TO 
 179 
 
 Hydraulic tables, 
 efficients, and 
 
 
 CO 
 
 form.il 
 
 
 Southern Branch 
 of the 
 
 University of California 
 
 Los Angeles 
 
 Form L-l 
 
 TC,
 
 This book is DUE on last date stamped below 
 
 19281 
 
 AW? 5
 
 HYDRAULIC 
 TABLES, COEFFICIENTS, 
 
 AND 
 
 FORMULA 
 
 FINDING THE DISCHARGE OF WATER FROM ORIFICES, 
 NOTCHES, WEIRS, PIPES, AND RIVERS. 
 
 BY 
 
 JOHN NEVILLE, C.E., M.R.I.A., 
 
 COUNTY SCBVEYOR OF LOUTH, AND OP THE COUNTY OF THE TOWN OF DBOGHEDA. 
 
 46475 
 
 " It ought to be more generally known, that theory is nothing more than the conclusions 
 of reason from numerous and accurately-observed phenomena, and the deduction of the laws 
 which connect causes with effects; that practice is the application of those general truths and 
 principles to the common affairs and purposes of life; and that Science is the recorded 
 experience and discoveries of mankind, or, as it has been well defined, 'the knowledge of 
 many, orderly and methodically digested and arranged, so as to become attainable by one." " 
 American Quarterly Review. 
 
 LONDON : 
 JOHN WEALE, 59, HIGH HOLBORN. 
 
 1853. 
 SAN FRANCISCO E. J. MUYGRIDGB,
 
 LONDON : 
 WOODFALL AND KINDER, PRINTERS,
 
 TC 
 IT? 
 
 CONTENTS. 
 
 PAGE. 
 
 INTRODUCTION . . . . . . . . vii 
 
 SECT. 
 
 ^ I. Application and use of the Tables, Formulae, &c. . 1 
 II. Formulae for the Velocity and Discharge from Orifices, 
 Weirs, and Notches. Coefficients of Velocity, Contrac- 
 tion, and Discharge. Practical Remarks on the Use of 
 the Formulas . . " . . . . . 17 
 III. Experimental Results. Coefficients of Discharge . . 31 
 Vj IV. Variations in the Coefficients from the Position of the 
 Orifice. General and Partial Contraction. Velocity of 
 Approach. Central and Mean Velocities ... 51 
 V. Submerged Orifices and Weirs. Contracted River Chan- 
 nels 71 
 
 VI. Short Tubes, Mouth-pieces, and Approaches. Coefficients 
 of Discharge for Simple and Compound Short Tubes. 
 
 "Shoots .' 78 
 
 VII. Lateral Contact of the Water and Tube. Atmospheric 
 
 Pressure. Head measured to the Discharging Orifice. 
 
 Coefficient of Resistance. Formula for the Discharge 
 
 from a Short Tube. Diaphragms. Oblique Junctions. 
 
 Formula for the Time of the Surface sinking a Given 
 
 Depth 89 
 
 VIII. Flow of Water in uniform Channels. Mean Velocity. 
 Mean Radii and Hydraulic Mean Depths. Border. 
 Train. Hydraulic Inclination. Effects of Friction. 
 Formulae for calculating the Mean Velocity. Applica- 
 tion of the Formulae and Tables to the Solutions of 
 three useful Problems . ' , . . . . 99 
 IX. Best forms of the Channel. Regimen .... 127 
 X. Effects of Enlargements and Contractions. Backwater 
 
 Weir Case. Long and Short Weirs .... 136 
 XI. Bends and Curves. Branch Pipes. Different Losses of 
 Head. General Equation for finding the Velocity. 
 Hydrostatic and Hydraulic Pressure. Piezometer. 
 Catchment Basins. Rain Fall per annum. Water 
 
 Power . . ' 150 
 
 A 2
 
 IV CONTENTS. 
 
 PAGE. 
 
 NOTE A. Discharge from one Vessel or Chamber into another. 
 
 Lock-chambers 169 
 
 TABLE. 
 
 I. Coefficients of Discharge from Square and Differently Pro- 
 portioned Rectangular Lateral Orifices in Thin Verti- 
 cal Plates 174 
 
 II. For Finding the Velocities from the Altitudes and the 
 
 Altitudes from the Velocities 176 
 
 III. Square Roots for finding the Effects of the Velocity of 
 
 Approach when the Orifice is small in proportion to the 
 Head. Also for finding the Increase in the Discharge 
 from an Increase of Head. See p. 55 . . . . 186 
 
 IV. For Finding the Discharge through Rectangular Orifices ; 
 
 in which n = ~ Also for Finding the Effects of the 
 
 Velocity of Approach to Weirs, and the Depression on 
 
 the Crest. See p. 55 188 
 
 V. Coefficients of Discharge for different Ratios of the Chan- 
 nel to the Orifice 192 
 
 VI. The Discharge over Weirs or Notches of One Foot in 
 
 Length in Cubic Feet per Minute .... 198 
 
 VII. For Finding the Mean Velocity from the Maximum Velo- 
 city at the Surface, in Mill Races, Streams, and Rivers 
 with uniform Channels ; and the Maximum Velocity 
 from the Mean Velocity. See p. 101 . . . . 204 
 VIII. For Finding the Mean Velocities of Water flowing in 
 
 Pipes, Drains, Streams, and Rivers . . . . 205 
 IX. For Finding the Discharge in Cubic Feet, per Minute, 
 when the Diameter of a Pipe, or Orifice, and the Velo- 
 city of Discharge, are known ; and vice versd . . 218 
 X. For Finding the Depths on Weirs of different Lengths, the 
 Quantity discharged over each being supposed constant. 
 
 See pages 149 and 150 220 
 
 XI. Relative Dimensions of equal Discharging Trapezoidal 
 
 Channels, with slopes from to 1, up to 2 to 1 . . 221 
 XII. Discharges from the Primary Channel in the first column 
 
 of TABLE XI. ... : . ... . . 222 
 
 XIII. The Square Roots of the Fifth Powers of Numbers for 
 finding the Diameter of a Pipe, or Dimensions of a 
 Channel from the Discharge, or the reverse ; showing 
 the relative discharging powers of Pipes of different 
 Diameters, and of any similar Channels whatever, closed 
 or open. (See pages 13, 123, 124, &c.) . . .224
 
 ERRATA. 
 
 Page 4, line 11, for "3807" read " '3807." 
 Page 6, line 10, for " 1-497" read " 14-97.' 
 line 15, for "1-3089" read " 18-089." 
 line 16, for " -1745" read "1745." 
 line 17, for " -0131" read " -131." 
 line 18, for "-0004" read " -004." 
 line 19, for " 1-4969" read " 14-969." 
 line 22, for " 1-497" read " 14-97." 
 for "1-485" read "14-85." 
 Page 9, line 24, for "1732-2" read "17322." 
 Page 13, line 15, for " TABLE IV." read " TABLE XIII." 
 
 lines 30 and 31, for "EXAMPLE 9, page 6" read "EXAMPLE IX., 
 
 page 109. 
 
 Page 14, line 5 from the bottom, for "formulae" read " formula." 
 Page 20, line 13, for "or = '617oR" read " or = '7854 o K." 
 
 Page 24, line 2, for -" read "-". 
 
 Page 25, line 8, for " 3-4478" read " "34478." 
 
 Page 29, line 15, for ".? ^2?' read "- */2J 
 
 3 
 
 Page 36, in note, for " -637" read " -636." 
 Page 142, line 9, for " proposition" read "proportion."
 
 INTRODUCTION. 
 
 IN preparing the following little work, we had three objects 
 in view: first, a collection of useful hydraulic formulae; 
 secondly, a collection of experimental results, and coeffi- 
 cients ; and, thirdly, a collection of useful practical tables, 
 some calculated entirely from the formulae and experiments, 
 and others for the purpose of rendering the calculations more 
 easy. 
 
 The TABLES at the end of the volume are all original, with 
 the exception of TABLE I., which contains the well-known 
 coefficients of PONCELET and LESBROS ; but those are newly 
 arranged, the heads reduced to English inches, and the coeffi- 
 cients for heads measured over and back from the orifice, 
 placed side by side, for more ready comparison. The coeffi- 
 cients in the small Tables throughout the work have been all 
 calculated by us from the original experiments ; the formulas 
 have been carefully examined, and the continental ones re- 
 duced to English measures some of them, as will be seen, 
 for the first time. No labour has been spared in preparing 
 the TABLES, and they are all purely hydraulic, though some 
 of them are capable of being otherwise applied. We have 
 filled no gap by the introduction of Tables applicable to other 
 subjects, and in every-day use. 
 
 The correction of some of the experimental formulae, par- 
 ticularly the continental ones, as printed in some English 
 books, cost us some labour. Even Du Buat's well-known 
 formula is frequently misprinted ; and in a late hydraulic 
 
 work, x/^ "1, one of the factors, is printed \/d '1 in 
 every page where it is quoted. It is not always that such 
 mistakes can be avoided, but experimental formulae are so 
 often copied from one work into another without sufficient
 
 Vlll INTRODUCTION. 
 
 examination, that an error of this kind frequently becomes 
 fixed ; and when applied to practical purposes erroneous for- 
 mulae get the correct ones into disrepute. See note to for- 
 mula (91), p. 113. 
 
 The TABLES of velocities and discharges over weirs and 
 notches have been calculated for a great number of coeffi- 
 cients to meet different circumstances of approach and over- 
 fall, and for various heads from th of an inch up to 6 feet. 
 TABLE II. embodies the velocities acquired by falling bodies 
 under the head of " theoretical velocity," and the velocities, 
 suited to various coefficients, for heads up to 40 feet. 
 
 The formulae for calculating the effects of the velocity of 
 approach to orifices and weirs, and the necessary corrections 
 for the ratio of the channel to the orifice, as well as TABLE V., 
 we believe to be original. They will be found of much value 
 in determining the proper coefficients suited to various ratios. 
 The remarks throughout SECTION IV. are particularly appli- 
 cable to the proper use of this TABLE. 
 
 TABLE VII. of surface and mean velocities will be found to 
 vary from those generally in use, and to be much more correct, 
 and better suited for practical purposes, particularly as applied 
 to finding the mean velocities in rivers. 
 
 We have extended TABLE VIII. so as to make it directly 
 available for hydraulic mean depths, from T ^th of an inch to 
 12 feet, and for various hydraulic inclinations, even up to ver- 
 tical, for pipes. The fall in rivers seldom exceeds 2 or 3 feet 
 per mile, or the velocity 5 or 6 feet per second. The exten- 
 sion of the table for great inclinations, and consequently great 
 velocities, was made for purposes of calculation, and to 
 include pipes. It must be understood throughout this TABLE 
 that the velocities are those which continue unchanged for 
 any length of channel, viz., when the resistance of friction 
 is equal to the acceleration of gravity, the moving water and 
 channel being then in train. Several of Du Buat's experi- 
 ments were made with small vertical pipes. Thfs TABLE is 
 equally applicable to pipes and rivers, and gives directly 
 either the hydraulic inclination, the hydraulic mean depth, or 
 the velocity when any two of them are known.
 
 INTRODUCTION. IX 
 
 Hydraulic formulae have been frequently rendered unneces- 
 sarily complex, and unsuited for practical application, by 
 combining them with those of mere mensuration in order to 
 find the discharge. We have therefore given formulae for 
 finding the mean velocity principally, unless in a few in- 
 stances, as in orifices near the surface, where the discharge it- 
 self is first necessary to find the mean velocity : this once 
 determined, the calculation of the discharge becomes one of 
 simple mensuration. 
 
 We have preferred giving the mean velocity to the dis- 
 charge itself in TABLE VIII., because, while an infinite 
 number of channels having the same hydraulic inclination (5) 
 and the same hydraulic mean depth (r) must have the same 
 velocity (V), yet the sectional areas, and consequently the 
 discharges, may vary upwards from 6'2832r 2 , the area of a 
 semicircular channel, to any extent; and the operation of 
 multiplying the area by the mean velocity, to find the dis- 
 charge, is so very simple that any tabulation for that purpose 
 is unnecessary. Besides this, the banks of rivers, unless 
 artificially protected, remain very seldom at a constant slope, 
 and therefore any TABLES of discharge for particular side 
 slopes are only of use so far as they apply to hypothetical 
 cases. Indeed we have seen, in new river cuts, the banks, 
 cut first to a given slope, alter very considerably in a few 
 months ; while the necessary regimen between the velocity of 
 the water and the channel was in the course of being esta- 
 blished. The velocity suited to the permanency of any pro- 
 posed river channels, though too often entirely neglected, is 
 the very first element to be considered. 
 
 For circular pipes, however, TABLE IX. gives the discharge 
 in cubic feet per minute when the velocity in inches per second 
 is known, or found from TABLE VIII., and is calculated for 
 pipes from Jth of an inch up to 12 inches in diameter. 
 TABLE XII. gives also the discharges in cubic feet per minute 
 from the different equivalent river channels in TABLE XI. 
 
 TABLE X., for finding the heads on weirs of different 
 lengths, TABLE XI., of equal discharging river channels, and 
 TABLE XII., of the actual discharges from the equivalents in
 
 X INTRODUCTION. 
 
 TABLE XL, will be found of great practical value when new 
 weirs and water-cuts have to be made. TABLES XI. and XII. 
 are equally applicable to channels having side slopes, the 
 widths being then the mean or central widths : seepages 124, 
 125, and 126. 
 
 When the discharge and fall are known, and the hydraulic 
 mean depth and the dimensions of any channel have to be 
 determined, Problem III., page 122, as illustrated in EX- 
 AMPLE 17, page 13, gives a new and perhaps the most 
 practically useful solution yet published. Tables XL, XII., 
 and XIII. are particularly applicable to this problem. 
 
 A uniform notation is preserved throughout the work, so 
 that the different experimental formulae can be compared 
 without any further reduction. The letter h is used in every 
 instance for the head, c for the coefficient, r for the mean 
 radius or hydraulic mean depth, and s for the sine of the 
 hydraulic inclination, unless it be otherwise stated. In order 
 to designate particular values, the primary letters have depo- 
 nent or initial letters below to explain them. Thus h t is 
 the head to the top of an orifice, h b the head at the bottom, 
 A w the head on a weir, h f the head due to friction, c d the 
 coefficient of discharge, c v the coefficient of velocity, c c the 
 coefficient of contraction, &c. When the whole head is made 
 up of different elements, such as the portions due to friction, 
 velocity, contractions, bends, &c., it is expressed by the 
 capital letter H. 
 
 Some writers and engineers appear to confound the inclina- 
 tion of a pipe, simply so called, or the head divided by the 
 length, with the hydraulic inclination ; and consequently have 
 fallen into error in applying such of the known formulae as 
 take into consideration only the head due to the resistance of 
 friction. When pipes are of considerable length, and the 
 water is supplied from a reservoir at one end, the inclination, 
 found as above, and the hydraulic inclination, may be taken 
 equal to each other without sensible error; but for shorter 
 pipes, of say up to 100 feet long, or even longer, the greater 
 number of formulae, as Du Buat's and others, do not directly 
 apply ; and it is necessary to take into consideration the head
 
 INTRODUCTION. XI 
 
 due to the orifice of entry, the velocity in the tube, and also 
 to the impulse of supply when there are junctions. These 
 separate elements, and their effects, will be considered in the 
 following pages ; but it will be of use to refer here to some 
 late experiments, and the imperfect application of formulae to 
 them, first premising that a pipe may be horizontal, or even 
 turn upwards, and yet have a considerable hydraulic in- 
 clination. 
 
 Mr. Pro vis's valuable experiments* with IJ-inch pipes, 
 from 20 to 100 feet long, have been used in a recent work 
 for the purpose of testing the accuracy of Du Buat's and 
 some other formulas ; but the head divided by the length is 
 assumed to be the hydraulic inclination throughout, and no 
 allowance is made for the head due to the orifice of entry and 
 velocity in the pipe. Of course the writer's conclusions are 
 erroneous. We have shown, SECTION I., page 12, how nearly 
 the formulae and experiments agree. 
 
 The formulas appear to have been also misunderstood by the 
 surveyor who experimented for the General Board of Health ; 
 for the inclination of the pipe in itself is assumed to be the 
 hydraulic inclination, and no allowance is made for the head 
 due to the impulse of supply. We quote from the CIVIL 
 ENGINEER AND ARCHITECT'S JOURNAL, Vol. XV., page 366, 
 in which it is stated that "the chief results as respect the 
 house drains are thus described in the examination of the 
 surveyor appointed to make the trials."*t" 
 
 " What quantity of water would be discharged through a 3-inch pipe 
 on an inclination of 1 in 120 1 Full at the head, it would discharge 
 100 gallons in three minutes, the pipe being 50 feet in length. This is 
 with stone-ware pipe manufactured at Lambeth. This applies to a pipe 
 receiving water only at the inlet, the water not being higher than the 
 head of the pipe. 
 
 " What water was this ? Sewage-water of the full consistency, and it 
 was discharged so completely that the pipe was perfectly clean. 
 
 * Transactions of the Institution of Civil Engineers, Vol. II., pp. 201 
 210. 
 
 t Minutes of Information with reference to Works for the removal of 
 Soil, Water, or Drainage, <fcc., &c. Presented to both Houses of Parlia- 
 ment, 1852.
 
 Xll INTRODUCTION. 
 
 " At the same inclination what would a 4-inch pipe discharge with the 
 same distances 1 Twice the amount (that I found from experiment) ; 
 or, in other words, 100 gallons would be discharged in half the time. 
 This likewise applies to a pipe receiving water only at the inlet, and of 
 not greater height than the head. In these cases the section of the 
 stream is diminished at the outlet to about half the area of the pipe. 
 
 " Before these experiments were made, were there not various hypo- 
 thetical formulae * proposed for general use ? Yes. 
 
 " What would these formulae have given with a 3-inch pipe, and at an 
 inclination of 1 in 100 ? and what was the result of your experiments 
 with the 3-inch pipe ? The formulae would give 7 cubic feet, the actual 
 experiment gave 111 cubic feet ; converting it into time, the discharge, 
 according to the formulae, compared with the discharge found by actual 
 practice, would be as 2 to 3. 
 
 " How would it be with a 4-inch pipe ? The formulae would give about 
 14-7 cubic feet per minute, whereas practice gave 23 cubic feet per 
 minute. 
 
 " Take the case of a 6-inch pipe of the same inclination ? The results, 
 according to Mr. Hawkesley's formula, would be 40J cubic feet per 
 minute ; from experiment it was found to be 63^ cubic feet per minute. 
 
 " Then with respect to mains and drainage over a flat surface, the re- 
 sult of course becomes of much more value, as the difference proved by 
 actual practice increases with the diminution of the inclination 1 Cer- 
 tainly, to a very great extent. For example, the tables give only 14'2 
 cubic feet per minute as the discharge from a pipe 6 inches diameter, 
 with a fall of 1 in 800 ; practice shows that, under the same conditions, 
 47'2 cubic feet will be discharged. 
 
 " Will you give an example of the practical value of this when it is 
 required to carry out drainage works over a very flat surface ? An in- 
 clination of 1 in 800 gives only 14 cubic feet per minute, according to 
 theory, while, according to actual experiment, and with the same incli- 
 nation, 47 cubic feet are given. 
 
 " Then this difference may be converted either into a saving of water 
 to effect the same object, or into power of water to remove feculent mat- 
 ter from beneath the site of any houses or town ? It may be so. 
 
 " And also the power of small inclinations properly managed ? Yes ; 
 
 * It is a mistake to call those formulae hypothetical, unless so far as 
 the hypothesis is founded on facts. Every formula with which we are 
 acquainted is founded on experiments, and has been deduced from them, 
 but those formulae are too often hypothetically applied to short tubes 
 without the necessary corrections. It will be seen from SECTION VIII. 
 that the experiments from which the formulae there given were derived, 
 were in every way greatly more extensive than those made by the direc- 
 tions of the Board of Health.
 
 INTRODUCTION. 
 
 Xlll 
 
 for example, if it was required to construct a water course that should 
 discharge, say 200 feet per minute, the formula would require an incli- 
 nation of 1 in 60 = 2 inches in 10 feet ; whereas, experiment has shown 
 that the same would be discharged at an inclination of 1 in 200 = inch 
 in 10 feet, thus effecting a considerable saving in excavation, or a smaller 
 drain would suffice at the greater inclination." 
 
 We have extracted and tabulated the results given above, 
 in the following Table, and also eight of the experiments 
 made for the Metropolitan Commissioners of Sewers*; and 
 assuming for the present, with the surveyor examined by the 
 Commissioners, that the inclinations of the pipes and hy- 
 draulic inclinations of the formulae are the same, which is 
 incorrect, we give the calculated discharges, found by means 
 of TABLES VIII. and IX., in the last column of the Table. 
 
 Diameter of 
 pipe 
 in inches. 
 
 Inclination of 
 pipe. 
 
 Discharge in 
 cubic feet 
 per minute 
 by experiment. 
 
 Hypothetical 
 discharge by 
 Du Buat's 
 formula. 
 
 3 
 
 1 in 120 
 
 5-3 
 
 6-6 
 
 4 
 
 1 in 120 
 
 10-7 
 
 14 
 
 3 
 
 in 100 
 
 11-2 
 
 7'5 
 
 4 
 
 in 100 
 
 23 
 
 15-6 
 
 6 
 
 in 100 
 
 63-5 
 
 43-8 
 
 6 
 
 in 800 
 
 47-2 
 
 13-3 
 
 6 
 
 Iin60 
 
 75 
 
 59-3 
 
 6 
 
 1 in 100 
 
 63 
 
 438 
 
 6 
 
 1 in 160 
 
 54 
 
 33-4 
 
 6 
 
 1 in 200 
 
 52 
 
 29-2 
 
 6 
 
 1 in 320 
 
 49 
 
 21-8 
 
 6 
 
 1 in 400 
 
 48-5 
 
 19-6 
 
 6 
 
 1 in 800 
 
 47-2 
 
 13-3 
 
 6 
 
 Level 
 
 46 
 
 00 
 
 Du Buat's formula, therefore, gives larger results than the 
 experiments in the two first cases, because the water received 
 at one end only barely filled it, and the pipe was not full at 
 
 Adcock's Engineer's Pocket Book, 1852, pp. 261 and 262.
 
 XIV INTRODUCTION. 
 
 the lower end ; but less in the others. If in these the head 
 due to the impulse of entry, at the upper end, and at the 
 side junctions, were known, and the proper hydraulic inclina- 
 tion determined by the experiments, the formulae would be 
 found to give larger approximate results in every case, as 
 might have been expected from the sewage- water used. In 
 the last eight experiments it is stated*, that "the water 
 was admitted at the head of the pipe, and at Jive junctions 
 or tributary pipes on each side, so regulated as to keep the 
 main pipe full," and that " without the addition of junctions 
 the transverse sectional area of the stream of water near the 
 discharging end was reduced to one-fifth of the corre- 
 sponding area of the pipe, and that it required a simple 
 head of water of about 22 inches to give the same result 
 as that accruing under the circumstances of the junctions." 
 It is also stated, that " in the case of the 6-inch pipe, which 
 discharged 75 cubic feet per minute, the lateral streams had 
 a velocity of a few feet per minute." 
 
 Now, the head of "about 22 inches" is wholly neglected 
 in the foregoing calculations, though in a pipe 100 feet long 
 it would be equal to an inclination of 1 in 55 ! It however 
 includes three elements at least, viz. the portion due to the 
 orifice of entry, the portion due to the velocity in the pipe, 
 and the portion due to friction. Let us assume the case of 
 the horizontal pipe, which discharged 46 cubic feet per 
 minute^. This is equal to a mean velocity of 46'9 inches 
 per second; with this velocity, we find from TABLE VIII. 
 the hydraulic inclination of a 6-inch pipe to be 1 in 94, and, 
 therefore, the head due to friction in a pipe 100 feet long 
 is 12'7 inches. Assuming the coefficient for the orifice of 
 entry and velocity to be '815, we also find from TABLE II. a 
 head of 4J inches due to these. We then have, 
 
 * Adcock's Engineer's Pocket Book, 1852, pp. 261 and 262. 
 
 t The horizontal pipe would discharge equally at both ends, unless 
 there was a head of water at either, or an equivalent in the velocity 
 of approach. Of course, a smaller pipe with a fall, must be better than 
 the larger one with none at all, in preventing deposits.
 
 INTRODUCTION. XV 
 
 Head due to the velocity and orifice of entry 4*25 inches 
 Head due to the resistance of friction . . 12*70 
 Radius of pipe 3*00 
 
 Total . . . 19-95 
 
 which is about 2 inches less than the observed head : this, 
 however, is not stated definitely. It is therefore evident, 
 that the formula gives, if anything, larger results than 
 these experiments*, as. might have been expected, instead 
 of less in the ratio of 2 to 3, as is stated in the Report. 
 
 Wherever junctions are applied, as in the examples above 
 referred to, the formulae in general use require correction; 
 for the quantity of water then flowing below each junction 
 is increased. A certain amount of error is, perhaps, in- 
 separable from every calculation of this kind ; but before we 
 condemn formulae deduced from experiment by men every 
 way qualified for the task, it would be well that we should 
 learn to understand and properly apply them. 
 
 The diameter of a short pipe gives in itself the means of 
 increasing very considerably the surface inclination of the 
 fluid stream, by reducing the section at the lower end. If 
 we assume a horizontal pipe 50 feet long and 6 inches in 
 diameter, we perceive, that if the receiving end be full, and 
 the discharging end one-third full, this inclination will be 
 
 6 2 1 
 
 ' =: ; and that the discharging end cannot be 
 
 50 x 12 150' 
 kept full unless a head of several inches be maintained at 
 the receiving end, or an equivalent from a lateral supply. 
 When the pipe is about two diameters long it becomes a 
 short tube; and when the length vanishes, the transverse 
 section becomes, simply, a discharging orifice. 
 
 We have been led into the foregoing remarks, not from 
 any desire to find fault with a Report containing so much 
 valuable information as the one referred to, but for the pur- 
 
 * This is also true of the other formula), for finding the discharge 
 from pipes, given in this work.
 
 XVI INTRODUCTION. 
 
 pose of defending from unmerited reproach, in a Blue Book, 
 the researches in this department, of 
 
 " Those dead but sceptred sovereigns who still jule 
 Our science from their urns " 
 
 Du Buat, Young, Eytelwein, Prony, and others. 
 
 We do not pretend to any particular accuracy in the 
 sketches scattered throughout the work; they are only in- 
 tended to illustrate the text, and were sketched while writ- 
 ing it, without further aim; neither -do we pretend to have 
 entered fully into the principles or practice of hydraulics, our 
 object being to select, construct, and arrange useful hydraulic 
 formulae, experiments, and Tables for the use of all classes of 
 engineers. We make, however, no apology for preferring 
 formulas, in their simplicity, to any written rules which may 
 be deduced from them, as being in every way more general, 
 concise, and elegant. In conclusion, it is hoped that any 
 errors of consequence in the work, will be found corrected 
 in the errata. 
 
 Baden Place, Dundalk, 
 January, 1853.
 
 ON THE 
 
 DISCHARGE OF WATER 
 
 FROM 
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 SECTION I. 
 
 APPLICATION AND USE OF THE TABLES, FORMULAE, &C. 
 
 To find the velocity of a falling body from the height 
 fallen, or the height fallen from the velocity. 
 
 RULE. MULTIPLY THE SQUARE ROOT OF THE HEIGHT 
 IN INCHES BY 27'8, AND THE PRODUCT WILL BE THE 
 VELOCITY IN INCHES. To FIND THE HEIGHT FROM THE 
 VELOCITY, SQUARE THE VELOCITY IN INCHES AND DIVIDE 
 THE SQUARE BY 772*84, THE QUOTIENT WILL BE THE 
 HEIGHT IN INCHES. See equation (1). TABLE II., column 1, 
 will give the velocity from the height, found in the column 
 of " altitudes," or the height from the velocity, directly. 
 
 EXAMPLE 1. What is the velocity acquired by a heavy 
 body falling \th of an inch ? In the Table opposite to th 
 of an inch, found in the column headed "altitudes h," we 
 find 9*829 in column 1, for the required velocity in inches 
 per second. 
 
 EXAMPLE 2. What is the velocity acquired by a fall of 
 1 1 feet 3 inches ? Opposite to 1 1 feet 3 inches, as before, 
 we find 323*007 inches, for the velocity required. 
 
 EXAMPLE 3. What height must a heavy body fall from 
 to acquire a velocity of 40 feet per second? Here 40 feet 
 equal 486 inches, opposite to the nearest number to which, 
 found in column 1, we find 25 feet 6 inches for the re- 
 quired fall. In this example, the nearest number to 486 
 
 B
 
 2 THE DISCHARGE OF WATER FROM 
 
 found in the Table is 486-301. The difference '301 cor- 
 responds, very nearly, to |ths of an inch in altitude, and, 
 therefore, the true head according to the rule would be 
 25' 6f"; but for all practical purposes the difference is 
 immaterial. 
 
 By means of TABLE II. we can find directly, or by simple 
 interpolation, the velocity due to all heights from y^- part of 
 an inch up to 40 feet, and the heights from the velocities. 
 For a greater height than 40 feet it may be divided by 4, 9, 
 or some square number s 2 , and the velocity found for the 
 quotient, from the Table, multiplied by 2, 3, or s, the square 
 root of the divisor, will give the velocity required. 
 
 EXAMPLE 4. What is the velocity acquired by a fall of 
 
 45 
 45 feet ? = 11' 3", the velocity corresponding to which, 
 
 found from the Table, is 323-007. Hence, 
 
 323-007 x v/4 = 323-007 x 2 = 646" -014 = 53' 10"-014 
 is the velocity per second required. The reverse of this 
 example is equally simple. 
 
 Columns 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 in the Table, 
 give the values of \/2gh multiplied by the coefficients 
 therein stated. These columns will be found of great prac- 
 tical use in finding the mean velocities in the vena-contracta, 
 in the orifice, and in short tubes ; and consequently also in 
 finding the mechanical force and discharge. An examination 
 of the coefficients in the small Tables in SECTION III., 
 and also those in TABLES I. and V., at the end of the work, 
 will show how much they vary; but those most generally 
 useful and their products by the theoretical velocity due to 
 different heads, up to 40 feet, are given in the columns 
 referred to. 
 
 EXAMPLE 5. What is the discharge from an orifice 
 4 inches by 8 inches, the centre sunk 20 feet below the 
 surface of a reservoir? From TABLE II., we find 430-676 
 inches equal 35'89 feet for the theoretical velocity of dis- 
 charge : hence, -JL. x 35-89 = -|- x 35'89 = 7'976 cubic 
 144 y
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. .3 
 
 feet per second is the theoretical discharge. If the discharge 
 takes place through a thin plate, or if the inner arrisses next 
 the water in the reservoir be perfectly square, and the water 
 in flowing out does not fill the passage so as to convert the 
 orifice into a short tube, the coefficient will be found from 
 TABLE I. to be '603. The true discharge then is 7-976 x 
 603 = 4-809 cubic feet per second. 
 
 For the determination of the coefficient suited to any 
 particular orifice, and the circumstances of its position, we 
 must refer generally to the following pages. If in the 
 example just given, the arrisses next the reservoir were 
 rounded into the form of the contracted vein, see Fig. 4, 
 the coefficient would increase from '603 to '974 or '956, for 
 a passage not exceeding a couple of feet in length. With 
 the former the discharge would be 7'976 x '974 = 7-769 
 cubic feet, and with the latter 7-976 x -956 =7*625 cubic feet. 
 We may find from TABLE II. the latter results otherwise. 
 With a head of 20 feet and the coefficient -974, the velocity 
 
 is 419'48 inches = 34-957 feet; hence, the discharge is 
 
 fy 
 
 ^ X 34-957 = 7-768 cubic feet. With a coefficient of -956, 
 
 2 
 the velocity is 411-73 inches =: 34-31 feet, and - x 34*31 = 
 
 7-624 cubic feet. These results are the same, practically, as 
 those previously found. 
 
 If the inner arrisses be square, and the passage out be 
 from 18 inches to 2 feet long, the orifice will be converted 
 into a short tube, the coefficient for which is -815. With 
 this coefficient, and a head of 20 feet, we find as before, 
 from TABLE II., the mean velocity of discharge equals 
 351 inches = 29*25 feet; hence, the discharge now is 
 2 
 n X 29-25 = 6-5 cubic feet per second. 
 
 y 
 
 The velocities in indies per second, given in TABLES II. 
 and VIII., or elsewhere in the following pages, may be 
 converted into velocities in feet per minute, by multiplying 
 
 , , ,60 
 
 by 5, equal . 
 
 B 2
 
 4 THE DISCHARGE OF WATER FROM 
 
 EXAMPLE 6. The discharge from a small orifice having 
 its centre placed 10 feet below the surface of a reservoir 
 is 18 feet per minute, what will be the discharge from the 
 same orifice at a depth of 17 feet ? The discharges will 
 be to each other as \/T6 : \/\l, or as 1 : \/\'1\ or > fr m 
 TABLE III., as 1 : 1-3038, whence we get the discharge 
 sought equal 1-3038 x 18 = 23-4684 cubic feet. 
 
 EXAMPLE 7. What is the value of the expression 
 
 c d 1 1 "*" 5 ^~~r f i n Aquation (45), when c d = '617, and 
 \. tn c ,jJ 
 
 m == 2 ? Here we have 
 
 ' 4--6\7*~ 3^6193 ~' W52 ' 
 whence the expression becomes equal to '617 (1-1052)* equal, 
 from TABLE III., -617 x 1-0513 = -649, the value sought. 
 TABLE V. contains the values of this expression for various 
 values of c d and m, which latter, m, stands for the ratio of 
 the channel to an orifice ; and we can immediately find from 
 it, opposite 2 in the first column, and under the coefficient 
 617 in the sixth column, -649, the value sought. When the 
 head due to the pressure, and the velocity of approach, are 
 both known, we can determine the new coefficient of dis- 
 charge by the above expression, and thence the discharge 
 itself. The coefficient suited to the velocity of approach is, 
 however, to be found directly in TABLE V. 
 
 EXAMPLE 8. What is the discharge from an orifice 
 17 inches long and 9 inches deep, having the upper edge 
 placed 4 inches below the surface, and the lower edge 
 13 inches ? The expression for the discharge is 
 
 2 
 
 3 
 
 equation (43), in which we must take d = 9 inches ; h t = 9 
 inches; A = 17x9=153 square inches; and </2gd, found 
 
 from TABLE II. = 83-4 inches. We have, also, -^ = 1 = -444, 
 and hence the value of 
 
 (1-444)' - (-444)' = (from TABLE IV.) 1-44.
 
 - x -617 x 27/8 x 17 x {13* - 
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. O 
 
 Assuming the coefficient of discharge to be '617, we then 
 have the discharge in cubic inches per second equal to 
 
 | x 153 x 83-4 x -617 x 1-44 = 
 3 
 
 | x 12760-2 x -88848 = 7558. 
 3 
 
 Consequently, == 4 - 374 is the discharge in cubic feet 
 
 i7 2o 
 
 per second. From equation (6), we get the discharge 
 equal to 
 
 2 
 
 3 
 
 But 13* 4' = 46-872 - 8, from TABLE IV., equal to 
 38-872, whence the discharge is 
 
 ? x -617 x 27-8 x 17 x 38-872 = 11-4351 x 17 x 38-872 = 
 3 
 
 194-3967 x 38-872 = 7557 cubic inches = 4-374 cubic feet, 
 the same as before. 
 
 It is shown in equation (31), that by using the mean 
 depth for orifices near the surface, the discharge will ap- 
 proximate very closely to the true discharge, and that even 
 for weirs the error will not exceed 6 per cent. The discharge 
 is then expressed by '617 Jig x 8% x 9 X 17 = (from 
 TABLE II.) 50-01 x 153 = 7651-53 cubic inches = 4-427 
 cubic feet per second. The head to the centre of the orifice 
 is here 8 inches, and the depth of the orifice 9 inches, 
 therefore, in equation (31), h d very nearly; and, therefore, 
 this result must be multiplied by "989, as shown in that 
 equation; then -989 x 4-427 = 4'378 cubic feet, which gives 
 a result differing from those otherwise found, by a very small 
 quantity, which, practically, is of no value. By means of 
 TABLE VI. the discharge from rectangular orifices near the 
 surface can be found with very great facility. 
 
 We may always find the discharge from an orifice near 
 the surface with sufficient accuracy by taking the head to 
 the centre, as if it were sunk to a considerable depth, and
 
 THE DISCHARGE OF WATER FROM 
 
 then applying the corrections given in equation (31); or if 
 the orifice be circular, those given in equation (28). 
 
 EXAMPLE 9. What is the discharge from a circular 
 orifice 4 inches in diameter, having its centre placed 4 inches 
 below the surface, when the coefficient of discharge is "617? 
 The area of the orifice is 4 x 4 x '7854 = 12-566 square 
 inches. The velocity at the mean depth of 4 inches, with a 
 coefficient of '617, is 34'31 inches, whence the discharge is 
 12-566 x 34-31 = 431-139 cubic inches = -2496 cubic feet 
 per second, or 1*497 cubic feet per minute. By means of 
 TABLE IX. the discharge in cubic feet per minute can be 
 found very readily when the velocity, 34'31 inches per second, 
 is known. Thus, 
 
 Inches. Cubic feet. 
 
 For a velocity of 30-00 the discharge is 1-3089 
 
 4-0 '1745 
 
 -30 -0131 
 
 01 -0004 
 
 34-31 1-4969 
 
 By applying the coefficient found from equation (28), which 
 is '992, when the depth at the centre is twice the radius, as 
 it is in this example, we get -992 x 1-497 = 1-485 for the 
 discharge in cubic feet per minute. 
 
 The application of TABLE VI. will enable us to find the 
 discharge from rectangular orifices near the surface very 
 quickly. Resuming " EXAMPLE 8," the discharge may be 
 found from this Table for each foot in length of the orifice, 
 as follows. The discharge in cubic feet per minute, when the 
 coefficient is -617 for a notch 1 foot long and 13 inches 
 deep, is 223-323; and for a notch of 4 inches deep, 38-116; 
 therefore, the discharge from an orifice 9 inches deep, with 
 the upper edge 4 inches below the surface, is 223*323 
 38-116 = 185-207 cubic feet per minute. But as the length 
 
 of the orifice is 17 inches, this must be multiplied by , and 
 
 the product 262-377 is the discharge in cubic feet per minute ; 
 this is equal to a discharge of 4-373 cubic feet per second,
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 7 
 
 which agrees with that before found. This is the simplest 
 way of finding the discharge from rectangular orifices near 
 the surface. 
 
 EXAMPLE 10. Wliat is the discharge in cubic feet per 
 minute, from an orifice 2 feet 6 inches long and 1 inches 
 deep, the upper edge being 3 inches below the surface, and 
 the coefficient of discharge *628 ? From TABLE VI. we 
 find the discharge from a notch 1 foot long and 10 inches 
 deep to be 153*353, and for a notch 3 inches deep, 25' 199. 
 The difference, or 128' 154, multiplied by 2, will be the 
 discharge required; viz. 2 x 128' 154 = 320'385 cubic feet 
 per minute. 
 
