138-5 0-385 121-8 0-462 115-4 0-500 97-3 0-654 2 130-7 0-307 114-1 0-369 107-7 400 89-6 0-523 3 128-2 0-282 115-1 0-338 105-1 0-366 87-0 0-479 4 126-9 0-269 110-2 0-320 103-8 0-349 85-7 0-457 5 126-1 0-261 109-4 0-313 103-0 339 84-9 0-444 6 125-6 0-256 108-9 0-307 102-5 0-332 84-4 0-435 7 125-2 0-252 108-5 0-302 102-1 0-328 84-0 0-428 8 124-9 0-249 108-3 0-299 101-8 0-324 83-8 0-424 9 124-7 0-247 108-0 0-297 101-6 0-321 83-5 0-420 0010 124-5 0-245 107-9 0-295 101-5 319 83-4 0-417 20 123-8 0-238 107-1 0-285 100-7 0-309 82-6 0-404 30 123-5 0-235 106-8 0-282 100-4 0-306 82-3 0-400 40 123-4 0-234 106-7 0-281 100-3 0-304 82-2 0-398 50 123-3 0-233 106-6 0-280 100-2 0-303 82-1 0-396 60 123-3 0-233 106-6 0-279 100-2 0-302 82-1 0-395 70 123-2 0-232 106-5 0-279 100-1 0-301 82-0 0-395 80 123-2 0-232 106-5 0-278 100-1 0-301 82-0 0-394 90 123-2 0-232 106-5 0-278 100-1 0-301 82-0 0-394 0-0100 123-15 0-231 106-48 0-278 100-06 0-301 81-97 0-393 0-0200 123-08 0-230 106-41 0-277 99-99 0-300 81-90 0-392 0-0300 123-05 0-230 106-38 0-277 99-96 0-299 81-87 0-392 0-0400 123-04 0-230 106-37 0-276 99-95 0-299 81-86 0-392 0500 123-03 0-230 106-36 0-276 99-94 0-299 81-85 0-391 0-0600 123-03 0-230 106-36 276 99-94 0-299 81-85 0-391 0-0700 123-02 0-230 106-35 0-276 99-93 0-299 81-84 0-391 0-0800 123 02 0-230 106-35 0-276 99-93 0-299 81-84 0-391 0-0900 123 02 0-230 106-35 0-276 99-93 0-299 81-84 0-391 o-iooo 123-01 0-230 106-34 0-276 99-92 0-299 81-83 0-391 CO 123-00 0-230 106-33 0-276 99-91 0-299 81-82 0-391 722 THE NEW FOBMTTLA FOE MEAN VELOCITY. [CHAP. II m = 025 n = 030 Inclination J. 2 X z X 0-000000 00 00 00 00 0-000001 1613 39 325 1606-3 47-190 3 579 7 13 492 573-0 16-190 5 373 8 325 366-3 9-990 7 284 4 6 111 277-8 7-333 0-000010 218 4 450 211-3 5-340 15 166 3 3 157 159-7 3-790 20 140 5 2 512 133-8 3-015 25 125 2 215 118-3 2-550 30 114 7 1 867 108-0 2-240 35 107 3 1 682 100-6 2-019 40 101 7 1 544 95-1 1-852 45 97 4 1 436 90-8 1-723 50 94 1 350 87-3 1-620 55 91 2 1 280 84-5 1-535 60 88 8 1 221 82-2 1-465 65 86 8 1 171 80-2 1-405 70 85 1 1 128 78-5 1-354 75 83 7 1 092 77-0 1-310 80 82 4 1 059 75-7 1-271 85 81 2 1 031 74-6 1-237 90 80 2 1 005 73-6 1-206 95 79 3 983 72-6 1-180 0-000100 78 5 962 71-8 1-155 150 73 3 833 66-7 1-000 200 70 7 769 64-1 0-922 300 68 2 704 61-5 0-845 400 66 9 672 60-2 0-806 500 66 1 652 59-4 0-783 600 65 6 640 58-9 0-767 700 65 2 630 58-5 0-756 800 64 9 623 58-3 0-748 900 64 7 618 58-0 0-741 o-ooi 64 55 614 57-88 0-736 0-002 63 77 594 57-10 0-713 0-003 63 52 588 56-85 0-705 0-004 63 39 585 56-72 0-702 0-005 63 31 583 56-64 0-699 0-006 63 26 581 56-59 0-698 0-007 63 22 580 56-55 0-697 0-008 63 19 ■580 56-52 0-696 0-009 63 17 579 56-50 0-695 0-01 63 15 ■579 56-48 0-694 0-02 63 08 577 56-41 0-692 0-03 63 05 •576 56-38 0-691 0-04 63 04 •576 56-37 0-691 0-05 63 03 •576 56-36 0-691 00 63-00 0-575 56-33 0-690 CH4P. H.J FLOW JN OPEN CHANNELS IN EARTH. 73 31. The Transformation op the Einal Formula from Metrical into Swiss, English, and other Measures. The general formula for coefficients of mean velocity as deduced in the preceding paragraph, is z v x where c = ^jjj the terms of which are ,IS I m Z=a+ n+J ""= (" + ?)"■ In these formulae v is the mean velocity of discharge ; c is the coefficient of mean velocity ; R is the hydraulic mean radius ; J is the sine of the inclination of the water surface or fall in a length of 1 ; n is the natural coefficient, or coefficient dependent on the nature of the surface of the soil, or material over which the water flows ; a, I, and m are constant coefficients, determined from experimental observation in the mode already shown. The expression giving the value of c in a single equation is I m a + Z + T n J and this is applicable to measures of any description that may be employed in the formula 74 THE NEW FORMULA FOE MEAN VELOCITY. [OHAP, II. For metrical measures, the values of a, I, and m have been found to be respectively 23, 1, and 0-00155; and n for metrical as well as for all other measures has been found to varynGetween 0-008 and 0-050. The local values of n for various rivers, streams, and canals, have been already given in the table at pages 67 to 69, paragraph 28. Its general values, as suited to ordinary application, are 0-009 Well-planed timber. ( , 0-010 PlasterinpTireoement. o^a,), jfriiV^^w A'*-/; '*," u ' <^\«-vftO y^-j 0-011 Plaster in cement, with one-third sand. i \ ~ i ok ~ AK»XaipA (tP^ 0-012 Unplaned timber. M- V..\^~ : --^> tf' *'^V i^M/;. M vmauu> v^ i t. f ooi3 AgMar ^ brickwork ^ U/(i 0W*U C c*-'j [' \X '■ V<- }V!. 1 + (23 + ' To transform this equation so as to be suitable to values of R and v in other measures, the constant coefficients a, I, m, require new values {n remaining the same), which will be obtained by multiplying those given for metrical measures by the square root of the ratio that the unit of the new svstem bears to the unit of the metrical system, or metre.. CHAP. II.] FLOW IN OPEN CHANNELS IN EARTH. 75 The square roots of these ratios for the most useful and most general systems are : 1. Metrical measures 2. English and Kussian feet . . 3. Austrian feet 4. Prussian feet 5. Swiss and Baden feet 3-333 The equation for each of these sorts of measures then becomes as follows : (2) For English and Eussian feet, r '^A<^ [ ,, „ 1-811 . 0-00281 1 *tU\&st sUfui.OYH Ratio. Square Root 1-000 1-000 3-281 1-811 3-163 1-779 3-186 1-785 3-333 1-826 l+(41-6 + (3) For Austrian feet, 41-6 + Q-Q0281 \ n ., 1-779 0-00276 41 + + = — 1 + ( 41 + (4) For Prussian feet, 0-00276 ) — VK.J" u *\*a»A\ ^ Vf.