 EXAMPLE 11. The size of a channel is 2'75 times the 
 size of an orifice, what is the coefficient of discharge when 
 that for a very large channel in proportion to the orifice 
 is '628 ? We find from TABLE V. the coefficient sought to be 
 645, when the approaching water suffers full contraction. By 
 attending to the auxiliary Tables in the text, we find, for this 
 
 case, , = = '36. We must, therefore, multiply 
 
 channel 275 " J 
 
 2*75 by '857, which gives 2 - 36 for the ratio of the mean 
 velocities in the orifice and approaching it. With this new 
 value of the ratio of the channel to the orifice, we find, as 
 before, the value of the coefficient from TABLE V. to be 
 651. The remarks throughout the work, with the aux- 
 iliary tables, will be found of much use in determining 
 the coefficients for different ratios of the channel to the 
 orifice, notch, or weir, and the corrections suited to each. 
 If in this example we were considering, other things being 
 the same, the alteration in the coefficient for a notch, or weir, 
 it would be found from the Table, column 4, to be "672 
 instead of '645 found in column 3, for an orifice sunk some 
 depth below the surface. For the corrections suited to mean 
 and central velocity, and to the nature of the approaches, we 
 must refer to the body of this work and the auxiliary tables. 
 EXAMPLE 12. What is the discharge over a weir 50 feet 
 long; the circumstances of the overfall, crest, and ap- 
 proaches, being such that the coefficient of discharge is '617,
 
 8 THE DISCHARGE OF WATER FROM 
 
 when ihe head measured from the water in the weir basin, 
 6 feet above the crest, is 17 inches? TABLE VI. will give 
 the discharge in cubic feet per minute, over each foot in 
 length of weir, for various depths up to 6 feet. It is divided 
 into two parts ; the first for " greater coefficients," viz. -667 
 to "617; and the second for "lesser coefficients," viz. '606 to 
 518. The coefficient assumed being '617, we find the dis- 
 charge over 1 foot in length, with a head of 17 J inches, to 
 be 348*799 cubic feet per minute ; hence the required dis- 
 charge is 50 x 348-799 = 17439-95 cubic feet. 
 
 The determination of the coefficient suited to the circum- 
 stances of the overfall, crest, approaches, and approaching 
 section, will be found discussed elsewhere through this 
 work. The valuable Table derived from Mr. Blackwell's 
 experiments will also be of use ; but the heads being taken 
 at a much greater distance back from the crest than is 
 generally usual, the coefficients taken from it for heads 
 greater than 5 or 6 inches, will be found under the true ones 
 for heads measured immediately at or about 6 feet, above 
 the crest. For heads measured on the crest, the small 
 Table of coefficients in SECTION III., applicable to the pur- 
 pose, will be of use. 
 
 EXAMPLE 13. What is the mean velocity in a large 
 channel, when the maximum velocity along the central line 
 of the surface is 31 inches per second? TABLE VII. gives 
 25*89 inches for the required velocity, and for smaller chan- 
 nels 24'86 inches. In order to find the mean velocity at the 
 surface from the maximum central velocity, the latter must 
 be multiplied by '914. 
 
 The velocity at the surface is best found by means of a 
 floating hollow ball, which just rises out of the water. The 
 velocity at a given depth is best found by means of two 
 hollow balls connected with a link, the lower being made 
 heavier than the upper, and both so weighted by the ad- 
 mission of a certain quantity of water that they shall float 
 along the current, the upper one being in advance but nearly 
 vertical over the other. The velocity of both will then be the 
 velocity at half the depth between them. The velocity at the
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 9 
 
 surface, found by means of a single ball, being also found, the 
 velocity lost at the half depth is had by subtracting the 
 common velocity due to the linked balls from that of the 
 single ball at the surface. The velocity at any given depth is 
 then easily found by a simple proportion ; but the result will 
 be most accurate when the given depth is nearly half the 
 distance between the balls, which distance can never exceed the 
 depth of the channel. Pitofs tube, JVoltmanri's tachometer, 
 the hydrometric pendulum, the rheometer, and several other 
 hydrometers, have been used for finding the velocity; but 
 these instruments require certain corrections suited to each 
 separate instrument, as well as kind of instrument, and are 
 not so correct or simple for measuring the velocity in open 
 channels as a ball and linked balls. 
 
 EXAMPLE 14. What is the discharge from a river having 
 a surface inclination of 18 inches per mile, or 1 in 3520, 
 40 feet wide, with nearly vertical banks, and 3 feet deep ? 
 The area is 40 x 3 = 120 feet, and the border 40 + 2 x 3 
 
 1 20 
 = 46 feet ; therefore the hydraulic mean depth is 
 
 40 
 
 = 2-61 feet = 2 feet 7-3 inches*. With this and the in- 
 clination we find from TABLE VIII. 28-27 + 2-75 x 
 
 6 
 
 = 28-87 inches per second = 28'87 x 5 = 144-35 feet per 
 minute for the mean velocity; hence we get 144*35 x 120 
 = 1732-2 cubic feet per minute for the required discharge. 
 For channels with sloping banks we have only to divide the 
 border, which is always known, into the area for the hydraulic 
 mean depth, with which, and the surface inclination, we can 
 always find the velocity by TABLE VIII., and thence the 
 discharge. Unless the banks of rivers be protected by stone 
 pavement or otherwise, the slopes will not continue per- 
 manent; it is therefore almost useless to give the discharges 
 for channels of particular widths and side slopes. When the 
 mean velocity is once known, the remaining calculations are 
 
 * For greater hydraulic depths than 144 inches, the extent of tlie 
 TABLE, divide by 9, and find the corresponding velocity. This multiplied 
 by 3 will be the velocity sought.
 
 10 THE DISCHARGE OF WATER FROM 
 
 those of mere mensuration, and they should be made 
 separately. 
 
 EXAMPLE 15. The diameter of a very long pipe is 1 
 inch, and the rate of inclination, or whole length of the pipe 
 divided by the whole fall, is 1 in 71^-; what is the discharge 
 in cubic feet per minute? The hydraulic mean depth, 
 
 1*5 3 
 
 or mean radius, is rz '375 inches = _ inch. Conse- 
 4 8 
 
 quently we find from TABLE VIII. the velocity in inches per 
 second equal to 25'09 - 1-92 x = 25-09 - -29 = 24'80. 
 
 The discharge in cubic feet per minute for a l|-inch pipe is 
 now found most readily by means of TABLE IX., as follows: 
 
 Inches. Cubic feet. 
 
 For a velocity of 20-0 the discharge is 1-227 
 4-0 -245 
 
 8 -049 
 
 1-521 
 
 Whence the discharge in cubic feet per minute is 1'521. 
 
 For short pipes, of 100 or 200 feet in length and 
 under, the height due to the velocity and orifice of entry 
 must be deducted from the whole height to find the proper 
 hydraulic inclination, and also the height due to bends, curves, 
 cocks, slides, and erogation. The neglect of these corrections 
 has led some writers into mistakes in applying certain formulae, 
 and in testing them by experimental results obtained with 
 short pipes. We shall now apply the TABLES to the deter- 
 mination of the discharge from short pipes, and compare the 
 results with experiment, referring generally to equation (153) 
 and the remarks preceding it for a correct and direct solution. 
 
 EXAMPLE 16. What is the discharge in cubic feet per 
 minute from a pipe 1 00 feet long, with a fall or head of 
 35 inches at the lower end, when the diameter is 1^ inch? 
 Find also the discharge from pipes SO feet, 60 feet, 40 feet, 
 and 20 feet, of the same diameter and having the same 
 head. If the water be admitted by a stop-cock at the upper 
 end, the coefficient due to the orifice of entry will probably 
 be about -75 or less, -815 being that for a clear entry to
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 11 
 
 a short cylindrical tube. The approximate inclination is 
 
 100 x 12 = 1 in 34-3; but as a portion of the fall must be 
 35 
 
 absorbed by the velocity and orifice of entry, we may assume 
 for the present that the inclination is 1 in 35. With this 
 
 1- 3 
 
 inclination and the mean radius = inches, we find the 
 
 mean velocity from TABLE VIII. to be 38*06 inches. Now 
 when the coefficient due to the orifice of entry and velocity is 
 75, we find from TABLE II. the head due to this velocity to 
 be 3f- inches nearly, whence 35 3-| =. 3 If 31 '625 inches 
 
 is the height due to friction, and 1QQ x 12 equals 1 in 37-9, 
 
 31*ti25 
 
 the inclination, very nearly. With this new inclination we 
 find, as before, from TABLE VIII. the mean velocity of dis- 
 charge to be now 36*35 inches ; and by repeating the opera- 
 tion we shall find the velocity to any degree of accuracy in 
 accordance with the table, and the shorter the pipe is, the 
 oftener must it be repeated. The height due to 36*35 inches 
 taken from TABLE II. as before, with a coefficient of *750, is 
 3^ 3*125 inches. The corrected fall due to the friction is 
 
 now 35 - 3*125 = 31'875, and ]20 equal 1 in 37*6, the 
 
 31*875 
 
 corrected inclination. With this inclination we find the cor- 
 rected velocity to be now 36*53 inches per second. It is not 
 necessary to repeat the operation again. The discharge 
 determined from TABLE IX. is as follows: 
 
 Inches. Cubic feet. 
 
 For a velocity of 30*00 the discharge is 1*841 
 
 6*00 *368 
 
 50 -031 
 
 _03 *002 
 
 36-53 2-242 
 
 The experimental discharge found by Mr. Provis was 2*264 
 cubic feet per minute in one experiment, and 2*285 in 
 another. The discharge from the shorter pipes may be 
 found in a similar manner, and we place the results alongside
 
 12 
 
 THE DISCHARGE OF WATER FROM 
 
 the experimental ones given in the work referred to below 
 in the following short table : 
 
 EXPERIMENTAL AND CALCULATED DISCHARGES FROM SHORT PIPES. 
 
 
 
 *S.j 
 
 1 
 
 ?ll 
 
 I. 
 
 1. 
 5 
 
 3 
 
 S B 
 
 1 
 
 -2 &*> 
 
 It 
 
 \v 
 |. a 
 
 .S 
 1 
 
 s t^ 
 111 
 
 <9 
 
 II 
 > 
 
 III 
 
 H* 
 
 ^2 
 I 1 
 
 u 
 
 w.l 
 
 11 
 
 3? 
 
 11 
 
 3 
 
 n 
 
 
 
 w 
 
 
 
 
 
 100 
 
 35 
 
 2-275 
 
 37-082 
 
 3| 
 
 311 
 
 37-6 
 
 36-53 
 
 2-242 
 
 80 
 
 35 
 
 2-500 
 
 40-750 
 
 3| 
 
 3H 
 
 30-8 
 
 41-18 
 
 2-521 
 
 60 
 
 35 
 
 2-874 
 
 46-846 
 
 5 
 
 30 
 
 24-0 
 
 48-02 
 
 2-946 
 
 40 
 
 35 
 
 3-504 
 
 57-115 
 
 7i 
 
 27| 
 
 17-5 
 
 58-50 
 
 3-590 
 
 20 
 
 35 
 
 4-528 
 
 73'801 
 
 1ft 
 
 22| 
 
 10-7 
 
 78-61 
 
 4-824 
 
 The velocities in the fourth column have been calculated 
 by the writer from the observed quantities discharged, from 
 which the height due to the orifice of entry and velocity in 
 column 5 is determined, and thence the quantities in the 
 other columns as above shown. The differences between the 
 experimental and calculated results are not large, and had we 
 used a lesser coefficient than '750 for the velocity, stop-cock, 
 and orifice of entry, say '715, the calculated results, and 
 those in all of Mr. Pro vis's experiments in the work referred 
 to, would be nearly identical t. 
 
 EXAMPLE 17. It is proposed to supply a reservoir near 
 the town of Drogheda with water by a long pipe, having an 
 inclination of 1 in 480, the daily supply to be 80,000 cubic 
 feet; what must the diameter of the pipe be? The discharge 
 
 * " Transactions of the Institution of Civil Engineers," vol. ii. p. 203. 
 " Experiments on the Flow of Water through small Pipes." By W. A. 
 Provis. 
 
 t In a late work, " Researches in Hydraulics," the author is led into 
 a series of mistakes as to the accuracy of Du Buat's and several other 
 formulae, from neglecting to take into consideration the head due to the 
 velocity and orifice of entry when testing them by the experiments above 
 referred to.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 13 
 
 80 000 
 per minute must be ' = 56 cubic feet, nearly. Assume 
 
 a pipe whose "mean radius" is 1 inch, or diameter 4 inches, 
 and the velocity per second found from TABLE VIII. will be 
 14-41 inches. We then have from TABLE IX., 
 
 Inches. Cubic feet. 
 
 For a velocity of 10*00 a discharge of 4-363 
 
 4-00 1-745 
 
 40 -175 
 
 01 -004 
 
 14-41 6-287 
 
 The discharge from a pipe 4 inches in diameter would be 
 therefore 6-287 cubic feet per minute. We then have 
 
 4* : d* : : 6'287 : 56, or 1 : d* : : '196 : 56 : : 1 : 286; 
 therefore d* = 286, and d = 9'61 inches, nearly, as found 
 from TABLE IV., &c. This is the required diameter. It is 
 to be observed that this diameter will not agree exactly with 
 Du Buat's formula, because by it the discharges are not 
 strictly as d^, but in practice the difference is immaterial. 
 
 EXAMPLE 18. The area of a channel is 50 square feet, 
 and the border 20'6 feet; the surface has an inclination of 
 4 inches in a mile; what is the mean velocity of discharge? 
 
 t 50 = 2-427 feet = 29'124 inches is the hydraulic mean 
 
 depth; and we get from TABLE VIII. 12-03 - l ' 3Q * * 876 
 
 =: 12*03 "19 = 11'84 inches per second for the required 
 velocity. Though this velocity will be found under the true 
 value for straight clear channels, it will yet be more correct 
 for ordinary river courses, with bends and turns, of the di- 
 mensions given, than the velocity found from equation (114). 
 For a straight clear channel of these dimensions, Watt found 
 the mean velocity to be 13'35 inches. See EXAMPLE 9, 
 page 6. 
 
 EXAMPLE 19. A pipe 5 inches in diameter , 14637 feet in 
 length) has a fall of 44 feet; what is the discharge in cubic
 
 14 THE DISCHARGE OF WATER FROM 
 
 feet per minute'? The inclination is 14637 = 332-7, and 
 
 mean radius = 1J. We then find from TABLE VIII. the 
 
 41 v 4-8 
 
 velocity equal to 19'81 + * * = 19'81 + -16 = 19'97, 
 12'o 
 
 or 20 inches per second very nearly; and by TABLE IX. the 
 discharge in cubic feet per minute is, as before found to be, 
 13-635. 
 
 EXAMPLE 20. What is the velocity of discharge from a 
 pipe or culvert 4 feet in diameter, having a fall of 1 foot to 
 
 a mile ? Here s = J an ^ r ] foot - We then find the 
 
 velocity of discharge from TABLE VIII. to be 14'09 inches 
 equal 1'174 feet per second. By calculating from the different 
 formulae referred to below, we shall find the velocities as 
 follows : 
 
 Velocity 
 
 in feet. 
 
 Reduction of Du Buat's formula ....... equation (81) 1-174 
 
 Girard's do. (Canals with aquatic plants) (86) -521 
 
 Prony's do. (Canals) ....... (88) 1-201 
 
 Prony's do. (Pipes) ....... (90) 1-257 
 
 Prony's do. (Pipes and Canals) ... (92) 1-229 
 
 Eytelwein's do. (Rivers) ..... (94) 1-200 
 
 Eytelwein's do. (Rivers) ..... (96) 1-285 
 
 Eytelwein's do. (Pipes) ..... (98) 1-364 
 
 Eytelwein's do. (Pipes) ..... (99) 1-350 
 
 Dr. Young's do ......... (104) 1-120 
 
 D'Aiibuisson's do. (Pipes) ..... (109) 1-259 
 
 D'Aubuisson's do. (Rivers) .... (111) 1-199 
 
 The writer's do. (Clear straight Channels) (114) 1-268 
 
 Weisbach's do. (Pipes) ...... (119) 1-285 
 
 We have calculated this example from the several formulas 
 above referred to, whether for pipes or rivers, in order that 
 the results may be more readily compared. The formulae 
 from which the velocity and tables for the discharge in 
 rivers are usually calculated is, for measures in feet, 
 v = 94- 17 \/rs. This gives the mean velocity, in the 
 foregoing example, equal to 1'295 feet per second, which is
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 15 
 
 certainly much too high if bends and curves be not allowed 
 for separately. For all practical purposes the result of 
 Du Buat's general formula, equation (81), should be adopted ; 
 and we have, accordingly, preferred calculating the results in 
 TABLE VIII. from it, notwithstanding the greater additional 
 labour required in the calculations, than from any other. 
 
 Dr. Young's formula gives less results for rivers and large 
 pipes than Du Buat's, but they are too small unless where 
 the curves and bends are numerous and sudden. Girard's 
 formula (86) is only suited for canals containing aquatic 
 plants, and must be increased by about '7 foot per second to 
 find the velocity in clear channels. A knowledge of various 
 formulae, and their comparative results, applied to any par- 
 ticular case, will be found of great value to the hydraulic 
 engineer, and the differences in the results show the probable 
 amount of error. 
 
 EXAMPLE 21. Water flowing down a river rises to a 
 height of 10 inches on a weir 62 feet long; to what height 
 will the same quantity of water rise, on a weir similarly 
 
 62 
 circumstanced, 12Q feet long? ='517, nearly. In 
 
 TABLE X. we find, by inspection, opposite to '517, the ratio of 
 the lengths, the coefficient '644, rejecting the fourth place of 
 decimals; whence 10|- x '644 = 6'76 inches, the height 
 required. When the height is given in inches it is not 
 necessary to take out the coefficient to further than two 
 places of decimals. 
 
 EXAMPLE 22. The head on a weir 220 feet long is 
 6 inches; what will the head be on a weir 60 feet long, 
 similarly circumstanced, the same quantity of water flowing 
 
 over each ? ~ = '273. As this lies between -27 and '28, 
 
 we find from TABLE X. the coefficient '4208; hence 
 
 =. 14'26 inches, the head required. 
 
 TABLE X. will be found equally applicable in finding the 
 head at the pass into weir basins and in contracted water 
 channels. See SECTION X.
 
 16 THE DISCHARGE OF WATER FROM 
 
 EXAMPLE 23. A river channel 40 feet wide and 4'5 feet 
 deep is to be altered and widened to 70 feet; what must the 
 depth of the new channel be so that the surface inclination 
 and discharge shall remain unaltered? In "TABLE XL, OF 
 
 EQUAL DISCHARGING RECTANGULAR CHANNELS," W6 find 
 
 opposite to 4'54, in the column of 40 feet widths, 3 in the 
 column of 70 feet widths, which is the depth required. 
 
 EXAMPLE 24. It is necessary to unwater a river channel 
 7Qfeet wide and 1 foot deep, by a rectangular side cut 1 Qfeet 
 wide; what must the depth of the side cut be, the surface 
 inclination remaining the same as in the old channel? In 
 TABLE XI. we find 4'5 feet for the required depth. When 
 the width of a channel remains constant, the discharge 
 varies as \/r s X d, in which d is the depth ; and when the 
 width is very large compared with the depth, the hydraulic 
 mean depth r approximates very closely to the depth d, and 
 
 d = r; consequently the discharge varies as d* x * , and 
 when it is given d* must vary inversely as * , or more 
 generally dr* must vary inversely, as * , when the width and 
 discharge remain constant. 
 
 In narrow cuts for unwatering, it is prudent to make the 
 depth of the water half the width of the cut very nearly, 
 when local circumstances will admit of these proportions * j 
 for then a maximum effect will be obtained with the least 
 possible quantity of excavation ; but for rivers and permanent 
 channels the proper relation of the depth to the width must 
 be regulated by the principles referred to in SECTION IX. 
 
 TABLE XI. is equally applicable, whether the measures be 
 in feet, yards, or any other standards. 
 
 * Section IX.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 17 
 
 SECTION II. 
 
 FORMULA FOR THE VELOCITY AND DISCHARGE FROM ORI- 
 FICES, WEIRS, AND NOTCHES. - COEFFICIENTS OF VELOCITY, 
 CONTRACTION, AND DISCHARGE. - PRACTICAL REMARKS ON 
 THE USE OF THE FORMULA. 
 
 The quantity of water discharged in a given time through 
 an aperture of a given area in the side or bottom of a vessel, 
 is modified by different circumstances, and varies more or 
 less with the form, position, and depth of the orifice ; but the 
 discharge may be easily found, when we have determined the 
 velocity and the contraction of the fluid vein. 
 VELOCITY. 
 
 If g be the velocity acquired by a heavy body falling from 
 a state of rest for one second, in vacuo, then it has been 
 shown by writers on mechanics, that the velocity v per second 
 acquired by falling from a height h, will be 
 
 The numerical value of g varies with the latitude ; we shall 
 assume 2g =. 772'84 inches = 64-403 feet. These will give 
 for measures in inches, 
 
 v = 27-8 vT,* and h = 
 and for measures in feet, 
 
 = 8-025 v/T, and Ass 
 
 * The velocities for different heights are given in the column num- 
 ber 1, TABLE II. 
 
 t The force of gravity increases with the latitude, and decreases with 
 the altitude above the level of the sea, but not to any considerable 
 extent. If X be the latitude, and h the altitude, in feet, above the sea 
 level, then we may, generally, take 
 
 g = 32-17 (1 -0029 cos 2X) x ( I \ 
 
 in which B, the radius of the earth at the given latitude is equal to 
 20887600 (1 + -0016 cos 2 X). 
 
 C
 
 18 THE DISCHARGE OF WATER FROM 
 
 COEFFICIENT OF VELOCITY. 
 
 Let the vessel ABCD, Fig. 1, be filled with water to the 
 level EF: then it has been found by experiment that the 
 velocity of discharge through a small orifice o, in a thin 
 plate, at the distance of half the diameter outside it, will be 
 very nearly that due to a heavy body falling freely from the 
 height h, of the surface of the water EF, above the centre 
 
 of the orifice. The velocity of discharge determined by the 
 equation v = \/2gh, for falling bodies, is, therefore, called 
 the theoretical velocity. If we now put v d for the actual 
 mean velocity of discharge, and c v for its ratio to the theo- 
 retical velocity v, we shall get v d = c v v ; and by substituting 
 for v, its value \/2ffh. 
 (2.) v d = c v v/2p; 
 
 c v is termed the coefficient of velocity; its numerical value 
 at half the diameter from the orifice is about '974, and, 
 consequently, 
 
 tf d = -974 v/20A. 
 This for measures in inches becomes 
 
 v d = 27-077 -v/A,* 
 and for measures in feet 
 
 v d = 7-816 v/A. 
 
 * The velocities for different heights calculated from this formula, 
 are given in the column numbered 2, TABLE II.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 19 
 
 The orifice o, is termed an horizontal orifice in Fig. 1, and 
 in Fig. 2 a vertical or lateral orifice. When small, each is 
 found to have practically the same velocity of discharge, 
 when the centres of the contracted sections are at the same 
 depth, h, below the surface; but when lateral orifices are 
 large, or rather deep, the velocity at the centre is not, even 
 practically, the mean velocity; and in thick plates and 
 modified forms of adjutage, the mean velocities are found 
 to vary. 
 
 VE.VA-CONTRACTA AND CONTRACTION. 
 
 It has been found that the diameter of a column issuing 
 from a circular orifice in a thin plate, is contracted to very 
 nearly eight-tenths of the whole diameter at the distance 
 of the radius from it, and that at this distance the con- 
 traction is greatest. The ratio of the diameter of the orifice 
 to that of the contracted vein, vena-contracta, is not always 
 found constant by the same or different experimentalists. 
 
 Newton makes it 1: -841, and, therefore, that of the areas as 1 -707 
 
 (7156 
 (622 
 
 802 1 -6432 
 
 8 1 -64 
 
 (81 f-656 
 
 (-818 " L \-669 
 
 816 1 -667 
 
 798 1 -637 
 
 64 
 617 
 
 Poleni 
 
 Borda 
 Michellotti 
 
 Bossut 
 
 841, and, therefore, that of the areas as 1 
 -846 
 
 -7854 
 
 DuBuat 
 
 Venturi 
 
 Eytelwein 
 
 Bayer 
 
 Bayer's value for the contraction has been determined on 
 the hypothesis, that the velocities of the particles of water 
 as they approach the orifice from all sides, are inversely as 
 the squares of their distances from its centre ; and the cal- 
 culations made of the discharge from circular, square, and 
 rectangular orifices, on this hypothesis, coincide pretty closely 
 with experiments. 
 
 FORM OF THE CONTRACTED VEIN. 
 
 Let o li = d, Fig. 3, be the diameter of an orifice ; then at 
 the distance RS = the contraction is found to be greatest; 
 
 c 2
 
 20 
 
 THE DISCHARGE OF WATER FROM 
 
 Fig. 3 
 
 we shall assume the contracted diameter or = '7854 d. If 
 we suppose the fluid column between OK and or to be so 
 reduced, that the curve lines Rr and oo shall become arcs 
 
 of circles, then it is easy 
 to show from the proper- 
 ties of the circle, that the 
 radius cr must be equal 
 to 1'22 d. The mean ve- 
 locity in the orifice, OR, 
 is to that in the vena- 
 contracta, or, as '617 : 1; 
 and the mouth piece, 
 Rroo, Fig. 4, in which OJOZZ^OR, and or = '617 OR, will 
 give for the velocity of discharge at o, r, 
 
 v d = -974 \/2#A = 7-816 ^/~h, 
 
 in feet very nearly. In 
 speaking of the velocity 
 of discharge from orifices 
 in thin plates, we always 
 take it to be the velocity 
 in the vena-contracta, and 
 not that in the orifice it- 
 
 Fig-. 4 
 
 self, unless in TABLE II., where the mean velocity in the 
 latter, as representing c d \/2gh, is also given. 
 
 COEFFICIENTS OF CONTRACTION AND DISCHARGE. 
 
 If we put A for the area of the orifice OR, Fig. 3, and 
 c c x A for that of the contracted section at or, then c c is 
 called the coefficient of contraction. The velocity of dis- 
 charge u d is equal to c v \/2gh, equation (2). If we mul- 
 tiply this by the area of the contracted section c c x A, we 
 shall get for the discharge 
 
 D = c v x c c x A \/2gh.* 
 
 * The expression c v c c \/2yA = Cd \/2yA is the coefficient of the area 
 A, and, consequently, represents the mean velocity in the orifice, the 
 coefficient of which is, therefore, equal to Cd- The values of c d 
 for different heights and coefficients, are given in TABLE II,
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 21 
 
 It is evident A \/2gh would be the discharge if there were 
 no contraction and no change of velocity due to the height h ; 
 C Y x c c is therefore equal to the coefficient of discharge. If 
 we call the latter c d , we shall have the equation 
 (3.) c d = c v x c c , 
 
 and hence we perceive that the coefficient of discharge is 
 equal to the product of the coefficients of velocity and con- 
 traction. In the foregoing expression for the discharge D, h 
 must be so taken, that the velocity at that depth shall be the 
 mean velocity in the orifice A. In full prismatic tubes the 
 coefficients of velocity and discharge are equal to each other. 
 
 MEAN AND CENTRAL VELOCITY. 
 
 In order to find the mean velocity of discharge from an 
 orifice, it is, in the first instance, necessary to determine the 
 velocity due to each point in its surface, and the discharge 
 itself; after which, the mean velocity is found by simply 
 dividing the area of the orifice into the discharge. The 
 velocity due to the height of water at the centre of a circular, 
 square, or rectangular orifice, is not the mean velocity, nor 
 is the latter in these, or other figures, that at the centre 
 of gravity. When, however, an orifice is small in proportion 
 to its depth in the water, the velocity of efflux determined 
 for the centre approaches very closely to the mean velocity ; 
 and, indeed, at depths exceeding four times the depth of the 
 orifice, the error in assuming the mean velocity to be that at 
 the centre of the orifice is so small as to be of little or 
 no practical consequence. It is, therefore, for greater sim- 
 plicity, the practice to determine the velocity from the depth 
 h of the centre of the orifice ; and the coefficients of dis- 
 charge and velocity in the following pages have been cal- 
 culated from experiments on this assumption, unless it be 
 otherwise stated. 
 
 DISCHARGES THROUGH ORIFICES OF DIFFERENT FORMS IN 
 THIN PLATES. 
 
 The orifices which we have to deal with in practice are 
 square, rectangular, or circular; and sometimes, perhaps, 
 triangular or quadrangular in form. It will be necessary to
 
 22 
 
 THE DISCHARGE OF WATER FROM 
 
 give here the theoretical expressions for the discharge and 
 velocity for each kind of form, but as the demonstrations are 
 unsuited to our present purposes we shall omit them. 
 
 TRAPEZOIDAL ORIFICES WITH TWO HORIZONTAL SIDES. 
 
 Put d for the vertical depth of an orifice, h t for the altitude 
 of pressure above the upper side, and h b for the altitude 
 above the lower side, we then get 
 
 h b - h, = d. 
 Let us also represent the upper side of the orifice A or c, 
 
 Fig. 5, by l t) and the lower side by Z b , and put 
 
 + 4, - 
 
 Now, when l t = Z b , the trapezoid becomes a parallelogram 
 whose length is / and depth d; and putting h for the depth 
 to the centre of gravity, we get the equation 
 
 The general expression for the discharge, D, through a tra* 
 pezoidal orifice, A, is 
 
 (4.) D = c d v/27 x | l b hl - l t hl + (/ t - / b ) ^ 
 
 in which e d is the coefficient of discharge; and when the 
 smaller side is uppermost as at c, 
 
 (5.) D = c d
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 23 
 
 PARALLELOGRAMIC AND RECTANGULAR ORIFICES. 
 
 When l t = l b = I, the orifice becomes a parallelogram, or 
 a rectangle, B, and we have for the discharge 
 
 (6.) DZZ^X/^X |/{4 -A*}. 
 
 NOTCHES. 
 
 When the upper sides of the orifices A, B, and c, rise to 
 the surface as at AO, B O , and C , h t becomes nothing, and we 
 get as h b = d : for the trapezoidal notch A with the 
 larger side up, 
 
 (7.) D = c dV / %x 
 
 for the trapezoidal notch, c , with the smaller side up, 
 (8.) D = 
 
 and for a parallelogramic or rectangular notch, B O , 
 (9.) D = c d JTg x /d 8 - = 
 
 It is easy to perceive that the forms of equations (4) and 
 (5), and also of equations (7) and (8), are identical. The 
 values for the discharge in equations (6) and (9) are equally 
 applicable, whether the form of the orifice be a parallelogram 
 or a rectangle, the only difference being in the value of the 
 coefficient of discharge, c d , which becomes slightly modified 
 for each form of orifice. 
 
 TRIANGULAR ORIFICES WITH HORIZONTAL BASES AND 
 RECTILINEAL ORIFICES IN GENERAL. 
 
 When the length of the lower side, l b = 0, the orifice 
 becomes a triangle, D, Fig. 6, with the base upwards.
 
 24 
 
 THE DISCHARGE OF WATER FROM 
 
 In this case, equation (4) becomes 
 (10.) D = . 
 
 which gives the discharge through the triangular orifice, D. 
 
 When l t = 0, in equation (5), the orifice becomes a triangle 
 F, with the base downwards; in this case, we find for the 
 value of the discharge, 
 
 (11.) D = 
 
 As any triangular orifice whatever can be divided into two 
 others by a line of division through one of the angles parallel 
 to the horizon; and as the discharge from the triangular 
 orifice D or F is the same as for any other on the same base 
 and between the same parallels, we can easily find, by such a 
 division, the discharge from any triangle not having one side 
 parallel to the horizon. 
 
 If the triangle F be raised so that the base shall be on the 
 same level with the upper side of the triangular orifice D ; if, 
 also, the bases be equal, and also the depths, we shall find, 
 by adding equations (10) and (11), and making the necessary 
 changes indicated by the diagram, 
 
 (12.) D = c d 
 
 (hi + Af - 2 x 
 
 for the discharge from a parallelogram E with one diagonal 
 horizontal. If the orifices D, F, and E rise to the surface of 
 the water, as at D O , E O , and F O , we shall then have for the dis- 
 charge from the notch D O ,
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 25 
 
 (13.) D = < 
 
 for the discharge from the notch F O , 
 
 (14.) D = c d \/2g x jT/ b d . 
 
 and for the discharge through the notch E O , 
 
 (15.) D = Qv2 x - - 
 
 -9752 /rfl 
 
 When the parallelogram E O becomes a square I =z2d, and 
 hence, 
 
 (16.) D = 
 
 '9752 1* 
 
 x 3'4478 1*. 
 
 The foregoing equations will enable us to find an expression 
 for the discharge from any rectilineal orifice whatever, as it 
 can be divided into triangles, the discharge from each of which 
 can be determined, as already shown in the remark following 
 equation (11). The examples which we have given will be 
 found to comprehend every form of rectilineal orifice which 
 occurs in practice ; but for the greater number of orifices, sunk 
 to any depth below the surface, the discharge will be found 
 with sufficient accuracy by multiplying the area by the velocity 
 due to the centre. 
 
 CIRCULAR AND SEMICIRCULAR ORIFICES. 
 
 The discharge through circular and semicircular orifices in 
 thin plates can only be represented by means of infinite series. 
 Let us represent by ^ the sum of the series 
 
 Fig. 7
 
 26 THE DISCHARGE OF WATER FROM 
 
 fi /i i\ /i ivr! /i i s swi 1 3\^_ 
 
 l2~\2"4V Vi'4/7 Vice's; U'i'e/ A 4 
 
 '4'6'8'10'12/ \2'4'6'8M 8 
 Let us also represent by s. 2 the sum of the series 
 
 l 1VL , /"l 1 3 
 3-1416 
 
 then the discharge from the semicircle G, Fig. 7, with the 
 diameter upwards and horizontal, is 
 
 (17.) D = c d \/2ffh x 3-1416/- 2 (s, + sj. 
 
 And the discharge from the semicircle i, with the diameter 
 downwards and horizontal, is 
 
 (18.) D = c d ^27^ x 3-1416 r 2 (^ 2 ). 
 
 If we put A for the area, we shall also have for the discharge 
 from a circle H, 
 
 (19.) D = c dV /27A x 2 AS,. 
 
 In each of these three equations (17), (18), and (19), h is the 
 depth of the centre of the circumference below the surface, 
 and r the radius. 
 
 When the orifices rise to the surface, we have for the 
 discharge from a semicircular notch G O , with the diameter 
 horizontal and at the surface, 
 
 (20.) D = c d v/2</r X -9586 r 2 = c d \/2gr x -6103 A; 
 
 when the circumference of the semicircle is at the surface, 
 and the diameter horizontal, as at i , 
 
 (21.) D = c d ^/2^ x ^(v/K8-7)r t 3S<? dN /2^x-7324Aj 
 
 when the horizontal diameter of the semicircle is uppermost, 
 and at the depth r below the surface, 
 
 (22.) D = C A v/577 x 1-8667 r 2 = c d \/lgr x 1-1884 A;
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 27 
 
 and when the circumference of the entire circle is at the 
 surface, as at H O , 
 
 (23.) D = C A v/2- x 3-0171 r 2 = c d </^g~r x -9604 A. 
 
 If we desire to reduce equations (20), (21), and (22), to 
 others in which the depth h of the centre of gravity from the 
 
 surface is contained, we have only to substitute .4944 ^ or r m 
 
 equation (20), and we shall get, for the discharge from a 
 semicircle with the diameter at the surface, 
 
 (24.) D = C A \/Tgh x -9367 A : 
 
 also, by substituting .cygfi for r in equation (21), we get, 
 
 for the discharge from a semicircle when the circumference is 
 at the surface and the diameter horizontal, 
 
 (25.) D = c d s/Tgh x -9653 A ; 
 
 and when the horizontal diameter is uppermost, and at the 
 
 depth r below the surface r = - ,- and 
 
 1-4244 
 
 (26.) D = c d </2~ih x -9957 A. 
 
 As A stands for the area of the particular orifice in each of 
 the preceding expressions for the discharge, it must be taken 
 of double the value in equation (23), for instance, where it 
 stands for the area of a circle to what it is in equations 
 (20), (21), or (23), where it represents only the area of a 
 semicircle. 
 
 MEAN VELOCITY. 
 
 The mean velocity is easily found by dividing the area into 
 the discharge per second given in the preceding equations. 
 
 For instance, the mean velocity in the example represented in 
 
 2 
 equation (9) is equal 3 c d \^2 g d, which is had by dividing 
 
 the area I d into the discharge ; and in like manner the mean 
 velocity in equation (23) is '9604 C A
 
 28 
 
 THE DISCHARGE OF WATER FROM 
 
 PRACTICAL REMARKS. CIRCULAR ORIFICES. 
 
 It has been shown, equation (19), that, for the discharge 
 from a circle, we have 
 
 D r= c d ^/2 g h x 2 A s lt 
 
 in which h is the depth of the centre, A the area, and * A the 
 sum of the series 
 
 1 l\ll fl 1 3 5\ /l l 3\_^ - 
 
 *4/)F ~ U-4-6-8J U'i'S/'V . 
 
 and it has also been shown, equation (23), that, when the 
 circumference touches the surface, this value becomes 
 
 D = c d 
 
 -9604 A. 
 
 Now when h is very large compared with r, it is easy to per- 
 ceive that 2 Si = 1, and hence 
 
 (27.) D == c d v/2 $r A x A. 
 
 As this is the formula commonly used for finding the dis- 
 charge, it is clear, if the coefficient c d remain constant, that 
 the result obtained from it for D would be too large. The 
 differences, however, for depths greater than three times the 
 diameter, or 6 r, are practically of no importance ; for, by 
 calculating the values of the discharge at different depths, we 
 shall find, when 
 
 h r, that D =: c d 
 5r 
 
 (28.) 
 
 A-- 
 
 - 2 
 
 7r 
 
 D=C d 
 
 = r, D = 
 A = 3r, D = C A 
 
 = 5r, D = 
 = 6r, D = c d
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 29 
 
 These results show very clearly that, for circular orifices, 
 the common expression for the discharge c d \S%gh A is abun- 
 dantly correct for all depths exceeding three times the 
 diameter, and that for lesser depths the extreme error cannot 
 exceed four per cent, in reduction of the quantity found by 
 this formula. We shall show, hereafter, when discussing the 
 value of c d , that from the sinking of the surface, and perhaps 
 other causes, the discharge at lesser depths is even larger 
 than that exhibited by the expression c d ^/2ffhA, the value 
 of c d being found to increase as the depths h decrease. 
 
 RECTANGULAR ORIFICES. 
 
 It has been shown, equation (6), that the discharge from 
 rectangular orifices with two sides parallel to the surface is 
 expressed by the equation 
 
 D r= c d x <*/1g x I \h$ hi }, 
 
 in which I is the horizontal length of the orifice, h b the depth 
 of water on the lower, and h t the depth on the upper, side. 
 As it is desirable in practice to change this form into a more 
 simple one, in which the height h of the centre and depth d 
 
 of the orifice only are included, we then have h b = h + - 
 
 and h t = h -. By substituting these values of h b and h t 
 
 in the foregoing equations, and developing the result into a 
 series, the terms of which, after the third, may be neglected, 
 and putting A for the area I d, we shall find 
 
 (29.) D = c d \/2ffh x A |l <jg ^}very nearly. 
 We have therefore for the accurate theoretical discharge 
 
 (30.) D =
 
 30 
 
 THE DISCHARGE OF WATER FROM 
 
 for the approximate discharge 
 
 jl - 4 pj; 
 
 and for the discharge by the common formula 
 D = c d \/2gh x A. 
 
 When the head (h) is large compared with (d) the height of 
 the orifice, each of the three last equations gives the same 
 value for the discharge; but as the common expression 
 c d \/2ghA. is the most simple; and as the greatest possible 
 error in using it for lesser depths does not exceed six per 
 cent., viz. when the orifice rises to the surface and becomes a 
 notch, it is evidently that formula best suited for practical 
 purposes. The following Table will show more clearly the 
 differences in the results as obtained from the true, the 
 approximate aud the common formulas, applied to lesser 
 heads. 
 
 (31.) 
 
 1 
 
 2 
 
 3 
 
 *=Jr D = ^ 
 
 '2^Ax-9428A. D = c dV /277 
 
 \ x -9583 A. 
 
 
 x -9693 A. 
 
 x -9733 A. 
 
 A = ^> ,, 
 
 x -9796 A. 
 
 x '9815A. 
 
 h _l_d 
 
 x -9854A. n 
 
 x -9864 A. 
 
 h =; d, 
 
 , x -9890A. 
 
 x -9896 A. 
 
 A- 3rf 
 
 * 2~ 
 
 x -9953 A. 
 
 x -9954A. 
 
 h = 2d, 
 
 x -9974A. 
 
 x -9974A. 
 
 * = / 
 
 x -9983 A. 
 
 x -9983 A. 
 
 A = 3d, 
 
 x -9988 A. 
 
 x -9988A. 
 
 h - 7d 
 h--, 
 
 x -9991 A. 
 
 x -9991 A. 
 
 h = 4 d, 
 
 x -9994A. 
 
 x -9994A. 
 
 A = lOd, 
 
 x -9999 A. 
 
 x -9999A.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 31 
 
 In the foregoing Table the first column contains the head at 
 the centre of the orifice expressed in parts of its height d; 
 the second contains the values of the discharges according to 
 equation (30); and the third column contains the approximate 
 values determined from equation (29), the results in which are 
 something larger than those in column 2, derived from the 
 correct formula. The numerical coefficients of A, at every 
 depth, are less in both than unity, the constant coefficient 
 according to the common formula. The latter, therefore 
 (as in circular orifices), gives results exceeding the true ones, 
 but the excess is inappreciable at greater depths than h = 3d, 
 and for lesser depths than this the error cannot exceed six per 
 cent. It may be useful to remark here, that when the orifice 
 rises to the surface the centre of mean velocity is at four- 
 ninths of the depth, and the centre of gravity at two-thirds 
 of the depth from the surface. The former fraction is the 
 square of the latter. 
 
 SECTION III. 
 
 EXPERIMENTAL RESULTS. COEFFICIENTS OF DISCHARGE. 
 
 We have heretofore dwelt but very partially on the numerical 
 values of the general coefficient of discharge C A . In order to 
 determine its value under different circumstances more par- 
 ticularly, it will be now necessary to consider some of the 
 experiments which have been made from time to time. These 
 do not always give the same results, even when conducted 
 under the same circumstances and by the same parties, and 
 there appears to exist a certain amount of error, more or less, 
 inseparable from the subject. The experiments with orifices 
 in thin plates afford the most consistent results; but even 
 here the differences are sometimes greater than might be 
 expected. In many of the earlier experiments the value of 
 the coefficient c d comes out too large, which arises, very 
 probably, from the orifices experimented with not being in
 
 32 
 
 THE DISCHARGE OF WATER FROM 
 
 thin plates, and partaking, more or less, of the nature of a 
 short tube or mouth-piece with rounded arrisses, which we 
 shall see gives larger coefficients than- simple orifices. When 
 an orifice is in the bottom of a vessel, it would appear more 
 correct to measure the head from the surface to the vena- 
 contracta than from the surface to the orifice itself; and as 
 any error in measuring the head in any experiment must 
 affect the value of the coefficient derived from such experi- 
 ment, so as to increase it when the error is to make the head 
 less, and vice versd, it appears that heads measured to an 
 orifice in the bottom of a vessel, and not to the vena-contracta, 
 must give larger coefficients from the experimental results 
 than, perhaps, the true ones. The coefficients in the following 
 pages have been almost all arranged and calculated, by the 
 writer, from the original experiments. 
 
 In 1739 Dr. Bryan Robinson made some experiments on 
 the discharge through small circular orifices, from one- tenth 
 to eight-tenths of an inch in diameter, with heads of two and 
 four feet*, which give the following coefficients. 
 
 COEFFICIENTS FROM DK. B. ROBINSON'S EXPERIMENTS. 
 
 Heads. 
 
 T\5inch 
 diameter. 
 
 ft inch 
 diameter. 
 
 inch 
 diameter. 
 
 ^sinch 
 diameter. 
 
 2 feet head 
 
 768 
 
 767 
 
 761 
 
 728 
 
 4 feet head 
 
 768 
 
 774 
 
 765 
 
 742 
 
 These results are pretty uniform, and the values from which 
 they are derived are said to be " means taken from five or six 
 experiments;" as values of C A they are, however, too high. 
 The apparatus made use of is not described ; but it is probable, 
 from the results, that the plate containing the hole or orifice 
 was of some thickness, and that the inner arriss was slightly 
 rounded. There is here, however, a very perceptible increase 
 in the coefficients for the smaller orifices, but none for the 
 smaller depth. 
 
 * Helsham's Lectures, p. 390. Dublin, 1739.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 33 
 
 In a paper in the Transactions of the Royal Irish Academy, 
 vol. ii. p. 81, read March 1st, 1788, Dr. Mathew Young de- 
 termines the value of the coefficient for an orifice ^V mcn m 
 diameter, with a mean head of 14 inches, to be '623. The 
 manner in which this value is determined is very elegant, viz. 
 by comparing the observed with the theoretical time of the 
 water in the vessel sinking from 16 inches to 12 inches. 
 
 The following experiments by Michelotti, with circular 
 orifices from 1 to about 3 inches diameter, and with from 6 
 to 23 feet heads, give for the mean value c d = '613 ; and for 
 square orifices of from 1 to 9 square inches in area, at like 
 depths, the mean value of c d = '628. The experiments are 
 given in French feet and inches, according to which standard 
 we have, in feet, D =: I'll A \/Jt X t, t being the time in 
 seconds*. As the time of discharge in these experiments 
 varies from ten minutes to an hour, and as the depths are 
 considerable, the results must be looked upon as pretty accu- 
 rate ; and it is worthy of note that here the coefficients are 
 larger for square than for circular orifices. 
 
 * The value of */2ffh, equation (1), for measures in French feet, is 
 7'77 s/ k, and for measures in French inches, 26-9 V h, g being equal to 
 30-2 feet, or 362-4 inches, French measure. One French foot is equal to 
 1-06578 English feet, and the inches preserve the same proportion. The 
 resulting coefficients must be the same, whatever standards we make our 
 calculations from. Many of the most valuable formulae and experi- 
 ments in hydraulics are given in French measures of the old style. As 
 our object, however, in the present section is simply to determine from 
 experiment the relation of the experimental to the theoretical dis- 
 charge, it is not necessary to reduce the experiments to other measures 
 than those in the original; but the value of the force of gravity, g, 
 must of course be taken in those measures with which the experiments 
 were made. In the French decimal, or modern style, the metre is equal 
 to 3-2809 English feet, or 39-371 inches. The tenth part of a metre 
 is the decimetre, and the tenth part of the decimetre is the centimetre, 
 as the names imply.
 
 34 
 
 THE DISCHARGE OF WATER FROM 
 
 COEFFICIENTS FROM MICHELOTTl's EXPERIMENTS. 
 
 Description and 
 size of orifice, in 
 French inches. 
 
 Depth of the 
 centre of the 
 orifice, in 
 French feet 
 
 Quantity 
 discharged, 
 in cubic feet. 
 
 Time of 
 discharge, 
 in seconds. 
 
 Theoretical 
 time, 
 calculated 
 from 
 
 t D . 
 
 Resulting 
 coefficients 
 of discharge. 
 
 7-77 A V h 
 
 Square orifice, 
 3" X 3" 
 
 6-613 
 6-852 
 11-676 
 11-818 
 21-691 
 21-715 
 
 463-604 
 566-458 
 516-785 
 612-118 
 415-437 
 499-222 
 
 600 
 
 720 
 510 
 600 
 300 
 360 
 
 371-3 
 445-6 
 311-4 
 366-6 
 
 183-7 
 220-6 
 
 619 
 619 
 610 
 611 
 612 
 613 
 
 Mean value of the coefficient ; square orifice 3" x 3" 
 
 
 614 
 
 
 Square orifice, 
 2" X 2" 
 
 6-625 
 11-426 
 21-442 
 
 329-806 
 423-465 
 385-333 
 
 900 
 900 
 600 
 
 594- 
 580-4 
 385-7 
 
 660 
 645 
 643 
 
 Mean value of the coefficient ; square orifice 2" x 2" 
 
 
 649 
 
 
 Square orifice, 
 1" X 1" 
 
 6-757 
 11-889 
 21-507 
 
 158-549 
 163-792 
 562-944 
 
 1800 
 
 1440 
 3600 
 
 1585- 
 880-6 
 2249-9 
 
 628 
 612 
 625 
 
 Mean value of the coefficient; square orifice 1" x 1" 
 
 
 621 
 
 
 Circular orifice, 
 3" diameter 
 
 6-694 
 11-590 
 21-611 
 
 542-85 
 570-972 
 521-299 
 
 900 
 720 
 480 
 
 550-1 
 439-6 
 293-8 
 
 611 
 610 
 612 
 
 Mean value of the coefficient ; circular orifice 3" diameter -611 
 
 Circular orifice, 
 2" diameter 
 
 6-785 
 11-722 
 21-903 
 
 488-687 
 589-535 
 575-486 
 
 1800 
 1680 
 1200 
 
 1108-1 
 1016-4 
 725-9 
 
 616 
 605 
 605 
 
 Mean value of the coefficient ; circular orifice 2" diameter -609 
 
 Circular orifice, 
 1" diameter 
 
 6-875 
 11-743 . 
 22-014 
 
 247-354 
 324-11 
 444-535 
 
 3600 
 3600 
 3600 
 
 2227- 
 2233- 
 2237-2 
 
 619 
 620 
 621 
 
 Mean value of the coefficient ; circular orifice 1" diara 
 
 eter 
 
 620 

 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 35 
 
 The experiments made by the Abbe Bossut, contained in 
 the following table, give the mean value of c d , for both circular 
 and square orifices, equal to '616 nearly; and it may be per- 
 ceived that, for the small depth in the last experiment, the 
 coefficient rises so high as '649. These and other experiments 
 
 COEFFICIENTS FROM BOSSUl's EXPERIMENTS. 
 
 Description, position, and size of 
 orifice, in French inches. 
 
 Depth of 
 the centre 
 of the 
 orifice, in 
 French 
 inches. 
 
 Number 
 of French 
 cubical 
 inches 
 discharged 
 per minute. 
 
 Theoretical 
 discharge 
 per minute, 
 
 D = l6l4A*/Jl. 
 
 Resulting 
 coefficients. 
 
 Horizontal and circular, 5" diameter 
 
 140-832 
 
 2311 
 
 3760-8 
 
 614 
 
 Horizontal and circular, 1" diameter 
 
 140-832 
 
 9281 
 
 15043-3 
 
 617 
 
 Horizontal and circular, 2" diameter 140-832 
 
 37203 
 
 60173-1 
 
 618 
 
 Horizontal and rectangular,!" x 4" 
 
 140-832 
 
 2933 
 
 4788-4 
 
 613 
 
 Horizontal and square, 1" X 1" - 
 
 140-832 
 
 11817 
 
 19153-7 
 
 617 
 
 Horizontal and square, 2" X 2" ... 
 
 140-832 
 
 47361 
 
 76614-6 
 
 618 
 
 Lateral and circular, 5" diameter... 
 
 108- 
 
 2018 
 
 3293-3 
 
 613 
 
 Lateral and circular, 1" diameter... 
 
 108- 
 
 8135 
 
 13173-3 
 
 617 
 
 Lateral and circular, \' diameter... 
 
 48- 
 
 1353 
 
 2195-5 
 
 616 
 
 Lateral and circular, 1" diameter. . . 
 
 48- 
 
 5436 
 
 8782-2 
 
 619 
 
 Lateral and circular, 1" diameter... 
 
 0-5833 
 
 628 
 
 968- 
 
 649 
 
 led the Abbe to construct a table of the discharges, at dif- 
 ferent depths, from a circular orifice 1 inch in diameter, from 
 which we have determined the following table of coefficients. 
 
 COEFFICIENTS DEDUCED FROM BOSSUl's EXPERIMENTS. 
 
 Heads, 
 infect. 
 
 Coefficients. 
 
 Heads, 
 in feet. 
 
 Coefficients. 
 
 Heads, 
 in feet. 
 
 Coefficients. 
 
 1 
 
 621 
 
 6 
 
 620 
 
 11 
 
 619 
 
 2 
 
 621 
 
 7 
 
 620 
 
 12 
 
 618 
 
 3 
 
 621 
 
 8 
 
 619 
 
 13 
 
 618 
 
 4 
 
 620 
 
 9 
 
 619 
 
 14 
 
 618 
 
 5 
 
 620 
 
 10 
 
 619 
 
 16 
 
 617 
 
 These increase, as the orifice approaches the surface, from 
 '617 to '621 ; and at lesser depths than 1 foot other experi- 
 
 D 2
 
 36 
 
 THE DISCHARGE OF WATER FROM 
 
 ments show an increase in the coefficient up to '650. The 
 experiments of Poncelet and Lesbros show, however, a re- 
 duction in the coefficients for square orifices 8" x 8" as they 
 approach the surface from "601 to '572. 
 
 Brindley and Smeaton's experiments, with an orifice 1 inch 
 square placed at different depths, give a mean value for C A of 
 
 COEFFICIENTS CALCULATED FROM BRINDLEY AND SMEATON'S EXPERIMENTS. 
 
 1 foot head : orifice 1" x 1" : coefficient -639 \ 
 
 2 feet head : orifice 1" X 1" : coefficient -635 j 
 
 3 feet head : orifice 1" x 1" : coefficient -648 > mean " 
 
 4 feet head : orifice 1" x 1" : coefficient '632 
 
 5 feet head : orifice 1" X 1": coefficient -632 
 
 6 feet head : orifice J" X J" : coefficient -577 
 
 637. The last experiment, with an orifice only % inch by 
 inch, gives so small a coefficient as '557 placed at a depth 
 of 6 feet ! 
 
 For notches 6 inches wide and from 1 to 6j inches deep, 
 Brindley and Smeaton's experiments give the mean value of 
 
 COEFFICIENTS FOR NOTCHES, CALCULATED FROM BRINDLET AND SMEATON'S 
 
 EXPERIMENTS. 
 
 Ratio of the 
 to the tepth. 
 
 Size of 
 notches 
 in inches. 
 
 Coefficients. 
 
 Ratio of the 
 length 
 to the depth. 
 
 Size of 
 notches 
 in inches. 
 
 Coefficients. 
 
 92 to 1 
 
 6x6J 
 
 633 
 
 3-7 to 1 
 
 6 x 1| 
 
 638 
 
 1-07 to 1 
 
 6X6| 
 
 571 
 
 4-4 to 1 
 
 6 X If 
 
 654 
 
 1-2 tol 
 
 6x5 
 
 609 
 
 4-8 to 1 
 
 6 X 1J 
 
 681 
 
 1-92 to 1 
 
 6 X 3| 
 
 602 
 
 6- tol 
 
 6X 1 
 
 713 
 
 2-4 tol 
 
 6 X 2f s * 
 
 636 
 
 Mean 
 
 value. 
 
 637 
 
 c d = '637. The coefficient of discharge for notches and 
 orifices appear to differ as little from each other as those for 
 either do in themselves. The results also show a general 
 though not uniform increase in the coefficients for smaller 
 depths. 
 
 Du Buat's experiments with notches 18'4 inches long, 
 
 * This depth is misprinted 2 T 5 5 inches in the Encyclopaedias, the 
 resulting coefficient for which would be '568 instead of '637, as above, 
 for a depth of 2^ inches.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 37 
 
 give the mean value of c d = "632, which differs very little 
 from the mean value determined from Brindley and Smeaton's 
 experiments. 
 
 COEFFICIENTS FOR NOTCHES, CALCULATED FROM DTJ BUAT*S EXPERIMENTS. 
 
 Ratio of the 
 length 
 to the depth. 
 
 Size of 
 notches 
 in inches. 
 
 Coefficients. 
 
 Ratio of the 
 length 
 to the depth. 
 
 Size of 
 notches 
 in inches. 
 
 Coefficients. 
 
 2-72 to 1 
 3-94 to 1 
 
 18-4 X 6753 
 18-4 x 4-665 
 
 630 
 627 
 
 5-75 to 1 
 10-3 to 1 
 
 18-4 X 3-199 
 18-4 X 1'778 
 
 624 
 648 
 
 Poncelet and Lesbros' experiments give the coefficients 
 in the following table, for notches 8 inches wide ; the mean 
 
 COEFFICIENTS FOR NOTCHES, BY PONCELET AND LESBROS. 
 
 Ratio of the 
 length 
 to the depth. 
 
 Size of 
 notches 
 hi inches. 
 
 Coefficients. 
 
 Ratio of the 
 length 
 to the depth. 
 
 Size of 
 notches 
 in inches. 
 
 Coefficients. 
 
 9 tol 
 
 8x9 
 
 577 
 
 3-33 to 1 
 
 8 X 2-4 
 
 601 
 
 1 tol 
 
 8x8 
 
 585 
 
 5 tol 
 
 8 Xl'6 
 
 611 
 
 1-3 tol 
 
 8x6 
 
 590 
 
 6-7 tol 
 
 8 X 1-2 
 
 618 
 
 2 tol 
 
 8x4 
 
 592 
 
 10 to 1 
 
 8 XO-8 
 
 625 
 
 2-5 tol 
 
 8 X 3-2 
 
 595 
 
 20 tol 
 
 8 X 0-4 
 
 636 
 
 value of all the coefficient in these experiments is '603. 
 Here the coefficients increase in every instance as the depth 
 decreases, or as the ratio of the length of the notch to its 
 depth increases. We shall have to refer to the valuable 
 experiments made at Metz, on the discharge from differently- 
 proportioned orifices immediately. 
 
 Rennie's experiments for circular orifices at depths from 
 
 COEFFICIENTS FOR CIRCULAR ORIFICES, FROM RENNIE'S EXPERIMENTS. 
 
 Heads at the 
 centre of the 
 orifice in feet. 
 
 iinch 
 diameter. 
 
 i inch 
 diameter. 
 
 i inch 
 diameter. 
 
 1 inch 
 diameter. 
 
 Mean 
 values. 
 
 1 
 
 671 
 
 634 
 
 644 
 
 633 
 
 646 
 
 2 
 
 653 
 
 621 
 
 652 
 
 619 
 
 636 
 
 3 
 
 660 
 
 636 
 
 632 
 
 628 
 
 639 
 
 4 
 
 662 
 
 626 
 
 614 
 
 584" 
 
 621 
 
 Means 
 
 661 
 
 629 
 
 635 
 
 616 
 
 635 
 
 46475
 
 38 
 
 THE DISCHARGE OF WATER FROM 
 
 1 foot to 4 feet, and of diameters from % inch to 1 inch, 
 give the following coefficients. Here the increase in the 
 coefficients for lesser orifices and at lesser depths exhibits 
 itself very clearly, notwithstanding a few instances to the 
 contrary. The mean value of the coefficient c d derived 
 from the whole, is '635. For small rectilineal orifices the 
 coefficients were as follows : 
 
 COEFFICIENTS FOR RECTANGULAR ORIFICES, FROM RENNIE's EXPERIMEHT8. 
 
 Heads at the 
 centre of gravity, 
 in feet. ' 
 
 Square orifice, 
 1 inch x 1 inch. 
 
 fe 
 
 I|S- 
 
 Rectangular 
 orifice, longer side 
 horizontal, 
 li"x|". 
 
 Equilateral 
 triansle 
 of 1 square inch, 
 with base down. 
 
 Equilateral 
 triangle 
 of 1 square inch, 
 with base up. 
 
 1 
 
 617 
 
 617 
 
 663 
 
 
 
 596 
 
 2 
 
 635 
 
 635 
 
 668 
 
 
 
 577 
 
 3 
 
 606 
 
 606 
 
 606 
 
 
 
 572 
 
 4 
 
 693 
 
 593 
 
 593 
 
 593 
 
 593 
 
 Means 
 
 613 
 
 613 
 
 632 
 
 593 
 
 585 
 
 The most valuable series of experiments of which we are 
 possessed are those made at Metz, by Poncelet and Lesbros. 
 These were made with orifices eight inches wide, nearly, and 
 of different vertical dimensions placed at various depths 
 down to 10 feet. The discrepancies as to any general law 
 in the relation of the different values of the coefficient of 
 discharge c d to the size and depth of the orifice in the pre- 
 ceding experiments, have been remedied to a great extent by 
 these. They give an increase of the coefficients for the 
 smaller and very oblong orifices as they approach the surface, 
 and a decrease under the same circumstances in those for 
 the larger square and oblong orifices. There are a few 
 depths where maximum and minimum values are obtained : 
 we use the terms "maximum and minimum values" for 
 those which are greater in the one case and less in the other 
 than the coefficients immediately before and after them, and 
 not as being numerically the greatest or least values in the 
 column. We have marked with a *, in the arrangement of
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 39 
 
 the coefficients, TABLE I., these maximum and minimum 
 values. The heads given in this table are measured to the 
 upper side of the orifices, and by adding half the depth (d) 
 to any particular head, we obtain the head at the centre. 
 
 As a perceptible sinking of the surface takes place in 
 heads less than from five to three times the depth of the 
 orifice, the coefficients are arranged in pairs, the first column 
 containing the coefficients for heads measured from the still 
 water surface some distance back from the orifice, and the 
 second those obtained when the lesser heads, measured 
 directly at the orifice, were used. A very considerable in- 
 crease in the value of the coefficients for very oblong and 
 shallow small orifices, may be perceived as they approach the 
 surface, and the mean value for all rectilinear orifices at 
 considerable depths, seems to approach to "605 or '606. 
 
 We have shown, equation (29), that the discharge is 
 
 approximately, in which d is the depth of the orifice, and h 
 the head at its centre. Now it is to be observed, that it is 
 not the value of C A simply, which is given in TABLE I., but 
 
 the value of C A jl - - 2 [> the coefficient of A ^/2gh. 
 
 The coefficients in the table are, therefore, less than the 
 coefficient of discharge, strictly so called, by a quantity 
 
 equal to -^ .. The value of this expression is in general 
 yo h 
 
 very small, and it is easy to perceive from the first of the 
 expressions in equation (31), that it can never exceed 4J-per 
 cent., or '0417 in unity. If we wish to know the discharge 
 from an orifice 4 inches square = 4" x 4", with its centre 
 4 feet below the surface, which is equivalent to a head of 
 3 feet 10 inches at the upper side, we find from the table the 
 
 value of c d 1 - = '601 ; hence we get
 
 40 
 
 THE DISCHARGE OF WATER FROM 
 
 D = -601 x A \/2(/h = -601 x - x 8-025 x 2 = 
 
 9 
 
 601 x I x 16-05 = I x 9-646 = 1-072 
 
 cubic feet per second. In the absence of any experiments 
 with larger orifices, we must, when they occur, use the co- 
 efficients given in this table ; and, in order to do so with 
 judgment, it is only necessary to observe the relation of the 
 sides and heads. For example, if the size of an orifice be 
 16" x 4", we must seek for the coefficient in that column 
 where the ratio of the sides is as four to one, and if the head 
 at the upper side be five times the length of the orifice, we 
 shall find the coefficient "626, which in this case is the same 
 for depths measured behind, or at the orifice. For lesser 
 orifices, the results obtained from the experiments of Mi- 
 chelotti and Bossut, pages 34 and 35, are most applicable; 
 and also the coefficients of Rennie, pages 37 and 38. It is 
 almost needless to observe, that all these coefficients are only 
 applicable to orifices in thin plates, or those having the outside 
 arrises chamfered, as in 
 Fig. 8. Very little de- 
 pendance can be placed 
 on calculations of the 
 quantities of water dis- 
 charged from other ori- 
 fices, unless where the 
 coefficients have been al- 
 ready obtained by ex- 
 periment for them. If the inner arris next the water be 
 rounded, the coefficient will be increased. 
 
 NOTCHES AND WEIRS. 
 
 We have already given some coefficients, pages 36 and 37, 
 derived from the experiments of Du Buat, Brindley and 
 Smeaton, and Poncelet and Lesbros, for finding the dis- 
 charge over notches in the sides of large vessels; and it does 
 not appear that there is any difference of importance between
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 41 
 
 these and those for orifices sunk some depth below the 
 surface, when the proper formula for finding the discharge 
 for each is used. If we compare Poncelet and Lesbros' co- 
 efficients for notches, page 37, with those for an orifice at 
 the surface, TABLE I., we perceive little practical difference 
 in the results, the heads being measured back from the 
 orifice, unless in the very shallow depths, and where the 
 ratio of the length to the depth exceeds five to one. The 
 depths being in these examples less than an inch, it is 
 probable that the larger coefficients found for the orifice 
 at the surface, arise from the upper edge attracting the fluid 
 to it and lessening the effects of contraction. Indeed, the 
 results obtained from experiments with very shallow weirs, or 
 notches, have not been at all uniform, and at small depths 
 the discharge must proportionably be more affected by move- 
 ments of the air and external circumstances than when the 
 depths are considerable. We shall see that in Mr. Black- 
 well's experiments the coefficient obtained for depths of 
 1 and 2 inches was '676 for a thin plate 3 feet long, while 
 for a thin plate 10 feet long it increased up to '805. 
 
 The experiments of Castel, with weirs up to about 30 inches 
 long, and with variable heads of from 1 to 8 inches, lead 
 to the coefficient '597 for notches extending over one-fourth 
 of the side of a reservoir; and to the coefficient '664 when 
 they extend for the whole width. For lesser widths than 
 one-fourth, the coefficients decrease down to "584 ; and for 
 those extending between one-third of, and the whole width, 
 they increase from -600 to '665 and -680. Bidone finds 
 c d *620, and Eytelwein c d = '635. It will be perceived 
 from these and the foregoing results, that the third place 
 of decimals in the value of c d , and even sometimes the 
 second, is very uncertain ; that the coefficient varies with the 
 head and ratio of the notch to the side in which it is placed; 
 and we shall soon show that the form and size of the weir, 
 weir-basin, and approaches, still further modify its value. 
 
 When the sides and edge of a notch increase in thickness, 
 or are extended into a shoot, the coefficients are found to 
 reduce very considerably; and for small heads, to an extent
 
 42 
 
 THE DISCHARGE OF WATER FROM 
 
 beyond what the increase of resistance, from friction alone, 
 indicates. Poncelet and Lesbros found, for orifices, that 
 the addition of a horizontal shoot, 21 inches long, reduced 
 the coefficient from '604 to '601, with a head of 4 feet; but 
 for a head of only 4 inches, the coefficient fell from '572 
 to "483, the orifice being 8" x 8". For notches 8 inches 
 wide, with a horizontal shoot 9 feet 10 inches long, the 
 coefficient fell from -582 to '479, for a head of 8 inches; and 
 from -622 to '340, for a head of only 1 inch. Castel found 
 also, for a notch 8 inches wide with a shoot 8 inches long 
 attached and inclined at an angle of 4 18', that the mean 
 coefficient for heads from 2 to 4 inches was only '527*. 
 
 We have obtained the following table of coefficients from 
 some experiments made by Mr. Ballard, on the river Severn, 
 near Worcester, "with a weir 2 feet long, formed by a 
 board standing perpendicularly across a trough." f The 
 
 COEFFICIENTS FOR SHORT WBIR3 OVER BOARDS. 
 
 Heads measured on the crest. 
 
 Depths 
 in inches. 
 
 Coefficients. 
 
 Depths 
 in inches. 
 
 Coefficients. 
 
 Depths 
 in inches. 
 
 Coefficients. 
 
 1 
 
 762 
 
 3 
 
 801 
 
 5 
 
 733 
 
 H 
 
 662 
 
 H 
 
 765 
 
 H 
 
 713 
 
 1| 
 
 673 
 
 >i 
 
 748 
 
 5k 
 
 735 
 
 If 
 
 692 
 
 3| 
 
 740 
 
 5| 
 
 729 
 
 2 
 
 684 
 
 4 
 
 759 
 
 6 
 
 727 
 
 s* 
 
 702 
 
 4* 
 
 731 
 
 7 
 
 716 
 
 2* 
 
 756 
 
 *1 
 
 744 
 
 8 
 
 726 
 
 2* 
 
 786 
 
 4! 
 
 745 
 
 Mean 
 
 732 
 
 heads or depths were here measured on the weir, and hence 
 the coefficients are larger than those found from heads mea- 
 sured back to the surface of still water. 
 
 Experiments made at Chew Magna, in Somersetshire, by 
 Messrs. Blackwell and Simpson, in 1850 J, give the following 
 coefficients. 
 
 * Traite Hydraulique, par D'Aubuisson, pp. 46, 94 et 95. 
 t Civil Engineer and Architect's Journal for 1851, p. 647. 
 J Civil Engineer and Architect's Journal for 1851, pp. 642 and 645
 
 ORIFICES, WEIRS^ PIPES, AND RIVERS. 
 
 43 
 
 COEFFICIENTS DERIVED FROM THE EXPERIMENTS OF BkACKWELL AND SIMPSON. 
 
 Heads 
 in inches. 
 
 Coefficients. 
 
 Heads 
 in inches. 
 
 Coefficients. 
 
 Heads 
 in inches. 
 
 Coefficients. 
 
 1 to J 
 
 591 
 
 <J 
 
 743 
 
 6 
 
 749 
 
 1 tol re 
 
 626 
 
 4ft 
 
 760 
 
 6& 
 
 748 
 
 2to2i 
 
 682 
 
 *l 
 
 741 
 
 6 T ' 6 to 6} 
 
 747 
 
 2' 
 
 665 
 
 4i 7 5 
 
 750 
 
 611 
 
 772 
 
 21 
 
 670 
 
 *i 
 
 725 
 
 7i 
 
 717 
 
 2j 
 
 655 
 
 5 
 
 780 
 
 8 
 
 802 
 
 2| 
 
 653 
 
 *ft 
 
 781 
 
 8to8} 
 
 737 
 
 2{i 
 
 654 
 
 KB 
 
 749 
 
 811 
 
 750 
 
 3 to 3^ 
 
 725 
 
 5&to5^ 
 
 751 
 
 9 
 
 781 
 
 4 
 
 745 
 
 5}f 
 
 728 
 
 Mean 
 
 723 
 
 " The overfall bar was a cast-iron plate 2 inches thick, 
 with a square top." The length of the overfall was 10 feet. 
 The heads were measured from still water at the side of 
 the reservoir, and at some distance up in it. The area of 
 the reservoir was 21 statute perches, of an irregular figure, 
 and nearly 4 feet deep on an average. It was supplied from 
 an upper reservoir, by a pipe 2 feet in diameter and of 
 19 feet fall; the distance between the supply and the weir 
 was about 100 feet. The width of the reservoir as it ap-
 
 44 THE DISCHARGE OP WATER FROM 
 
 preached the overfall was about 50 feet, and the plan and 
 section, Fig. 9, of the weir and overfall in connection with it, 
 will give a fair idea of the circumstances attending the 
 experiments. For heads over 5 inches the velocity of ap- 
 proach to the weir was " perceptible to the eye," though its 
 amount was not determined. We perceive that the co- 
 efficient (derived from two experiments) for a depth of 
 8 inches is "802, while the coefficient (derived from three 
 experiments) for a depth of 7f inches is '717, and for 
 depths from 8 to 8|| inches the mean coefficient is '743 : as 
 all the attendant circumstances appear the same, these dis- 
 crepances and others must arise from the circumstances of 
 the case: perhaps the supply, and, consequently, the velocity 
 of approach, was increased while making one set of experi- 
 ments, without affecting the still water near the side where 
 the heads appear to have been taken. By comparing the 
 results with those obtained by one of the same experi- 
 menters, Mr. Blackwell, on the Kennet and Avon Canal, 
 we shall immediately perceive that the velocity of approach, 
 and every circumstance which tends to alter and modify it, 
 has a very important effect on the amount of the discharge, 
 and, consequently, on the coefficient. 
 
 The experiments made by Mr. Blackwell, on the Kennet 
 and Avon Canal, in 1850*, afford very valuable instruction 
 as the form and width of the crest were varied, and brought 
 to agree more closely with actual weirs in rivers than the 
 thin plates or boards of earlier experimenters. We have 
 calculated and arranged the coefficients in the following table 
 from these experiments. The variations in the values for 
 different widths of crest, other circumstances being the same, 
 are very considerable ; and the differences in the coefficients, 
 at depths of 5 inches and under, for thin plates and crests 
 2 inches wide, are greater than mere friction can account 
 for ; and greater also than the differences at the same depths 
 between the coefficients for crests 2 inches wide, and 3 feet 
 wide. 
 
 * Civil Engineer and Architect's Journal, 1851, p. 642.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 45 
 
 I 
 
 Crests 3 feet wide. 
 
 dj 
 
 ""1 
 
 s | 
 
 . o . eo . b- 
 
 . ,_! . -<t* . o ; ; ; ; 
 
 III 
 
 OOt- I \j-lr-l |?OOOO ' \ 
 
 II 
 
 |- 
 
 to 
 
 <M t- . t^ t^- . O O t 
 
 11 
 
 i 
 
 ?O CO 
 
 3 IS : S fc : g : : : 
 
 
 
 Is 
 i 
 
 o <o 
 ip ip 
 
 i- i-i ?o . eo T-I (N 
 
 US -^tl T5 S -rt< ^( 
 
 fi 
 
 (M (M 
 
 i-H O5 O5 l-l OO O O 
 
 3 JJ *r g 00 t-. t- ; j ; 
 
 
 1 
 g 
 
 S 
 
 -2 
 s 
 
 & 
 
 J 
 
 m 
 iflt 
 
 IO t>- 
 
 
 : g fe ::::::: 
 
 n 
 
 
 IO *O 
 
 ? s 
 
 1115111?? : 
 
 it 
 
 s s 
 
 -*t< us 
 
 
 l>.iSrHOOOOOOiOiCO 
 
 
 1! 
 
 t- O5 
 
 ^ to 
 
 COO50OCOt>-COO^t< 
 O-^OOOli-IOOi-( 
 
 ipipipiO(r)<ooo 
 
 j 
 
 1 
 .S 
 En 
 
 I! 
 
 OJ CO 
 
 
 S g : : S S : : : 
 
 || 
 
 s s 
 
 g ^ g :::::: 
 
 
 
 uioAjasaj am ui 
 
 UIOJJ p.UHKlMIH 
 
 'saqoui ui spsan 
 
 ----.-.. s s s
 
 46 
 
 THE DISCHARGE OF WATER FROM 
 
 The plan and section, Fig. 10, will give a fair idea of the 
 approach to, and nature of the overfall made use of in these 
 
 Fig. 10 
 
 The dotted lines on 
 Plan shew the sub- 
 merged masonry ap- 
 pearing at C in Section. 
 
 Scale of Section in Feet. 
 
 experiments. The area of the reservoir was 2A. IR. 30p., 
 and the head was measured from the surface of the still 
 water in it, which remained unchanged between the be- 
 ginning and end of each experiment. The width of the 
 approach AB from the reservoir was about 32 feet; the 
 width at ab about 13 feet, below which the waterway 
 widened suddenly, and again narrowed to the length of the 
 overfall. The depth in front of the dam appears to have 
 been about 3 feet; the depth on the dam, next the overfall, 
 about 2 feet; and the depth on the sunk masonry in the 
 channel of approach, about 18 inches. Altogether, the cir- 
 cumstances were such as to increase the amount of re- 
 sistances between the reservoir, from which the head was 
 measured, and the overfall, particularly for the larger heads, 
 and we accordingly see that the coefficients become less for 
 heads over six inches, with a few exceptions. The measure- 
 ments of the quantities discharged appear to have been 
 made very accurately, yet the discharges per second, with the 
 same head and same length of overfall, sometimes vary ;
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 47 
 
 for instance, with the plank 2 inches thick and 10 feet long, 
 the discharge per second for 4 inches head varies from 
 6'098 cubic feet to 6-491 cubic feet, or by about one-sixteenth 
 of the whole quantity. Most of the results, however, are 
 means from several experiments. The quantities discharged 
 varied from one- tenth of a cubic foot to 22 cubic feet per 
 second, and the times from 24 to 420 seconds. If we com- 
 pare the coefficients for a plank 10 feet long and 2 inches 
 thick in the foregoing table with those for the same overfall 
 at Chew Magna, we shall immediately perceive how much 
 the form of the approaches affects the discharge. Indeed, 
 were the area of the reservoir at Chew Magna even larger 
 than that for the Kennet and Avon experiments, it would 
 be found, notwithstanding, that the coefficients in the former 
 would still continue the larger, though not fully as large as 
 those found under the particular circumstances. 
 
 HEAD, AND FROM WHENCE MEASURED. 
 
 By referring to TABLE I., we shall see that there is a 
 difference in the coefficients as obtained from heads measured 
 on or above the orifice. This difference is greater in notches, 
 or weirs, than in orifices sunk below the surface; and when 
 the crest of a weir is of some width, the depths upon it 
 vary*. In the Kennet and Avon experiments, the heads 
 measured from the surface of the water in the reservoir, and 
 the depths at the " outer edge " (by which we understand 
 the lower edge) of the crest were as follows : 
 
 * There is a very important omission in almost all the preceding 
 experiments on weirs and notches. In Fig. 10, for instance, it would 
 have been necessary to have obtained the heads at AB and a b in each 
 experiment, above the crest, and also the head on and a few feet above 
 the crest itself. These would indicate the resistances at the different 
 passages of approach, and enable us to calculate the coefficients cor- 
 rectly, and thereby render them more generally applicable to practical 
 purposes. The coefficients in the table at page 45, are not as valuable 
 as they otherwise would be from this omission. The level of still water 
 near the banks is below that of the moving water in the current, 
 therefore, heads measured from still water must give larger coefficients 
 than if taken from the centre of the current. This may account, to 
 some extent, for the larger coefficients in the table at p. 43.
 
 THE DISCHARGE OF WATER FROM 
 
 DIFFERENCE OF HEADS MEASURED ON AND ABOVE WEIRS. 
 
 
 Heads on crests 
 
 Heads on 
 
 II 
 
 2 inches thick. 
 
 crests 3 feet wide. 
 