\y&j>* ? S tor HUL^aufA tA oJLujouu/ = Wet yvfc^Cwm) j; = / „ 1-785 0-00277 41 + + = n J + (41- Q- 00277 \ n VR J* (5) For Swiss feet, 42 + 1-826 ,0-00283 1+42 + 0-00283 Ve j* This mode of reduction may be similarly applied to any other unit of measurement whatever. If the values of the coefficients and terms, c, x, and z, obtained through calcu- lations in metrical measures require adaptation to other measures, they will in the same way require multiplying by 76 THE NEW FOKMULA FOB MEAN TELOCITY. [CHAP. II.. the square root of the ratio that the new unit bears to the metre. Thus if c the coefficient obtained for metrical measures either from a diagram or from tables or calcu- lation is 50*00, its value for English measures will be 50 x 1"811 = 90 "55, if. we retain the same general formula v = o \ZbTT. In actual practice, however, the general formula v = o x 100 ^/R J is more convenient for English measures, as it affords a ready mode of at once reducing the number of cyphers in the term J; in this case then the corresponding coefficient would be - 9055, or more simply 0-91: It will have been noticed that the earlier tables in this work from the beginning up to page 42, par. 20, are in Swiss measures ; and that all the later tables from that page to the end are in metrical measures. The former are principally tables of observed results, from Switzerland as well as elsewhere, and of reductions of Bazin's calculated co- efficients arranged for purposes of comparison; as then these are never required by the hydraulic engineer as working tables for purposes of calculation ; and as the Swiss is nearly equal to the English foot, no object would have been gained by reducing these tables into metrical measures in this translation, except an appearance of uniformity. As, however, there might be an occasional case in which a reduction of coefficients from Swiss into other measures might be required, we annex the following factors of re- duction, which can be applied in the mode already de- scribed. Ratio, Square Root. 1. Metrical measures 3-000 0-546 2. English and Russian feet 0-9843X 0-992 3. Austrian feet 0-9489 0-974 4. Prussian feet 0-9558 0-977 5. Swiss and Baden feet 1-000 1-000 The following tables, for facilitating conversion of metrical into English measures, may also be occasionally of use. CHAP, n.] FLOW IN OPEN CHANNELS IN EAETH. 77 ? » o w i*l o Iz M 3 R £ 'A ■w < s» III gq M M a H 'S 4 M g w o <— i(NcoTtHiracoi>QoaiO ©rHrHtNCOCOH^HHlrtCO OOOOOOOOOO o £ t*-rtHrH00CDC0Ot^H-r-iHHa0(MCDC5CQt> i-ICOTtHCOQ0O3i-HC0THCD r-1 t-t rH rH d £ s a H fil H COCOCiCD(MOi»0(NCOlO t^-OlMOCOlOCOr-IOOCO TH 00 CO l> IN CD H O O tH CO (N C5 lO (N OD O pH CO -* iHi-H(MCQC0tH»O»OC0 "2 s 3 = | ■a £3 1-H (NlOt*OSH-*,{DQOr-(M OSOOC-tOCOO-^CQCOCN Mh-HiOOJCOt^HWOl ir3©COrHCDCNI>CQ00C0 NOI>OlN»Ot>0(M10 HHHH(NCN(N is ■a iH(MCO-*iOCDt>OOOiO rH 1 1 to 2 1 rHCOHHCOt^OSOCNH-firj CN^tOQOOlMlOIXJJH CDC0OaiO(N ©rHrHtNCOCQHHHHiOCO 1 I>-THi-H05C0e0OQ0i0CM O5oaoscooooooor-i>c* ■^OiTHOSHHOSHHOiHKOi OOHH«lNC0«'*^ o 1 00 CD"lO WHOMCDlOW HMOt^OHIM'flCOQO COCDTHNOOJh-lOCOH M >0 CO ih rH CO O) (N W CO i CQCOCnCWCOCiCNCOOaCN HHH«IMCNCO 1 HNC0rHi0C0t>M0)O B) s a 1 Cil>C0THCOCNOCOt^tO COt>COlO-*COCM001OT OH(MCO^IOCD|>|>CO CDCMOOtHOCOCNQOt^O 1 8 a S rHCtt^HcboQCi-H^^HCO i— I r-t i-H rH i 11 0<-H(NCOTH»OCDCDr^CO HCMCC^IOC0l>»CJ5O H(MC0^10COI>C0 01H OOOOOOOOOrH r 6 CM^«OCOO(MtHCOCOO i-H rH rH i-H rH CMI> OOrHrHCNtNCOCOHiHH CQCDOS0 OOOrHrHrHCNCNtMCO i 3 1 iH(MC0^ir5CDt>C0C71O rH 78 THE NEW FORMULA FOB MEAN TELOCITY. [OHAP. II. O m w a p on ■4 H 3 CO a o H a 1 a a o H 0003 5 tO«QOWHl>COOCO(N '<*OOQOOCOt>WI>'-«CO OOHH(NNMCO'*'H i— I »H i— 1 i— 1 l-t Cfl s .8 a o 1 1 i eM^f-Oi-H^COOOi-ICO COCOOCMCOOeNlOOSCNi ThOOWt-i-HOOrHOOCO iOOCDHt^NCO«»^ i-H«^l>OOaQOSO 1—1 ■S 2 5 S ■S P H COCQCO^OSCOi-lrHTHOS MOO^t*r-(lOW(NlO WOCOHCD(NI>NCOW ^OJCOMNt^HCDOlO OOi-lrHOOOSO r—l I 11. ■n o o x S 3 sis ggo. HO COCC05CD0«D10tHtHM TdOO!Nt0 10 03COI>H COCOOCOt>OOTt>OrH OOHHHNJJINMCO feS'l" 1 £ V B'fu a tr. ? S o (D(NOO-*ONCOOiiOH Tf'MCDi-C0O© S u ai .A u M *" tt ,S CmS 53 o

i- OCOt^COCDlO^COCOtM (MiOQOHtHC-OMWOJ rtHrllN(NlNIN -- u OOiO«OCOCD«OC3iCO i>n«HOJ(>'omooo ttlO'^CSCOOOCOOOCOO i-(CO-*COI>OSO(MCOiO I— 1 I— 1 1-M 1— 1 -3 THOOOiO l-H CHAP. n.J PLOW IN OPEN CHANNELS IN EARTH. 79 o < ft a P CO ft o si Is M3 fico p ° OOHHH«(NMeO« ! S a I OCM»rtt-0(NiO l-t r-l r-l r-l ©3 CN <£l 'i' pj « is Eh r« a. as QQ 5 CO OOHHNINNMeC^ t*^HXWlNOJ(DWO t>WMO00CDWHCil> H (N W tHW W l> CO Ol O l iHGOOSO i-H Is II ■^OOWCOO-^COtNCOO COCDlOM(NOGOI>W^ irji-tt-COCSlOfHlXNCO t-l rH i-H rH CN CN I>CD»OTHTHCO»050 1-^ 1 li Si m 1* iS iflOWON onoooi-icocoosN^t^ iH i-l i-H (H 0^|>H-* COt^OrHO'-H-^OOr-liO i-lrHi-lCMCNCNCOCQ i-l(N«TH»OCOt>CX3 05 r-l i w - s % c £ a 3 ■a 3 n CO t* O tH t- O-M t* O ■* cccoomcDO«y5om CD(NOilOHQO-*Oh.eQ eooo^oOHiooiiso OOHHHNIMINWCO i i i t>lO(MOSCD^OSCOCOCO i-ICOWCOCOOrHCOiOt- THCOWCDOlOQCOh-H i-lrtCN(M«CO-*-^ o . ii o COCDOHM^COO-rHtXTJ 0DCD-^HC0rHCi00CD-H« OOOfHrHr-ii-lOKNCS OOOOOOOOOO oS iH«C0-*>0«)t>C0 0)O i-H 80 THE NEW FOBMULA FOE MEAN TELOCITY. [CHAP. II. Continued. I ton per linear inch = 2-5798 tpnneaux per linear centimetre. 1 pound per square foot = 420 "941 kilogrammes per square centimetre. 1 cwt. per square foot = 47142 kilogrammes per square centimetre. 1 tonneau per linear centimetre = 0-3876 tons per linear inch. 1 kilogramme per square centimetre = 0-002 374 pounds per square foot. 1 kilogramme per square centimetre = 0-000 021 cwt. per square foot. 1 quintal = 100 kilogrammes = 0-1 tonneau = 0-0984 ton. = 1-9684 cwt. = 220-4621 pounds. Mbastjees of Wateb Supply. A Waterlog in Cubic Feet per Acre of A Watering in Cubic Metres per Hectare of A Watering in Cubic Metres per Hectare of A Watering in Cubic Feet per Acre of 1000 = 11-44 100 = 8739 2000 = 22-88 200 = 17479 3000 = 34-32 300 = 26218 4000 = 45-76 400 = 34958 5000 = 57-20 500 = 43697 6000 = 68-64 600 = 52437 7000 = 80-08 700 = 61176 8000 = 91-52 800 = 69916 9000 = 102-96 900 = 78655 10000 = 114-40 1000 = 87395 A -watering of 1000 cubic yards per acre = one of 308-9 cubic metres per hectare. A watering of 1000 cubic metres per hectare = one of 3236*8 cubic yards per acre. A supply of 0-01 cubic foot per second per acre = one of 0-1144 litre per second per hectare. A supply of 1-00 litre per second per hectare = one of 0-0874 cubic foot per second per acre. 1 hectare = 10 000 square metres. 1 litre = 0-001 cubic metre. CHAP. H.J FLOW IK OPEN CHANNELS IN BABTH. 81 Measures of Heat. *l-» i > " ^ nUs3^ Old . Fahrenheit. Centigrade. Ke— &ZSs£. Old Fahrenheit Centigrade.' Reaumur. Improved Fahrenheit. -13 -25 -20 -45 99-5 37-5 30 67-5 -10 -23-3 -18-6 -42 100 37-8 30-2 68 -8 -22-2 -17-8 -40 102 38-9 31-1 70 —4 -20 -16 -36 104 40 32 72- -17-8 -14-2 -32 110 43-3 34-7 78 2 - 16-7 -13-3 - 30 112 44-4 35-6 80- 9-5 -12-5 -10 -22-5 120 48-9 39-1 88 10 -12-2 -9-8 -22 122 50 40 90- 12 -111 -8-9 -20 130 54-4 43-6 98 14 - 10 -8 -18 132 55-6 44-4 100- 20 -6-6 -5-3 -12 140 60 48 108 22 -5-5 -4-5 - 10 142 61-1 48-9 110- 30 - 1-1 -0-9 -2 144-5 62-5 50 112-5 32 150 65-6 52-4 118- 152 66-7 53-3 120 Freezin g point. 