 JJ 
 8 
 
 i 
 
 f 
 
 H 
 
 *t 
 
 ih 
 
 ft 
 
 fl 
 
 fl, 
 
 zl 
 
 I 2 
 
 1 
 
 1 
 
 li 
 
 i!= 
 
 If- 
 
 II 
 
 II 
 
 S* 
 
 IF 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 ... 
 
 J 
 
 i 
 
 7s 
 
 T 5 S 
 
 2 
 
 ... 
 
 
 i 
 
 
 
 
 i 
 
 re 
 
 8 
 
 ... 
 
 lit 
 
 
 
 ... 
 
 U to li 
 
 
 ... 
 
 4 
 
 3 to 2ft 
 
 3* 
 
 if 
 
 If 
 
 li 
 
 
 
 1| 
 
 5 
 
 3 
 
 H 
 
 2i 
 
 If 
 
 ... 
 
 
 U 
 
 ... 
 
 6 
 
 4f 
 
 4 i 
 
 2| 
 
 
 ... 
 
 2| 
 
 2| 
 
 2J 
 
 7 
 
 ... 
 
 
 ... 
 
 2J 
 
 2| 
 
 
 
 
 s 
 
 g. 
 
 
 
 
 
 
 34 
 
 , 
 
 Q 
 
 
 
 
 
 
 41 
 
 
 2 
 
 10 
 
 
 ... 
 
 
 ... 
 
 
 
 4 
 
 
 No intermediate heads are given, but those registered point 
 out very clearly the great differences which often exist be- 
 tween the heads measured on a weir, or notch, and those 
 measured from the still water above it ; and how the form of 
 the weir itself, as well as the nature of the approaches, alters 
 the depth passing over. On a crest 2 feet wide, with 
 14 inches depth on the upper edge, we have found that the 
 depth on the lower edge is reduced to 11 inches. The head 
 taken from 3 to 20 feet above the crest, where the plane of 
 the approaching water surface becomes curved, is that in 
 general which is best suited for finding the discharge by 
 means of the common coefficients, but a correct section of 
 the channel and water-line, showing the different depths 
 upon and for some distance above the crest, is necessary in all 
 experiments for determining accurately by calculation the 
 value of the coefficient of discharge c d . 
 
 Du Buat, finding the theoretical expression for the dis- 
 charge through an orifice of half the depth h,
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 49 
 
 . . = 1^x1 {**-(*)*> 
 
 equation (6) 
 
 x 1 - 4 = -646 x Ih 
 
 to agree pretty closely with his experiments, seems to have 
 assumed that the head h was reduced to 5- in passing over. 
 
 This is a reduction, however, which never takes place unless 
 with a wide crest and at its lower edge, or where the head h 
 is measured at a considerable distance above the weir, and 
 a loss of head due to the distance and obstructions in 
 channel takes place. When there is a clear weir basin 
 immediately above the weir, we have found that, putting h 
 for the head measured from the surface in the weir basin 
 and h w for the depth on the upper edge of the weir, that 
 
 (32.) A-A w = -14v/A; 
 
 for measures in feet, and 
 (33.) h - A w = -48 v/^ 
 
 for measures in inches. The comparative values of h and 
 7* w depend, however, a good deal on the particular circum- 
 stances of the case. Dr. Robinson found* A = 1*111 A w , 
 when h was about 5 inches. The expressions we have given 
 are founded on the hypothesis, that h 7i w is as the velocity 
 of discharge, or as the *J h nearly. For small depths, 
 there is a practical difficulty in measuring with sufficient 
 accuracy the relative values of h and A w . Unless for very 
 small heads the sinking will be found in general to vary from 
 
 yrr to -j, and in practice it will always be useful to observe 
 
 the depths on the weir as well as the heads for some 
 distances (and where the widths contract) above it. 
 
 In order to convey to our readers a more definite idea of 
 the differences between the coefficients for heads measured 
 
 * Proceedings of the Royal Irish Academy, vol. iv. p. 212. 
 
 E
 
 50 THE DISCHARGE OF WATER FROM 
 
 at the weir, or notch, and at some distance above it, we 
 shall assume the difference of the heads h h w = ; 
 
 hence h = h w and h w = r h. 
 
 r r + 1 
 
 Now the discharge may be considered as that through an 
 orifice whose depth is h w with a head over the upper edge 
 
 equal to h A w = -j hence from equation (6) the discharge 
 is equal to 
 
 and substituting for 7t* its value ( r -^- A w J , we shall find 
 the value of 
 (34.) D = -| 
 
 As the value of the discharge would be expressed by 
 
 2 / 
 
 -^Ihy, \/2gh w X c d 
 
 if the head h A w were neglected, it is evident the coefficient 
 is increased, under the circumstances, from c d to 
 
 or, more correctly, the common formula has to be multiplied 
 by (l + _ V ( V, to find the true discharge, and 
 
 the value of this expression for different values of =: n 
 will be found in TABLE IV. If we suppose that 
 
 and we find from the table ( 1 +-V - (-Y = M221.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 51 
 
 Now if we take the value of c d for the full head h to be -628, 
 we shall find 1 "1221 x '628 r= '705, rejecting the latter figures, 
 for the coefficient when the head is measured at the orifice; 
 
 and if _ = - =. n, we should find in the same manner 
 
 r 10 
 
 the new coefficient to be 1-2251 x '628 = -769 nearly. 
 The increase of the coefficients determined, page 42, from 
 Mr. Ballard's experiments is, therefore, evident from prin- 
 ciple, as the heads were taken at the notch ; and it is also 
 pretty clear that, in order to determine the true discharge, 
 the heads both on, at, and above a weir should be taken. 
 
 SECTION IV. 
 
 VARIATIONS IN THE COEFFICIENTS FROM THE POSITION OF THE 
 ORIFICE. GENERAL AND PARTIAL CONTRACTION. VELO- 
 CITY OF APPROACH. CENTRAL AND MEAN VELOCITIES. 
 
 A glance at Table I. will show us that the coefficients 
 increase as the orifices approach the surface, to a certain 
 depth dependent on the ratio of the sides, and that this 
 increase increases with the ratio of the length to the depth. 
 Some experimenters have found the increase to continue 
 uninterrupted for all orifices up to the surface, but this seems 
 to hold only for depths taken at or near the orifice when it is 
 square or nearly so. It has also been found that the co- 
 efficient increases as the orifice approaches to the sides or 
 bottom of a vessel : as the contraction becomes imperfect the 
 coefficient increases. The lateral orifices A, B, c, D, E, F, G, 
 H, i, and K, Fig. 11, have coefficients differing more or less 
 from each other. The coefficient for A is found to be larger 
 than either of those for B, c, E, or D ; that for G or K larger 
 than that for H or i; that for H larger than that for i; and 
 that for F, where the contraction is general, least of all. The 
 contraction of the fluid on entering the orifice F removed 
 from the bottom and sides is complete ; it is termed, therefore, 
 
 E2
 
 52 
 
 THE DISCHARGE OF WATER FROM 
 
 Fig". 11 
 
 
 general contraction ; that at the orifices A, E, G, H, i, K, and 
 D, is interfered with by the sides ; it is therefore incomplete, 
 and termed partial contraction. The increase in the co- 
 efficients for the same-sized orifices at the same mean depths 
 may be assumed as proportionate to the length of the 
 perimeter at which the contraction is partial, or from which 
 the lateral flow is shut off; for example, the increase for the 
 orifice G is to that for H as cd + de : de; and in the same 
 manner the increase for G is to that for E as c d + d e : c d. If 
 we put n for the ratio of the contracted portion cde to the 
 entire perimeter, and, as before, c d for the coefficient of 
 general contraction, we shall find the coefficient of partial 
 contraction to be equal to 
 
 (35.) 
 
 + -09 n 
 
 for rectangular orifices. The value of the second term *09 n 
 is derived from experiments. If we assume *617 for the mean 
 value of c d , we may change the expression into the form 
 (1 + -146)c d . When n = i, this becomes 1*036 c d ; when 
 n = , it becomes 1*073 e d ; and when n = f, contraction is 
 prevented for three-fourths of the perimeter, and the coefficient 
 for partial contraction becomes M09c d . The form which we 
 have given equation (35) is, however, the simplest ; but the 
 value of n must not exceed f . If in this case c d = *617, the 
 coefficient for partial contraction becomes '617 + -09 x f 
 = *617 + "067 = '684. Bidone's experiments give for the 
 coefficient of partial contraction (1 + *152w)c d ; and Weis- 
 bach's (1 + -132w)c d .
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 53 
 
 VARIATION IN THE COEFFICIENTS FROM THE EFFECTS OF 
 THE VELOCITY OF APPROACH. 
 
 Heretofore we have supposed the water in the vessel to be 
 almost still, its surface level unchanged, and the vessel conse- 
 quently large compared with the area of the orifice. When 
 the water flows to the orifice with a perceptible velocity, the 
 contracted vein and the discharge are both found to be 
 increased, other circumstances being the same. If the area 
 of the vessel or channel in front exceed thirty times that of 
 the orifice, the discharge will not be perceptibly increased by 
 the induced velocity in the conduit ; but for lesser areas of 
 the approaching channel corrections due to the velocity of 
 approach become necessary. It is clear that this velocity 
 may arise from either a surface inclination in the channel, an 
 increase of head, or a small channel of approach. 
 
 We get equation (6) for the discharge from a rectangular 
 
 Fig. 12 
 
 orifice A, Fig. 12, of the length /, with a head measured from 
 still water 
 
 x *- 
 
 in which h b and h t are measured at some distance back from 
 the orifice, as shown in the section. The water here, however, 
 must move along the channel towards the orifice with con- 
 siderable velocity. If A be the area of the orifice, and c the 
 area of the channel, we may suppose with tolerable accuracy
 
 64 THE DISCHARGE OF WATER FROM 
 
 that this velocity is equal to v , in which v represents the 
 
 mean velocity in the orifice. If we also represent by v & the 
 velocity of approach, we get the equation 
 
 (36.) v a = x v ot 
 
 and consequently the height (A a ) due to it is 
 
 (37.) *. = ? X ^T- 
 
 The height A a may be considered as an increase of head, 
 converting h b into h b + A a , and A t into h t + A a . The discharge 
 therefore now becomes 
 
 (38.) D = I c d I v/^{(A b + h^ - (h, + htf} ; 
 which, for notches or weirs, is reduced to 
 (39.) D 
 
 as A t then vanishes. As D is also equal to A x v of equation 
 (37) may be changed into 
 
 (40.) *. = J?x^. 
 
 If this value for A a be substituted in equations (38) and (39), 
 the resulting equations will be of a high order and do not 
 admit of a direct solution; and in (38) and (39), as they 
 stand, A a involves implicitly the value of D, which we are 
 seeking for. By finding at first an approximate value for the 
 velocity of approach, the height A a due to it can be easily 
 found, equation (37) ; this height, substituted in equation (38) 
 or (39), will give a closer value of D, from which again a 
 more correct value of A a can be determined j and by repeating 
 the operation the values of D and h A can be had to any degree 
 of accuracy. In general the values found at the second 
 operation will be sufficiently correct for all practical purposes.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 55 
 
 It has been already observed that, for orifices, it is advisable 
 to find the discharge from a formula in which only one head, 
 that at the centre, is made use of; and though TABLE IV., as 
 we shall show, enables us to calculate the discharge with 
 facility from either formula, it will be of use to reduce 
 equation (38) to a form in which only the head (h) at the 
 centre is used. The error in so doing can never exceed six 
 per cent, even at small depths, equation (31), and this is more 
 than balanced by the observed increase in the coefficients for 
 smaller heads. 
 
 The formula for the discharge from an orifice, h being the 
 head at the centre, is 
 
 D = c^^/lgh x A; 
 
 and when the additional head A a due to the velocity of 
 approach is considered, 
 
 D = <? d </2g(h + 7i a ) x A, 
 which may be changed into 
 
 (41.) 
 
 Equation (39), for weirs, may be also changed to the form 
 (42.) D=- 
 
 this is similar in every way to the equation 
 (43.) .- 
 
 for the discharge from a rectangular orifice whose depth is d, 
 with the head h t at the upper edge. TABLE III. contains the 
 
 values of ! 1 + ~r j- in equation (41), and TABLE IV. the 
 values of ( 1 + T^ (i~^\ in equation (42), or the similar 
 
 expression in (43), -JT or r being put equal to n; and we per- 
 ceive that the effect of the velocity of approach is such as to
 
 56 THE DISCHARGE OF WATER FROM 
 
 increase the coefficient c d to C A -I 1 + -r [ for orifices sunk 
 some distance below the surface, and into 
 
 .{('+*-()'} 
 
 for weirs when h A is the height due to the velocity of 
 approach, h the depth of the centre of the orifice, and h b the 
 head on the weir. A few examples, showing the application 
 of the formulae (41), (42), and (43), and the application of 
 TABLES I., II., III., and IV. to them, will be of use. We 
 shall suppose, for the present, the velocity of approach v a to 
 be given. 
 
 EXAMPLE I. A rectangular orifice, 12 inches wide by 
 4 inches deep, has its centre placed 4 feet below the surface, 
 and the water approaches the head with a velocity of 
 28 inches per second; what is the discharge ? For an orifice 
 of the given proportions, and sunk to a depth nearly four 
 times its length, we shall find from TABLE I. 
 
 =f621 nearly. 
 
 As the coefficient of velocity, equation (2), for water flowing 
 in a channel is about '956, we shall find, column No. 3, 
 TABLE II., the height 7* B == !- n: 1-125 inch, nearly cor- 
 responding to the velocity 28 inches. Equation (41), 
 
 now becomes 
 
 D = 12 x 4 v/27fc x -621 {l + ~- 5 }*. 
 
 We also find ^/'Zgh =s 192-6 inches when h i= 48 inches, 
 TABLE II.; therefore 
 
 D = 12 x 4 x 192-6 x -621 (l + ^j* 
 
 I 48 ) 
 
 = 9244-8 x -621 {l + -0234}* = 9244-8 x -621 x 1-0116,
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 57 
 
 (as {1-0234}* = 1-0116 from TABLE III.) = 9244'8 x -628 
 nearly = 5805-7 cubic inches = 3'306 cubic feet per second. 
 Or thus: The value of -621 x (1-0234) being found equal 
 
 628, D = A x -628 ^/2g x 48. Now for the coefficient 
 628, and h = 48 inches, TABLE II. gives us -628 </2g x 48 
 = 120-96 inches; hence we get D = 12 x 4 x 120*96 
 = 5806-08 cubic inches = 3'306 cubic feet, the same as 
 before, the difference -38 in the cubic inches being of no 
 practical value. If the centre of the orifice were within 
 1 foot of the surface, the effect of the velocity of approach 
 would be much greater ; for then 
 
 CA x |l + | a |* = (from TABLE I.) -623 jl + V 
 
 = (from TABLE III.) -623 x 1-047 = '652 instead of -628. 
 In this case the discharge is D == 12 x 4 x "652 \/2g x 12 
 = 12 x 4 x -652 x 96-3 (from TABLE II.) = 12 x 4 x 62-8 
 = 3014-4 cubic inches = 1'744 cubic feet per second. Or 
 we may find the value of -652 ^/2gh directly from Table II. 
 thus : 
 
 The value of -628 </2g x 12 = 60-48 -628 
 The value of -666 </2g x 12 = 64-14 -652 
 
 38 : 3-66 ::~24:2'31. 
 
 Hence '652 </2gh = 60'48 + 2-31 = 62-79, and the dis- 
 charge = 12 x 4 x 62-79 =: 3013-92 cubic inches = 1-744 
 cubic feet per second, the same as before. 
 
 EXAMPLE II. A rectangular notch, 7 feet long, has a head 
 of 8 inches measured at abmit 4 feet above the orifice, and 
 the water approaches the head with a velocity of 16 inches 
 per second ; what is the discharge ? For a still head we shall 
 assume e d = '628 in this case, and we have from equation (42) 
 
 As in the last example, we shall find from TABLE II. (hj the
 
 58 THE DISCHARGE OF WATER FROM 
 
 height due to the velocity of approach (16^ inches) to be 
 - = 0*375 inch, assuming the coefficient of velocity to be 
 956. We have, therefore, A a = -375, h b = 8, c d = -628, and 
 A = 7 x 12 x 8, or for measures in feet ~r *= '047, h b = -, 
 
 ">b 
 
 and A = 7 x -; hence 
 
 x I x -628 {(1*047)* - (*047)*}. 
 
 The value of (1-047)* (*047)* will be found from TABLE IV. 
 equal to 1*0612; the value of A/ 2g x will be found from 
 
 TABLE II. equal to 6*552, viz. by dividing the velocity 78*630, 
 to be found opposite 8 inches, by 12; hence 
 
 D=!x7x?x 6*552 x -628 x 1*0612 
 
 3 3 
 
 = - x 7 x 4*368 x *628 x 1*0612 
 
 3 
 = |x 7 x 4*368 x *666 nearly 
 
 3 
 
 = 2 x 7 x 2-909 = 7 x 1*939 
 
 3 
 
 = 13*573 cubic feet per second = 814-38 cubic feet per 
 minute. Or thus: From TABLE VI. we find, when the 
 coefficient is "628, the discharge from a weir 1 foot long, with 
 ahead of 8 inches, to be 109*731 cubic feet per minute. 
 
 The discharge for a weir 7 feet long, when j^ = *047 is 
 
 therefore 109*731 x 7 x 1*0612 = 815*12 cubic feet per 
 minute. The difference between this value and that before 
 found, 814*38 cubic feet, is immaterial, and has arisen from 
 not continuing all the products to a sufficient number of 
 places of decimals. 
 
 We have, in equations (36) and (37), pointed out the 
 relations between the channel, orifice, velocity of approach, 
 and velocity in the orifice, viz.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 59 
 
 A A 2 v* D 2 Vl 
 
 =- x v , and A a = ? x =^-, m which A a = 
 
 (neglecting the coefficient of velocity '974 or -956). As v is 
 the velocity in the orifice, 
 head h + # a , and therefore 
 
 the velocity in the orifice, must be the velocity due to the 
 
 1 
 
 < x 4 
 
 We have hence 
 
 (44.) 
 
 substituting this value in equations (41) and (42), there 
 results 
 
 (45.) A 
 
 in which m = , for the discharge from an orifice at some 
 depth, and for the discharge from a weir, 
 
 (46.) D = | 
 
 The two last equations give the discharge when the ratio of 
 
 rf-i 
 
 the channel to the orifice - = m is known, and also when 
 
 A 
 
 the whole quantity of water passing through the orifice, 
 that due to the velocity of approach as well as the pressure, 
 suffers a contraction whose coefficient is c d . When e d = 1, 
 equation (45) may be changed into
 
 60 THE DISCHARGE OF WATER FROM 
 
 This is the equation of Daniel Bernoulli, which is only a 
 particular case of the one we have given. 
 
 If we put = j^Lp the values of (l + ^A 
 
 and of 1 + ri - ^f' can be easi| y liad 
 
 from TABLES III. and IV. We have, however, calculated 
 TABLE V. for different ratios of the channel to the orifice, 
 and for different values of the coefficient of discharge. This 
 table gives at once the values of 
 
 as new coefficients, and the corresponding value of 
 
 Aa_ 4 
 
 h-m*-cl' 
 
 It is equally applicable, therefore, to equations (41) and (42) 
 as to equations (45) and (46). For instance, we find here at 
 once the value of 628 {(1'047)^ - (-047)^} in EXAMPLE II., 
 
 p. 58, equal to -666, as ^ = -047, and the next value to it 
 h 
 
 for the coefficient '628, in the table, is '046, opposite to which 
 we find '666, the new coefficient sought. The sectional area 
 of the channel in this case, as appears from the first column, 
 must be about three times that of the weir or notch. 
 
 TABLE V. is calculated from coefficients c d in still water, 
 which vary from '550 to 1. Those from '606 to "650, and 
 the mean value '628 are most suited to practice. When the 
 channel is equal to the orifice, the supply must equal the 
 discharge, and for open channels, with the mean coefficient 
 628, we find accordingly from the table the new coefficient 
 1-002 for weirs, or 1 very nearly as it should be. We also 
 find, in the same case, viz. when A = c, and c d = '628, that 
 for short tubes, Fig. 13, the re- 
 sulting new coefficient becomes 
 807. This, as we shall after- 
 wards see, agrees very closely 
 with the experimental results. 
 When the coefficients in still
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 61 
 
 water are less than '628, or more correctly '62725, the orifice, 
 according to our formula, cannot equal the channel unless 
 other resistances take place as from friction in tubes longer 
 than one and a half or two diameters, or in wide crested 
 weirs; and for greater coefficients the junction of the short 
 tube with the vessel must be rounded, Fig. 14, on one or 
 more sides; and in weirs or 
 notches the approaches must 
 slope from the crest and ends 
 to the bottom or sides, and the 
 overfall be sudden. The con- 
 verging form of the approaches 
 must, however, increase the ve- 
 
 Fig.U 
 
 locity of approach; and therefore # a is greater than x v 
 
 when c is measured between ro and RO, Fig. 14, to find the 
 discharge, or new coefficient of an orifice placed at r o. 
 
 As the coefficients in TABLE V. are calculated for orifices 
 at the end of short cylindrical or prismatic tubes at right 
 angles to the sides or bottom of a cistern, a correction is re- 
 quired when the junction is rounded off as at R o r o, Fig. 14. 
 When the channel is equal to the orifice, the new coefficient 
 in equation (45) becomes 
 
 = C A x 
 
 The velocity in the tube Fig. 14 is to that in the tube Fig. 13 
 
 
 as 1 to cA T - -yl nearly, or for the mean value c d =. '628, 
 
 1 ~" 
 
 as 1 to '807. Now, as - is assumed equal to in the cylin- 
 drical or prismatic tube, Fig. 13, - = ~ in the tube 
 
 Fig. 14 with the rounded junction, for t? a becomes rg^; 
 
 hence, in order to find the discharge from orifices at the end 
 of the short tube, Fig. 14, we have only to multiply the
 
 62 THE DISCHARGE OF WATER FROM 
 
 rf-1 
 
 numbers representing the ratio - in the first column, 
 
 TABLE V., by '807, or more generally by C A \ \ __ c * \ > and 
 find the coefficient opposite to the product. Thus if c d = *628, 
 
 we find, when- = 1, C A ]-, 5-! = -807 in the table. If, 
 
 A I ^d ) 
 
 c 
 again, we suppose - = 3, then 3 x '807 = 2-421, the value 
 
 of for the tube Fig. 14, and opposite this value of -, taken 
 
 in column 1, we shall find '651 for the new coefficient. For 
 the cylindrical or prismatic tube, Fig. 13, the new coefficient 
 would be -642. 
 
 DIFFERENT EFFECTS OF CENTRAL AND MEAN VELOCITIES. 
 
 There is, however, another circumstance to be taken into 
 consideration, and which we shall have to refer to more par- 
 ticularly hereafter ; it is this, that the central velocity directly 
 facing the orifice is also the maximum velocity in the tube, 
 and not the mean velocity. The ratio of these is as 1 : -835 
 
 nearly ; hence, in the above example, where - = 3, we get 
 
 3 x -835 = 2-505 for the value of j in column 1, TABLE V., 
 
 opposite to which we shall find '649, the coefficient for an 
 orifice of one-third of the section of the tube when cylin- 
 drical or prismatic, Fig. 13 ; and 3 x '835 x '807 = 2'02 
 nearly, opposite to which we shall get '661 for the coefficient 
 when the orifice is at the end of the short tube, Fig. 14, with 
 
 J 
 
 a rounded junction. We have, therefore, - X '835 equal to 
 the new value of for finding the discharge from orifices at 
 
 A 
 
 p 
 
 the end of cylindrical or prismatic tubes, and x '835 x '807
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 63 
 
 o c 
 
 := - x '67 nearly for the new value of - when finding the 
 
 A. A. 
 
 discharge from orifices at the end of a short tube with a 
 rounded junction, nearly. 
 
 The ratio of the mean velocity in the tube to that facing 
 the orifice cannot be less than '835 to 1, and varies up to 
 1 to 1 ; the first ratio obtaining when the orifice is pretty 
 small compared with the section of the tube, and the other 
 when they are equal. If 
 we suppose the curve D c, 
 whose abscissae (A b) repre- 
 sent the ratio of the ori- 
 fice to the section of the 
 tube, and whose ordinates 
 (b c) represent the ratio of 
 the mean velocity in the 
 tube to that facing the orifice, to be a parabola, we shall find 
 the following values : 
 
 Fig 15 
 
 Ratios 
 
 Values of 
 
 - = do. 
 
 
 1 
 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 
 8 
 
 9 
 
 1-0 
 
 165 
 163 
 158 
 150 
 139 
 124 
 106 
 084 
 059 
 031 
 000 
 
 Values of 
 be. 
 
 835 
 837 
 842 
 850 
 861 
 876 
 894 
 916 
 941 
 969 
 1-000 
 
 These values of b c are to be multiplied by the corresponding 
 
 c 
 ratio in order to find a new value, opposite to which will be 
 
 found, in the table, the coefficient for orifices at the ends of
 
 64 
 
 THE DISCHARGE OF WATER FROM 
 
 short prismatic or cylindrical tubes ; and this new value again 
 
 -i . i 
 
 multiplied by '807, or more generally by c d 1 1 __ c * f > will 
 
 give another new value of -, opposite to which, in the table, 
 
 will be found the coefficient for orifices at the ends of short 
 tubes with rounded junctions. 
 
 EXAMPLE III. What 
 is the discharge from an 
 orifice A, Fig. 16, 2 feet 
 long by 1 foot deep, the \ 
 
 value of ~ being 3, and 
 
 the depth of the centre of 
 A 1 foot 6 inches below 
 the surface? We have 
 
 D t = 2 x 1 x 
 
 117-945 
 12 
 
 (TABLE II.) = 2 x 9'829 = 19-658 cubic feet per second for 
 the theoretical discharge. From the foregoing table the co- 
 efficient for the mean velocity, facing the orifice, is about '86; 
 
 c 
 hence - x '86 = 3 x -86 = 2'58. If we take the coefficient 
 
 from TABLE I., we shall find it (opposite to 2, the ratio of the 
 length of the orifice to its depth) to be '617 ; and, for this co- 
 efficient, opposite to 2-58, in TABLE V., or the next number to 
 it, we find the required coefficient -636 ; hence the discharge is 
 636 x 1 9-658 = 12-502 cubic feet per second. If we assume 
 the coefficient in still water to be -628, then we shall obtain the 
 new coefficient -647, and the discharge would be -647 x 19*658 
 = 12-719 cubic feet. If the junction of the tube with the cistern 
 be rounded, as shown by the dotted lines, we have to multiply 
 
 2-58 by -807, which gives 2-08 for the new value of -, opposite 
 
 which we shall find, in TABLE V., when the first coefficient is 
 628, the new coefficient -659 ; and the discharge in this case 
 would be -659 x 19-658 = 12-955 cubic feet per second.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 65 
 
 It is not necessary to 
 take out the coefficient of 
 mean velocity facing the 
 orifice to more than two 
 places of decimals. For 
 sluices in streams and mill 
 races, Fig. 17, the mean co- 
 efficient -628 in still water 
 may be assumed, and thence the new coefficient suited to the 
 
 ratio may be found, as in the first portion of EXAMPLE III. 
 
 EXAMPLE IV. What is the discharge through the aper- 
 ture A, equal 2 feet by 1 foot, when the channel is to the 
 orifice as 3'375 to 1, and the depth of the centre is l'25foot 
 below the surface, taken at about 3 feet above the orifice? 
 Here the coefficient of the approaching velocity is '85 nearly, 
 
 85 = 2-87; and as 
 
 whence the new value of - is 3*375 x 
 
 c d = -628, we shall get from TABLE V. the new coefficient '644. 
 Hence 
 
 =2 X 1 X 
 
 107-669 
 12 
 
 x -644 (TABLE II.) = 2 x 8-972 x -644 
 
 = 17-944 x -644 = 11-556 cubic feet per second. 
 
 Weisbach finds the discharge, by an empirical formula, to be 
 11-31 cubic feet. If the coefficient be sought in TABLE I., 
 we shall find it -617 nearly, from which, in TABLE V., we 
 shall find the new coefficient to be "632 ; hence 17-944 x "632 
 = 1 1-341 cubic feet per second. If the coefficient -6225 were 
 used, we should find the new coefficient equals "638, and the 
 discharge 11 -468 cubic feet. Or thus: The ratio of the 
 
 head at the upper edge to the depth of the orifice is = -75, 
 
 and from TABLE IV. we find (1-75) (-75) J = 1-6655. 
 Assuming the coefficient to be -644, we find from TABLE VI. 
 the discharge per minute over a weir 12 inches deep and 
 
 1 foot long to be 208 ' 650 + 205 ' 119 = 206-884 cubic feet nearly;
 
 66 THE DISCHARGE OF WATER FROM 
 
 and as the length of the orifice is 2 feet, we have 
 2X206-884X1-6655 . 
 
 the correct theoretical discharge for the coefficient "644, and 
 less than the approximate result, 11*556 cubic feet above 
 found, by only a very small difference. The velocity of 
 approach in this example must be derived from the surface 
 inclination of the stream. 
 
 For notches or Poncelet weirs the approaching velocity is a 
 maximum at or near the surface. If the central velocity at the 
 surface facing the notch be 1, the mean velocity from side to 
 side will be '914. We may therefore assume the variation of 
 the central to the mean velocity to be from 1 to '91 4 ; and hence 
 the ratio of the mean velocity at the surface of the channel 
 to that facing the notch or weir cannot be less than '914 to 1, 
 and varies up to 1 to 1 ; the first ratio obtaining when the 
 notch or weir occupies a very small portion of the side or 
 width of the channel, and the other when the weir extends 
 for the whole width. Following the same mode of calculation 
 as at p. 63, Fig. 15, we shall find as follows : 
 
 Ratio of the Values of Values of 
 
 width of the notch j /, r 
 
 i t ,1 f Q G+ O Cj 
 
 to the width of -n* i c TV i 
 
 the channel. Fl g- 15 Fl g 15 
 
 -086 -914 
 
 1 -085 -915 
 
 2 -083 -917 
 
 3 -078 -922 
 
 4 -072 -928 
 
 5 -064 -936 
 
 6 -055 -945 
 
 7 -044 -956 
 
 8 -031 -969 
 
 9 -016 -984 
 
 1-0 '000 1-000 
 
 These values of b c are to be used as before in order to find 
 
 the value of , opposite to which in the tables, and under the 
 heading for weirs, will be found the new coefficient.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 67 
 
 EXAMPLE V. The length of a weir is 1 Ofeet ; the width of 
 the approaching channel is 20 feet ; the head, measured about 
 6 feet above the weir, is 9 inches ; and the depth of the channel 
 3 feet : what is the discharge ? Assuming the circumstances 
 of the overfall to be such that the coefficient of discharge for 
 heads, measured from still water in a deep weir basin or 
 reservoir, will be -617, we find from TABLE VI. the discharge 
 to be 128-642 x 10 = 1286-42 cubic feet per minute; but 
 from the smallness of the channel the water approaches the 
 
 weir with some velocity, and C = 1Q x 3 =: 8. We have also 
 
 the width of the channel equal to twice the width of the weir, 
 and hence (small table, p. 30) 8 x '936 = 7-488 for the new 
 
 c 
 value of -. From TABLE V. we now find the new coefficient 
 
 ' 622 + ' 624 = -623, and hence the discharge is 
 2 
 
 1286-42 X '623 __ 
 
 617 
 
 Or thus : As the theoretical discharge, TABLE VI., is 2084*96 
 cubic feet, we get 2084-96 x -623 = 1298-93, the same as 
 before. In this example, however, the mean velocity ap- 
 proaching the overfall bears to the mean velocity in the 
 channel a greater ratio than 1 : '936, as, though the head is 
 pretty large in proportion to the depth of the channel, the 
 
 ratio of the sections - = -g is small. We shall therefore be 
 more correct by finding the multiplier from the small table, 
 p. 63. By doing so the new value of - is 8 x -838 = 6-704. 
 
 From this and the coefficient "617 we shall find, as before 
 from TABLE V., the new coefficient to be -627 ; hence we get 
 2084-96 x -627 = 1307-27 cubic feet per minute for the 
 discharge. 
 
 The foregoing solution takes for granted that the velocity 
 of approach is subject to contraction before arriving at the 
 overfall or in passing through it; now, as this reduces the 
 mean velocity of approach from 1 to -784, when the coeffi- 
 
 F 2
 
 68 THE DISCHARGE OF WATER FROM 
 
 cient for heads in still water is '617, we have to multiply 
 the value of y = 6-704, last found, by '784, and we get 
 
 A 
 
 6-704 x -784 = 5'26 for the value - due to this correction, 
 
 from which we find the corresponding coefficient in TABLE V. 
 to be -629, and hence the corrected discharge is 2084-96 x '629 
 = 1311-44 cubic feet. 
 
 It is to be remarked that the value of - in TABLE V. is 
 
 A 
 
 simply an approximate value for the ratio of the velocity in 
 the channel facing the orifice to the velocity in the orifice 
 itself; and the corrections applied in the foregoing examples 
 were for the purpose of finding this ratio more correctly than 
 
 c 
 
 the simple expression gives it. The following auxiliary 
 
 table will enable us to find the correction, and thence the 
 new coefficient, with facility. Thus, if the channel be five 
 times the orifice, and a loss in the approaching velocity takes 
 
 AUXILIAKY TABLE TO TABLE V. 
 
 It 
 
 Il* 
 
 Multipliers for finding the new values of - in TABLE V., 
 
 If 
 
 -.Sic c 
 
 when the water approaches without contraction or loss of 
 velocity. 
 
 "o 
 
 It! 8 
 
 Coeffic'. 
 
 Coeffic'. 
 
 Coeffic'. Coeffic'. Coeffic'. 
 
 Coeffic'. 
 
 Coeffic'. 
 
 S 2 
 
 I-" 58 
 
 639 
 
 628 
 
 617 
 
 606 
 
 595 
 
 584 
 
 573 
 
 
 
 835 
 
 69 
 
 67 
 
 65 
 
 64 
 
 62 
 
 60 
 
 68 
 
 1 
 
 837 
 
 70 
 
 68 
 
 66 
 
 64 
 
 62 
 
 60 
 
 59 
 
 '2 
 
 842 
 
 70 
 
 68 
 
 66 
 
 64 
 
 62 
 
 61 
 
 59 
 
 3 
 
 850 
 
 71 
 
 69 
 
 67 
 
 65 
 
 63 
 
 61 
 
 59 
 
 4 
 
 861 
 
 72 
 
 69 
 
 68 
 
 66 
 
 64 
 
 62 
 
 60 
 
 5 
 
 876 
 
 73 
 
 71 
 
 69 
 
 67 
 
 65 
 
 63 
 
 61 
 
 6 
 
 894 
 
 74 
 
 72 
 
 70 
 
 68 
 
 66 
 
 64 
 
 62 
 
 7 
 
 916 
 
 76 
 
 74 
 
 72 
 
 70 
 
 68 
 
 66 
 
 64 
 
 8 
 
 941 
 
 78 
 
 76 
 
 74 
 
 72 
 
 70 
 
 68 
 
 66 
 
 9 
 
 969 
 
 81 
 
 78 
 
 76 
 
 74 
 
 72 
 
 70 
 
 68 
 
 1-0 
 
 1-000 
 
 831 
 
 807 
 
 784 
 
 762 
 
 740 
 
 719 
 
 699
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 place equal to that in a short cylindrical tube, we get 
 5 x -842 = 4-210 for the new value of -, opposite to which, 
 
 in TABLE V., will be found the coefficient sought. If the 
 coefficient for still water be '606, we shall find it to be -612 
 for orifices and '623 for weirs. But when the water ap- 
 proaches without loss of velocity, we find from the auxiliary 
 table '64 for the multiplier instead of '842, and consequently 
 
 the new value of becomes 5 x 
 
 64 = 3-2, from which we 
 
 shall find '617 to be the new coefficient for orifices and '636 
 for weirs. The auxiliary table is calculated by multiplying 
 
 the numbers in column 2 (see p. 63) by c d I > , which 
 
 will be found for the different values of c d in the table, viz. 
 639, -628, -617, -606, -595, -584, and -573, to be -831, -807, 
 784, -762, -740, -719, -699 respectively, as given in the top 
 and bottom lines of figures. 
 
 In weirs at right angles to channels with parallel sides, the 
 sectional area can never equal that of the channel unless it be 
 measured at or above the 
 point A, where the sinking 
 of the overfall commences ; 
 and unless also the bed 
 c D and surface A B have 
 the same inclination. In 
 all open channels, as mill 
 races, streams, rivers, the supply is derived from a surface 
 inclination A B, and this inclination regulates itself to the 
 discharging power of the overfall. When the overfall and 
 channel have the same width, and it is considerable, we have, 
 as shall appear hereafter, 91 \Jhs for the mean velocity in 
 
 the channel, where h is the depth in feet and s the rate of 
 
 2 
 
 inclination of the surface A B. We have also g </2yk for the 
 
 theoretical velocity of discharge at the overfall, of equal depth 
 with the channel, and, when both velocities are equal,
 
 70 THE DISCHARGE OF WATER FROM 
 
 2 / ^^ r "=5-35 x /A = 91 
 
 from which we find 
 
 the inclination of B A when the supply is equal to the theo- 
 retical discharge at the overfall. If the coefficient at the 
 overfall were '628, or, which is nearly the same thing, if a 
 large and deep weir basin intervene between the weir and 
 
 channel, Fig. 19, the velocity of approach would be destroyed, 
 and we should have 
 
 5-35 x -628 v/Tss 3-36 v/T= 91 v/^T; 
 
 and thence the inclination of A B 
 
 _ J_ 
 * "" 734 
 
 very nearly. When we come to discuss the surface inclination 
 of rivers, we shall see that the conditions here assumed and 
 the resulting surface inclinations would involve a considerable 
 loss of head. If the quantity discharged under both circum- 
 stances be the same, and h be the depth in the first case, 
 Fig. 18, we shall then have the head in the latter case, 
 
 Fig. 19, equal (3^5) h = 1'36 h very nearly, from which 
 
 and the surface inclination the extent of the backwater may 
 be found with sufficient accuracy. When, in Fig. 19, the 
 
 inclination of A B exceeds -L, the head at a must exceed the 
 
 depth of the river above A. We must refer to pages further 
 on for some remarks on the backwater curve.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 71 
 
 SECTION V. 
 
 SUBMERGED ORIFICES AND WEIRS. CONTRACTED 
 RIVER CHANNELS. 
 
 The available pressure 
 at any point in the depth 
 of the orifice A, Fig. 20, 
 is equal to the difference 
 of the pressures on each 
 side. This difference is 
 equal to the pressure due 
 to the height h, between the water surfaces on each side 
 of the orifice ; in this case, the velocity is 
 
 (47.) v 
 
 and the discharge 
 (48.) D 
 
 in which, as before, I is the length, and d the depth of the 
 rectangular orifice A. 
 
 When the orifice is 
 partly submerged, as in 
 Fig. 21, we may put 
 h b h = d. 2 for the sub- 
 merged depth, and h 
 h t = d if the remaining 
 portion of the depth; 
 whence d l + d. 2 =. d is the entire depth. The discharge 
 through the submerged depth d z is c d td 2 \/2gh, and the 
 discharge though the upper portion d t is 
 
 whence the whole discharge assuming the coefficient of 
 discharge c d is the same for the upper and lower depths is 
 
 (49.) 
 
 D =
 
 72 THE DISCHARGE OF WATER FROM 
 
 We may, however, equation (31), assume that 
 
 very nearly, and hence 
 (50.) 
 
 D = 
 
 As h t + = h -^- , this equation may be changed into 
 
 (51.) D = 
 
 In either of these forms the values of 
 
 c d 
 
 A - , and 
 
 can be had from TABLE II., and the value of the discharge 
 D thence easily found. 
 
 When the water approaches the orifice with a determinate 
 velocity, the height h & due to that velocity can be found 
 from TABLE II., and the discharge is then found by sub- 
 stituting h + h & and h t + h & for h and h t in the above 
 equations. 
 
 In the submerged weir, 
 Fig. 22, h becomes equal 
 to d lt and h t 0; the 
 discharge, equation (49), 
 then becomes 
 
 (52.) D = c d ld, 
 
 When the water approaches with a velocity due to the 
 height 7t a , then h becomes h + 7t a , /t t = 7* a , and equation (49) 
 becomes 
 
 (53.) D =
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 73 
 
 In the improvement of the navigation of rivers, it is some- 
 times necessary to construct weirs so as to raise the upper 
 waters by a given depth d^. The discharge D is in such 
 cases previously known, or easily determined, and from the 
 values of rf x and D we can easily determine, equation (52), 
 the value of 
 
 (54.) 
 
 or, by taking the velocity of approach into account, 
 
 (55.) ,= D <* + '-*! 
 
 This value of d 2 must be the depth of the top of the weir 
 below the original surface of the water, in order that this 
 surface should be raised by a given depth d^ When h & is 
 small compared with d. 2 , we may take 
 
 (d, + A.) = - x <i - _ in equation (55) 
 
 EXAMPLE VI. A river whose width at the surface is 
 70 feet, whose hydraulic mean depth is 4*4 feet, and whose 
 cross sectional area is '325 feet, has a surface inclination 
 of 1 foot per mile ; to what depth below, or height above 
 the surface must a weir at right angles to the channel 
 be raised, so that the depth of water immediately above 
 it shall be increased by 3 feet ? 
 
 When the hydraulic mean depth is 4'4 feet, and the fall 
 per mile 1 foot, we find from TABLE VIII. that the mean 
 velocity of the river is 29*98 or 30 inches very nearly per 
 second. The discharge is, therefore, 325 x 2 = 812-5 cubic 
 feet per second, or 48750 cubic feet per minute. Hence, 
 
 -7Q- = 696-4 cubic feet must pass over each foot in length 
 
 of the weir per minute. Assuming the coefficient C A = "628 
 in the first instance, we find from TABLE VI. the head 
 passing over a weir corresponding to this discharge to be 
 27-4 inches; but as the head is to be increased by 3 feet, or
 
 74 THE DISCHARGE OF WATER FROM 
 
 42 inches, it is clear that the weir must be perfect; that is, 
 have a clear overfall, and rise 42 27'4 = 14 - 6 inches over 
 the original water surface. In order that the weir may be 
 submerged, or imperfect, the head could not be increased 
 by more than 27'4 inches. Let us, therefore, assume in the 
 example, that the increase shall be only 18 instead of 
 42 inches; the weir then becomes submerged, and we have, 
 from equation (54), 
 
 d z = 696 ' 4 - 4- x 18" (as I = 1 foot). 
 
 628x/18"x 
 
 The value of the first part of this expression is found from 
 TABLE VI. or TABLE II. equal to 
 
 = 1-88 feet = 22-56 inches ; 
 
 12 x 1 x 370-341 370 ' 341 
 
 18 2 
 
 hence, 22-56 - = 10-56 inches is the values of d a ; that is, 
 3 
 
 the submerged weir must be built within 10*56 inches of the 
 surface to raise the head 18 inches above the former level. 
 If, however, the velocity of approach be taken into account, 
 
 we shall find this velocity equals = 2 feet per second 
 
 430 
 
 very nearly; and the height, or value of A a , due to this 
 velocity, taken from TABLE II., is -j= '75 inches nearly; 
 therefore, from equation (55), 
 
 696-4 _ _2_ x (18-75)1 - (-76)* 
 
 V/T8-75 
 
 = (from TABLE VI.) 
 
 = *= 1*84 feet = 22-08 inches; 
 
 393-75 
 
 * This is found from TABLE II. more readily.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 75 
 
 also, 
 
 2_ (18-75)* (-75)' _ 2^ 
 
 v/18-75 
 
 = T x 18-76 -r x 
 
 
 -65 
 4-33 
 
 = 12-5 - -1 = 12-4. 
 
 Hence d 2 = 22-08 12-4 = 9'68 inches, or about 1 inch 
 less than the value previously found from equation (54) . The 
 mean coefficient of discharge was here assumed to be *628. 
 Experiments on submerged weirs show that the value of 
 c d varies up to -8, but as this coefficient would reduce the 
 value of d 2 , or the depth of the top of the weir below the 
 surface, it is safer (where a given depth above a weir must 
 be obtained) to use the lesser and ordinary coefficients of 
 perfect weirs, with a clear overfall, for finding the crest levels 
 of submerged weirs, when it is necessary to construct them. 
 If the coefficient '8 were used in the previous calculation, we 
 should have found 
 
 = -628 x 22-08 _ m = ^ _ ^ = 4 . g3 .^ 
 
 'O 
 
 or not much more than half the previous value; but this 
 would only increase the whole height of the weir by 
 9-68 - 4- 93 = 4-75 inches. 
 
 As D= -gc A l\/2ff~{(d l + h$ A!} for a perfect weir 
 with a free overfall, it is clear that when D is greater than 
 
 the weir is imperfect or submerged. 
 
 CONTRACTED RIVER CHANNELS. 
 
 When the banks of a river, whose bed has a uniform 
 inclination, approach each other, and contract the width of 
 
 igr-23
 
 76 THE DISCHARGE OF WATER FROM 
 
 the channel in any way, as in Fig. 23, the water will rise 
 at the contracted portion, A, until the increased velocity of 
 discharge compensates for the reduced cross section. If we 
 put, as before, d t for the increase of depth at the contracted 
 width, and d 2 for the previous depth of the channel, we shall 
 find the water passing through the lower depth, d 2 , equal 
 to c A ld. 2 \/2gd i , in which I is the width of the contracted 
 channel at A, and the water overflowing through d L equal to 
 
 c^ldi \flgd^ and hence the whole discharge through A is 
 3 
 
 (56.) D = Cl 
 
 When our object is to find the width I of the contracted 
 channel, so that the depth of water in the upper stretch 
 shall be increased by a given depth d u we have 
 
 (57.) l = - 
 
 When the velocity of approach is large, or the height h a due 
 to it a large portion of d lt it must not be neglected. In this 
 case, as before, we find the discharge through the depth 
 d 2 equal to c d ld 2 ^/2g(d 1 + A a ); and the discharge through 
 the depth d l equal to -| c d l \/2g{(d, + AJ 3 - 7*J}j and 
 hence the whole discharge is 
 
 (58.) D = c A l ^/2j{d 2 (d, + /O* + 4 [ft + /O 1 - 4]}; 
 from which we find 
 
 (59.) I = 
 
 + ftj + [ (d, + hrf -/,}]} 
 
 If the projecting spur or jetty at A be itself submerged, 
 these formulae must be extended; the manner of doing so, 
 however, presents no difficulty, as it is only necessary to find 
 the discharges of the different sections according to the pre- 
 ceding formulas and add them together; but the resulting 
 formula so found is too complicated to be of any practical 
 value.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 77 
 
 HEADS ABOVE BRIDGES. 
 
 Equations (56), (57), (58), and (59), are applicable to the 
 cases of contraction in river channels caused by the con- 
 struction of bridge-piers and abutments, when the. width I is 
 put for the sum of the openings between them. The value 
 of the coefficient C A will depend on the peculiar circumstances 
 of each case ; we have seen that it rises as high as '8, in 
 some cases of submerged weirs, and for cases of contracted 
 channels it rises sometimes as high as '956, particularly when 
 they are analogous to those for the discharge through mouth- 
 pieces and short tubes. When the heads of the piers are 
 square to the channel, the coefficient may be taken at about 
 8; when the angles of the cut-waters or sterlings are obtuse, 
 it may be taken at about '9; and when curved and acute, 
 at -97. With this coefficient, a head of 1| inch will give 
 a velocity of very nearly 36 inches, or 3 feet per second ; 
 but as a certain amount of loss takes place from the velocity 
 of the tail-water being in general less than that through the 
 arch, from obstructions in the passage, and from square-headed 
 and very short piers, the coefficient may be so small in some 
 cases as '628, which would require a head of 4J inches to 
 obtain the same velocity. This head is to the former as 
 17 to 7. The selection of the proper coefficient suited to any 
 particular case is, therefore, a matter of the first importance 
 in determining the effect of obstructions in river channels: we 
 shall have to recur to this subject again, but it is necessary 
 to observe here, that the form of the approaches, the length 
 of the piers compared with the distance between them, or 
 span, and the length and form of the obstruction compared 
 with the width of the channel, must be duly considered 
 before the coefficient suited to the particular case can be 
 fixed upon. Indeed, the coefficients will always approxi- 
 mate towards those, given in the next section, for mouth-pieces, 
 shoots, and short tubes similarly circumstanced. For some 
 further remarks on contracted channels, see SECTION X.
 
 78 
 
 THE DISCHARGE OF WATER FROM 
 
 SECTION VI. 
 
 SHORT TUBES, MOUTH-PIECES, AND APPROACHES. COEFFI- 
 CIENTS OF DISCHARGE FOR SIMPLE AND COMPOUND SHORT 
 TUBES. SHOOTS. 
 
 The only orifices we have heretofore referred to were those 
 in thin plates or planks, with a few incidental exceptions. 
 It has been shown, page 20, Fig. 4, that a rounding off', next 
 the water, of the mouth-piece increases the coefficient; and 
 when the curving assumes the form of the vena-contracta, 
 the coefficient increases to '974, or nearly unity. The dis- 
 charge from a short cylindrical tube A, Fig. 24, whose length 
 
 is from one and a half to three times the diameter, is found 
 to be very nearly an arithmetical mean between the theoretical 
 discharge and the discharge through a circular orifice in a 
 thin plate of the same diameter as the tube, or '814 nearly. 
 If, however, the inner arris be rounded or chamfered off 
 in any way, the coefficient will increase until, in the tube B, 
 Fig. 24, with a properly-rounded junction, the coefficient be- 
 comes unity very nearly. In the conical short tubes c and D 
 the coefficients are found to vary according to some function 
 of the converging or diverging angles o, o, and according as we 
 take the lesser or greater diameter to calculate from. When 
 the length of the tube exceeds three times the diameter, the 
 friction of the water against the sides should be taken into 
 account. As for orifices in thin plates, so also for short tubes, 
 the coefficients are found to vary according to the depth of
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 79 
 
 the centre below the surface of the water, and to increase as 
 the depths and diameter of the tube decrease. Poleni was 
 the first who remarked that the discharge through a short 
 tube was greater than that through a simple orifice of the 
 same diameter, in the proportion of 133 to 100, or as '617 to 
 821. 
 
 CYLINDRICAL SHORT TUBES, A, FIG. 24. 
 
 The experiments of Bossut, as reduced by Prony, give the 
 following coefficients, at the corresponding depths, for a 
 cylindrical tube A, Fig. 24, 1 inch in diameter and 2 inches 
 
 COEFFICIENTS FOR SHORT TUBES, FROM BOSSUT. 
 
 Heads, 
 in feet. 
 
 Coefficients. 
 
 Heads, 
 in feet. 
 
 Coefficients. 
 
 Heads, 
 in feet. 
 
 Coefficients. 
 
 1 
 
 818 
 
 6 
 
 806 
 
 11 
 
 805 
 
 2 
 
 807 
 
 7 
 
 806 
 
 12 
 
 804 
 
 3 
 
 807 
 
 8 
 
 805 
 
 13 
 
 804 
 
 4 
 
 807 
 
 9 
 
 805 
 
 14 
 
 804 
 
 5 
 
 806 
 
 10 
 
 805 
 
 15 
 
 803 
 
 long. The depths are given in Paris feet in the original, but 
 the coefficients remain the same practically for depths in 
 English feet. 
 
 Venturi's experiments give a coefficient *823 for a short 
 tube A, 1 inch in diameter and 4^ inches long, at a depth 
 of 2 feet 8 inches, the coefficient through an orifice in a thin 
 plate of the same diameter and at the same depth being -622. 
 We have calculated these coefficients from the original ex- 
 periments. The measures in those were Paris feet and inches, 
 from which the calculations were directly made; and as the 
 difference in the coefficient for small changes of depth or 
 dimensions is immaterial or vanishes, as may be seen by the 
 foregoing small table, and as 1 Paris inch or foot is equal to 
 1*0658 English inches or feet, the former measures exceed 
 the latter by only about iV tn - We may therefore assume 
 that the coefficient for any orifice, at any depth, is the same,
 
 80 
 
 THE DISCHARGE OF WATER FROM 
 
 whether the dimensions be in Paris or English feet or 
 inches. This remark will be found generally useful in the 
 consideration of the older continental experiments, and will 
 prevent unnecessary reductions from one standard to another 
 where the coefficients only have to be considered. 
 
 The mean value derived from the experiments of Miche- 
 lotti, at depths from 3 to 20 feet, and with short tubes A 
 from inch to 3 inches in width, is c d == '814. Buff's 
 experiments* give the following results for a tube T 3 of an 
 
 BUFF'S COEFFICIENTS FOR SMALL SHORT TUBES. 
 
 Head 
 in inches. 
 
 Coefficient. ^^ 
 
 Coefficient. 
 
 Head 
 in inches. 
 
 Coefficient. 
 
 H 
 
 855 
 
 6 
 
 840 
 
 23 
 
 829 
 
 2| 
 
 861 
 
 14 
 
 840 
 
 32 
 
 826 
 
 inch wide and T 5 7 of an inch long, nearly. The increase for 
 smaller tubes and for lesser depths appears by comparing 
 these results with the foregoing, and from the results in 
 themselves, generally. Weisbach's experiments give a mean 
 value for c d = '815, and for depths of from 9 to 24 inches 
 the coefficients -843, -832, '821, -810 respectively, for tubes 
 
 To' To' To' an ^ To ^ an * nc ^ Wl de, ^ ne l en gth of each tube 
 being three times the diameter. D'Aubuisson and Castel's 
 experiments with a tube '61 inch diameter and 1*57 inch 
 long, give *829 for the coefficient at a depth of 10 feet. 
 
 We have calculated the coefficients in the following two 
 short tables, from Rennie's experiments with glass orifices 
 and tubes, Table 7, p. 435, Philosophical Transactions for 
 1831. The form of the orifices, or length of the shorter 
 tubes is not stated, but it is probable, from the results, that 
 the arrises of the ends were in some way rounded off; it is 
 stated they were " enlarged." Indeed, the discharges from 
 the short tube, or orifice of ^ inch diameter exceed the theo- 
 
 * Annalen der Physik und Chemie von Poggendorff, 1839. 
 p. 243. 
 
 Band 46,
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 81 
 
 retical ones in the proportion of 1-261 to 1, and 1'346 to 1. 
 These results could not have been derived from a simple 
 cylindrical tube, but might have arisen from the arrises being 
 more or less rounded at both ends, and the orifice partaking 
 of the nature of a compound tube, which may be constructed, 
 as we shall hereafter shew, so as to increase the theoretical 
 discharge from 1 up to T553. The resulting coefficients 
 
 COEFFICIENTS FOE SHORT TUBES, THE ENDS ENLARGED. 
 
 Head 
 in feet. 
 
 iinch 
 diameter. 
 
 J Inch 
 diameter. 
 
 3 inch 
 diamete . 
 
 1 inch 
 ameter. 
 
 1 
 
 1-231 
 
 831 
 
 766 
 
 912 
 
 2 
 
 1-261 
 
 839 
 
 820 
 
 920 
 
 3 
 
 1-346 
 
 838 
 
 821 
 
 880 
 
 4 
 
 1-261 
 
 831 
 
 829 
 
 991 
 
 for the |- and f inch tubes, approach very closely to those 
 obtained by other experimenters, but those for the inch tube 
 are too high, unless the arris at the ends was also rounded. 
 The coefficients derived from the experiments with a cylin- 
 drical glass tube 1 foot long, as here given, are very vari- 
 able; like the others they are, however, valuable, as ex- 
 
 COEFFICIENTS DERIVED FROM EXPERIMENTS WITH A GLASS TUBE ONE FOOT LONG. 
 
 Heads 
 in feet. 
 
 i inch 
 diameter. 
 
 I inch 
 diameter. 
 
 3 inch 
 diameter. 
 
 1 inch 
 diameter. 
 
 1 
 
 892 
 
 703 
 
 691 
 
 760 
 
 2 
 
 914 
 
 734 
 
 718 
 
 749 
 
 3 
 
 931 
 
 723 
 
 709 
 
 777 
 
 4 
 
 914 
 
 725 
 
 677 
 
 815 
 
 hibiting the uncertainty attending " experiments of this 
 nature," and the necessity for minutely observing and re- 
 cording every circumstance which tends to alter and modify 
 them. Indeed, for small tubes, a very slight difference in 
 the measurement of the diameter must alter the result a good 
 deal, particularly when it is recollected that measurements 
 are seldom taken more closely than the sixteenth of an inch, 
 
 a
 
 82 THE DISCHARGE OF WATER FROM 
 
 unless in special cases. As the author, however, states, 
 p. 433 of the work referred to, that the " diameters of the 
 tubes at their extremities were carefully enlarged to prevent 
 wire edges from diminishing the sections;" this circumstance 
 alone must have modified the discharges, and would account 
 for most of the differences. 
 
 The coefficient for rectangular short tubes differs in no 
 way materially from those given for cylindrical ones, and 
 may be taken on an average at *814 or '815. 
 
 SHORT TUBES WITH A ROUNDED MOUTH-PIECE, B, FIG. 24. 
 
 When the junction of a short tube with a vessel takes the 
 form of the contracted vein, Figs. 3 and 4, page 20, the mean 
 value of the coefficient C A = '956, and the actual discharge is 
 found to be from 93 to 98 per cent, of the theoretical dis- 
 charge. Wiesbach, for a tube 1 inch long and -^ inch 
 diameter, rounded at the junction, found at 1 foot deep 
 C A = -958, at 5 feet deep c d = '969, and at 10 feet deep 
 c d = '975. These experiments shew an increase of co- 
 efficient in this particular case for an increase of depth. Any 
 other form of junction than that of the contracted vein, will 
 reduce the discharge, and the coefficients will vary from 
 807 to '974, according to the change from the cylindrical 
 to the properly curved form, beyond which the coefficient 
 again decreases from '974 to '807. The coefficients derived 
 from Venturi's experiments will be given hereafter. 
 
 SHORT CONICAL CONVERGENT TUBES, C, FIG. 24. 
 
 The experiments of D'Aubuisson and Castel lead to the 
 following coefficients of discharge and velocity* from a coni- 
 cally convergent tube c at a depth of 10 feet. We have 
 interpolated the original angles and coefficients so as to render 
 the table more convenient to refer to for practical purposes 
 than the original. The diameter of the tube at the smaller 
 or discharging orifice in the experiments was '61 inches, and 
 
 * Traite d'Hydraulique, Paris, p. 60.
 
 ORIFICS, WEIRS, PIPES, AND RIVERS. 83 
 
 COEFFICIENTS FOR CONICAL CONVERGES! TDBK8. 
 
 Converging 
 angle o. 
 
 Coefficient 
 of discharge. 
 
 Coefficient 
 of velocity. 
 
 Converging 
 angle o. 
 
 Coefficient 
 of discharge. 
 
 Coefficient 
 of velocity. 
 
 1 
 
 858 
 
 858 
 
 14 
 
 943 
 
 964 
 
 2 
 
 873 
 
 873 
 
 16 
 
 937 
 
 970 
 
 3 
 
 908 
 
 908 
 
 18 
 
 931 . 
 
 971 
 
 4 
 
 910 
 
 909 
 
 20 
 
 922 
 
 971 
 
 5 
 
 920 
 
 916 
 
 22 
 
 917 
 
 973 
 
 6 
 
 925 
 
 923 
 
 26 
 
 904 
 
 975 
 
 8 
 
 931 
 
 933 
 
 30 
 
 895 
 
 976 
 
 10 
 
 937 
 
 950 
 
 40 
 
 869 
 
 980 
 
 12 
 
 942 
 
 955 
 
 50 
 
 844 
 
 985 
 
 the length of the axis 1'57 inch; that is, the length was 
 2'6 times the smaller diameter of the tube. The coefficient 
 was '829 for the cylindrical. tube, i. e. when the angle at o was 
 nothing. The angle of convergence o determines, from the 
 proportions, the length of the inner and longer diameter of 
 the tube. The coefficients of discharge increase up to '943 
 for an angle of 13^ or 14 degrees, after which they again 
 decrease ; but the coefficients of velocity increase as the 
 angle of convergence o increases from '829, when the angle 
 is zero up to '985 for an angle of 50 degrees. 
 
 When D is the discharge and A the area of the section, we 
 have, as before shown, D c d A \/2gh; but as, in conically 
 convergent or divergent tubes the inner and outer areas (or, 
 as they may be called, the receiving and discharging sections) 
 vary, it is clear that (the discharge being the same, and also 
 the theoretical velocity \/2gh) the coefficient c d must vary 
 inversely with the sectional area A, and that c d A must be 
 constant. For the coefficients tabulated, the sectional area 
 used is that at the smaller end of a convergent tube c. 
 
 For a short tube c, whose length is -92 inch, lesser 
 diameter 1'21 inch, and greater diameter 1-5 inch, we have 
 found, from Venturi's experiments, that c d rz '607 if the 
 larger diameter be used in the calculation, and c d = '934 
 
 G 2
 
 84 THE DISCHARGE OF WATER FROM 
 
 when the lesser diameter is made use of, the discharge taking 
 place under a pressure of 2 feet 85- inches. 
 
 The earlier experiments of Poleni, when reduced, furnish 
 us with the following coefficients : A tube 7 '67 inches long, 
 2'167 inches diameter at each end, c d '854; the like tube 
 with the inner or receiving orifice increased to 2| inches, 
 c d = -903 ; increased to 3'5 inches, C A =: '898 ; increased to 
 5 inches, c d = '888 ; and increased to 9'83 inches, C A = -864. 
 The depth or head was 21 P 33 inches, the discharging orifice 
 2*167 inches diameter, and the length 7 '67 inches, in each 
 case. 
 
 In the conically divergent tube D, Fig. 24, the coefficient 
 of discharge is larger than for the same tube c, convergent, 
 when the water fills both tubes, and the smaller areas, or 
 those at the same distances from the centres o o, are made 
 use of in the calculations. A tube whose angle of con- 
 vergence o is 5 nearly, with a -head of from 1 to 10 feet, 
 whose axial length is 3^ inches, smaller diameter 1 inch, and 
 larger diameter 1'3 inch, gives, when placed as at c, '921 
 for the coefficient; but when placed as at D, the coefficient 
 increases to '948. In the first case the smaller area, used in 
 both calculations, being the receiving, and in the other the 
 discharging, orifice. The coefficient of velocity is, however, 
 larger for the tube c than for the tube D, and the discharging 
 jet of water has a greater amplitude in falling. The effects 
 of conically diverging tubes will, however, be better perceived 
 from the experiments on compound short tubes. 
 
 COMPOUND ADJUTAGES AND SHORT TUBES. 
 
 If the tube A, Fig. 24, be pierced all round with small holes 
 at the distance of about half its diameter from the reservoir, 
 the discharge will be immediately reduced in the proportion 
 of '814 to '617. Venturi found the reduction for a tube 
 1 inch diameter and 4 inches long, at a depth of 2 feet 
 10 inches, as 41 to 31, or as -823 to -622. As long as one 
 hole remained open, the discharge continued at the same 
 reduced rate ; but when the last hole was stopped, the
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 85 
 
 discharge again increased to the original quantity. If a 
 small hole be pierced in a tube 4 diameters long, at the 
 distance of 1 or 2 diameters at farthest from the junction, 
 the discharge will remain unaffected. This shows that the 
 contraction in the cylindrical tube extends only a short 
 distance from the junction, probably 1^- or 1 diameter, in- 
 cluding the whole curvature of the contraction. 
 
 The contraction at the entrance into a tube from a reservoir 
 accounts for the coefficients for a short tube A, Fig. 24, and 
 the short tubes, diagrams 1 and 2, Fig. 25, being each the 
 
 same decimal nearly, when OR : or :: I : '8, or when or is 
 not less than o R x -79, and is at the distance of nearly 
 
 from OR. The form of the junction oorR remaining as we 
 have described it, the following coefficients will enable us to 
 judge of the discharging powers of different short mouth- 
 pieces. They have been deduced and calculated principally 
 from Venturi's experiments*. 
 
 These coefficients show very clearly that any calculations 
 from the mere head and size of the orifice, without taking 
 into consideration the form of the discharging tube and its 
 connection with the reservoir, are very uncertain; and that 
 the discharge can only be correctly obtained when all the 
 circumstances of the case, including the form of the dis- 
 charging orifice and its approaches, are duly considered. 
 
 * See Nicholson's translation of Venturi's Experimental Inquiries, 
 published in the "Tracts on Hydraulics," London, 1836. The coeffi- 
 cients in the Table, next page, have been all calculated for the first 
 time by us.
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE OF COEFFICIENTS FOR MOUTH-PIEOKS AND SHORT TUBES. 
 
 Description of orifice, mouth-piece, or short tube. 
 
 Coefficients 
 for the 
 diameter 
 
 O R. 
 
 Coefficients 
 for the 
 diameter 
 or. 
 
 1. An orifice 1 J inch diameter in a thin plate 
 
 622 
 
 974 
 
 2. A cylindrical tube 1$ inch diameter and 4 inches 
 long, A, Fig. 24 
 
 823 
 
 823 
 
 ' 3. A cylindrical tube, B, Fig. 24, having the junction 
 rounded, as in Fig. 4, page 20 
 4. A short conical convergent mouth-piece, c, Fig. 24, 
 
 611 
 607 
 
 956 
 934 
 
 5. The like tube divergent, with the smaller diameter 
 at the junction with the reservoir ; length Scinches, 
 lesser diameter 1 inch, and greater diameter 
 1-3 inch 
 
 561 
 
 948 
 
 6. The tube, OOUVTR, diagram 2, Fig. 25, when 
 OR = 1 inch, or = 1-21 inch, uv = 1-21 inch, 
 and o u = r v = 2 inches, the cylindrical portion 
 being shown by dotted lines 
 
 600 
 
 923 
 
 7. The same tube when o u = 11 inches 
 
 567 
 531 
 
 873 
 817 
 
 8. The tube, oossitrR, diagram 2, Fig. 25, in 
 which OR = S<=.ST = I inch, from o to s 
 If inch, and ss = 3 inches, gives the same coeffi- 
 cient as the cylindrical tube, result No. 2 (see 
 No. 18), viz 
 
 823 
 
 1-266 
 
 9. The tube, diagram 1, Fig. 25, OR = 1J inch 
 10. The same tube, having the spaces oso and rtR 
 between the mouth-piece oorfi and the cylindrical 
 tube OST R open to the influx of the water 
 11. The double conical tube, oosirR, diagram 3, 
 Fig. 25, when OR = ST = !| inch, or = l'21 
 inch, oo = '92 inches, and o s = 4 - l inch 
 12. The like tube when, as in diagram 4, Fig. 25, 
 oorR = oBtr, and oos = TS4 inch 
 13. The like tube when ST=1'46 inch, and os 
 2-17 inches 
 
 804 
 
 785 
 
 928 
 823 
 823 
 
 1-237 
 1-209 
 
 1-428 
 1-266 
 1-266 
 
 14. The like tube when s T = 3 inches, and o a = 9 J 
 
 911 
 
 1-400 
 
 15. The like tube when os = 6J inches, and ST 
 enlarged to 1'92 inch 
 
 1-020 
 
 1-569 
 
 16. The like tube when ST = 2 inches, and os 
 12i inches 
 
 1-215 
 
 1-855 
 
 17. A tube, diagram 5, Fig. 25, when o* = r< = 3 
 inches, or = st = 1-21 inch, and the tube oSTr 
 the same as described in No. 1 1, viz. s T = 1 J inch, 
 
 895 
 
 1-377 
 
 18. The tube, diagram 2, Fig. 25, when ST is enlarged 
 to 1'97 inch, and *s to 7 inches, the other di- 
 mensions remaining as in No. 8 
 19. When the junction of osrt with * s T t, diagram 2, 
 Fig. 25, is improved, the other parts remaining as 
 described in No 8 
 
 945 
 850 
 
 1-454 
 1-309 
 
 
 847 
 
 1-303 
 
 
 

 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 87 
 
 When a tube similar to diagram 5, Fig. 25, has the 
 junction oorR rounded, as in Fig. 4, page 20, the outer 
 extremity s t s T, such that s t or, s s 9 s t, and the 
 diameter ST = 1-8 times the diameter s t, with a short 
 central cylindrical piece orst between, the coefficient of 
 discharge corresponding to the diameter or rs will increase 
 
 1'493 
 to 1'493 or 1'555; that is, the discharge is -^^ = 2'4, or 
 
 7g22~ = 2*5 times as much as through an orifice (whose 
 
 diameter is or) in a thin plate, and -^^ =2 T9 times as much 
 as through a short cylindrical tube A, Fig. 24, whose diameter 
 is also or. Venturi was of opinion that this discharge con- 
 tinued even when the central cylindrical portion orst was of 
 considerable length ; but this was a mistake, as the maximum 
 discharge is obtained when it is reduced so that corn and 
 sstT shall join, as in diagram 3, Fig. 25. We see from 
 
 No. 15 of the foregoing coefficients that ^2" = ^'52 and 
 
 822 ~~ ^ ^ are > P erna P s > nearer to the maximum results 
 obtainable in comparing the discharge from a compound tube 
 oosTrR, diagram 3, Fig. 25, with 
 those through an orifice in a thin 
 plate, and through a short cylin- 
 drical tube. When the form of the 
 tube becomes curvilineal through- 
 out, as in Fig. 26, ST = 1'Sor and os ^ 9 or, the coeffi- 
 cient suited to the diameter or will be 1*57 nearly, and the 
 
 discharge will be ^2 = 2'52 times as much as through an 
 orifice or in a thin plate. 
 
 The whole of the preceding coefficients have been deter- 
 mined from circumstances in which the coefficient for an 
 orifice in a thin plate was '622, and for a short cylindrical 
 tube '822 or '823. When the circumstances of head and 
 approaches in the reservoir are such as to increase or decrease 
 those primary coefficients, the other coefficients for compound 
 adjutages will have to be increased or decreased propor- 
 tionately.
 
 88 THE DISCHARGE OF WATER FROM 
 
 On examining the foregoing results, it appears sufficiently 
 clear that the utmost effect produced by the formation of the 
 compound mouth-piece o o a T r R, with the exception of 
 No. 16, is simply a restoration of the loss effected by con- 
 traction in passing through the orifice o R in a thin plate, 
 and that the coefficient 2'5 applied to the contracted 
 section at or is simply equal to the theoretical discharge, 
 or the coefficient unity, applied to the primary orifice o R ; 
 for as, orifice OR : orifice or :: 1 : '64, very nearly, when 
 o o r R takes the form of the vena-contracta, and the coeffi- 
 cient of discharge for an orifice or in a. thin plate is '622, we 
 get the theoretical discharge through the orifice o R to the 
 actual discharge through an orifice o r, so is 1 to '622 x '64, 
 so is 1 : -39808 :: 1 : -4 very nearly; and as '4 x 2-5 = 1, it 
 is clear that the form of the tube o o s T r R, when it produces 
 the foregoing effect, simply restores the loss caused by con- 
 traction in the vena-contracta. Venturi's sixteenth experi- 
 ment, from which we have derived the coefficients in No. 1 6, 
 gives the coefficient 1*215 for the orifice OR. This indicates 
 that a greater discharge than the theoretical, through the 
 receiving orifice, may be obtained. It is, however, observable 
 that Venturi, in his seventh proposition, does not rely on this 
 result, and Eytelwein's experiments do not give a larger 
 coefficient than 2'5 applied to the contracted orifice or, 
 which, we have above shown, is equal to the theoretical 
 discharge through o R. 
 
 SHOOTS. 
 
 When the sides and under edge of an orifice or notch increase in 
 thickness, so as to be converted into a shoot or small channel, open at 
 the top, the coefficients reduce very considerably, and to an extent 
 beyond what the increased resistance from friction, particularly for small 
 depths, indicates. Poncelet and Lesbros * found for orifices 8" X 8", that 
 the addition of a horizontal shoot 21 inches long reduced the coefficient 
 from -604 to -601, with a head of about 4 feet ; but for a head of 4 inches 
 the coefficient fell from -572 to '483. For notches 8" wide, with the 
 addition of an horizontal shoot 9' 10" long, the coefficient fell from -582 
 to -479 for a head of 8* ; and from -622 to '340 for a head of 1". Castel 
 also found for a notch 8" wide, with the addition of a shoot 8" long, 
 inclined 4 18', the mean coefficient for heads from 2" to 4J", to be 
 527 nearly. 
 
 * Traite d'llydraulique, pp. 46 et 94.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 89 
 
 SECTION VII. 
 
 LATERAL CONTACT OF THE WATER AND TUBE. ATMOSPHERIC 
 PRESSURE. HEAD MEASURED TO THE DISCHARGING ORI- 
 FICE. COEFFICIENT OF RESISTANCE. FORMULA FOR THE 
 
 DISCHARGE FROM A SHORT TUBE. DIAPHRAGMS. OB- 
 LIQUE JUNCTIONS. FORMULA FOR THE TIME OF THE 
 SURFACE SINKING A GIVEN DEPTH. 
 
 The contracted vein or is 
 about *8 times the diameter 
 OR; but it is found, notwith- 
 standing, that water, in pass- 
 ing through a short tube of 
 not less than 1 \ diameter in 
 length, fills the whole of the 
 discharging orifice s T. This 
 is partly effected by the outflowing column of water carrying 
 forward and exhausting a portion of the air between it and 
 the tube, and by the external air then pressing on the column 
 so as to enlarge its diameter and fill the whole tube. When 
 once the water approaches closely to the tube, or is caused to 
 approach, it is attracted and adheres with some force to it. 
 The water between the tube and the vena-contracta is, how- 
 ever, rather in a state of eddy than of forward motion, as 
 appears from the experiments, with the tube, diagram 2, 
 Fig. 25, giving the same discharge as the simple cylindrical 
 tube. If the entrance be even contracted by a diaphragm, as 
 at o R, Fig. 27, the water will generally fill the tube if it be 
 only sufficiently long. Short cylindrical tubes do not fill 
 when the discharge takes place in an exhausted receiver ; but 
 even diverging tubes, D, Fig. 24, will be filled, in the air, when 
 the angle of divergence o does not exceed 7 or 8 degrees, and 
 the length be not very great nor very short. 
 
 When a tube is fitted, with any inclination, to ihe bottom 
 or side of a vessel, it is found that the discharge is that due 
 to the head measured from the surface of the water to the
 
 90 THE DISCHARGE OP WATER FROM 
 
 lower or discharging extremity of the tube. It must, how- 
 ever, be sufficiently long, and not too long, to get filled 
 throughout. Guiglielmini was the first who referred this 
 effect to atmospheric pressure, but the first simple explanation 
 is that given by Dr. Mathew Young, in the Transactions oj' 
 the Royal Irish Academy, vol. 7, p. 53. Venturi also, in his 
 fourth proposition, gives a demonstration. 
 
 The values of the coefficients for short cylindrical tubes, 
 which we have given above, have been derived from experi- 
 ment. Coefficients which agree pretty closely with them, and 
 which are derived from the coefficients for the discharge 
 through an orifice in a thin plate, may, however, be calculated 
 as follows : Let c be the area of the approaching section, 
 Fig. 27, A the area of the discharging short tube, and a the 
 area of the orifice o R which admits the water from the vessel 
 into the tube : also put, as before, h for the head measured 
 from the surface of the water to the centre of the tube, and 
 diaphragm OR; v for the velocity of discharge at s T ; v & for 
 the velocity of approach in the section c towards the 
 diaphragm OR; and c c for the coefficient of contraction in 
 passing from o R to or ; then we have c x r a = A.V, the con- 
 tracted section or c c a, and consequently the velocity at 
 
 the contracted section = r= c v& . Now a head equal to 
 a c c ac c 
 
 is necessary to change the velocity v a into v if there be no loss 
 in the passage ; but as the water at the contracted section o r, 
 
 moving with a velocity , strikes against the water be- 
 tween it and T 8, moving, from the nature of the case, with a 
 slower velocity, a certain loss of effect takes place from the 
 impact. If this be sudden (as it is generally supposed) it is 
 then shown by writers on mechanics that a loss of head equal 
 
 to that due to the difference of the velocities r, before
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 91 
 
 and after the impact must take place. This loss of head is 
 therefore equal to 
 
 (-- 
 
 \ac c 
 
 whence we must have the whole head, 
 
 (60.) 
 
 h 
 
 from which we find the velocity 
 (61.) v-^/~g 
 
 Now, as v =z ^/2ffh would be the velocity of discharge 
 were there no resistances, or loss sustained, it is evident that 
 
 i Y 
 
 is the coefficient of velocity. 
 
 When the diameter of the diaphragm OR becomes equal to 
 the diameter ST of the tube, A = a, and as the coefficient 
 of velocity becomes equal to the coefficient of discharge 
 when there is no contraction, we get in this case the co- 
 efficient of discharge 
 
 <> 
 
 and when the approaching section c is very large compared 
 with the area A, 
 
 (63.) 
 
 If c c = '64, we shall find from the last equation c d = *872 ; 
 if c c = '601, Cd = -833; if c c = -617, c d = -847; and if
 
 92 THE DISCHARGE OF WATER FROM 
 
 e c = '621, c d = '856. These results are in excess of those 
 derived from experiment with cylindrical short tubes, per- 
 fectly square at the ends and of uniform bore. As some 
 loss, however, takes place in the eddy between or and the 
 tube, and from the friction at the sides, not taken into 
 account in the above calculation, it will account for the 
 differences of not more than from 4 to 6 per cent, between 
 the calculation and experiment. If c c be assumed for cal- 
 culation equal '590, then C A =. *821 ; and as this result agrees 
 very closely with the experimental one, c c should be taken 
 of this value in using the above formulae for practical 
 purposes. 
 
 COEFFICIENT OF RESISTANCE. LOSS OF MECHANICAL POWER 
 IN THE PASSAGE OF WATER THROUGH THIN PLATES AND 
 PRISMATIC TUBES. 
 