158 70 56 126- 35 1-7 1-3 3 160 71-1 56-9 128 40 4-4 3-6 8 162 72-2 57-8 130- 42 5-5 4-5 10 167 75 60 135 50 10 8 18 170 76-7 61-3 138- 52 11 1 8-9 20 172 77-8 62-2 140 54-5 12-5 10 22-5 176 80 64 144- 60 12-6 12-4 28 180 82-2 65-8 148 62 16-7 13-3 30 182 83-3 66-7 150- 68 20 16 36 189-5 87-5 70 157 5 70 211 16-9 38 190 87-8 70-2 158- 72 22-2 17-8 40 192 88-9 711 160 77 25 20 45 194 90 72 162- 80 26-7 21 >3 48 200 93-3 74-7 168 82 27-8 22-2 50 202 94-4 75-6 170- 86 30 24 54 212 100- 80 180 90 32-2 25-8 58 92 33-3 26-7 60 Boilin ; point. 82 THE NEW POEMULA FOR MEAN VELOCITY. [chap. n. 02 Ph on « i-h p fe r/5 «J tn W a •— n EC o fc 3 S H C5 M Ph EH OQ «1 □Q |-1 O pH O a i O M « > 02 g I M w ^ i-3 < > M 55 s l-H CD 3 EH w Cm O CO CO a » pp a> ■ M ■ OS CO cq OS rH i-t CM © © CO © 00 © o OS CM B OS in CO to CM CO rH T-H CM OS l-H CD CO rH i-H © © co © (M CO ITS 00 o O 1 — i CO rH rH CO t- »o i-H CO i- CM o CO CO (M CM CM (M C-l CO CM D o i-H +5 t*. CO rH CO OS CO rH rH (M CM CSI CO CO lO t> © *o = 1 CO o i-H OS CO CO 00 OS CM t> '" H w-g i-H w f-M i-H o © © © '© l-H © i-H O CO CO rH i-H o . OS (M t- © © © © rH o CM CO 1-t CO t- O © © © Q] '© 00 in OS I—I CO rH © CO © :? rH &M (M CO OS oc ITS iG CM « g a o OS OS QC oc cc OC OS c t> © H P CO- 1— i—< M rn < S IO ■5L* rH OS CM CN O" rf c CO CT CO © .£&• o o CO a- a l> rf OS CN rH (M "3)4) a H t- to r- Tf- i- i-l a © l> CO © I> o o CO OS ta -+ CO O CM © © CO rH CO CO C rH CO CO © CO CM CO t> tfi »rs ri 00 CO rH O Si CM o o OS OS OS OS OS © © © i-H CO i-H pH © © © © © ^ O i-H 3 ^_ H-a at O 1 i cm O CO 1 60 S I H- c «£ -4- C ■5 .1 1 c 1 c E a 1 T3 1 -^> O o c < Ph o ,o * so .9 "55 a | 1 > 1 h X c p. DO 1 1 1 -a P "4 a: Ph A Ph □Q <5 rH cm CO rH iC «o i> oc © c l-H (M 1 l-C r* 4 P CHAP. D.J FLOW IN OPEN CHANNELS IN BAETH. 83 Equivalents of Fobeign Measures of Length. Miles. In Local Measures. Number in a degree of latitude. English Statute Miles. Kilo- metres. The geographical mild of England j and America, and nautical mile) of all nations ) 6076-98 ft. 60- 1-1509 1-8516 English statute mile since 1824 . . 5280 ft. 69-06 1- 1-6089 Old English mile, now used on\ 5000 ft. 72-93 0-9470 1-5236 6720 ft. 54-26 1-2728 2-0477 5952 ft. 61-26 1-1273 1-8137 Kilometre of France, Italy, Spain,\ 1000 m. 11110 6216 1- Prussian and Danish post mile .. 24000 ft. 14-75 4-6816 7-5322 24000 ft. 14-65 4-7136 7-5836 3500 ft. 104-18 0-6629 10664 13-33 5-1806 8-3350 10- 6-9055 111100 36000 ft. 10-4 6-6395 10-6827 Belgian, Dutch, and Polish mile.. 20- 3-4527 5-5550 Wurtemberg geographical mile . . 26000 ft. 15- 4-6036 7-4067 14815 ft. 25- 2-7622 4-4440 Bavarian mile of A nspach .. - 28800 ft. 12-87 5-3666 8-6342 18000 ft. 20-58 3-3564 5-4000 60- 1-1509 1-8516 Greek stadium (modern) 112-16 0-6156 0-9905 Arabian and Egyptian mile .. 6000 ft. 57-88 11933 1-9200 6236 ft. 54- 1-2788 2-0574 5000 ft. 79-86 0-8650 1-3917 66-66 10361 1-6670 360 paces. 199-72 3458 5563 4 1i. 49-93 1-3831 2-2253 G 2 84 THE NEW FOKMULA FOR MEAN VELOCITY. [CHAP. II. Equivalents op Foreign Measures op Subfaoe. ACBES. English and American acre Irish acre Scotch acre . . - French hectare Russian dessatina Prussian morgen Wurtemberg morgen .. Baden morgen Amsterdam morgen Polish morgow Hanoverian morgen .. Austrian jochart Tyrolese jauchart Swiss (Vaud) juchart . . . Norman journal Bavarian tagwerk Swedish tunnland Danish toende-hartkorn . Piedmontese giornata . . Venetian migliajo Tuscan saccata Roman pezza Arabian feddan Portuguese geira Spanish cuadra cuadrada .. Spanish fan egada In Local Measures. English Acres. French Hectares. Acre-side in English Feet. ft» ft. 43 560 sq. ft. 70 560 sq. ft. 55 353 sq. ft. 10 000 sq. m. 2 400 sq.sash 25 920 sq. ft. 38 400 sq.ft. 40 000 sq. ft. 101 400 sq. ft. 67 500 sq. 30 720 sq. 57 600 sq. ft. 36 000 sq. ft. 50 000 sq. ft. 77 440 sq. ft. 40 000 sq. ft. 56 000 sq. ft. 224 000 sq.ft. 14400 sq.ft. 25 000 sq. ft. 16 500 sq. br. 52900 sq. pal. 57 600 sq. ft. 4 840 sq. va. 22 500 sq. va. 82 944 sq. ft. 1" 1-6199 1-2708 2-4725 2-4954 0-6313 0-7793 o'-8901 2-0095 1-3843 0-6476 1-4230 0-8900 1-1126 2-0204 0-8425 1-2203 5-4557 0-9398 0-7474 1-3895 0-6529 1-4584 1-4480 3-9600 1-5888 0-404 44 0-655 11 0-513 92 1- 1-092 50 0-255 32 315 17 0-360 00 0-812 71 0-559 87 0-26192 0-575 54 0-359 94 0-450 00 0-817 15 0-340 73 0-493 53 2-206 49 0-380 09 0-302 30 56197 0-264 07 0-589 82 0-585 64 1-603 56 0-642 56 208-7 265-6 235-3 328-2 343 165-7 184- 1 196-9 295-7 245-4 167-7 249-0 196-5 220-1 296-7 191-6 230-6 487-3 202-1 180-1 245-7 168-6 251-9 251-3 415-3 262-8 CHAP. II.] FLOW IN OPEN CHANNELS IN EABTH. 85 Equivalents of Fobeign Measubes op Capacity. Wei and Dbx Measures. Gallons. Litres. Side of Cube In English Feet. English Imperial gallon of 10 lbs. water, 277 ■ 274 oub. inches . . 1- 4-54 0-543 Old English wine gallon (American) 231 cub. inches . . 0-833 3-78 0-511 Old English beer gallon, 1-017 4-62 0-549 French litre, 1 cub. decimetre .. •220 1- 0-328 2-708 12-30 0-756 Prussian anker, f of a scheffel . . 7-564 34-35 1-065 8-242 37 43 1-096 8-641 39-24 1-114 8-387 38-09 1-102 12-774 58-01 1-263 15-066 68-42 1-340 64-721 293-93 2-189 8-918 40-50 1-125 1-154 5-24 0-569 Portuguese almude (Lisbon) .. 3-642 16-54 0-835 Spanish arroba (Castillo) 3-554 16-14 0-828 English Imperial bushel, 8 Bushels. V Litres. 36-33 Side of Cube In English Feet. 1-087 Winchester bushel (American) .. 0-969 35-22 1-074 French hectolitre 2-7522 100- 1-523 5-772 209-73 1-948 1-512 54-96 1-246 Danish sMeppe 0-478 17-39 0-849 6-119 222-35 1-986 4-878 177 23 1-842 0-275 10- 0-707 1-693 61-49 1-293 1-962 73-25 1-371 Portuguese fanga (Lisbon) 1-488 54-08 1-239 Spanish fanega (Castillo) .. 1-572 57-15 a-262 86 THE NEW FORMULA FOR MEAN VELOCITY. [CHAP. II. Equivalents of Fobeign Measures of Weight. Pounds ahd Toss. English pound avoirdupois . . nearly English pound troy . . . . nearly Old English and Scotch pound nearly French kilogramme . . . . exactly Prussian and Wurtemberg pound Danish and Norwegian pound Swiss (Vaud) pound Austrian and Bavarian pound Eussian pound Swedish skalpund Portuguese arratel Spanish libra (Castple) English and American hundredweight French quintal Zollverein centner Prussian centner Austrian centner .Russian berkowitz Danish centner Swedish centner Portuguese quintal Spanish quintal (Castille) English and American ton French tonneau German ton (Hamburg) Russian ton Portuguese ton Spanish tonelada Equivalent in Distilled Water according to Local Measure.' English Grains. ■£% of a cub. ft. ■fa of a cub. ft. -fa of a cub. ft. 1 cub. decim. r of a cub. ft. - of- a cub. ft. In Local lbs. 112 100 kilog. 100 110 100 400 100 120 128 100 2240 1000 kilog. 2000 2400 1728 2000 7000 5760 7600 15432 7217 7707 7716 8642 6317 6535 7083 7099 English lbs. 112 220-46 110-23 113-43 123-46 361-01 110-10 112-05 129-53 101-42 2240- 2204-6 2135 ■» 2166-0 1748-5 2028-2 French Grammes. 453-6 373-2 492-3 1000- 467-7 499-4 500- 560- 409-4 423-5 459- 460- Kilo- grammes. 50-80 100- 50- 51-45 56- 163-76 49-94 50-82 58-75 46-00 1016-05 1000- 968-80 982-53 793-15 920-05 chap. n.j flow in open channels in eabth. 87 34. The application op the new Formula to the Calculation of Dischaeges in open Channels in Eaeth, and the use of the Tables and Diagram. The following tables of velocities and discharges in open channels in earth, having an object similar to those of Claudel for pipes, are intended principally for determining the dimensions of cross-section (the depth and bottom width) of any canal designed to carry a previously fixed amount of discharge with a given velocity under limited conditions of inclination. As in these we treat only of canals and channels in earth, and not of those in masonry, brickwork, or timber, we can confine ourselves to the three following grades of roughness of surface of cross-section, indicated by the three values of n, - 025, - 030, and 0'035 in our formula for metrical measures : 23+ l + ^00155 n J 1 + (23+ -H" Ve7j First class. — Perfectly clear and well maintained channels in loamy earth, free from irregularities, and without stones, silt, or. weeds, in which n = 0"025. Second class. — Channels, rather defectively maintained, having slight irregularities, as well as gravel, stones, and weeds occasionally, in which n = 0*030. Third class. — Very defectively maintained channels with great irregularities, and having grass, weeds, and large stones, in which n = ■ 035. Although these grades are rather distant from each other, they will, in practice, be found to be close enough to render any intermediate degrees needless. We had at one time intended to include the results for these three classes in one table, but have since preferred the arrangement we have 88 THE NEW FORMULA FOB MEAN VELOCITY. [CHAP. II. adopted, of making three separate tables, one for each class, as more convenient. These tables are directly applicable to only one form of section, that shown in Figure 1, Plate I., a trapezoid with side slopes of 1^ to 1 ; for this the true velocities and discharges are given direct ; for the other forms of section, shown in Figure 2, the rectangle or the trapezoid with side slopes of 1 to • 5, 1 to 1, 1 to 2, and 1 to 3, the velocities and discharges given for the original type of section must be reduced or modified by applying the percentages given in the additional small table constructed for that purpose, which immediately follows them. The following example will illustrate this method of reduction. Example. — A channel of the "first class, for which n = ■ 025, having a fall of 1 per thousand, a hottom width of 5 metres^ and a depth' of - 8 metre, will have its side slopes altered from 1£ to 1, to 1 to 1, what will be the effect on the velocity and on the discharge ? An inspection of the additional table shows that the velocity given for the first case must be increased by 0'3 per cent, to obtain that for the second, and the discharge reduced by 9 • 1 per cent., the new velocity and discharge becoming v = 0*910 -| Ynn = 0" 913 metre per second. q = 4 '513 ■ — - = 4* 102 cubic metres per second. It is generally found that in such cases the percentages of velocity and of discharge vary principally with the depth of channel and are not much affected by varying either the bottom width or the inclination. For other sections not comprised in these tables, for which a percentage of reduction cannot be conveniently calculated, the coefficient corresponding to the special case CHAP. H.] I-LOW IN OPEN CHANNELS IN EAKTH. 89 under consideration may be obtained from the tables of coefficients, one of which accompanies and precedes those of velocity and discharge in each of the three classes ; this coefficient can then be applied in the formula, v = c JTIJ and the Telocity and the discharge can then be calculated in the ordinary way. The values of the expression VRJ have been tabulated by Mr. Kutter, but have been omitted in the ' Cultur-Ingenieur ' for want of space ; these, however, may be obtained from tables of other writers oh hydraulics. For most ordinary purposes, however, this mode of determination will only be required for checking the velocities and dis- charges obtained direct from the tables. Before using, as intended, the tables for reading off velo- cities and discharges, it will, of course, be necessary to decide whether the case under consideration is more nearly suited to the first, the second, or the third class, for which separate tables are given, or, in other words, whether the coefficient indicating the nature of the surface on which the water acts in the channel is nearer to 0" 025, to 0' 030, or to 0'035. Most cases fall in the second class, and intermediate classes are rarely required in actual practice. After deciding this point, and on referring to the tables, two quantities will be found to correspond to each inclination or fall per thousand and each bottom width ; the upper of these, in thinner type, is the mean velocity of discharge per second in metres, the lower, in thicker type, the discharge per second in cubic