 The coefficients of contraction, velocity, and discharge have 
 been already defined. The coefficient of resistance is the 
 ratio of the head due to the resistance, to the theoretical 
 head due to the discharging or final velocity. If v be this 
 
 velocity, the theoretical head due to it is ; and if c r be 
 the coefficient of resistance, then the head due to the resist- 
 
 ance itself is. from our definition, c r x ^. Now if c, be 
 
 20 
 the coefficient of velocity, the theoretical velocity of discharge 
 
 must be , and the head due to it is equal - - ; but 
 <?/ c*x2/ 
 
 as the theoretical head due to v is ^ , we have 
 
 20 
 
 <2g 
 
 for the height due to the resistance; and, therefore, from our 
 definition 
 
 (64.) c r = 1 - 1 ;
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 from which we find 
 
 (65.) c v = 
 
 93 
 
 These equations enable us to calculate the coefficient of re- 
 sistance from the coefficient of velocity, and vice versd. If 
 c v = 1 , C T = 0, as it should be. The following short Table, 
 calculated from equation (65), will be of use. 
 
 COEFFICIENTS OF VELOCITY AND RESISTANCE*. 
 
 Coefficient 
 of velocity. 
 
 Coefficient 
 of resistance. 
 
 Coefficient 
 of velocity. 
 
 Coefficient 
 of resistance. 
 
 Coefficient 
 of velocity. 
 
 Coefficient 
 of resistance. 
 
 990 
 
 020 
 
 910 
 
 208 
 
 830 
 
 452 
 
 970 
 
 063 
 
 890 
 
 263 
 
 820 
 
 488 
 
 950 
 
 109 
 
 870 
 
 320 
 
 814 
 
 508 
 
 930 
 
 156 
 
 850 
 
 383 
 
 810 
 
 525 
 
 The coefficient of velocity for an orifice in a thin plate, or 
 for a mouth-piece, Fig. 4, is '974; while that for a short 
 prismatic tube, A, Fig. 24, is '814 nearly. The coefficient of 
 resistance in the former case is '054, and in the latter "508; 
 there is, therefore, 9'4 times as great a loss of mechanical 
 power in the passage through short prismatic tubes, as 
 through orifices in thin plates or tubes with a rounded 
 junction, as in Fig. 4, the quantities of water discharged and 
 the discharging velocities being the same. 
 
 If the quantities discharged and the heads be the same in 
 both cases, then we shall have 
 
 x -814 3 
 
 that is, 
 
 
 x -974 
 v 2 
 
 - equal the head ; 
 
 663 x 2g -949 x 
 
 , or -949^ = -663 vl ; 
 
 whence we get v\ '698 vl and 1% 1'431 v\ for the relation 
 of the discharging velocities, v from an orifice, and v t from 
 a short tube. The height due to the resistance is, therefore, 
 
 ( _ 1 ) .^L for short prismatic tubes, and 
 V-814 2 / 2a 
 
 * See the Tables of resistances, discharge, and contraction, pp. 96 and 98.
 
 94 THE DISCHARGE OF WATER FROM 
 
 / 1 _ j\ 1-431 vj 
 
 for orifices in thin plates. These are to each other as '508 
 to '054 x 1-431, or as 5'08 to '773; that is to say, the loss 
 of mechanical power arising from the resistance in passing 
 through short tubes is 15^ times as great as when the 
 water passes through thin plates or mouth-pieces, as in 
 Fig. 4; and the discharging mechanical power in plates, is 
 to that in tubes as 1-431 to 1, or as 1 : '698, the heads and 
 quantities discharged being the same. 
 
 The whole loss of mechanical power in the passage is 
 5'4 per cent, for the plates, and about 51 per cent, for 
 short tubes. If the loss compared with the whole head be 
 
 sought, we get, when v is the discharging velocity, for 
 
 the theoretical velocity due to the head in short tubes, and 
 
 its square V _. =. v is as the whole head ; therefore, 
 *o 14 *OOo 
 
 the whole head is to the head due to the discharging velocity, 
 
 as to # 2 , or as 1 to '663, and as -508 is the coefficient 
 663 
 
 of resistance* for the discharging velocity, '508 x '663 
 = '337 is the coefficient of resistance due to the whole 
 head ; this is equal to a loss of 34 per cent, nearly, or about 
 one-third. In like manner, we find -974 2 x '054 = '0512 
 for the coefficient when the discharge takes place through 
 thin plates, or 5 per cent, of the whole head. 
 
 DIAPHRAGMS. 
 
 When a diaphragm, o R, Fig. 27, is placed at the entrance 
 of a short tube, we have shown, page 91, that a loss of 
 
 head equal ? takes place when v is the dis- 
 charging velocity, whence the coefficient of resistance is 
 * Table, p. 93.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 95 
 
 equal to ( 1 ) ,* according to our definition. The 
 \ac c I 
 
 coefficient of construction, c c , as we have before shown, 
 page 92, should be taken equal to '590 in the application of 
 formula (63) ; and, as it must also be taken equal to about 
 621 when the area of the tube A is very large compared 
 with the area a of the orifice o R in the diaphragm, we may 
 
 assume that when is equal to 
 
 ' 10' 10* 10' 10' 10' 10' 10' 10' 10 
 
 and 1 successively, the coefficient c c must be taken equal 
 to -621, -618, '615, -612, -609, -606, -603, -600, -597, -593 
 and -590, in the same order. As the approaching section c 
 may be considered exceedingly large, the value of the co- 
 efficient of discharge or velocity, as the tube ORST is sup- 
 posed full, in equation (61), becomes 
 
 | 1 1* 
 
 (66.) 
 
 and the coefficient of resistance 
 
 (67.) c r =(-!_.- l 
 
 \ac c 
 
 from which equations and the above values of e c , corre- 
 sponding to , we have calculated the following values of 
 
 A 
 
 the coefficients of discharge and resistance through the tube 
 ORST, Fig. 27. 
 
 * When the alteration in the velocity passing through a diaphragm 
 is sudden, we must reject the hypothesis of D'Aubuisson, " Traite d'Hy- 
 draulique," p. 238, and adopt that of Navier, taking the loss of head to 
 correspond to the square of the difference and not to the difference of 
 the squares of the velocities in and after passing the orifice. The 
 coefficient of contraction must, however, be varied to suit the ratio of 
 the channels, as in this and the following pages.
 
 96 THE DISCHARGE OF WATER FROM 
 
 COEFFICIENTS OF CONTRACTION, DISCHARGE, AND RESISTANCE FOR DIAPHRAGMS. 
 
 Ratio 
 
 i 
 
 1 
 
 I 
 
 Ratio 
 
 1 
 
 1 
 
 i 
 
 a 
 
 f ^ 
 
 3 " 
 
 d </ 
 
 a 
 
 i u 
 
 jg -O 
 
 tf3 - u 
 
 T 
 
 3 
 
 1 
 
 8 
 
 o 
 
 A 
 
 
 a " 
 
 I 
 
 o-o 
 
 621 
 
 000 
 
 infinite. 
 
 0-6 
 
 603 
 
 493 
 
 3-115 
 
 o-i 
 
 618 
 
 066 
 
 231- 
 
 0-7 
 
 600 
 
 587 
 
 1-907 
 
 0-2 
 
 615 
 
 139 
 
 50-8 
 
 0-8 
 
 597 
 
 675 
 
 1-198 
 
 0-3 
 
 612 
 
 219 
 
 19-8 
 
 0-9 
 
 .593 
 
 753 
 
 762 
 
 0-4 
 
 609 
 
 307 
 
 9-6 
 
 1-0 
 
 590 
 
 821 
 
 483 
 
 0-5 
 
 606 
 
 399 
 
 5-3 
 
 
 
 
 
 In this table c c is the coefficient of contraction, C A the co- 
 efficient of discharge, suited to the larger section of the 
 pipe A, at ST; and c r the coefficient of resistance. The 
 discharge is found from equation (61), as c is here very large 
 compared with A, to be 
 
 x _ f ' )' 
 
 (67A.) DSAv/20* , , / A ,\H 
 
 /A \ 2 
 
 The coefficient of resistance, c r) is here equal ( 1 ) , 
 
 \ac c 
 
 and the coefficient of discharge C A r= ..* 
 
 The tube must be so placed, that the water, after passing 
 the diaphragm, shall fill it; for instance, between two cis- 
 terns, when the height h must be measured between the 
 water surfaces, or when the tube is sufficiently long to be 
 filled ; in this case, however, the height must be determined 
 from the discharging velocity, as a portion of the head is 
 required to overcome the friction, which we shall have im- 
 mediately to refer to. 
 
 * For the loss sustained by contraction in the bore of a pipe by a 
 diaphragm, see further on.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 97 
 
 The table shows that the head due to the resistance is 
 5*3 times that due to the discharging velocity, when the 
 area of the diaphragm is half the area of the tube ; that is, 
 the whole head required is 6'3 times that due to the velocity, 
 and that the coefficient of discharge is reduced to '399. In 
 order to find the coefficients suited to the smaller area of the 
 orifice in the diaphragm OR, when it is to be used in calcula- 
 tions of the discharge, we have only to divide the numbers 
 
 corresponding to into those of C A , opposite to them in the 
 
 table. Thus, when = '8, we have the coefficient of dis- 
 
 675 
 charge suited to the area a, equal -^- = -844, and so of 
 
 other values of the ratio . The coefficients in the opposite 
 page are for the larger orifice A. 
 
 TUBES OBLIQUE AT THE JUNCTION. 
 
 Fig.28 
 
 When a tube is at- 
 tached obliquely, as in 
 Fig. 28, we have found 
 that if the number of 
 degrees in the angle TOS, 
 formed by the direction 
 of the tube os, with the 
 perpendicular OT, be re- 
 presented by <p, then '814 *0016p will give the coefficient of 
 discharge corresponding to the obliquely attached short tube in 
 the Figure. This formula is, however, empirical, but it is simple, 
 and agrees pretty closely with experimental results. As the 
 
 coefficient of resistance is equal _ 1, equation (64), we 
 
 have here c t =: 
 
 (814- -0016 ?)* "~ l ; from these e( l uations 
 we have calculated the following Table :
 
 THE DISCHARGE OF WATER FROM 
 
 COEFFICIENTS OF DISCHARGE AND RESISTANCE FOR OBLIQUE JUNCTIONS. 
 
 in degrees. 
 
 of discharge. 
 
 Coefficient 
 of resistance. 
 
 in degrees. 
 
 Coefficient 
 of discharge. 
 
 Coefficient 
 of resistance. 
 
 
 
 814 
 
 508 
 
 35 
 
 758 
 
 740 
 
 5 
 
 806 
 
 539 
 
 40 
 
 750 
 
 778 
 
 10 
 
 798 
 
 569 
 
 45 
 
 742 
 
 816 
 
 15 
 
 790 
 
 602 
 
 50 
 
 734 
 
 856 
 
 20 
 
 782 
 
 635 
 
 55 
 
 726 
 
 897 
 
 25 
 
 774 
 
 669 
 
 60 
 
 718 
 
 940 
 
 30 
 
 766 
 
 704 
 
 65 
 
 710 
 
 984 
 
 The coefficient of resistance for a tube at right angles to the 
 side, is to the like coefficient when it makes an angle of 
 45 degrees as '508 to *816, or as 1 to 1*6 nearly; and the 
 loss of head is greater in the same proportion. If the short 
 tube be more than three or four diameters in length, friction 
 will have to be taken into account. 
 
 FORMULA FOR FINDING THE TIME THE SURFACE OF WATER 
 IN A CISTERN TAKES TO SINK A GIVEN DEPTH. 
 
 In experiments for find- 
 ing the value of the coeffi- 
 cients of discharge, one of 
 the best methods is to ob- 
 serve the time the water 
 discharged from the orifice 
 takes to sink the surface in 
 a prismatic cistern a given 
 depth ; the ratio of the observed to the theoretical time will 
 then give the coefficient sought. A formula for finding the 
 theoretical time is, therefore, of much practical value. In 
 Fig. 29, the time of falling from st to ST, in seconds, is 
 
 (68.) 
 
 {(A 
 
 4-0125 a 
 in which a is the area of the orifice OR, and A the area of
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 99 
 
 the prismatic vessel at st or ST; this formula is for measures 
 in feet. For measures in inches, we have 
 
 EXAMPLE VII. A cylindrical vessel 5'74 inches in dia- 
 
 meter has an orifice '2 inch in diameter at a depth of 
 
 16 inches below the surface, measured to the centre ; it is 
 
 found that the water sinks 4 inches in 51 seconds; what is 
 
 the coefficient of discharge ? 
 
 The theoretical time, t, is found from equation (69), equal 
 
 17-6566 31-8 
 
 = . 556 x -5359 = 31-8 seconds; hence, -^- = -624 is 
 
 the coefficient sought. When the orifice OR and the horizontal 
 
 A ST 2 
 
 section of the vessel are similar figures, is equal j; and 
 
 therefore, for circular cisterns and orifices, it is unnecessary to 
 introduce the multiplier "7854. 
 
 SECTION VIII. 
 
 FLOW OF WATER IN UNIFORM CHANNELS. MEAN VELOCITY. 
 
 MEAN RADII AND HYDRAULIC MEAN DEPTHS. BORDER. 
 
 TRAIN. HYDRAULIC INCLINATION. EFFECTS OF FRIC- 
 TION. FORMULA FOR CALCULATING THE MEAN VELOCITY. 
 
 APPLICATION OF THE FORMULAE AND TABLES TO THE 
 
 SOLUTIONS OF THREE USEFUL PROBLEMS. 
 
 In rivers the velocity is a maximum along the central line 
 of the surface, or, more correctly, over the deepest part of 
 the channel; and it decreases thence to the sides and bottom : 
 but when backwater arises from any obstruction, either a 
 
 * See NOTE A for formula: for finding the time of filling Lock- 
 chambers. 
 
 H 2
 
 100 THE DISCHARGE OF WATER FROM 
 
 submerged weir, Fig. 22, or a contracted channel, Fig. 23, 
 the velocity in the channel approaching the obstruction is a 
 maximum at the depth of the backwater below the surface, 
 and it decreases thence to the surface, sides, and bottom. 
 When water flows in a pipe of any length, the velocity at 
 the centre is greatest, and it decreases thence to the sides 
 or circumference of the pipe. If the pipe be supposed di- 
 vided into two portions in the direction of its length, the 
 lower portion or channel will be analogous to a small river 
 or stream, in which the velocity is greatest at the central 
 line of the surface, and the upper portion will be simply the 
 lower reversed. A pipe flowing full may, therefore, be 
 looked upon as a double stream, and we shall soon see that 
 the formulas for the discharge from each kind are all but 
 identical, though a pipe may discharge full at all inclinations, 
 while the inclinations in rivers or streams, having uniform 
 motion, never exceed a few feet per mile. 
 
 MEAN VELOCITY. 
 
 It is found, by experiment, that the mean velocity is nearly 
 independent of the depth or width of the channel, the central 
 or maximum velocity being the same. From a number of 
 experiments, Du Buat derived empirical formulas equivalent 
 to 
 
 = v - v* + - v = vi-l 2 and v = v + l 2 
 
 in these equations v is the mean velocity, v the maximum 
 velocity, and v b the velocity at the sides, or bottom, ex- 
 pressed in French inches. Tables calculated from these for- 
 mulas do not give correct results for measures in English 
 inches, though they are those generally adopted. Disregarding 
 the difference in the measures, which are as 1 to 1*0678, it 
 will be found that, in the generality of channels, the mean 
 velocity is not an arithmetical mean between the velocity at 
 the central surface line and that at the bottom, though nearly 
 so between the mean bottom and mean surface velocities. 
 Dr. Young*, modifying Du Buat's formula, assumes for 
 * Philosophical Transactions, 1808, p. 487.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 101 
 
 English inches that v + v* = v, and hence v=.v + ^ (v + )* 
 This gives results very nearly the same as the other formula 
 for v, but something less, particularly for small surface velo- 
 cities. For instance, Du Buat's formula gives '5 inch for the 
 mean velocity when the central surface velocity is 1 inch, 
 whereas Dr. Young's makes it '38 inch. For large velocities 
 both formulas agree very closely, disregarding the difference 
 between the measures, which is only 7 per cent. They are 
 best suited to very small channels or pipes, but unless at mean 
 velocities of about 3 feet per second, they are wholly inappli- 
 cable to rivers. 
 
 Prony found, from Du Buat's experiments, that for measures 
 
 / 2-37187 + v\ ,, . 
 m metres v =. ( - ^ Jv. I his, reduced for measures 
 
 \ O " 1 OO 1 -c* *f~ V/ 
 
 in English feet, becomes 
 
 = 
 
 and for measures in English inches, 
 
 (7i.) , = ( 93 ' 39 + MV. 
 
 V 124-14 + v/ 
 
 For medium velocities v := '81 v. The experiments from 
 which these formulae were derived were made with small 
 channels. We have calculated the values of v from that of 
 v, equation (71), and given the results in columns 3, 6, and 9, 
 in TABLE VII. Ximenes, Funk, and Brunnings' experiments 
 in larger channels give the mean velocity at the centre of the 
 depth equal '914 v, when the central surface velocity is v ; but 
 as the velocity also decreases in nearly the same ratio from the 
 centre to the sides of the channel, we shall get the mean 
 velocity in the whole section equal -914 x '914 v = -835 v; 
 and hence we have, for large channels, 
 (72.) v = -835 v. 
 
 We have also calculated the values of v from this formula, 
 and given the results in columns 2, 5, and 8 of TABLE VII. 
 This table will be found to vary considerably from those cal- 
 culated from Du Buat's formula in French inches, hitherto 
 generally used in this country, and much more applicable for 
 all practical purposes.
 
 102 
 
 THE DISCHARGE OF WATER FROM 
 
 MEAN RADIUS. - HYDRAULIC MEAN DEPTH. - BORDER. 
 COEFFICIENT OF FRICTION. 
 
 If, in the diagrams 1 and 2, 
 Fig. 30, exhibiting the sections 
 of cylindrical and rectangular 
 tubes filled with flowing water, 
 the areas be divided respectively 
 by the perimeters A c B D A and 
 ABDCA, the quotients are termed "the mean radii" of the 
 tubes, diagrams 1 and 2, and the perimeters in contact with 
 the flowing water are termed " the borders." In the diagrams 
 3 and 4, the surface A B is not in contact with the channel, and 
 the length of the bed and sides A c D B become " the border" 
 " The mean radius" is equal to the area ABDCA divided by 
 the length of the border AC D B. " The hydraulic mean depth" 
 is the same as " the mean radius" this latter term being 
 perhaps most applicable to pipes flowing full, as in diagrams 
 1 and 2 ; and the former to streams and rivers which have a 
 surface line A B, diagrams 3 and 4. We shall, throughout the 
 following equations, designate the value of the mean radius, 
 
 hydraulic mean depth, or quotient, ^ ^ r BDCA >* 
 
 the 
 
 letter r, only remarking here thatybr cylindrical pipes flowing 
 full) or rivers with semicircular beds, it is always equal to 
 half the radius, or one-fourth of the diameter. 
 
 Du Buat was the first to observe that the head due to the 
 resistance of friction for water flowing in a uniform channel 
 increased directly as the length of the channel I, directly as 
 the border, and inversely as the area of the cross-flowing 
 
 * M. Girard has conceived it necessary to introduce the coefficient of 
 correction 1/7 as a multiplier to the border for finding r, to allow for the 
 increased resistance from aquatic plants; so that, according to his 
 reduction, 
 
 area 
 ~ 1-7 border' 
 
 See Rennie's First Report on Hydraulics as a Branch of Engineering ; 
 Third Report of the British Association, p. 1G7. Also equation (85), p. 112.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 103 
 
 section *, very nearly ; that is, as . It also increases as the 
 
 square of the velocity, nearly ; therefore the head due to the 
 
 v 2 / v 2 / 
 
 resistance must be proportionate to o If c t x <5> ~ = h& 
 
 then c { is the coefficient for the head due to the resistance of 
 friction, as h { is the head necessary to overcome the friction, 
 c { is therefore termed " the coefficient of friction " 
 
 HYDRAULIC INCLINATION. TRAIN. 
 
 If I be the length of a pipe or channel, and h t the height 
 due to the resistance of friction of water flowing in it, then y f 
 is the hydraulic inclination. In Fig. 31 the tubes A B, c D, of 
 
 the same length /, and whose discharging extremities B and D are 
 on the same horizontal plane B D, will have the same hydraulic 
 inclination and the same discharge, no matter what the actual 
 inclinations or the depth of the entrances at A and c may be, 
 so they be of the same kind and bore j and as the velocities in 
 A B and c D are the same, the height h due to them must be the 
 same when the circumstances of the orifices of entry A and c 
 are alike. We have the whole head H = h + h t (see pp. 89 
 and 90). The hydraulic inclination is not therefore the whole 
 head H, divided by the length / of the pipe, as it is some- 
 times mistaken, but the height h { (found by subtracting 
 the height h, due to the entrance at A or c, and the velocity 
 in the pipe, from the whole height) divided by the length /. 
 When the height h is very small compared with the whole 
 height H, as it is in very long tubes with moderate heads, 
 
 * Pitot had previously, in 1726, remarked that the diminution arising 
 from friction in pipes is, azteris paribw, inversely as the diameters.
 
 104 
 
 THE DISCHARGE OF WATER FROM 
 
 fit 
 
 y may be substituted for y without error j but for short pipes 
 
 up to 100 feet in length the latter only should be used in 
 applying Du Buat's and some other formulae ; otherwise the 
 results will be too large, and only fit to be used approximately 
 in order to determine the height h from the velocity of dis- 
 charge thus found. When the horizontal pipe c D, Fig. 32, is 
 
 equal in every way to the inclined pipe A B, and the head at A 
 is that due to the velocity in c D, the discharge from the pipe 
 A B will be equal to that from c D ; but a peculiar property 
 belongs to the pipe AB in the position in which it is here 
 placed ; for if we cut it short at any point e, or lengthen it to 
 any extent, to E, the discharge will remain the same and 
 equal to that through the horizontal pipe c D. The velocity 
 in A B at the angle of inclination ABC, when A c h {) and 
 A B = c D, is therefore such that it remains unaffected by the 
 length A E or A e, to which it may be extended or cut short ; 
 and at this inclination the water in the pipe A B is said to be 
 "in train" In like manner a river or stream is said to be 
 " in train " when the inclination of its surface bears such a 
 relation to the cross section that the mean velocity is neither de- 
 creased nor increased by the length of the channel ; and we per- 
 ceive from this that the acceleration caused by the inclination 
 is exactly counterbalanced by the resistances to the motion 
 when the moving water in a pipe or river channel is in train. 
 
 As h = (1 + c r ) o~ where c r is the coefficient of the height 
 due to the resistance at the orifice of entry A or c, and 
 h t = c f o , vve therefore get
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 105 
 
 U 2 IV* / / 
 
 (73.) H = (1 + C + C f X = 
 
 and hence we find the mean velocity of discharge 
 
 (74.) v : 
 
 When A is small compared with h f , or, which comes to the 
 same thing, 1 + C T small compared with c f x - > 
 
 lv* 
 (75.) H = c f x 2^, 
 
 and 
 
 (76.) 
 
 c t 
 
 H 
 
 If, in the last equation, we substitute s for y, equal the sine 
 of the angle of inclination ABC, we then have 
 
 (77.) v = 
 
 i f ) 
 
 An average value of c f for all pipes with straight channels is 
 0069914, from which we find equation (77) becomes, for 
 measures in feet, 
 
 (78.) v = 96 v/ 
 
 As the mean value of the coefficient of resistance c r for the 
 entrance into a tube is -508, and as 2g := 64'403, and 
 c { = '0069914, equation (74), for measures in feet, becomes 
 
 64-403 H 1* 
 
 1-508 + -0069914-^J ' 
 
 nr H 
 
 + -0001 085 /) ' 
 This, multiplied by the section, gives the discharge.
 
 106 THE DISCHARGE OF WATER FROM 
 
 DU BUAT S FORMULA. 
 
 The coefficient of friction c f is not, however, constant, as it 
 varies with the velocity. That which we have just given 
 answers for pipes when the velocity is from 18 to 24 inches 
 per second. For pipes and rivers it is found to increase as 
 the velocity decreases; that is, the loss of head is propor- 
 tionately greater for small than for large velocities. Du Buat 
 found the loss of head to be also greater for small than large 
 channels, and applied a correction accordingly in his formula. 
 This, expressed in French inches, is 
 
 297 (r -0-1) 
 (80.) v = ^ , rf- - 4 _-3(r 4 -0-l), 
 
 (-) -hyp.log.( 7 + l-6) 
 
 maintaining the preceding notation, in which s =. ~r. In this 
 
 formula O'l, in the numerator of the first term, is deducted as 
 a correction due to the hydraulic mean depth, as it was found 
 that 297 (r 4 O'l) agreed more exactly with experiment 
 
 than 297 r* simply. The second term, hyp. log. ( - + 1-6 J, 
 
 of the denominator is also deducted to compensate for the 
 observed loss of head being greater for less velocities, and the 
 last term '3 (r 4 O'l) is a deduction for a general loss of 
 velocity sustained from the unequal motions of the particles 
 of water in the cross section as they move along the channel. 
 These corrections are empirical; they were, however, deter- 
 mined separately, and after being tested by experiment, 
 applied, as above, to the radical formula v = 297 \/rs. 
 
 Du Buat's formula was published in his Principes 
 d'Hydr antique, in 1786. It is, as we have seen, partly 
 empirical, but deduced by an ingenious train of reasoning 
 and with considerable penetration from about 125 experi- 
 ments, made with pipes from the 19th part of an inch to 
 18 inches in diameter, laid horizontally, inclined at various 
 inclinations, and vertical ; and also from experiments on open 
 channels with sectional areas from 19 to 40,000 square inches,
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 107 
 
 and inclinations of from 1 in 112 to 1 in 36,000. The lengths 
 of the pipes experimented with varied from 1 to 3, and from 
 3 to 3600 feet. 
 
 In several experiments by which we have tested this 
 formula, the resulting velocities found from it were from 
 1 to 5 per cent, too large for small pipes, and too small for 
 straight rivers in nearly the same proportion. As the experi- 
 ments from which it was derived were made with great care, 
 those with pipes particularly so, this was to be expected. 
 Experiments with pipes of moderate or short lengths should 
 have the circumstances of the orifice of entry from the 
 reservoir duly noted ; for the close agreement of this 
 formula with them must depend a great deal, in such pipes, 
 on the coefficient due to the height h, which must be deducted 
 
 from the whole head H before the hydraulic inclination, y s, 
 
 can be obtained ; but for very long pipes and uniform channels 
 this is not necessary. 
 
 Under all the circumstances, and after comparing the 
 results obtained from this and other formulae, we have pre- 
 ferred calculating tables for the values of v from this formula 
 reduced for measures in English inches, which is 
 
 306-596 (f* - -1032) 
 
 ?; = 7TV~ -77- -r - -2906 (f* - -1032), 
 
 (-) -hyp.log.(-+l-6) 
 
 or more simply, 
 
 307 (r* - -1) 
 (81.) " = 
 
 / 
 -) -hyp.log.( 7 +l-6) 
 
 This gives the value of v a little larger than the original 
 formula, but the difference is immaterial. For measures in 
 English feet it becomes 
 
 88-51 (r* - -03) 
 (82.) y= -j - - -84 (** - -03). 
 
 The results of equation (81) are calculated for different
 
 108 THE DISCHARGE OF WATER FROM 
 
 values of * and r, and tabulated in'TABLE VIIL, the first 
 eight pages of which contain the velocities for values of r 
 varying from ^th inch to 6 inches ; or if pipes, diameters from 
 J inch to 2 feet, and of various inclinations from horizontal 
 to vertical. The five last pages contain the velocities for 
 values of r from 6 inches to 12 feet, and with falls from 
 6 inches to 12 feet per mile. 
 
 EXAMPLE VI II. A pipe, 1^ inch diameter and 100 feet 
 long, has a constant head of 2 feet over the discharging ex- 
 tremity ; what is the velocity of discharge per second ? 
 
 H 3 . , , 100 CA 1 . 
 
 The mean radius r rz-f - inches, and - = 50 = -. is 
 
 48 % s 
 
 the approximate hydraulic inclination. At page 2 of TABLE 
 VIIL, in the column under the mean radius -, and opposite to 
 
 the inclination 1 in 50, we find 30*69 inches for the velocity 
 sought. This, however, is but approximative, as the head 
 due to the velocity should be subtracted from the whole head 
 of 2 feet, before finding the true hydraulic inclination. This 
 head depends on the coefficient of resistance at the entrance 
 orifice, or the coefficient of discharge for a short tube. In 
 all Du Buat's experiments this latter was taken at "8125, but 
 it will depend on the nature of the junction, as, if the tube 
 runs into the cistern, it will become as small as '715; and, if the 
 junction be rounded into the form of the contracted vein, it will 
 rise '974, or 1 nearly. In this case, the coefficient of discharge 
 may be assumed '815*, from which, in TABLE II., we find the 
 
 head due to a velocity of 30'69 inches to be 1 1 = 1'87 inch 
 nearly, which is the value of h; and hence, H h = h f = 24 
 
 - 1-87 = 22-13 inches; and L - ^ * o 12 =54-2 = \ the 
 hf 2Z' lo 
 
 hydraulic inclination more correctly. With this new in- 
 clination and the mean radius , we find the velocity by 
 
 interpolating between the inclinations 1 in 50 and 1 in 60, 
 
 given in the Table to be 30'69 1-34 = 29-35 inches per 
 
 * See EXAMPLE 16, pp. 10. 12.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 109 
 
 second. This operation may be repeated until v is found to 
 any degree of accuracy according to the formula; but it is, 
 practically, unnecessary to do so. If we now wish to find 
 the discharge per minute in cubic feet, we can easily do so 
 from TABLE IX., in which for an inch and a half pipe we get 
 
 Inches. Cubic feet. 
 
 For a velocity of 20'00 per second, 1-22718 per minute. 
 9-00 -55223 
 
 30 -01841 
 
 04 , -00245 .. 
 
 29-34 1-80027 
 
 The discharge found experimentally by Mr. Provis, for a 
 tube of the same length, bore, and head, was 1'745 cubic foot 
 per minute. 
 
 If we suppose the coefficient of discharge due to the 
 orifice of entry and stop-cock in Mr. Provis's 208 experi- 
 ments* with 1 inch lead pipes of 20, 40, 60, 80, and 
 100 feet lengths, to be -715, the results calculated by the 
 tables will agree with the experimental results with very 
 great accuracy, and it is very probable from the circum- 
 stances described, that the ordinary coefficient '815 due to 
 the entry was reduced by the circumstances of the stop- 
 cock and fixing to about '715; but even with -815 for the 
 coefficient, the difference between calculation and experiment 
 is not much, the calculation being then in excess in every 
 experiment, the average being about 5 per cent., and not so 
 much in the example we have given. 
 
 TABLE VIII. will give the velocity, and thence the discharge 
 immediately, for long pipes, and TABLE X. enables us to cal- 
 culate the cubic feet discharged per minute, with great facility. 
 For river?, the mean velocity, and thence the discharge, is also 
 found with quickness. See also TABLES XL, XII., and XIII. 
 
 EXAMPLE IX. A watercourse is 7 feet wide at the 
 bottom, the length of each sloping side is 6'Sfeet, the width 
 at the surface is 18 feet, the depth 4 feet, and the inclina- 
 
 * Proceedings of the Institution of Civil Engineers, pp. 201, 210, 
 vol. ii.
 
 110 THE DISCHARGE OP WATER FROM 
 
 tion of the surface 4 inches in a mile ; what is the quantity 
 flowing down per minute ? 
 
 (18 + 7)x- 50 
 Here 7 + 2x g.g = ^ = 2 ' 4272 feet = 29 ' 126 inches 
 
 = r, is the hydraulic mean depth; and as the fall is 4 inches 
 per mile, we find at the llth page of TABLE VIII., the ve- 
 locity v = 12*03 '16 = 11 '87 inches per second; the dis- 
 charge in cubic feet per minute is, therefore, 
 
 50 x H|? x 60 = 2967-5. 
 
 Watt, in a canal of the fall and dimensions here given, 
 found the mean velocity about 13g inches per second. This 
 corresponds to a fall of 5 inches in the mile, according to 
 the formula. 
 
 In one of the original experiments with which the formula 
 was tested on the canal of Jard, the measurements accorded 
 
 very nearly with those in this example, viz. - = 15360, and 
 
 r =. 29*1 French inches; the observed velocity at the surface 
 was 15'74, and the calculated mean velocity from the formula 
 11-61 French inches*. TABLE VII. will give 12-29 inches for 
 the mean velocity, corresponding to a superficial velocity of 
 15*74 inches. This shews that the formula also gives too 
 small a value for v in this case, by about tS-th of the result, 
 
 it being about -^ part in the other. The probable error 
 
 in the formula applied to straight clear rivers of about 
 2 feet 6 inches hydraulic mean depth is nearly -jVth or 8 per 
 cent, of the tabulated velocity, and this must be added for the 
 more correct result; the watercourse being supposed nearly 
 straight and free from aquatic plants. 
 
 Notwithstanding the differences above remarked on, we 
 are of opinion that the results of this formula, which we 
 
 * These measures reduced to English inches, give r = 31-014, 
 v = 12-374 ; and the surface velocity 16-775 inches ; reduced for mean 
 velocity 13-101 inches.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. Ill 
 
 have calculated and tabulated, may be more safely relied 
 on as applied to general practical purposes than most of 
 those others which we shall proceed to lay before our readers. 
 Rivers or watercourses are seldom straight or clear from 
 weeds, and even if the sections, during any improvements, 
 be made uniform, they will seldom continue so, as "the 
 regimen" or adaptation of the velocity to the tenacity of 
 the banks, must vary with the soil and bends of the channel, 
 and can seldom continue permanent for any length of time 
 unless protected. From these causes a loss of velocity takes 
 place, difficult, if not impossible, to estimate accurately, but 
 which may be taken at from 10 to 15 per cent, of that in the 
 clear unobstructed direct channel; but, be this as it may, it is 
 safer to calculate the drainage or mechanical results obtainable 
 from a given fall and river channel, from formula which give 
 lesser, than from those which give larger velocities. This is 
 a principle engineers cannot too much observe. 
 
 We have before remarked, that for both pipes and rivers 
 the coefficient of resistance increases as the velocity de- 
 creases. This is as much as to say, in the simple formula for 
 the velocity, v =. m \/rs, that m must increase with v, and as 
 some function of it. This is the case in TABLE VIII., through- 
 out which the velocities increase faster than \J~r, the \/s, r 
 the \/7s. In all formulae with which we are acquainted but 
 Du Buat's and Young's, the velocity found is constant when 
 ^/rs or r x s is constant. In Du Buat's formula for r x s 
 constant, v obtains maximum values between r =: f inch 
 and r = 1 inch ; the differences of the velocities for different 
 values of r above 1 inch, r x s being constant, are not much. 
 We may always find the maximum value, or nearly so, by as- 
 suming r f inch, and finding the corresponding inclination 
 
 from the formula -^-?, which is equal to it. For example, 
 3 
 
 if r = 12 inches, and s = 1Q56o , the velocity is found equal 
 9'52 inches; but when rs is constant, the inclination s cor- 
 responding to r = I inch is ^l-?lL__ _. _!., f rom wn j c h
 
 112 THE DISCHARGE OF WATER FROM 
 
 we find from the table v 10'25 inches for the maximum 
 velocity, making a difference of fully 7 per cent. 
 
 When r "01 of an inch, or a pipe is -^jth part of an 
 inch in diameter, Du Buat's formula fails, but it gives 
 correct results for pipes th of an inch in diameter, and 
 two of the experiments from which it was derived were 
 made with pipes 12 inches long and only -j^th part of an 
 inch in diameter. 
 
 COULOMB having shown that the resistance opposed to a 
 disc revolving in water increases as the function av + bv* of 
 the velocity v, we may assume that the height due to the 
 resistance of friction in pipes and rivers is also of this form; 
 and that 
 
 (83.) h f = (av + bv 2 )-, 
 
 r 
 and, consequently, 
 
 (84.) rs = av + bv z . 
 
 GIRARD first gave values to the coefficients a and b. He 
 assumed them equal, and each equal to '0003104 for mea- 
 sures in metres, and thence the velocity in canals, 
 (85.) v = (3221-016 r* + -25)* - '5;* 
 
 which reduced for measures in English feet becomes 
 (86.) v = (10567-8 rs + 2-67) 1 1-64. 
 
 The value of a b = '0003104 was obtained by means of 
 twelve experiments by Du Buat and Chezy. Of course the 
 value is four times this in the original, as we use the mean 
 radius in all the formulae instead of the diameter. This 
 formula is only suited for very small velocities in canals 
 containing aquatic plants, and the velocity must be increased 
 by '7 foot per second for clear channels. 
 
 PRONY found from thirty experiments on canals, that 
 a = -000044450 and b = '0003093 14,f for measures in metres, 
 from which we find 
 (87.) v = (3232-96 rs + -00516)* - -0719; 
 
 * See Brewster's Encyclopedia, Article Hydrodynamics, p. 529. 
 t Recherches Physico-Mathematiques sur la Theorie des Eaux 
 Courantes.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 113 
 
 this reduced for measures in English feet is, 
 (88.) v = (10607-02 rs + -0556)* - -236;* 
 
 the velocities did not exceed 3 feet per second in the ex- 
 periments from which this was derived. 
 
 For pipes, Prony found,^ from fifty-one experiments made 
 by Du Buat, Bossut, and Couplet, with pipes from 1 to 
 5 inches diameter, from 30 to 17,000 feet in length, and one 
 pipe 19 inches diameter and nearly 4000 feet long, that 
 a = -00001733, and b = '0003483, from which values 
 
 (89.) v - (2871-09 rs + -0006192)* - -0249, 
 
 for measures in metres, and for measures in English feet, 
 
 (90.) v = (9419-75 rs + -00665)* - -0816. 
 
 Prony also gives the following formula applicable to pipes 
 and rivers. It is derived from fifty-one selected experiments 
 with pipes, and thirty-one with open channels : 
 (91.) v = (3041-47 rs + -0022065)* - -0469734,;}: 
 
 for measures in metres, which, reduced for measures in 
 
 English feet, is 
 
 (92.) v = (9978-76 rs + -02375)* -15412. 
 
 EYTELWEIN, following the method of investigation pursued 
 previously by Prony, found from a large number of experi- 
 ments, a -0000242651, and b = -000365543 in rivers, for 
 measures in metres; and, therefore, 
 
 * For canals containing aquatic plants, reeds, <kc., we must sub- 
 stitute for r. See note, p. 102. 
 
 t Recherches Physico-Mathematiques sur la Th^orie du Mouvement 
 des Eaux Courantes, 1804. 
 
 J Recherches Physico-Mathematiques sur la Th6orie des Eaux Cou- 
 rantes. A reduction of this formula into English feet is given at 
 page 6, Article Hydrodynamics, Encyclopedia Britannica ; at page 164, 
 Third Report, British Association, by Rennie ; and at pages 427 and 
 533, Article Hydrodynamics, Brewster's Encyclopedia. This reduction, 
 v= 0-1541 + (-02375+ 32806-6 rs)* is entirely incorrect; and, being 
 the same in each of those works, appears to have been copied one 
 from the other. 
 