metres corresponding to that velocity as well as to the inclination and the dimensions of cross section adopted. Should any case happen to comprise any intermediates between the values of the . dimensions or quantities, the velocities or discharges, given in the tables, there will be 90 THE NEW FORMULA FOR MEAN VELOCITY. [CHAP. II. no need to calculate them independently, they can easily be interpolated by proportionate differences which may be added or subtracted, as the limits within which the differences of the quantities given in the tables are kept are such as to allow this to be done with sufficient accuracy. The following examples will explain the use of the tables. Example 1. A channel is required to discharge 5 cubic metres per second with an inclination of • 008, or ■ 8 per thousand ; its section to be trapezoidal, with side slopes of 1 J to 1 ; and the highest water level in the canal is to be : 3 metre below the surface of the ground ; the soil is clay, with one-third sand and earth ; what will be the depth from the ground surface to the bottom of the channel ? The surface of the section being in smooth soil, and the channel being supposed to be kept in good order by yearly cleansing, the case may be considered as one of the first class. Now as with the given inclination several sections of different forms and dimensions may discharge the required quantity of water, it becomes a question whether greater depth and less bottom width or greater bottom width and less depth is to be preferred. The following are the tabular depths and bottom widths that 'will allow of the discharge of 5 cubic metres per second Depths - 8 metre. Bottom widths 6-3 metres. .. 1-0 » „ 4-0 „ and if we assume that a bottom width of 5 • metres would be the most convenient, the depth corresponding to this, obtained by proportionate differences, will be 0*91 metre, and the depth from ground level to the bottom of the canal will be 0-30 + 0-91 = 1-21 metres. Example 2. Kequired the mean velocity of discharge of a channel having an inclination of * 5 per thousand, and a bottom width of 10 metres, with side slopes of 1£ to 1, first, CHAP. II.] FLOW IK OPEN CHANNELS IN EARTH. 01 when the depth of water is 1 • 5 metres ; secondly, when it is 1 • 45 metres. The mean velocity for neither of these cases being given direct by the tables, an intermediate velocity has to be obtained by proportionate differences. Metres per second. The tabular velocity given for a depth of 1" i metres is . 0-971 And that for 1-6 „ 1-043 Henee that for a depth of 1-5 „ 1-007 For a depth of 1-45 metres, one-fourth the difference between the two tabular velocities will be added to the first of them ; thus the required velocity for that case will be 0-072 = 0-971 + — — = 0-989 metre per second. Example 3. A channel has to be conducted down sloping ground, whose soil is of suoh a quality as not to admit of a mean velocity of more than 1 metre per second without injury to its bed and banks. Its maximum discharge is to be • 5 cubic metre per second, its section trapezoidal, with a depth of water of - 4 metre, and side slopes of 1£ to 1 ; what will be the bottom width and the inclination of the channel ? In this case it would appear that the description of soil, and the probable necessity of the adoption of a curved course down the descent would place the example in the second class, but as the table for that class is still in the press we may, for convenience sake, make use of the table for the first class, which we have at hand, as, although the re- sults will differ, the mode of procedure will be exactly the same. Putting, therefore, the example in the first class, and using the portion of table corresponding to the given depth of water • 4 metre, we find that the following inclinations and 92 THE NEW FORMULA FOE MEAN VELOCITY. [CHAP, n. bottom widths are all applicable to the case as a discharge of '5 cubic metre per second. 0-2 per thousand inclination with 4-50 metres bottom width 0-3 „ „ 3-50 0-4 „ „ 3-00 0-5 „ „ 2-75 06 „ „ 2-50 0-7 „ „ 2-25 0-8 „ „ 2-00 0-9 „ „ 1-90 1-0 „ „ 1-80 12 „ „ 1-60 1-4 „ „ 1-45 1-6 „ „ 1-40 1-8 „ „ 1-00 In none of these cases does the mean velocity resulting exceed 1 metre per second, being ■ 250 in the first case and 0*780 in the last ; hence, as land may be saved by adopting the smallest bottom width of 1 • 00 metre with a fall of 2 ■ 8 per thousand, this will probably be the best in practice : or,' if preferred, a higher inclination and a narrower bottom width may be calculated. Example 4. What will be the mean velocity of discharge of a river, having an- inclination of water surface of 0-000040393, a sectional area of 1864*9 square metres, with a wetted perimeter of 514 ■ 2 metres ? To calculate this direct from the formula without the aid of the tables, the steps are as follows : The formula for mean velocity is where c =- 1 + 7S I m «-(« + £)« CHAP, n.] FLOW IN OPEN CHANNELS IN EARTH. 93 where for metrical measures a = 23, I = 1, m - 0-00155, and n lies between 0-008 and 0-050, remaining the same for all systems of measures. As in all cases it is necessary that the adopted value of n should be determined by comparison with observed results, and the degree of roughness of the surface of the channel acted on by the water fixed so as to be suitable to the case under consideration ; we will in this case assume a value of n of • 025, which is that suited to rivers and canals in very good order. Having then all the numerical values needful, we obtain „„ 1 0-00155 * = 23 + - + = — n J = 23 + 40 + 38-373 = 101-373. ... (23 + 2^)0-026, ,(?°±£-™)- 1-6843, and hence R _ 1864 9 _ 3.621 R = -Eu-2 ~ 3 621 ' 101-373 x , 1-5343 1 +-ro M VK V3-621 101-373 1-80631 = 56-122 but ■v/EJ = -/3-621 x 0-000 040 393 = 0-012 094 hence v= 56-122x0-012 094 =0-678 73 mfetre per second. The actually observed mean velocity of the Danube at Szob, of which this is an example, is 0-686 metre per 94 THE NEW FOBMTJLA FOB MEAN -VELOCITY. [CHAP. H. second; the small difference of 0*007 metre between the calculated and the observed velocity is due to our having assumed too high a value of n ; this, to be in accordance with the observed velocity, should be 0"0247 instead of 0-0250. In the case mentioned in the last example, as well as in all similar cases where the mean velocity has been actually observed, the value of the correct coefficient e may be cal- v culated by the formula o = =t==;, and the exact local value vRJ of the coefficient n by means of the formula »V£+i(7iry-i-^- A m A = a +J . In the same way, if any three of the four quantities E, J, c, n, be given, the fourth may be calculated by means of the above formula. Calculations of this nature, as shown in the last example, present no difficulty whatever ; a large number of such examples would, however, occupy a considerable amount of time, as each would have to be calculated separately. We therefore attach a diagram, Plate I., by means of which the values of coefficients c, corresponding to given values of R, J, and n, can be read off in a few seconds with the aid of a simple straight edge, or by which any one of the four quantities E, J, n, and c can be obtained from the remaining three, in any number of cases with the least possible ex- penditure of time and thought. In this diagram the diverging lines n, radiating from an origin or point where V R and E = 1 nidtre, indicate the grade of roughness of the surface of the channel, the curved CHAP. II.] FLOW IN OPEN CHANNELS IN EAETH. 95 lines indicate the degree of inclination J of the water surface ; the scale on the axis of abscissae denotes values of E in metres, and the scale of equal parts on the axis of ordinates gives values of the coefficient o. It is evident, therefore, that if a straight edge be laid across this diagram, in such a manner as to cut three of these lines in points corresponding to the three values given in any example, it will also cut the fourth line in a point, which will indicate to scale the value of the fourth required quantity. We recommend the employment of this diagram to all hydraulicians that make use of our formula. In bringing our work to a conclusion, we refer our readers for fuller information as to the derivation of our formula to the ' Zeitschrift des Oesterreichischen Ingenieur und Archi- tekten-vereins ' for 1869,* and express a hope that our formula may be universally employed. " See Extracts therefrom introduced in paragraph 27, pages 59 to 72. TABLES *. OP COEFFICIENTS OF MEAN VELOCITY, AND OP MEAN VELOCITIES AND OF DISCHAKGES PEK SECOND, FOB OPEN CHANNELS IN EAKTH, APPLICABLE TO RIVERS AND CANALS OP THREE CLASSES. Class I. — Those having their beds and banks in good order, and perfectly free from all irregularities, deposits of stone, and overgrowth. Class II. — Those with beds and banks in moderately good order in every respect. Class III. — Those with beds and banks in bad order, having irregularities and deposits of stone and pebbles, or much overgrown with vegetation. C a ) The quantities given in the following Tables are in metrical* measures, and are calculated according to the following formulas of G-anguillet and Kutter ; v = c/EJ 1 + . = i + 23 + °-°; 155 n J «-»(28 + -°° 1BB ) Where v is the mean velocity of discharge per second in metres, c is the coefficient of mean velocity, B is the hydraulic mean radius, J is the fall of the water-surface in a length of unity, n is the coefficient of roughness, having the fixed values of - 025 for channels of Class I., of 0-030 for those of Class II., and of 0'035 for Class III. The results are applicable to channels having side slopes t of 1^ to 1, having bottom-widths of from ■ 2 to 270 metres, to depths of water of from 0*2 to 6 metres, and to inclina- tions of from 0-000 02 to 0-003 00, or of 0-02 to 3-00 per thousand. * For conversion tables, see Paragraph No. 32 of the text, p I | '^ f An additional table enables the quantities to be reduced and applied to various forms of seotion. FIEST CLASS. EIVERS AND CANALS, HAVING THEIR BEDS AND BANKS IN GOOD ORDER, AND PERFECTLY FREE FROM ALL IRREGULARITIES, DEPOSITS OF STONE, AND OVERGROWTH. n = 0-025. ( iv ) CLASS I. (n = 0-025.) C Coefficients of Mean Velocity. Fob Values op E. Fall per thousand. 01 0-2 0-3 0-4 0-5 06 0-7 0-8 0-9 0-05 — 32-4 34-0 35-7 373 38-7 0-07 — — — — 330 34-6 36-1 37-5 38-8 01 m^L, r -25-0 28-5 31-0 33-2 35-0 365 37-8 390 0-2 20-6 26-2 29-3 31-8 33-8 35-5 36-9 38-0 390 0-3 21-3 26-5 29-6 32-2 34-2 35-6 36-9 38-0 390 0-4 21-5 26-7 29-8 32-3 34-3 35-8 370 38-0 39-0 0-5 21-7 26-8 30-0 32-4 34-3 35-8 37-1 38-1 391 0-6 21-8 26-9 30-0 32-5 34-4 35-8 37-1 38-1 391 0-7 21-9 27-0 30-1 32-5 34-4 35-8 37-1 38-1 39-1 0-8 22-0 27-1 30-2 32-5 34-5 35-9 37-2 38-2 39-1 0-9 22-0 27-2 30-3 32-6 34-5 35-9 37-2 38-2 391 1-0 22-0 27-2 30-3 32-6 34-5 35-9 37-2 38-2 39-1 Foe Values oe R. Fall pei- thousand. 2-6 2-8 3-0 3-2 3-4 3-6 3-8 4-0 4-2 0-02 — — — — — 60-7 61-7 62-5 63-3 0-03 57-4 58-3 59-0 59-7 0-05 51-0 51-9 52-7 53-4 54-1 54-8 55-4 56-0 56-5 0-07 50-0 50-7 51-5 52-1 52-6 53-3 53-7 54-2 54-7 01 49-0 49-7 50-3 50-8 51-3 51-8 52-4 52-8 532 0-2 47-7 48-2 48-7 49-2 49-6 50-0 50-4 50-8 512 0-3 47-4 48-0 48-4 48-8 49-1 49-5 49-9 50-2 50-5 0-4 47-1 47-7 48-1 48-5 48-9 49-3 49-8 50-1 50-4 05 46-9 47-4 47-8 48-2 48-6 49-0 49-3 49-6 49-9 0-6 46-8 47-3 47-7 48-1 48-5 48-9 49-1 49-4 49-7. 7 46-8 47-2 47-6 48-0 48-4 48-8 49-0 49-3 49-6 0* 46-7 47-1 47-5 47-9 48-3 48-7 49-0 49-3 49-6 0-9 46-7 47-1 47-4 47-8 48-2 48-6 48-9 49-2 49-5 10 46-7 47-0 47-4 47-8 48-2 48-6 48-9 49-2 49-5 The coefficients remain unaltered for steeper inclinations. ( v. ) CLASS I. (» = 0-025.) 0/ Coefficients of Mean Velocity. For Values or E. 1-0 1-2 1-4 1-6 1-8 20 2-2 2-4 Fall per thousand. 40- 421 43-8 45-2 46-6 47-9 49-0 50-0 0-05 40-0 42-0 43-3 44-7 46-1 47-2 48-2 49-1 0-07 40-0 41-7 43-0 44-3 45-5 46-5 47-4 48-3 0-1 40-0 41 "4 42-7 43-8 44-7 45-6 46-4 • 47-0 0-2 40 41-4 42-5 43-5 44-4 45'3 46-1 46-7 0-3 40-0 41-3 42 -4~ 43-4 44-4 45-2 45-9 46-5 0-4 40-0 41-3 42-4 43-4 44-3 450 45-7 46-3 0-5 400 41-3 42-4 43-4 44-3 45-0 45-7 46-2 0-6 40-0 41-3 42-4 43-4 44'3 45-0 45-7 46-2 0-7 40-0 413 42-4 43-4 44-3 45-0 45-7 46-1 0-8 40-0 41-3 42-4 43-4 44-3 450 45'7 46-1 0-9 40-0 41-3 42-4 43-4 44-3 45-0 45-7 46-1 1-0 For Values of E. 4-4 4-6 4-8 5-0 5-2 5-4 5-6 5-8 60 - Fall pel- thousand. 