 I
 
 114 THE DISCHARGE OF WATER FROM 
 
 (93.) v = (2735-66 rs + -001 102)* - -0332. * 
 
 This reduced for measures in English feet, is 
 
 (94.) v = (8975-43 rs + -01 18858)* - -1089. 
 
 He also shows, f that ffths of a mean proportional between 
 the fall in two English miles and the hydraulic mean depth, 
 gives the mean velocity very nearly. This rule for measures 
 in inches is equivalent to 
 
 (95.) ; = 324 v /^ ; 
 
 and for measures in feet 
 
 (96.) = 93-4 v/. 
 
 For the velocity of water in pipes, he found,;}: from the fifty- 
 one experiments of Du Buat, Bossut, and Couplet, that 
 a = -0000223, and b = '0002803, from which we get for 
 measures in metres, 
 
 (97.) v = (3567-29 rs + -00157)* - -0397 ; 
 
 which reduced for measures in English feet becomes 
 (98.) v = (1 1703-95 rs + -01698)* - -1303. 
 
 Another formula given by Eytelwein for pipes, which in- 
 cludes the head due to the velocity for the orifice of entry, is 
 
 in which h is the head, / the length, and d the diameter 
 of the pipe, all expressed in English feet. It must be here 
 mentioned, that much of the valuable information presented 
 by Eytelwein is but a modification of what Du Buat had pre- 
 viously given, and to whom for much that is attributed to the 
 former we are primarily indebted. 
 
 Dr. THOMAS YOUNG also derives his formula from the 
 supposition, that the head due to the resistance of friction 
 
 * Memoires de 1'Acadeinie de Berlin, 1814 et 1815. See Equation (110). 
 t Handbuch der Mechanik und der Hydraulik, Berlin, 1801. 
 I Memoires de 1'Academie des Sciences de Berlin, 1814 et 1815. 
 Philosophical Transactions for 1808.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 115 
 
 assumes the form of equation (83) ; calling the diameter of 
 a pipe d, he takes 
 
 h t =(av + bv*}-. 
 
 and the whole height H = h f + - , expressed in inches. 
 
 586 
 
 He found from some experiments of his own, those collected 
 by Du Buat, and some of Gerstner's, that 
 
 900 d* 
 
 (100.) a = -0000002 
 
 1136 
 
 inK _L 13-21 _,_ 1-0563 
 1085 + + -32 
 a a 
 
 and 
 
 (101.) 5 = -0000001 {413 + - 144 
 
 d d 12-8 
 then as jL = -00171, we get 
 
 HC? 
 
 (102.) v = 
 
 bl + -00171 d 
 
 al \ 2 )* al 
 
 i Z + -00341 d 
 
 When the length / of the pipe is very great compared with 
 the head due to the orifice of entrance and velocity, "00171 v*, 
 we have 
 
 (103.) *={^? + ^-}*- ; 
 
 or by substituting for its value s, equal the sine of the 
 inclination, 
 
 (104.) v = \ S A + }' . 
 
 } b 4i 2 / 26 
 
 The values of a and J are for measures in inches. For 
 most rivers, in which d must be taken equal 4r, he finds for 
 French inch measures, v = v/ 20000 ds; this reduced for 
 English inches is 
 
 i 2
 
 116 THE DISCHARGE OF WATER FROM 
 
 (105.) v = 292 v/r7; 
 
 which again reduced for feet measures, becomes 
 
 (106.) v = 84-3 v/r! 
 
 These latter values, for rivers, are even smaller than those 
 found from Du Buat's formula; less than the observed ve- 
 locities, and less than those found from any other formula, 
 with the exception of Girard's. The values of the coeffi- 
 cients a and b vary in this formula with the value of d~ 4r; 
 they are expressed generally in equations (101) and (102), 
 from which we have calculated the following Table for 
 different values of d and r. 
 
 An examination of this Table will show that a obtains a 
 minimum value when d is between 10 and 11 inches; and 
 b when the diameter is between and f of an inch. Now, it 
 
 appears from equation (102), that v increases with A / 
 
 bl 
 
 nearly, or, which is the same thing, as b decreases, there 
 must, c&teris paribus, be a maximum value of v for a 
 
 given value of , or rs, when d is between \ and f 
 
 inch ; but as has a minimum value when d is nearly 
 
 12 inches, the maximum value of v referred to will be found 
 between values of d from $ inch to 12 inches; in fact, when 
 dr= 10 inches nearly. We have already pointed out a similar 
 peculiarity in Du Buat's general theorem, at page 111. It 
 
 will not be necessary to take out the values of and -^ 
 
 2b 4b* 
 
 to more than one place of decimals. 
 
 The values of are also given in the following Table, and 
 2$ 
 
 may be used in equation (104) for finding the discharge from 
 long pipes. It is, however, necessary to remark, that this 
 equation is sometimes misapplied in finding the velocity from 
 short pipes, and those of moderate lengths. It is necessary
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 117 
 
 'B 
 
 . -. i - i - -r r. r. - _-. 
 
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 118 THE DISCHARGE OF WATER FROM 
 
 to use equation (102), which takes into consideration the 
 head due to the velocity and orifice of entry for such pipes. 
 
 SIR JOHN LESLIE states,* that the mean velocity of a 
 river in miles per hour, is -j-fths of the mean proportional 
 between the hydraulic mean depth and the fall per mile, 
 in feet. This rule is equivalent for measures in feet, to 
 
 (107.) t> = 100v/^; 
 
 and is most applicable to very large rivers and velocities. 
 
 D'AUBUISSON, from an examination of the results obtained 
 by Prony and Eytelwein, assumes^ for measures in metres 
 that a = -0000189, and b = '0003425 for pipes substituting 
 these in equation (84) and resolving the quadratic 
 
 (108.) v = (2919-71 rs + -00074)* - -027 ; 
 
 which reduced for measures in English feet becomes 
 (109.) v = (9579 r s + -00813)* - -0902. 
 
 For rivers he assumes with Eytelwein,;}; a = '000024123, 
 and b = -0003655, for measures in metres, and hence 
 
 (110.) v = (2735-98 rs + -0011)*- -033; 
 
 which for measures in English feet is 
 
 (111.) v = (8976-5 rs + -012) 4 - -109. 
 
 When the velocity exceeds two feet per second, he assumes, 
 from the experiments of Couplet, a = 0, and b = '00035875; 
 these values give 
 
 (112.) v = v / 2787-46>*; 
 
 for measures in metres, and 
 
 (113.) v = 95-6 v/rs = x/9145r* 
 
 for measures in English feet. Equations (110) and (111) 
 
 are the same as (93) and (94), found from Eytelwein's 
 
 values of a and 6, and it may be remarked that D'Aubuis- 
 
 son's equations for the velocity generally are but simple 
 
 modifications of Prony's and Eytelwein's. 
 
 * Natural Philosophy, p. 423. 
 
 t Trait6 d'Hydraulique, p. 224. 
 
 J Traite d'Hydraulique, p. 133. See Equation (93).
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 119 
 
 The values which we have found to agree best with ex- 
 periments on clear straight rivers are a = '0000035, and 
 b = '0001 150 for measures in English feet, from which we 
 find 
 
 (114.) v = (8695-6 rs + -00023)* - -0152, 
 
 which for an average velocity of 1 foot per second will give 
 v = 92'3 \/rs nearly, and for large velocities v 93'3<\/rs' > 
 for smaller velocities than lj foot per second, the coefficients 
 of ^/r s decrease pretty rapidly. This formula will be found 
 to agree more accurately with observation and experiment 
 than any other we know of. It is useful to remark in the 
 preceding formulae, that the second term and decimals 
 both tinder the vinculum may be entirely neglected without 
 error. 
 
 WEISBACH is perhaps the only writer who has modified the 
 form of the equation r s = a v + ft v 2 . In Dr. Young's for- 
 mula, a and b vary with r, but Weisbach assumes that 
 
 A f := ( a + ) - X , and finds from the fifty-one experi- 
 \ */ a 2ff 
 
 mentsof Couplet, Bossut and Du Buat, before referred to, one 
 experiment by Guemard, and eleven by himself, all with pipes 
 varying from an inch to five and a half inches in diameter, 
 and with velocities varying from 1^ inch to 15 feet per 
 second, that a = -01439, and b = '0094711 for measures in 
 metres ; hence we have for the metrical standard 
 
 (115.) A, 
 
 This reduced for the mean radius r is 
 
 (116.) h, = (-003597 + - 023678 )lx^ ; 
 
 \ ?i / r 2ff 
 
 from which we find for measures in English feet 
 
 (117.) h, = -003597 + - x 
 
 r
 
 120 
 
 and thence 
 (118.) 
 
 THE DISCHARGE OF WATER FROM 
 
 = (-003597 
 V 
 
 and by substituting for 2g, its value 64 - 403, 
 (119.) ..= (.00005585 
 
 In equation (117), ( -003597 + ' QQ42887 } = c t is the co- 
 X vi / 
 
 efficient of the head due to friction. The equation does not 
 admit of a direct solution, but the coefficient should be first 
 determined for different values of the velocity v and tabu- 
 lated, after which the true value of t> can be determined by 
 finding an approximate value and thence taking out the cor- 
 responding coefficient from the table, which does not vary to 
 any considerable extent for small changes of velocity. In the 
 following small table we have calculated the coefficients of 
 friction, and also those of v z , in equation (119), for different 
 values of the velocity v. 
 
 TABLE OF THE COEFFICIENTS OF FEIOTION IN PIPES. 
 
 Velocity 
 in feet 
 
 c f 
 
 c f 
 
 Velocity 
 in feet. 
 
 c f 
 
 c t 
 
 
 
 64-4 
 
 
 
 64-4 
 
 >0 
 
 Infinity. 
 
 Infinity. 
 
 1-75 
 
 006839 
 
 0001062 
 
 k l 
 
 017159 
 
 0002664 
 
 2 
 
 006629 
 
 0001029 
 
 2 
 
 '013186 
 
 0002047 
 
 2-5 
 
 006309 
 
 0000979 
 
 3 
 
 011427 
 
 0001774 
 
 3 
 
 006073 
 
 0000943 
 
 4 
 
 010378 
 
 0001611 
 
 3-5 
 
 005890 
 
 0000914 
 
 5 
 
 009662 
 
 0001500 
 
 4 
 
 005741 
 
 0000891 
 
 6 
 
 009133 
 
 0001418 
 
 5 
 
 005514 
 
 0000866 
 
 7 
 
 008723 
 
 0001354 
 
 6 
 
 005348 
 
 .0000830 
 
 8 
 
 008391 
 
 0001303 
 
 7 
 
 005218 
 
 0000810 
 
 -9 
 
 008117 
 
 0001260 
 
 8 
 
 005113 
 
 0000794 
 
 1-0 
 
 007886 
 
 0001224 
 
 10 
 
 004953 
 
 0000769 
 
 1-25 
 
 007433 
 
 0001154 
 
 16 
 
 004669 
 
 0000725 
 
 1-5 
 
 007098 
 
 '0001102 
 
 20 
 
 004556 
 
 0000707
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 121 
 
 If the reciprocal of ^-r-j* here found, be substituted in the 
 
 equation v == A/ - rs, we shall have the value of v. Ac- 
 
 cording to this table the coefficient of friction for a velocity 
 of seven inches is more than twice that for a velocity of 
 twenty feet. On comparing these coefficients and those for 
 pipes in the preceding formulae, with those for rivers of the 
 same hydraulic depth, we perceive that the loss from friction 
 is greatest in the latter, as might have been anticipated. 
 
 COEFFICIENTS DUE TO THE ORIFICE OF ENTRY. PROBLEMS. 
 
 Unless where otherwise expressed, the head due to the 
 velocity and orifice of entry is not considered in the pre- 
 ceding equations. In equation (74), where it is taken into 
 
 calculation generally, v = -L I \ in which 1 +c r 
 
 l T c t -r Cf x I 
 
 / 1 \ 2 
 is equal to ( J , c t being the coefficient of resistance due to 
 
 the orifice of entry, and c y the coefficient of velocity or dis- 
 charge from a short tube. If the tube project into the re- 
 servoir, and be of small thickness, c r will be equal *715 nearly, 
 and therefore c t = *956 ; if the tube be square at the junction, 
 the mean value of c v will be '814, and therefore c t = "508; 
 and if the junction be rounded in the form of the contracted 
 vein, c v is equal to unity very nearly, and C T =: 0. For other 
 forms of junction the coefficients of discharge and resistance 
 will vary between these limits, and particular attention must 
 be paid to their values in finding the discharge from shorter 
 tubes and those of moderate lengths, but in very long tubes 
 
 1 + c t becomes very small compared with c t x , and may 
 
 be neglected without practical error. These remarks are ne- 
 cessary to prevent the misapplication of the tables and for- 
 mulae, as the height due to the velocity and orifice of entry 
 is an important element in short tubes.
 
 122 ~ THE DISCHARGE OP WATER FROM 
 
 We have considered it unnecessary to give any formulae for 
 finding the discharge itself, because, the mean velocity once 
 determined, the calculation of the discharge from the area of 
 the section is one of simple mensuration ; and the introduc- 
 tion of this element into the three problems to which this 
 portion of hydraulic engineering applies itself, renders the 
 equations of solution complex, though easily derived ; and 
 presents them with an appearance of difficulty and want of 
 simplicity which excludes them, nearly altogether, from prac- 
 tical application. The three problems are as follows : 
 
 I. Given the fall, length, and diameter of a pipe or hy- 
 draulic mean depth of any channel, to find the discharge. 
 
 Here all that is necessary is to find the mean velocity of 
 discharge, which, multiplied by the area of the section (equal 
 d 2 X '7854 in a cylindrical pipe), gives the discharge sought. 
 TABLE VIII. gives the velocity at once for long channels. 
 TABLE IX. gives the discharge in cubic feet per minute for 
 different diameters of pipes, and velocities in inches per 
 second, when found from TABLE VIII. See also TABLES 
 XI. and XII. 
 
 II. Given the discharge, and cross section of a channel, 
 to find the fall or hydraulic inclination. 
 
 If the cross section be circular, as in most pipes, the hy- 
 draulic mean depth is one-fourth of the diameter ; in other 
 channels it is found by dividing the water and channel line of 
 the section, wetted perimeter, or border, into the area. The 
 velocity is found by dividing the area into the discharge, and 
 reducing it to inches per second ; then in TABLE VIII., under 
 the hydraulic mean depth, find the velocity, corresponding to 
 which the fall per mile will be found in the first column, and 
 the hydraulic inclination in the second. 
 
 III. Given the discharge, length, and fall, to find the 
 diameter of a pipe, or hydraulic mean depth and dimensions 
 of a channel. 
 
 This is the most useful problem of the three. Assume any 
 mean radius r & and find the discharge D a by Problem I. 
 We shall then have for cylindrical pipes
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 123 
 
 5 
 
 and as r a , D, and D a are known, r* becomes also known, 
 and thence r. TABLE XIII. will enable us to find r with 
 great facility. Thus, if we had assumed r a = 1 and found 
 D a = 15, D being 33, we then have 
 
 1 : r* : : 1 : ?i : : 1 : 2'2, therefore r* = 2'2; 
 
 15 
 
 and thence by TABLE XIII., r = 1'37, the mean radius 
 required, four times which is the diameter of the pipe. For 
 other channels, the quantity thus found must be the hydraulic 
 mean depth ; and all channels, however varied in the cross 
 section, will have the same velocity of discharge when the 
 fall, length, and hydraulic mean depth are constant. 
 
 In order to find the dimensions of any polygonal channel 
 whatever, which will give a discharge equal to D, we may 
 assume any channel similar to that proposed, one of whose 
 known sides is s a , and find the corresponding discharge, D a , 
 by Problem I., or from TABLES XI. and XII.; then, if we 
 call the like side of the required channel, s, we shall have 
 
 s = s a ( ) T , and thence the numerical value from TABLE 
 
 v D a ' 
 XIII. 
 
 As it frequently happens that a deposit in, and encrusta- 
 tion of, a pipe takes place from the water passing through 
 it, which diminishes the section considerably from time to 
 time, it is always prudent, when calculating the necessaiy 
 diameter, to take the largest coefficient of friction, c f , or to 
 increase its mean value by perhaps one half, when calculating 
 the diameter from the formulae. Some engineers, as D'Au- 
 buisson, increase the quantity of water by one half to find 
 the diameter; but much must depend on the peculiar cir- 
 cumstances of each case, as sometimes less may be sufficient, 
 or more necessary. The discharge increases in similar 
 figures, as f* or as d*, and the corresponding increase in the 
 diameter for any given or allowed increase in the discharge
 
 124 THE DISCHARGE OF WATER FROM 
 
 can be easily found by means of TABLE XIII., as shown 
 above. If we increase the dimensions by one-sixth, the dis- 
 charge will be increased by one-half nearly. 
 
 For shorter pipes, we have to take into consideration the 
 head due to the velocity and orifice of entry. Taking the 
 mean coefficient of velocity or discharge, we find, from 
 TABLE II., the head due to the velocity, if it be known; 
 this subtracted from the whole head, H, leaves the head, h { , 
 due to the hydraulic inclination, which is that we must make 
 use of in the table. If the velocity be not given, we can 
 find it approximately; the head found for this velocity, due 
 to the orifice of entry, when deducted, as before, will give a 
 close value of h { , from which the velocity may be determined 
 with greater accuracy, and so on to any degree of ap- 
 proximation. In general, one approximation to h { will be 
 sufficient, unless the pipes be very short, in which case it is 
 best to use equation (74). Example VIII., page 108, and 
 the explanation of the use of the tables, SECTION I., may 
 be usefully referred to. 
 
 TABLES XL, XII., and XIII. enable us to solve with 
 considerable facility all questions connected with discharge, 
 dimensions of channel, and the ordinary surface inclinations 
 of rivers. The discharge corresponding to any intermediate 
 channels or falls to those given in TABLES XI. or XII., will 
 be found with abundant accuracy, by inspection and simple 
 interpolation ; and in the same manner the channels from the 
 discharges. Rivers have seldom greater falls than those 
 given in TABLE XII., but in such an event we have only to 
 divide the fall by 4, then twice the corresponding discharge 
 will be that required. TABLE XIII. gives the comparative 
 discharging powers of all similar channels, whether pipes or 
 rivers, and the comparative dimensions from the discharges. 
 We perceive from it, that an increase of one-third in the 
 dimensions doubles, and a decrease of one-fourth reduces 
 the discharge to one-half. By means of this table, we can 
 determine by a simple proportion, the dimensions of any 
 given form of channel when the discharge is known. See 
 EXAMPLE 17, page 12.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 125 
 
 The mean widths in TABLES XI. and XII. are calculated 
 for rectangular channels, and those having side slopes of 
 1-J- to 1. Both these tables are, however, practically, equally 
 applicable to any side slopes from to 1 up to 2 to 1, 
 or even higher, when the mean widths are taken and not 
 those at top or bottom. A semihexagon of all trapezoidal 
 channels of equal area has the greatest discharging power, 
 and the semisquare and all rectangles exactly the same as 
 channels of equal areas and depths with side slopes of 
 1^ to 1. The maximum discharge is obtained between these 
 for the semihexagon with side slopes, of nearly to 1, but for 
 equal areas and depths the discharge decreases afterwards as 
 the slope flattens. The question of "HOW MUCH?" is here, 
 however, a very important one; for, as we have already 
 pointed out in equations (28) and (31), the differences for 
 any practical purposes may be immaterial. This is par- 
 ticularly so in the case of channels with different side slopes, 
 if, instead of the top or bottom, we make use of the mean 
 width to calculate from. We then have only to subtract the 
 ratio of the slope multiplied by the depth to find the bottom, 
 and add it to find the top. If the mean width be 50 feet, the 
 depth 5 feet, and the side slopes 2 to 1, we get 50 (2 x 5) 
 = 40 for the bottom, and 50 + (2 x 5) = 60 for the top width. 
 
 Side slopes of 2 to 1 present a greater difference from the 
 mean slope of 1^ to 1, than any others in general practice 
 when new cuts are to be made. A triangular channel 
 having slopes of 2 to 1, and bottom equal to zero, differs 
 more in its discharging power from the half square, equal to 
 it in depth and area, than if the bottom in each was equally 
 increased, yet even here it is easy to show that this maximum 
 difference is only 5 per cent. If the bottom be increased 
 so as to equal the depth, it is only 4 per cent.; when equal to 
 twice the depth, 3'8 per cent. ; and when equal to four times 
 the depth, to 2 per cent.; while the differences in the di- 
 mensions taken in the same order are only 2*2, T8, l - 5, and 
 0'8 per cent. For greater bottoms in proportion to the depth 
 the differences become of no comparative value. It there- 
 fore appears pretty evident, that TABLES XI. and XII. will
 
 126 THE DISCHARGE OF WATER FROM 
 
 be found equally applicable to all side slopes from to 1 
 up to 2 to 1, by taking the mean widths. When new cuts 
 are to be made, we see no reason whatever in starting from 
 bottom rather than mean widths, to calculate the other 
 dimensions; indeed, the necessary extra tables and calcu- 
 lations involved ought entirely to preclude us from doing so. 
 Besides, the formulae for finding the discharge vary in them- 
 selves, and for different velocities the coefficient of friction 
 also varies.* Added to which the inequalities in every 
 river channel, caused by bends and unequal regimen, pre- 
 cludes altogether any regularity in the working slopes and 
 bottom, though the mean width would continue pretty uniform 
 under all circumstances. 
 
 The quantities in TABLE XII. are calculated, from the 
 velocities found from TABLE VIII., to correspond to a channel 
 70 feet wide and of different depths, the equivalents to which 
 are given in TABLE XI. In order to apply these tables 
 generally to all open channels, the latter are to be reduced to 
 rectangular ones of the same depth and mean width, or the 
 reverse, as already pointed out. If the dimensions of the 
 given channel be not within the limits of TABLE XL, divide 
 the dimensions of the larger channels by 4, and multiply the 
 corresponding discharge found in TABLE XII. by 32; for 
 smaller channels, multiply the dimensions by 4 and divide 
 by 32. In like manner, if the discharge be given and ex- 
 ceed any to be found in TABLE XIII., divide by 32, and 
 multiply the dimensions of the suitable equivalent channel 
 found in TABLE XL by 4. If we wish to find equivalent 
 channels of less widths than 10 feet for small discharges, 
 multiply the discharge by 32 and divide the dimensions of 
 the corresponding equivalent by 4. Many other multipliers 
 
 * The coefficients of friction in rivers for velocities from 3 inches to 
 3 feet per second, varies from about '0082 to '0075 ; yet, strange to say, 
 most tables are calculated from one coefficient alone; or, rather, from 
 a formula equivalent to 94-17 (rs)*, which gives larger results than 
 either, see p. 14. Dimensions of channels calculated by means of this 
 formula are too small. In pipes the variation of the coefficients is 
 shown in the small table, p. 120.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 127 
 
 and divisors as well as 4 and 32 may be found from TABLE 
 XIII., such as 3 and 15'6, 6 and 88'2, 7 and 130, 9 and 243, 
 10 and 316, 12 and 499, &c. The differences indicated at 
 pages 111 and 112, must be expected in the application of 
 these rules, which will give, however, dimensions for new 
 channels which can be depended on for doing duty. 
 
 It will be seen from TABLE XIII. that a very small 
 increase in the dimensions increases the discharging power 
 very considerably. TABLE XII. also shows that a small 
 increase in the depth alone adds very much to the discharge. 
 If we express in this latter case a small increase in the depth, 
 
 d, by -, then it is easy to prove that the corresponding 
 
 increase in the velocity, v, will be -^-; and that in the 
 
 2n 
 
 discharge D, , if the surface inclination continue un- 
 2 n 
 
 changed ; but as it is always observable in rivers that the 
 surface inclinations increase with floods, the differences in 
 practice will be found greater than these expressions make it. 
 As in large rivers the surface inclinations are very small, four 
 times the amount will add veiy little to the sectional areas, 
 yet this increase will double fully the discharge, and we 
 thence perceive how rivers may be absorbed into others 
 without any great increase of the depths. 
 
 SECTION IX. 
 
 BEST FORMS OF THE CHANNEL. REGIMEN. 
 
 We have seen above, that the determination of the hy- 
 draulic mean depth does not necessarily determine the section 
 of the channel. If the form be a circle, the diameter is 
 four times the mean radius ; but, though this form be almost 
 always adopted for pipes, the beds of rivers take almost
 
 128 
 
 THE DISCHARGE OF WATER FROM 
 
 every curvilineal and trapezoidal shape. Other things being 
 
 the same, that form of a river channel, in which the area 
 
 divided by the border is a maximum, is the best. This is 
 
 a semicircle having the diameter for the surface line, and 
 
 in the same manner, half 
 
 the regular figures, an oc- 
 
 tagon, hexagon, and square, 
 
 in Fig. 33, are better forms 
 
 for the channel, the areas 
 
 being constant, than any 
 
 others of the same number 
 
 of sides. Of all rectangular 
 
 channels, Diagram 4, in which A BCD is half a square, is 
 
 the best cross section; and in Diagram 3, ACDB, half a 
 
 hexagon, is the best trapezoidal form of cross section. 
 
 When the width of the bottom, CD, Diagram 3, is given, 
 
 and the slope = n, then, in order that the discharge 
 may be the greatest possible, we must have 
 
 and 
 
 = {- - ^ - - }, 
 L2 (n a + 1)* n) 
 
 CD = n x ca 
 ca 
 
 in which c is the given area of the channel. As, however, we 
 have never known a river in which the slope of the natural 
 banks continued uniform, even though made so for any im- 
 provements, we consider it almost unnecessary to give tables 
 for different values of n. If, notwithstanding, we put p for 
 the inclination of the slope AC, equal angle CAa, we shall 
 
 find, as cot. <p = n, and 
 equations become 
 
 ** + 1 = - - > that the foregoing 
 
 CD
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 129 
 
 and 
 
 (121.) CD = -^ caxcot.?, 
 
 ca 
 
 which will give the best dimensions for the channel when the 
 angle of the slope for the banks is known. 
 
 When the discharge from a channel of a given area, with 
 given side slopes, is a maximum, it is easy to show that 
 
 THE HYDRAULIC MEAN DEPTH MUST BE HALF OF THE CENTRAL 
 
 OR GREATEST DEPTH. This simple principle enables us to con- 
 struct the best form of channel with great facility. Describe 
 any circle on the drawing-board; draw the diameter and 
 produce it on both sides, outside the circle; draw a tangent 
 to the lower circumference parallel to this diameter, and draw 
 the side slopes at the given inclinations, touching the circum- 
 ference also on each side and terminating on the parallel 
 lines: the trapezoid thus formed will he the best form of 
 channel, and the width at the surface will be equal to the 
 sum of the two side slopes. It is easy to perceive that this 
 construction may be, simply, extended for finding the best 
 form of a channel having any polygonal border whatever of 
 more sides than three and of given inclinations. 
 
 Commencing with the best form of channel, which in prac- 
 tice will have the mean width, about double the depth, an 
 equal discharging section of double the width of the first will 
 have the contents one-eleventh greater, and the depth less in 
 the proportion of 1 to 1*85. A channel of double the mean 
 width of the second must have the sectional area further in- 
 creased by about one-fifth, and a further decrease in the depth 
 from 1'67 to 1 nearly. The greater expense of the excava- 
 tion at greater depths will, in general, more than counter- 
 balance these differences in the contents of the channel. 
 When the banks rise above the flood line, and are unequal in 
 their section, the wider channel involves further upper extra 
 cutting, but there is greater capacity to discharge extra and 
 extraordinary flooding, the banks are less liable to slip or 
 give way, the slopes may be less, and the velocity being also 
 less, the regimen will, in general, be better preserved. 
 
 K
 
 130 
 
 THE DISCHARGE OF WATER FROM 
 
 
 Area in terms 
 of the depth d. 
 
 *53cOOOC<l^3o(Nl~-t CO 
 ^-t ^H Al <N <N <j* IH ' 
 
 tn of the top and 
 in multiplied by 
 he numbers cor- 
 the best form oi 
 
 
 5 
 
 
 S 9 "g 
 
 
 Ratio of top 
 depth/ 
 
 SSSSS-2S3S2 
 
 PJt- OO CO 00 <M 
 O CM OO O OO * 
 p cp op <p op * 
 
 C^CTC^OTOTM-^<MO 
 
 I'^l^ 
 
 
 | 
 
 
 ** j-. 3 
 
 I 
 
 1! 
 
 33 
 a 
 
 K 
 
 S23S2-2322-2 
 
 I1111SSII 
 
 (N TH 
 
 |l|i 
 
 i 
 
 s"! 
 
 
 |t||l 
 
 B 
 
 ill 
 n 
 
 COCOCOCOCQCOCOCOCOC^ 
 
 si^l 
 
 b 
 
 
 Top in terms of 
 the area. 
 
 |o [o |o lo lo fo lo fo lo 
 
 ^^^^^^i^0 
 
 iHl^l^C^C^CNCNCNTH" 
 
 111 
 
 
 
 I 
 
 
 l-=lil 
 
 
 
 c S 
 g| 
 
 *iOOCOi-lh-CNOOO 
 
 3 ||I| 
 
 M 
 
 r 
 
 ^Hoscpas^^epopoo 
 
 l|l|| 
 
 
 I 
 
 3 
 
 Depth in terms 
 of the area. 
 
 lo lo lo lo fo lo lo lo lo lo 
 
 S^S^SSoSH^ 
 
 " c-3^ g 
 
 
 1 
 
 
 '* 1 2 "8 1 
 
 
 11 
 
 r 
 
 mi mill 
 
 fill 
 
 
 "3 
 
 oio-j<-*neoeoeocq SO 
 m 
 
 B llll
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 131 
 
 When the sectional area is given, the above table shows 
 that the semicircle is the best discharging channel, and the 
 complete circle the worst; the latter is so, however, only 
 compared with the open channels given in the table, it being 
 the very best form for an enclosed channel flowing full. The 
 best form of channel is particularly suited for new cuts in 
 flat, marsh, callow, and fen lands, in which it is also often 
 advisable to cut them with a level bed, up from the dis- 
 charging point, in order to increase the hydraulic mean 
 depth, and consequently the velocity and discharge. 
 
 As the quantity of water coming down a river channel in 
 a season varies very considerably, we have observed it in 
 one case to vary from one to thirty, and occasionally in the 
 same channel from one to seventy-five, the proportion of the 
 water section to the channel itself must also vary, and those 
 relations of the depth, sides, and width to each other, above 
 referred to, cease to hold good and be the best under such 
 circumstances. If the object be to construct a mill race, 
 temporary drain for unwatering a river, or other small chan- 
 nel, in which the depth remains nearly constant, channels 
 of the form of a half hexagon, diagram 3, Fig. 33, will be, 
 perhaps, the best, if the tenacity of the banks permit the 
 slope ; but rivers, in which the quantity of water varies con- 
 siderably, require wider channels in proportion to the depth ; 
 and also, that the velocity be so proportioned to the tenacity 
 of the soil, or as it is termed " the regimen," that the banks 
 and bed shall not vary from time to time to any injurious 
 extent, and that any deposits made during their summer state, 
 and during light freshes, shall be carried off periodically by 
 floods. Another circumstance, also, modifies the effects of 
 the water on the banks. It is this, that at curves, and turns, 
 the current acts with greatest effect against the bank, con- 
 cave to the direc- 
 tion in which it is 
 moving ; deepening 
 the channel there ; 
 undermining also the 
 bank, as at A, Fig. 
 34; and raising the
 
 132 THE DISCHARGE OF WATER FROM 
 
 bed towards the opposite side B. The reflexion of the cur- 
 rent to the opposite bank from A acts also in a similar man- 
 ner, lower down, upon it; and this natural operation pro- 
 ceeds, until the number of turns, increased length of channel, 
 and loss of head from reflexion and unequal depths, brings 
 the currents into regimen with the bed and banks. At all 
 bends it is, therefore, prudent to widen the channel on the 
 convex side B, Fig. 34, in order to reduce the velocity of 
 approach, and if the bed be here also sunk below its natural 
 inclination, as we see it in most rivers at bends, the velocity 
 will be farther reduced, and the permanence of the bed be 
 better established. 
 
 The circumstances to be considered in deciding on the 
 dimensions and fall of a new river course, after the depth to 
 which the surface of the water is to be brought has been 
 decided on, are the following: 
 
 The mean velocity must not be too slow, or aquatic plants 
 will grow, and deposits take place, reducing the sectional 
 area until a new and smaller channel is formed within the 
 first with just sufficient velocity to keep itself clear. Thia 
 velocity should not in general be less than from ten to fourteen 
 inches per second. The velocity in a canal or river is increased 
 very considerably by cutting or removing reeds and aquatic 
 plants growing on the sides or bottom*. 
 
 The mean velocity must not be too quick, and should be 
 so determined as to suit the tenacity and resistance of the 
 channel, otherwise the bed and banks will change continually, 
 unless artificially protected ; it should not exceed 
 
 * M. Girard a fait observer, avec raison, que les plantes aquatiques, 
 qui croissent toujourS sur le fond et sur les berges des canaux, augment- 
 ent considerablement le perirnetre mouille, et par suite la resistance ; il 
 a rapelle que Dubuat, ayant rnesure la vitesse de 1'eau dans le canal du 
 Jard, avant et apres la coupe des roseaux dont il etait garni, avait trouve 
 un resultat bien moindre avant qu'apres. En consequence, il a presque 
 double la pente donnee par le calcul . . ." Traite d'Hydraulique, 
 p. 135. When the fall does not exceed a few inches per mile, the velocity, 
 as determined from the inclination, is very uncertain, and for this reason 
 it is always prudent to increase the depths and sectional areas of chan- 
 nels in flat lands, as far as the regimen will permit. In such cases the 
 section of the channel should approximate towards the best form. See 
 p. 130.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 133 
 
 25 feet per minute in soft alluvial deposits. 
 
 40 clayey beds. 
 
 60 sandy and silty beds. 
 
 120 gravelly. 
 
 180 strong gravelly shingle. 
 
 240 shingly. 
 
 300 shingly and rocky. 
 
 400 and upwards in rocky and shingly. 
 
 The fall per mile should decrease as the hydraulic mean 
 depth increases, and both be so proportioned that floods may 
 have sufficient power to carry off the deposits, if any, 
 periodically. The proportion of the width to the depth of 
 the channel should not be derived, for new cuts or river 
 courses, from any formula, but taken from such portions of 
 the old channel as approximate in depth and in the inclination 
 of the surface to that proposed. When the depth is nearly 
 half the width, the formula shows, cceteris paribus, that the 
 discharge will be a maximum ; but as (altogether apart from 
 the question of expense) the quantity of water discharged 
 daily, at different seasons, may vary from one to seventy, or 
 more, and " the regimen" has to be maintained, the best 
 proportion between the width and depth of a new cut 
 should be obtained, as we have stated, from some selected 
 portion of the old channel, whose general circumstances and 
 surface inclination approximate to those of the one proposed ; 
 and the side slopes of the banks must be such as are best 
 suited to the soil. The resistance of the banks to the current 
 being in general less than that of the beds, which get covered 
 with gravel, and the necessaiy provision required for floods, 
 appears to be the principal reason why rivers are in general 
 so very much wider than about twice the depth, the relation 
 which gives the minimum of friction. 
 
 For enclosed channels, the circular form of sewer flowing 
 full, will have the largest scouring power, at a given hvdraulic 
 inclination. For the area of the sections being the same, 
 the velocity in the circular channel will be a maximum. 
 When the supply is intermittent, and the channel too large, 
 the egg-shaped form with the smaller end for the bottom, 
 or the sides vertical with an inverted ridge tile or V bottom 
 for drains, will have a hydrostatic flushing power to remove
 
 134 THE DISCHARGE OF WATER FROM 
 
 soil and obstructions, which a cylindrical channel, only partly 
 full, does not possess ; because a given quantity of water rises 
 higher against the same obstruction, or obstacle, to the flow. 
 It must be confessed, however, that for small drains and 
 house-sewage, this gain is immaterial, and is at best but 
 effected by a sacrifice of space, material, and friction in the 
 upper part of drains, from 4 to 12 or 15 inches in diameter. 
 Besides this, the mere hydrostatic pressure is only intermit- 
 tent, and during an ordinary, or heavy, fall of rain, the hydro- 
 dynamic power is always more efficient in scouring properly- 
 proportioned cylindrical drains ; and the workmanship in the 
 form and joints is less imperfect than for more compound 
 forms, as those with egg-shaped and inverted tile bottoms. 
 The moulds and joints of cylindrical stone-ware drains, ex- 
 ceeding 15 inches in diameter, are seldom, however, perfect; 
 and the expense will exceed that of brick, stone, or other 
 sufficient drains in most localities. 
 
 As to the increased discharging power which it is asserted 
 by some, stone-ware cylindrical drains possess over other 
 ordinary drains, no doubt it is true for small sizes, because the 
 form, jointing, and surface are in general more smooth and 
 circular ; and for sewage matter*, the friction and adherence 
 to the sides and bottom is less ; any advantage from these 
 causes becomes, however, immaterial for the larger sizes, as 
 these can be constructed of brick abundantly perfect to any 
 form, and sufficiently smooth for all practical purposes, for in 
 the larger properly-proportioned sizes the same amount of 
 surface roughness opposed to the sewage matter is, compara- 
 tively, of no effect. The judicious inclination and form of 
 the bottom, and properly curved junctions, are the principal 
 points to be attended to. Smaller drains tile-bottomed, with 
 brick or stone sides, and flat-covered, have one great advan- 
 tage over circular pipesf. They can be opened up, for exa- 
 
 * Weisbach found the coefficient of resistance 1-75 times as great for 
 small wooden as for metallic pipes. All permeabk pipes present greater 
 resistance than impermeable ones ; hence the principal advantage derived 
 from glazing. 
 
 t Half-socket joints at bottom would remedy this imperfection in 
 small pipes, and they could be better laid and cemented. A semicircular 
 flange laid on at top would effectually protect the joint on the upper side.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 135 
 
 mination and repairs at any time with facility, and at the 
 smallest expense, but greater certainty must be attached to 
 the working of small stone-ware drains than to equal sized 
 small brick or stone drains, and they will be found, in general, 
 also the cheaper. This, however, will depend on the locality. 
 In order that different sections should have the same dis- 
 charging power, the inclination of the surface being the same, 
 the areas must be inversely as the square roots of the 
 hydraulic mean depths. The channel adcs, Fig. 35, will 
 have the same discharge as the channel A D c B if they be to 
 
 f ADCB )i ( adcB 
 each other as > 
 
 and hence the square root of the cube of the channel area, 
 divided by the border, must be constant. With a fall of one 
 or more feet to a mile, two channels, one 70 feet wide and 
 1 foot deep, and the other 20 feet wide and 2 feet deep, 
 will have the same discharge. If we put w for the width 
 and d for the depth of any rectangular channel, then 
 
 d*- 
 
 j 2 fj \ m * we therefore have the cubic equation 
 (122.) 
 
 for finding the depth d of any other rectangular channel 
 whose width is w, of the same discharging power. We have 
 calculated the depths d for different widths of channel from 
 this equation, assuming a width of 70 feet and different depths 
 to find m from. This table will be found sufficiently accurate 
 for all practical purposes, by taking a mean width, when the 
 banks are sloped. 
 