64-2 64-9 65-6 66-3 67-0 67-7 68-4 69-0 69-6 0-02 60-4 61-1 61-8 62-4 62-9 63-4 63-9 64-4 64-9 0-03 57-1 57-7 58-3 58-9 59-4 59-8 60-1 60-3 60-5 0-05 551 55-5 55-9 56-3 56-7 57-1 57-5 57- S 58-1 07 53-6 540 54-4 54-8 55-1 55-4 55-7 56-0 56-2 0-1 51-5 51-8 52- 1 52-4 52-7 53-0 53-2 534 53-6 0-2 50-8 51-1 51-4 51-7 52-0 52-2 52-4 52-5 52-6 0-3 50-7 51-0 51-2 51-4 51-6 51-8 52-0 52-2 52-3 0-4 50-2 50-5 50-8 510 51-2 51-4 51-6 51-8 52-0 0-5 500 50-3 50-6 50-8 51-0 51-2 51-4 51-6 51-8 0-6 " 49-9 50-2 50-4 50-6 50-8 51-0 51-2 51-4 51-6 0-7 49-9 50-1 503 50-5 50-7 50-9 51-1 51-3 51-5 0-8 49-8 50-0 50-2 50-4 50-6 50-8 51-0 51-2 51-4 0-9 49-8 50-0 50-2 50-4 50-6 50-8 51-0 51-2 51-4 1-0 The coefficients remain unaltered for steeper inclinations. 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RIVERS AND CANALS, WITH BEDS AND BANKS IN MODERATELY GOOD ORDER IN EVERY RESPECT. n = 0-030. ( Kv ) CLASS II. (n = 0-030.) COEFFICIENTS OF MEAN VELOCITY. Fob Values of R. Fall per thousand. 0-1 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 0-05 26-5 28-1 29-6 31-0 32-2 007 — — — — 27-0 28-5 29-9 31-2 32-3 01 15-5 200 23-0 25-2 27-3 28-9 30-3 31-4 32-4 0-2 16-5 21-0 23'8 26-0 27-8 29-2 30-4 31-4 32-4 0-3 17-0 21-3 24-2 26-3 28-2 29-4 30-5 31-5 32-5 0-4 17-2 21-5 24-3 26-4 28-2 29-4 30-5 31-5 32-5 0-5 173 21-6 24-3 26-5 28-2 29-4 30-6 31-6 32-5 0-6 17-4 21-7 24-4 26-5 28'3 29-5 30-7 31-6 32-5 0-7 175 21-8 24-5 26-6 28-3 29-5 30-7 31-6 32-5 0-8 17-6 21-9 24-6 26-6 28-4 29-6 30-8 31-7 32-5 ' 0-9 17-7 22-0 24-7 26-7 28-4 29-6 30-8 31-7 32-5 1-0 17'7 22-0 24-7 26-7 28-4 29-6 30'8 31-7 32-5 For Values of E. Fall per thousand. 2-6 2'8 3-0 3-2 3-4 3-6 3-8 4-0 4-2 0-02 51-8 52-7 53-5 54-3 0-03 49-6 50-3 51-0 51-7 0-05 43-5 44-3 45-0 45-7 46-4 47'0 47-6 48-1 48-6 0-07 42-6 43-3 44-0 44-7 45-2 45-8 46-2 46-7 47-2 0-1 41-7- 42 4 43 43-5 44-0 44-5 45-0 45-4 45-8 0-2 40-6 41-1 41-6 42-1 42-5 43-0 43-3 43-7 44-0 0-3 40-2 40-7 41-2 41-6 42-0 42-4 42-8 43-1 43-4 0-4 40-0 40-5 41-0 41-4 41-7 422 42-5 42-8 43-1 0-5 39-9 40-3 40-8 41-1 41-5 41-9 42-2 42-5 42-8 0-6 39-7 40'2 40-6 41-0 41-4 41-8 41-9 42-2 42-5 0-7 39'7 401 40-5 40-9 41-3 41-6 41-8 42-1 42-4 0-8 39-7 40-1 40-4 40-8 41-2 41-5 41-8 42-1 42-4 0-9 39-7 40-1 40-3 40-7 41-1 41-4 41-7 42-0 42-3 1-0 39-7 40-1 403 40-7 41-1 41-4 41-7 42-0 42-3 The coefficients remain unaltered for steeper inclinations, tfej^v / in, I b O c o ( Iv ) CLASS II. (n = 0-030.) O Coefficients of Mean Velocity. A- Fob Values of B. 1-0 1-2 1-4 1-6 1-8 2-0 2-2 2-4 Fall per thousand. 333 353 36-9 38-2 39-4 40-5 41-6 42-6 0-05 33-3 35-2 36-6 37-8 38-9 39-9 40-9 41-8 0-07 33-3 35-0 36-3 37-4 38-5 39-4 40'2 410 0-1 33-3 34-8 36-0 370 37-9 38-7 39-4 40-0 0-2 33-3 34-7 35-8 36-7 37-6 38-4 39- 1 39-7 0-3 33-3 34-7 35-8 36-7 375 38-3 39-0 39-5 0-4 33-3 34-7 35-7 366 37-4 38-1 38-8 39-4 0-5 33-3 •34-7 35-7 366 37-4 38-1 38-7 39-2 0-6 33-3 34-7 35'7 36-6 374 38-1 38-7 39 2 0-7 33-3 34-7 35-7 36-6 37-4 38-1 38-7 39-2 0-8 333 34-7 35-7 36-6 37-4 38-1 38-7 39-2 0-9 33-3 34-7 35'7 36-6 37-4 38-1 38-7 39-2 10 Fob Values of E. 4-4 4-6 4-8 5-0 5-2 5-4 5-6 5-8 6-0. _Fall per thousand. 55- 1 55-8 56-5 57-2 57-8 58-4 59-0 59-5 60-0 0-02 52-3 52-9 53-5 54-1 54-7 55-2 55-6 56-0 56-4 0-03 49-1 49-6 50-1 . 50 6 51-1 51-6 52-1 52-4 52-5 0-05 47' 6 48-0 48-4 48-8 49-2 49-6 49-9 50-2 50-5 0-07 46-2 46-6 46-9 47-2 47-5 47-8 48-1 48-4 48-6 01 44-3 44-6 44-9 45-2 45-5 45-8 46-0 46-2 46-4 0-2 43-7 40-0 44-3 44-5 44-7 44-9 45-1 45'3 45-5 03 43-4 43-8 44-0 44-2 44-4 44-6 44-8 45-0 45-2 0-4 43-1 43-4 43-7 43-9 44-1 44-3 44-5 44-7 44-9 0-5 42-8 43-1 43-4 43-6 43-8 44-0 44-2 444 44-6 0-6 42-7 43-0 43-2 43-4 43-6 43-8 44-0 44-2 44-4 0-7 42-7 42-9 43-1 43-3 43-5 43-7 439 44-1 443 0-8 42-6 42-8 43-0 43-2 43-4 43-6 43-8 44-0 44-2 0-9 42-6 42-8 430 43-2 43-4 43-6 43-8 44-0 44-2 1-0 The coefficients remain unaltered for steeper inclinations. 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EIVERS AND CANALS, WITH BEDS AND BANKS IN BAD ORDER, HAVING IRREGULARITIES AND DEPOSITS OF STONE, AND MUCH OVERGROWN WITH VEGETATION. » = 0-035. ( civ ) CLASS III. (n = 0-035.) Coefficients of Mean Velocity. Fob Values or E. Fall per thousand. o-i 0-2 0-3 0-4 0-5 0-6 0-05 — — — — 22-6 24-0 0-07 — — — — 22-8 24-3 0-1 12-8 16-7 19-3 21-3 23-0 24-5 0-2 13-6 17-5 20-0 22-0 23-5 24-8 0-3 14-0 17-8 20-2 22-1 23-8 24-9 0-4 14-1 18-0 20'3 22-2 23-9 25-0 0-5 14-2 18-1 20-4 22-3 24-0 25-1 0-6 14-3 18-2 20-5 22-3 24-0 25-1 0-7 14-4 18-3 20-5 22-4 24-0 25-2 0-8 14-5 18-4 20-6 22-4 24-0 25-2 0-9 14-5 18-4 20-6 22-4 24-0 25-2 10 14-5 18-4 20-6 22-4 24-0 25-2 Fob Values op B. Fall per thousand. 1-4 1-6 1-8 2-0 2-2 0-05 31-7 33-0 34-2 35-3 36-3 0-07 31-5 32-7 33-8 34-8 35-7 01 31-3 32-4 33-5 34-3 351 0-2 31-0 31-9 32-8 33-6 34-4 0-3 30-9 31-8 32-6 33-4 34-0 0-4 30-8 31-7 32-5 33-2 33-9 0-5 30-8 31-6 32-4 33-1 33-8 0-6 30-8 31-6 32-4 33-1 33-8 0-7 30-8 31-6 32-4 33-1 33-8 0-8 30-8 31-6 32-4 331 33-8 09 30-8 31-6 32-4 33-1 33-8 1-0 30-8 31-6 32-4 33-1 33-8 The coefficients remain unaltered for steeper inclinations. ( cv ) CLASS III. (n = 0-035.) Coefficients,, of Mean Velocity. A Foe Values of K. 0-7 0-8 0-9 1-0 1-2 Fall per thousand. 25-3 26-5 27-6 28-6 30-3 0-05 25-6 26-7 27-7 28-6 30'2 0-07 25-8 26-8 27-7 28-6 30-1 0-1 26-0 26-9 27-8 28-6 30-0 0-2 26-0 27-0 27-9 28-6 30-0 0-3 26- 1 27-1 27-9 28-6 30-0 0-4 26-1 27-1 27-9 28-6 30-0 0-5 26-2 27-1 27-9 28-6 30-0 0-6 26-3 27-1 27-9 28-6 30-0 0-7 26-3 27' 1 27-9 28-6 30-0 0-8 263 27-1 27-9 28-6 30-0 0-9 26-3 27-1 27-9 28-6 30-0 1-0 Fob Values of E. 2-4 2-6 2-8 3-0 3-2 Fall per thousand. 37-2 38-0 38-7 39-4 40-0 0-05 36-5 372 37-9 38-6 39-1 0-07 35-9 36-5 37-1 37'7 38-2 0-1 35-0 355 36-0 36-5 37-0 0-2 34-6 35-1 35-6 36-1 36-5 0-3 34-5 35-0 35-5 35-9 36-2 0-4 34-4 34-9 35-3 35-7 36-0 0-5 34-3 34-8 35-2 35-6 35-9 0-6 34-3 34-7 35-1 355 35-8 0-7 34-2 34-6 .351 35-4 35-7 0-8 34-2 34-6 '35-1 35-4 35-7 0-9 34-2 346 35- 1 35-4 35-7 10 The coefficients remain unaltered for steeper inclinations. 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CHAPTEE I. FLOW IN OPEN CHANNELS GENERALLY. PAGE 1. The formulas of D'Arcy and Bazin and of Humphreys and Abbot, for determining Mean Velocities of Discharge of Rivers and Canals . . 1 2. The previously accepted formula 2 3. The formulae of D'Arcy and Bazin 3 4. The formulae of Humphreys and Abbot 3 5. Practical Examination of these formulae, and Table of Discharges of Channels of High Inclination 4 6. Examination of the old-established formula and the new American one, with the view of applying Series of Coefficients to either of them as a basis 6 7. The Variation of the Coefficients e with the Inclination 9 8. The Employment of the formulas of D'Arcy and Bazin in con- structing a Series of Coefficients 10 9. Table of Calculated Coefficients of D'Arcy and Bazin applied to the Formula o = e J r s 14 10. Table of corresponding Coefficients experimentally obtained ■ ■ . . 22 11. Bemarks on the Series of Observations of D'Arcy and Bazin . . . . 26 12. The Coefficients of D'Arcy and Bazin for calculating Mean from Maximum Velocities 29 13. Table of those Coefficiente 30 14. Examples illustrating the application of the Table Xo. 9 . . . . 30 15. The formulae and categories of Gauckler 34 16. Table of the Coefficiente of Gauekler's First Formula 36 17. The F orma tion of a New and final set of Twelve Classes instead of the previous Categories 36 18. The Twelve Xew Classes of Coefficients 37 19. Table showing the Range of the observed Coefficients in these Classes 40 20. Determination of the Final Coefficients for these Classes in Metrical Measures 42 21. Table of Values of the New Coefficients c for the Formula r = e J r a 47 22. Table of ObserTcd Results with their corresponding Coefficients . . 48 PAGE ( oxl ) CHAPTEE II. FLOW IN OPEN CHANNELS IN EAKTH. 23. Various Formulas of Eytelwein Patzig, Hagen, Bomemann, Brunings, Bazin, Hagen (new), Humphreys and Abbot, for determining Discharges of Canals and Kivers in Earthen Channels 51 24. Table — Comparison of Besults of various Formulae 53 25. The Formula of Bornemann and Hagen 53 26. Safe Bottom Velocities of Dubuat, with Table 57 27. The Derivation of the New Formula for Coefficients of Mean Velocity 59 28. Table giving the Observed Values of the Coefficient », corresponding to their data of observation 67 I m 29. Table giving the Values of the Expressions a +— and j corre-_ sponding to various Values of m and of J 69 30. Table of the Values of the Expressions z and x, corresponding to different Values of n and J in the Formula 71 31. The Transformation of the Final Formula from Metrical into Swiss, English, and other Measures 73 32. Conversion Tables of the Translator 77 33. Tables of Equivalents of Foreign Measures, by the Translator . . 82 34. The Application of the New Formula to the Calculation of Dis- charges in Open Channels in Earth; and Explanation and Examples for the use of the Tables and Diagrams , 87 WOEKING TABLES. Class I. (n = - 025). Coefficients of mean velocity. Mean Velocities and Discharges per Second i to li Class H. (» = • 030). Coefficients of mean velocity. Mean Velocities and Discharges per Second liii to oi Class III. (» = • 035). Coefficients of mean velocity. Mean Velocities and Discharges per Second ciii to cxxxvii Supplenmetary Table of Percentages cxxxviii PLATES. Plate I. Figure 1. Type of Section adopted in the Working Tables. Figure 2. Types of Sections for which Percentages are given in the Subsidiary Table. Plate II. Diagram for obtaining Coefficients of Mean Velocity for Metrical and for English Measures. BY THE TRANSLATOR. HYDRAULIC MANUAL AND STATISTICS. In large 8vo, 550 pages. Price 28s. CONTENTS. PART I.— MANUAL. Chapter I. — Explanation of the Principles and Formulas adopted in Calculation and applied in the Working Tables. 1. Hydrodynamio Theories. 2. Notation and Symbols. 3. Rain- fall, Supply, and Flood Discharge. 4. Storage. 5. Discharges of Open Channels and Pipes. 6. Section of Channels and Pipes. 7. Other Theories of Flow. 8. Velocities in Section. 9. Bends and Obstructions. 10. Discharges of Sluices and Weirs. 11. Discharge from Basins, Locks, and Reservoirs. 12. Application of the Working Tables. pp. 1 to t 74 Chapter II. — On Field Operations and Gauging; with brief Accounts of the Methods adopted by various Hydraulioians. pp. 74 to 135 Chapter III.— Paragraphs on various Hydraulic Subjects. pp. 136 to 221 WORKING TABLES. PAGE Table I.— Gravity i II. — Catchment. — Parts 1, 2, 3 ii to iv III. — Storage and Supply. — Parts 1, 2 v and vii IV. — Flood Discharge. — Parte 1, 2 ix to xi V. — Velocities xii VI. — Slopes and Gradients. — Parts 1, 2, 3 xiii to xvii VII. — Rivers and Canals xviii to xxv , VIII. — Pipes and Sewebs. — Parts 1, 2, 3 xxvi to xxxiii IX. — Sluices and Weirs xxxvii to xlviii X. — Bends and Obstructions. — Parts 1, 2, 3 xlix to lii XI. — Equivalents. — Parts 1, 2, 3, 4, 5, 6 liv to lxvii XII. — Coefficients.— Parts 1, 2, 3, 4, 5, 6, 7 lxix to xcii MISCELLANEOUS TABLES AND DATA. Retaining Walls, and Weights of Material. — Trapezoidal Masonry Dams. — Thickness and Weight of Pipes. — Hydraulic Machines. — Indian Hydraulic Contrivances. — Constants of Labour for Earthwork, Bricklayer's Work, and Mason's Work.— Cartage Table. — Indian Coinage, Weights, and Measures i to xii k PAKT II.— HYDRAULIC STATISTICS. Gravity t- 1 -! Rivers C2Jto [4] Indian Rivebs -• [ 5 J to C 15 3 Bbief Accounts of Indian Rivers [16] to [27] Financial Statistics op Indian Canals [29] to [39] Irrigation Statistics op Indian Canals [40] to [46] Bbiep Accounts op Indian Canals [47] to [78] Statistics op Reservoirs and Dams [79] to [82] Financial Statistics op Indian Reservoirs [83] to [85] Bbiep Accounts op Indian Reservoirs [86] to [96] Waterworks op Indian Cities [97] to [109] Irrigated Cbops and Plantations and theib Watering .. [110] to [123] Indian Wateb Rates and Watebings [124] to [128] Descriptions and Analysis op Water and Silt [129] to [141] INDIAN METEOROLOGICAL STATISTICS. Season Rainfall (1) to (3) Mean Monthly Rainfall (4) to (24) Day Maximum Rainfalls and Special Rainfall Data . . . . (25) to (30) Humidity and Evaporation (31) to (42) Additional Meteorological Tables (43) to (53) General Remarks on the Meteorology of India (54) to (77) Extracts from Critical Notices. 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