 If the hydraulic inclinations vary, then the \/rs must be 
 inversely as the areas of the channels when ^/r s x channel 
 or the discharge is constant ; and if the area of the channel
 
 136 
 
 THE DISCHARGE OF WATER PROM 
 
 and discharge be each constant, r must vary inversely as * ; 
 and r s be also constant. For instance, a channel which has 
 a fall of four feet per mile, and a hydraulic mean depth of 
 one foot, will have the same discharge as another channel of 
 equal area, having a hydraulic mean depth of four feet and 
 a fall per mile of only one foot. 
 
 DISCHARGING} BEOTANGTJLAB CHANNELS*. 
 
 Values 
 of m. 
 
 The widths in feet are given in the top horizontal line, and the corresponding 
 depths in feet in the other horizontal lines. 
 
 70 
 
 60 
 
 50 
 
 40 
 
 35 
 
 30 
 
 25 
 
 20 
 
 15 
 
 10 
 
 8-7 
 24-6 
 
 0-25 
 0-5 
 
 
 
 55 
 
 30 
 62 
 
 35 
 73 
 
 40 
 80 
 
 45 
 89 
 
 52 
 1-02 
 
 68 
 1-19 
 
 71 
 1-48 
 
 98 
 2-04 
 
 45-0 
 
 75 
 
 82 
 
 94 
 
 1-10 
 
 1-20 
 
 1-35 
 
 1-56 
 
 1-82 
 
 2-28 
 
 3-22 
 
 69-0 
 
 1-0 
 
 1-10 
 
 1-26 
 
 1-48 
 
 1-62 
 
 1-81 
 
 2-10 
 
 2-46 
 
 3-11 
 
 4-50 
 
 96-9 
 
 1-25 
 
 1-39 
 
 1-58 
 
 1-86 
 
 2-04 
 
 2-28 
 
 2-65 
 
 3-12 
 
 3-98 
 
 5-89 
 
 126 
 
 1-5 
 
 1-67 
 
 1-90 
 
 2-24 
 
 2-46 
 
 2-75 
 
 3-20 
 
 3-80 
 
 4-88 
 
 7-31 
 
 158 
 
 1-75 
 
 1-95 
 
 2-22 
 
 2-62 
 
 2-88 
 
 3-23 
 
 3-75 
 
 4-50 
 
 5-80 
 
 8-86 
 
 193 
 
 2-0 
 
 2-23 
 
 2-54 
 
 3-00 
 
 3-31 
 
 3-72 
 
 4-32 
 
 5-22 
 
 6-78 
 
 10-50 
 
 267 
 
 2-5 
 
 2-79 
 
 3-18 
 
 3-76 
 
 4-16 
 
 4-70 
 
 5-50 
 
 6-68 
 
 8-84 
 
 14-00 
 
 349 
 
 3-0 
 
 8-35 
 
 3-84 
 
 4-54 
 
 5-04 
 
 572 
 
 6-69 
 
 8-22 
 
 11-03 
 
 17-68 
 
 437 
 
 3-5 
 
 3-91 
 
 4-50 
 
 5-33 
 
 5-95 
 
 6-75 
 
 7-93 
 
 9-82 
 
 13-32 
 
 21-68 
 
 531 
 629 
 
 4-0 
 4-5 
 
 4-48 
 5-05 
 
 514 
 5-79 
 
 6-13 
 6-95 
 
 6-85 
 
 7-75 
 
 7-81 
 8-90 
 
 9-21 
 10-50 
 
 11-48 
 13-19 
 
 15-75 
 18-22 
 
 26-00 
 30-36 
 
 732 
 
 5-0 
 
 5-62 
 
 6-45 
 
 7-75 
 
 8-66 
 
 10-00 
 
 11-79 
 
 14-96 
 
 20-80 
 
 35-00 
 
 839 
 
 5-5 
 
 6-18 
 
 7-12 
 
 8-57 
 
 9-62 
 
 11-10 
 
 13-24 
 
 16-77 
 
 23-47 
 
 39-81 
 
 951 
 
 6-0 
 
 6-75 
 
 7-80 
 
 9-40 
 
 10-60 
 
 12-22 
 
 14-65 
 
 18-65 
 
 26-25 
 
 44-86 
 
 SECTION X. 
 
 EFFECTS OF ENLARGEMENTS AND CONTRACTIONS. BACK- 
 WATER WEIR CASE. LONG AND SHORT WEIRS. 
 
 WHEN the flowing section in pipes or rivers expands or con- 
 tracts suddenly, a loss of head always ensues ; this is pro- 
 
 * This table is enlarged in TABLE XI., and is as equally applicable to 
 all other measures, inches, yards, fathoms, &c., as to feet.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 137 
 
 bably expended in forming eddies at the sides, or in giving 
 the water its new section. A side current, moving slowly 
 upwards, may be frequently observed in the wide parts of 
 rivers, when the channel is unequal, though the downward 
 current, at the centre, be pretty rapid ; and though we may 
 assume generally that the velocities are inversely as the sec- 
 tions, when the channels are uniform, we cannot properly do 
 so where the motions are so uncertain as those referred to. 
 When a pipe is contracted by a 
 diaphragm at the orifice of entry, 
 Fig. 27, we have seen (equation 
 60), that the loss of head is, 
 
 Fig-,36 
 
 (123.) 
 
 When the diaphragm is placed in a uniform pipe, Fig. 36, 
 then A =. c, and we get the loss of head 
 
 (124.) 
 
 and the coefficient of resistance 
 
 as in equation (67). The coefficient of discharge c d is here 
 equal to the coefficient of contraction c c , or very nearly. Now 
 we have shown in equation (45), and the remarks following 
 it, that the value of the coefficient of discharge, c d , varies ac- 
 
 cording to the ratio of the sections, , and in TABLE V. we 
 
 have calculated the new coefficients for different values of the 
 ratios, and different values of the primary coefficient <r d . If 
 we assume c d , when A is very large compared with , to be 
 628, we then find by attending to the remarks at pp. 61 and 
 
 62, that the different values of c d corresponding to '807 x , 
 taken from TABLE V., are those in columns Nos. 2 and 5 of
 
 138 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE OF COEFFICIENTS FOR CONTRACTION, BY A DIAPHRAGM IN A PIPB. 
 
 a 
 
 A 
 
 c d 
 
 c r 
 
 a 
 
 A 
 
 c d 
 
 c r 
 
 
 
 628 
 
 infinite 
 
 6 
 
 713 
 
 1-790 
 
 1 
 
 630 
 
 221-2 
 
 7 
 
 753 
 
 807 
 
 2 
 
 636 
 
 47-1 
 
 8 
 
 807 
 
 301 
 
 3 
 
 647 
 
 17'2 
 
 85 
 
 845 
 
 154 
 
 4 
 
 661 
 
 7-7 
 
 9 
 
 890 
 
 062 
 
 5 
 
 683 
 
 3-7 
 
 1 
 
 1-000 
 
 000 
 
 the above small table, the values of the coefficient of resist- 
 ance, in columns 3 and 6, being calculated from equation 
 (125) for the respective new values of the coefficient of dis- 
 charge thus found. The table shows that when the aperture 
 in a diaphragm is ^ths of the section of the pipe, that 47 
 times the head due to the velocity is lost thereby. If the 
 aperture in the diaphragm be rounded, the loss will not be so 
 great, as the primary coefficient e d will then be greater than 
 that due to an orifice in a thin plate : see coefficients, p. 96. 
 When there are a number of diaphragms in a tube, the loss 
 
 of head for each must be found separately, unless the ratio 
 
 be constant, and all added together for the total loss. If the 
 diaphragms, however, approach each other, so that the water 
 issuing from one of the orifices a, Fig. 36, shall pass into the 
 next before it again takes the velocity due to the diameter of 
 the pipe, the loss will not be so great as when the distance is 
 sufficient to allow this change to take place. This view is 
 fully borne out by the experiments of Eytelwein with tubes 
 1*03 inch in diameter, having apertures in the diaphragms 
 of '51 inch in diameter. 
 
 Venturi's twenty-fourth experiment, with tubes varying from 
 75 inch to '934 inch in diameter at the junction with the cistern, 
 so as to take the form of the contracted vein, and expanding 
 and contracting along the length from '75 to 2 inches and from
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 139 
 
 2 inches to '75 inch alternately, shows the great loss of head 
 sustained by successive enlargements and contractions of a 
 channel, even when the junction of the parts is gradual. 
 Calling the coefficient for the short tube, with a junction 
 of nearly the form of the contracted vein, 1, then the following 
 coefficients are derivable from the experiment : 
 
 Short tube with rounded junction 1* 
 
 One enlargement '741 
 
 Three enlargements '569 
 
 Five enlargements '454 
 
 Simple tube with a rounded junction of the same 
 length, 36 inches, as the tube with the five en- 
 larged parts -736 
 
 The head, in the experiment, was 32 inches. Venturi states 
 that no observable differences occurred in the times of dis- 
 charge when the enlarged portions were lengthened from 
 3| to 6^ inches. 
 
 With reference to this experiment, so often quoted, it is 
 necessary to remark that the diameters of the enlarged por- 
 tions were 2 inches each, while the lengths varied only from 
 3<L to 6^ inches, and consequently were at most only 3 th times 
 the diameter. Now with such a large ratio of the width to 
 the length of the enlarged 
 portions, a A B b, Fig. 37, it is 
 pretty clear that a good deal 
 of the head is lost by the 
 impact of the moving water 
 on the shoulders A and B. 
 That this is so is evident from the fact, stated by the experi- 
 menter, of the time of discharge remaining the same when 
 a A, in five different enlargements, was increased from 
 3^ to 6^ inches; though this must have lengthened the 
 whole tube from 36 to 50 inches*, thereby increasing the 
 loss from friction proportionately, but which happened to be 
 compensated for by the reduction in the resistances from 
 
 Fig. 37 
 
 * The dimensions throughout this experiment are given as in the 
 original, viz. in French inches.
 
 140 THE DISCHARGE OF WATER FROM 
 
 impact at A and B, and in the eddies, by doubling the lengths 
 from a to A. 
 
 If, however, the length from a to A be very large compared 
 with the diameter, and the junctions at a, A, B, and b, be well 
 grafted, less loss will arise from the enlargement than if the 
 smaller diameter continued all along uniform. The explanation 
 is clear, as the resistance from friction is inversely as the 
 square roots of the mean radii; and the length being the 
 same, the loss must be less with a large than a small 
 diameter. 
 
 These remarks, mutatis mutandis, apply equally to rivers 
 as to pipes. We have already, pp. 72 and 75, pointed out the 
 effects of submerged weirs and contracted river channels, and 
 given formulae for calculating them. 
 
 BACKWATER FROM CONTRACTIONS IN RIVERS. 
 
 A river may be contracted in width or depth by jetties or 
 weirs ; and when the quantity to be discharged is known, we 
 have given, in formulae (9), (55), and (57), equations from 
 which the increase of head may be determined. The effect 
 of a weir, jetty, or contracted channel of any kind, is to 
 increase the depth of water above; and this is sometimes 
 necessary for navigation purposes or to obtain a head for 
 mill power. When a weir is to rise over the surface, we can 
 easily find, from the discharge and length, the discharge per 
 minute over each foot of length, with which, and the coeffi- 
 cient due to the ratio of the sections, on and above the weir, 
 found from TABLE V., we can find the head from TABLE VI. 
 For submerged weirs and contracted widths of channel, the 
 head can be best calculated, by approximation, from the 
 equations above referred to. 
 
 The head once determined, the extent of the backwater is 
 a question of some importance. If F c o D, Fig. 38, be the 
 original surface of a river, and a A B F the raised surface by 
 backwater from the weir at a, then extent a F of this 
 backwater, in a regular channel, will be from 1*5 to 1'9 times 
 a c drawn parallel to the horizon to meet the original surface
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 141 
 
 Fig-. 38 
 
 in c. This rule will be found useful for practical purposes ; 
 but in order to determine more accurately the rise for a given 
 length, B! B 2 or ^ B, of the channel, it is necessary to com- 
 mence at the weir and calculate the heights from A to B, B to B O 
 and from B! to B 2 separately, the distance from A to B 2 being 
 supposed divided into some convenient number of equal 
 parts, so that the lengths A B, B B W &c., may be considered 
 free from curvature. Now, as the head A D is known, or may 
 be calculated by some of the preceding formulas, the section 
 of the channel at the head of the weir also becomes known, 
 and thence the mean velocity in it, by means of the discharge 
 over the weir. Putting A for the area of the channel at AH, 
 d for its depth A H, and v for the mean velocity ; also AJ for 
 the area of the channel at B i, d t for its depth, and v t for its 
 mean velocity ; b m the mean border between the sections at 
 
 A H and B i ; r m the mean hydraulic depth ; _ - the mean 
 
 velocity ; AV h; B o= ^ ; the sine of angle o D E =: s ; and 
 the length A B = D o nearly = I we get A x v r= A X x v^ and 
 
 r m =. 2 A > but as > * n passing from B to A, the velocity 
 
 t? 2 v 3 
 changes from v t to v, there is a loss of head equal ^ > 
 
 and if c { be the coefficient of friction, there is a loss of head 
 
 from this cause equal c f x 
 
 f m 
 
 ; hence the whole 
 change of head in passing from B to A is equal to 
 c f x x "Tjjrr -hfj But this change of head is 
 
 equal to BE AD = BO +OE AD = h L + Is h, whence 
 we get
 
 142 THE DISCHARGE OF WATER FROM 
 
 (126.) h l -h = d i -d = c { x x <*L t _. = _ i 8 . 
 
 I'm off 2g 
 
 Av A + A! 
 
 or as Vi = , and r m = TTT , we get, by a few reductions 
 and change of signs, 
 
 and therefore we get 
 
 A* -A? 
 
 ft hi - -j x 
 (128.) 1 
 
 from which we can calculate the length I corresponding 
 to any assumed change of level between A and B. Then, 
 by a simple proposition we can find the change of level for 
 any smaller length. To find the change of level directly 
 from a given length does not admit of a direct solution, for 
 the value of h h t in equation (127) involves A : , which de- 
 pends again on h h l} and further reduction leads to an 
 equation of a higher order; but the length corresponding 
 to a given rise, h 1} is found directly by equation (128). 
 
 When the width of the channel, w, is constant, and the 
 section equal to w x d nearly, the above equations admit 
 of a further reduction for A 1 = d L w and A = dw; by sub- 
 stituting these values in equation (127) it becomes, after a 
 few reductions, 
 
 (129.) h h i = d-d i 
 
 or, as it may be further reduced, 
 s - c ,x-^ 
 0300 -*,= - -4 
 
 ~
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 143 
 
 Now, we may take in this equation for all practical purposes, 
 d + d t x b m _ b 
 2,d\ d\ if) dtyD 
 approximately, b being the border of the section at AH; and 
 
 7,7 n 
 
 also, -^ - = -, approximately ; therefore, we shall have 
 d l d 
 
 b v* 
 
 s - c t x _ x _ 
 d w o 
 (131.) A-A, = 2" 
 
 l-x^- 
 
 d 2q 
 
 and 
 (132.) 
 
 Now, as - = , 2or = 64*4, and the mean value of the 
 rfw> r 
 
 coefficient of friction for small velocities c i = '0078, we 
 shall get 
 
 64-4 ds- -0078 -t; 2 
 
 (133.) 1 = A- 
 
 and 
 
 (134.) / = 
 
 64-4 rfs- -0078 
 
 very nearly. Having by means of these equations found 
 AB from BO or BE, and BO from AB, we can in the same 
 manner proceed up the channel and calculate BjC, B 2 c u &c., 
 until the points B, B U B 2 in the curve of the backwater shall 
 have been determined, and until the last nearly coincides 
 with the original surface of the river. When A ; = 0, we 
 shall have 
 
 64-4 ds- -0078 - 
 = 64-4 d-2t; 2 
 
 x L
 
 144 THE DISCHARGE OF WATER FROM 
 
 If we examine equation (134) it appears that when 
 
 r=2t> 2 , I must be equal to zero; or when = _^ , equal 
 
 2 64*4 
 
 the height due to the velocity v. When / is infinite, 64'4 d 
 must exceed 2v 2 , and 64'4 ds equal to -0078- v 2 ; 
 
 = v\ and v = 90'9 v/i7. 
 
 This is the velocity due to friction in a channel of the depth 
 ?, hydraulic mean depth r, and inclination s; and, as in 
 wide rivers r d nearly, v =. 90'9 \/ds, but when the 
 numerator was zero we had from it v = \/32'2d; equating 
 these values of v, we get s = '0039 = nearly : see p. 70. 
 
 Now, the larger the fraction s is, the larger will the velocity 
 v become ; and the larger v becomes, the more nearly, in 
 all practical cases, will the terms 
 
 64-4 d_ 2 1; 2 and 64-4 ds- -0078 -v\ 
 
 in the numerator and denominator of equation (134), approach 
 zero ; when 64-4 d 2 v z becomes zero first, / = ; when 
 
 64-4 ds -0078 v 2 becomes zero first, / equal infinity ; 
 and when they both become zero at the same time, l = h h L , 
 and * = , see p. 70 ; if s be larger than this fraction, the 
 
 numerator in equation (134) will generally become zero before 
 the denominator, or negative, in which cases / will also be zero, 
 or negative; and the backwater will take the form FC 2 b. 2 b l ba 1 a, 
 Fig. 38, with a hollow at c 2 . Bidone first observed a hollow, 
 
 as FC a 2 , when the inclination * was . When the inclina- 
 tion of a river channel changes from greater to less, the 
 velocity is obstructed, and a hollow similar to FC 2 i 2 some- 
 times occurs ; when the difference of velocity is considerable, 
 the upper water at b. z falls backwards towards c 2 and F, and 
 forms a bore, a splendid instance of which is the pororoca, 
 on the Amazon, which takes place where the inclination of
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 145 
 
 the surface changes from 6 inches to ^th of an inch per mile, 
 and the velocity from about 22 feet to 4 feet per second. 
 
 WEIR CASE, LONG AND SHORT WEIRS. 
 
 When a channel is of very unequal widths, above a weir, 
 we have found the following simple method of calculating 
 the backwater sufficiently accurate, and the results to agree 
 very closely with observation. Having ascertained the 
 surface fall due to friction in the channel at a uniform 
 mean section, add to this fall the height which the whole 
 quantity of water flowing down would rise on a weir 
 having its crest on the same level as the lower weir, and 
 of the same length as the width of the channel in the 
 contracted pass. The sum will be the head of water at 
 some distance above such pass very nearly. A weir was re- 
 cently constructed on the river Blackwater, at the bounds of 
 the counties Armagh and Tyrone, half a mile below certain 
 mills, which, it was asserted, were injuriously affected by 
 backwater thrown into the wheel-pits. The crest of the 
 weir, 220 feet long, was 2 feet 6 inches below the pit; the 
 river channel between varied from 50 and 57 feet to 123 feet 
 in width, from 1 foot to 14 feet deep; and the fall of the 
 surface, with 3 inches of water passing over the weir and 
 the sluices down, was nearly 4 inches in the length of half 
 a mile. Having seen the river in this state in summer, the 
 writer had to calculate the backwater produced by different 
 depths passing over the weir in autumn and winter, which 
 in some cases of extraordinary floods were known to rise to 
 3 feet. The width of the channel about 60 feet above the 
 weir averaged 120 feet. The width, 2050 feet above the 
 weir and 550 feet below the mills, was narrowed by a slip ifi 
 an adjacent canal bank, to 45 feet at the level of the top of 
 the weir, the average width at this place as the water rose 
 being 55 feet. The channel above and below the slip widened 
 to 80 and 123 feet. Between the mills and the weir there 
 were, therefore, two passes ; one at the slip, averaging 55 feet 
 wide; another above the weir, about 120 feet wide. As- 
 suming as above, that the water rises to the heights due 
 
 L
 
 146 
 
 THE DISCHARGE OF WATER FROM 
 
 to weirs 55 and 120 feet long, at these passes, we get, by 
 an easy calculation, or by means of TABLE X., the heads in 
 columns two and four of the following table, corresponding 
 to the assumed ones on the weir, given in the first column. 
 
 TABLE OF CALCULATED AND OBSERVED HEIGHTS ABOVE M'KEAJi's 
 WEIR ON THE RIVER BLACKWATER. 
 
 Heights at 
 M'Kean's 
 weir 220 feet 
 
 SS 
 
 Heights 60 feet above the 
 weir channel 120 feet wide. 
 
 Heights 2050 feet above the 
 weir channel 55 feet wide; 
 average. 
 
 Calculated 
 inches. 
 
 Observed 
 inches. 
 
 Calculated 
 inches. 
 
 Observed 
 inches. 
 
 1* 
 
 2i 
 
 24 
 
 
 
 H 
 
 3 
 
 41 
 
 
 7i 
 
 7 
 
 4 
 
 6 
 
 
 10 
 
 9 
 
 5 
 
 n 
 
 
 12| 
 
 Hi 
 
 6 
 
 9 
 
 9 
 
 15 
 
 16| 
 
 7 
 
 101 
 
 104 
 
 171 
 
 18| 
 
 8 
 
 12 
 
 
 20 
 
 204 
 
 9 
 
 18| 
 
 121 
 
 22 1 
 
 201 
 
 10 
 
 15 
 
 
 24i 
 
 20 
 
 11 
 
 16* 
 
 
 271 
 
 24 
 
 12 
 
 18 
 
 17 
 
 30J 
 
 31 
 
 13 
 
 19i 
 
 18i 
 
 32J 
 
 33 
 
 15 
 
 22J 
 
 21 
 
 37f 
 
 40 
 
 18 
 
 27 
 
 25 
 
 45i 
 
 46 
 
 21 
 
 31 i 
 
 294 
 
 53 
 
 54 
 
 24 
 
 36 
 
 34 
 
 60J 
 
 62 
 
 As the length of the river was short, and the hydraulic mean 
 depth pretty large, the fall due to friction for 60 feet above 
 the weir was very small, and therefore no allowance was made 
 for it ; even the distance to the slip was comparatively short, 
 being less than half a mile, and as the water approached it 
 with considerable velocity, this was conceived, as the observa- 
 tions afterwards showed, to be a sufficient compensation for 
 the loss of head below by friction. The observations were 
 made by a separate party, over whom the writer had no con-
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 147 
 
 trol, and it is necessary to remark, that with the same head of 
 water on the weir, they often differed more from each other 
 than from the calculation. This, probably, arose from the 
 different directions of the wind and the water rising during 
 one observation, and falling during another. 
 
 The true principle for determining the head at g, Fig. 39, 
 apart from that due to friction, is that pointed out at pages 
 75 and 76 ; but when the passes are very near each other, 
 or the depth c? 2 , Fig. 23, is small, the effect of the discharge 
 through d z is inconsiderable in reducing the head, as the con- 
 traction and loss of vis-viva are then large, and the head 
 d L becomes that due to a weir of the width of the contracted 
 channel at A, nearly. The reduction in the extent of the 
 backwater, by lowering the head on a longer weir, is found 
 by taking the difference of the amplitudes due to the heads 
 at </, Fig. 39, in both cases, as determined from equations (56,) 
 (128), et seq. This will seldom exceed a mile up the river, 
 as the surface inclination is found to be considerably greater 
 than that due to mere friction and velocity, and hence the 
 general failure of drainage works designed on the assump- 
 tion that the lowering of the head below, by means of long 
 weirs, extends its effects all the way up a channel. We must 
 nearly treble the length of a weir before the head passing 
 over can be reduced by one-half, TABLE X., even supposing 
 the circumstances of approach to be the same ; surely several 
 engineering appliances for shorter weirs, during periods of 
 flood, would be found more effective and far less expensive 
 than this alternative, with its extra sinking and weir basin. 
 
 The advocates for the necessity of weirs longer than the 
 width of the channel, for drainage purposes, must show that 
 the reduction of the head and extent of backwater above g, 
 Fig 39, is not small, and that the effects extend the whole 
 way up the channel, or at least as far as the district to be 
 benefited. Practice has heretofore shown, that long weirs 
 have failed (unless after the introduction of sluices or other 
 appliances) in producing the expected arterial drainage re- 
 sults, notwithstanding the increased leakage from increased 
 length, which must accompany their construction. 
 
 L 2
 
 148 THE DISCHARGE OF WATER FROM 
 
 Fig. 39 
 
 The deepening in the weir basin a b BE A is mostly of use in 
 reducing the surface inclination between a b and A B by in- 
 creasing the hydraulic mean depth; but, thereby, the velocity 
 of approach is lessened, and therefore the head at E increased. 
 When the length of a weir basin a E exceeds that point 
 where these two opposite effects balance each other, there 
 will be a gain by the difference of the surface inclinations in 
 favour of the long weir; but unless a E exceeds half a mile, 
 this difference cannot amount to more than 3 or 4 inches, un- 
 less the river be very small indeed : and if the channel be 
 sunk for the long weirs B A or b a lt it should also be sunk to 
 at least the same depth and extent for the short weirs B e, 
 b a, otherwise there is no fair comparison of their separate 
 merits. The effect of the widening between a b and A B, the 
 depth being the same, is also to reduce the surface inclination 
 from a to E; but, as before, unless E be of considerable 
 length, this gain will also be small. Now A B, at best, is but 
 a weir the direct width of the new channel at A B, and if the 
 length a E be considerable, we have an entirely new river 
 channel with a direct weir at the lower end, and the saving of 
 head effected arises entirely from the larger channel, with a 
 direct transverse weir at the lower end. 
 
 By referring to TABLE VII I., it will be found that for a 
 hydraulic mean depth of 5 feet a fall of 7^ inches per mile 
 will give a velocity of 2 feet per second : if we double the 
 depth, a fall of 4 inches will give the same velocity ; and for 
 a depth of only 2 feet 6 inches, a fall of 12^ inches is neces- 
 sary. This is a velocity much larger than we have ever ob-
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 149 
 
 served in a weir basin, yet we easily perceive that the 
 difference in the inclinations for a short distance, E a of a few 
 hundred feet, must be small. If one section be double the 
 other, the hydraulic mean depth remaining constant, the 
 velocity must be one-half, and the fall per mile one-fourth, 
 nearly. This would leave 7J 2 = 5| inches per mile, or 
 1 inch per 1000 feet nearly, as the gain with a hydraulic mean 
 depth of 5 feet for a double water channel. For greater depths 
 the gain would be less, and the contrary for lesser depths. 
 
 Is the saving of head and amplitude of backwater we have 
 calculated worth the increased cost of long weirs and the con- 
 sequent necessity and expense of sinking and widening the 
 channels for such long distances ? We think not; indeed, the 
 sinking in the basin immediately at the weir is absolutely 
 injurious, by destroying the velocity of approach, and in- 
 creasing the contraction. The gradual approach of the 
 bottom towards the crest, shown by the upper dotted line 
 b E in the section, Fig. 39, and a sudden overfall, will be 
 found more effective in reducing the head, unless so far as 
 leakage takes place, than any depth of sinking for only 60 
 or 80 feet above long weirs. 
 
 In most instances, the extra head will be only perceived by 
 an increased surface inclination, which may extend for a mile 
 or so up the channel, according to the sinking and widening. 
 
 It is a general rule that, for shorter weirs, the coefficients 
 of discharge decrease ; this arises from the greater amount of 
 lateral contraction, and is more marked in notches or Poncelet 
 weirs, than for weirs extending from side to side of the chan- 
 nel ; but for weirs exceeding 10 feet in length the decrease in 
 the coefficients from this cause is immaterial, unless the head 
 passing over bear a large ratio to the length ; and we even 
 see from the coefficients, page 45, derived from Mr. Black- 
 well's experiments, that with 10 inches head passing over a 
 2-inch plank, the coefficient for a length of 3 feet is '614; 
 for a length of 6 feet '539 ; and for a length of 10 feet '534; 
 showing a decrease as the weir lengthens, but which may, in 
 the particular cases, be accounted for. We have before re- 
 ferred to other circumstances which modify the coefficients, 
 yet we may assume generally, without any error of practical
 
 150 
 
 THE DISCHARGE OF WATER FROM 
 
 value, that the coefficients are the same for different weirs ex- 
 tending from side to side of a river. If, then, we put w and 
 w t for the lengths of two such weirs, we shall have the relation 
 of the heads d and d for the same quantity of water passing 
 over, as in the following proportion : 
 
 d : d, : : w\ : w* ', 
 and therefore 
 (135.) 
 
 By means of this equation we have calculated TABLE X., the 
 ratio being given in columns 1, 3, 5 and 7, and the value 
 
 H i j , or the coefficient by which d is to be multiplied, to 
 
 find c? x in columns 2, 4, 6, and 8. It appears also, that if we 
 take the heads passing over any weir in a river in an arith- 
 metical progression, the heads then passing over any other 
 weir in the same river must also be in arithmetical pro- 
 gression, unless the quantity flowing down varies from eroga- 
 tion or supply, such as drawing off by mill races, fyc. 
 
 SECTION XI. 
 
 BENDS AND CURVES. BRANCH PIPES. DIFFERENT LOSSES 
 OF HEAD. GENERAL EQUATION FOR FINDING THE VELO- 
 CITY. HYDROSTATIC AND HYDRAULIC PRESSURE. PIEZO- 
 METER. CATCHMENT BASINS. RAIN FALL PER ANNUM. 
 
 WATER POWER. 
 
 The resistance or loss of head due to bends and curves 
 has now to be considered. If we fix a bent pipe, F B c D E G, 
 Fig. 40, between two cisterns, so as to be capable of re- 
 volving round in collars 
 at F and G, we shall 
 find the time the water 
 takes to sink a given 
 distance from f to F in 
 the upper cistern the 
 same, whether the tube
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 151 
 
 occupy the position shown in the figure or the horizontal po- 
 sition shown by the dotted line vbcdeG. This shows that 
 the resistances due to friction and to bends are independent 
 of the pressure. If the tube were straight, the discharge 
 would depend on the length, diameter, and difference of level 
 between f and G, and may be determined from the mean ve- 
 locity of discharge, found from TABLE VIII. or equation (79.) 
 Here, however, we have to take into consideration the loss 
 sustained at the bends and curves, and our illustration shows 
 that it is unaffected by the pressure. 
 
 The experiments of Bossut, Du Buat, and others, show 
 that the loss of head from bends and curves like that from 
 friction increases as the square of the velocity ; but when 
 the curves have large radii, and the bends are very obtuse, the 
 loss is very small. With a head of nearly 3 feet, Venturi's 
 twenty-third experiment, when reduced, gives for a short 
 straight tube 15 inches long, and 1 inch in diameter, having 
 the junction of the form of the contracted vein, very nearly 
 873 for the coefficient of discharge. When of the same 
 length and diameter, but bent as in Diagram 1, Fig. 40, the 
 coefficient is reduced to '785; and when bent at a right angle 
 as at H, Fig. 40, the coefficient is further reduced to '560. In 
 these respective cases we have therefore, 
 
 1. v = '873 2h and h = 1-312 x - 
 
 v = -785 2h and h = 1-623 x 
 
 3. v = -560 ^/2^h, and h = 3-188 x | ; 
 
 showing that the loss of head in the tube H, Fig. 40, from 
 
 tf 
 the bend, is 1'876 x g , or nearly double the theoretical 
 
 head due to the velocity in the tube. The loss of head by 
 the circular bend is only '311^-, or not quite one-sixth of 
 the other.
 
 152 
 
 THE DISCHARGE OF WATER FROM 
 
 Du Buat deduced, from about twenty-five experiments, 
 that the head due to the resistance in any bent tube 
 ABCDEFGH, Diagram 1, Fig. 41, depends on the number 
 
 of deflections between the entrance at A and the departure at 
 H ; that it increases at each reflection as the square of the 
 sine of the deflected angle, A B R for instance, and as the 
 square of the velocity; and that if <p, ft, ft, ft, &c., be the 
 number of degrees in the angles of deflection at B, c, D, E, 
 &c., then for measures in French inches the height h b , due to 
 the resistances from curves, is 
 
 (136.) A b = 
 
 (sin. 2 <p + sin. 2 ft + sin. 2 ft + sin. 2 ft + &c.) 
 3000 
 
 which for measures in English inches becomes 
 
 tt 2 (sin. 2 p + sin. 2 ft + sin. 2 ft + sin. 2 ft + &c.) 
 (Id/.) n b 
 
 3197 
 
 and for measures in English feet, 
 
 (138.) h b = 
 
 v 9 (sin. 2 ? + sin.* ft + sin. 2 ft + sin. 2 ft +&c.) 
 - 
 
 266-4 
 
 or, as it may be more generally expressed for all measures, 
 (139.) A b = (sin.> + sin. 2 
 
 = -0^ '= -00375 in feet. 
 
 in which 
 
 The angle of deflection, in the experiments from which 
 equation (136) was derived, did not exceed 36. We have 
 already shown the loss of head from the circular bend in
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 153 
 
 diagram 1, Fig. 40, where the angle of deflection is nearly 
 
 i 
 45, to be 311 g- = -00483 v* t but as the sin. 45 = -707 
 
 sin. 2 45 = '5, we get -00483 v* == -00966 v* x sin. 2 45, or 
 more than two and a half times as much as Du Buat's 
 formula would give and if we compare it with Rennie's 
 experiments*, with a pipe 15 feet long, inch diameter, bent 
 into fifteen curves, each 3 inches radius, we should find the 
 formula gives a loss of head not much more than one-half 
 of that which may be derived from the observed change, '419 
 to '370 cubic feet per minute in the discharge. See p. 156. 
 
 Dr. Young -f- first perceived the necessity of taking into 
 consideration the length of the curve and the radius of cur- 
 vature. In the twenty-five experiments made by Du Buat, 
 he rejected ten in framing his formula, and the remaining 
 fifteen agreed with it very closely. Dr. Young finds 
 
 0000045 f * x v* 
 (140.) h b = - -^ 
 
 where <p is the number of degrees in the curve N p, diagram 
 2, Fig. 41, equal the angle NOP; f zz o N the radius of 
 curvature of the axis ; h b the head due to the resistance of 
 the curve, and v the velocity, all expressed in French inches, 
 This formula reduced for measures in English inches is 
 
 (141.) jt> = "0000044 >f .x^. 
 
 and for measures in English feet, 
 
 Equation (140) agrees to -g^-th of the whole with twenty of 
 Du Buat's experiments, his own formula agreeing so closely 
 with only fifteen of them. The resistance must evidently in- 
 crease with the number of bends or curves ; but when they 
 come close upon, and are grafted into, each other, as in 
 
 * Philosophical Transactions for 1831, p. 438. 
 
 t Philosophical Transactions for 1808, pp. 173-175.
 
 154 THE DISCHARGE OF WATER FROM 
 
 diagram 1, Fig. 41, and in the tube FBC DEG, Fig. 40, the 
 motion in one bend or curve immediately affects those in the 
 adjacent bends or curves, and this law does not hold. 
 
 Neither Du Buat nor Young took any notice of the rela- 
 tion that must exist between the resistance and the ratio of 
 the radius of curvature to the radius of the pipe. Weisbach 
 does, and combining Du Buat's experiments with some of 
 his own, finds for circular tubes, 
 
 (143.) /i^ifox {'131 + 1-847 (0}x;g ; 
 
 and for quadrangular tubes, 
 
 (144.) A b = - 
 
 in which <p is equal the angle N o P = N i R, diagram 2, 
 Fig. 41 ; d the mean diameter of the tube, and f the 
 
 radius N o of the axis. When exceeds "2, the value of 
 
 2f 
 
 131 + 1-847 (A] 1 exceeds -124 + 3-104 ( )*, and the 
 \2f / \ 2f / 
 
 resistance due to the quadrangular tube exceeds that due to 
 the circular one. We have arranged and calculated the follow- 
 ing table of the numerical values of these two expressions 
 for the more easy application of equations (143) and (144). 
 This table will be found of considerable use in calculating the 
 values of equations (143) and (144), as the second and fifth 
 
 columns contain the values of -131 + 1-847 (o~T> and the 
 
 third and sixth columns the values of -124 + 3-104 (^- V, 
 
 d 
 
 corresponding to different values of f j- ; and it is carried to 
 
 twice the extent of those given by Weisbach. 
 
 For bent tubes, diagrams 3, 4, and 5, Fig. 41, the loss of 
 head is considerably greater than for rounded tubes. If, as 
 before, we put the angle N i R = tp, i R being at right angles to 
 i o the line bisecting the angle or bend, we shall find, by
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 155 
 
 TABLE OF THE VALUES OF THE EXPRESSIONS 
 
 131 + 1-847 (j-\* and -124 + 3-104 (|-\l 
 
 d 
 
 *e 
 
 Circular 
 tubes. 
 
 Quadrangular 
 tubes. 
 
 d 
 2j 
 
 Circular 
 tubes. 
 
 Quadrangular 
 tubes. 
 
 i 
 
 131 
 
 124 
 
 .6 
 
 440 
 
 643 
 
 15 
 
 133 
 
 128 
 
 65 
 
 540 
 
 811 
 
 2 
 
 138 
 
 135 
 
 7 
 
 661 
 
 1-015 
 
 25 
 
 145 
 
 148 
 
 75 
 
 806 
 
 1-258 
 
 3 
 
 158 
 
 170 
 
 8 
 
 977 
 
 1-545 
 
 35 
 
 178 
 
 203 
 
 85 
 
 1-177 
 
 1-881 
 
 4 
 
 206 
 
 250 
 
 9 
 
 1-408 
 
 . 2-271 
 
 45 
 
 244 
 
 314 
 
 95 
 
 1-674 
 
 2-718 
 
 5" 
 
 294 
 
 398 
 
 1-00 
 
 1-978 
 
 3-228 
 
 decomposing the motion, that the head -~- becomes 
 
 o X cos. 2 2 <p from the change of direction ; and that a loss 
 of head 
 
 (145.) 
 
 A b = (1 - cos. 2 2 p) ~ = sin.' 2 p 
 
 must take place. When the angle is a right angle, as 
 in diagram 4, cos. 2 p == 0, and A b = ^~ ; that is to say, 
 
 the loss of head is exactly equal to the theoretical head. 
 When the angle or bend is acute, as in diagram 5, the loss of 
 
 head is(l + cozSZtp)^-- , for then cos. 2<p becomes negative. 
 
 Weisbach does not find the loss of head in a right angular 
 
 y 2 
 bend greater than *984 ^ 5 while Venturi's twenty-third ex- 
 
 periment, made with extreme care, p. 151, shows the loss to be 
 1*876 o~. When the pipes are long, however, the value of 
 
 * The values corresponding to - = '55 are -359 and '507 for cir- 
 cular and quadrangular tubes.
 
 156 THE DISCHARGE OF WATER FROM 
 
 v 2 . 
 2~ is in general small, and this correction does not affect the 
 
 final results in any material degree. 
 
 Rennie's experiments*, with a pipe 15 feet long, inch 
 in diameter, and with 4 feet head, give the discharge per 
 second 
 
 Cubic feet. 
 
 1. Straight .... ...... -00699 
 
 2. Fifteen semicircular bends . . . -00617 
 
 3. One bend, a right angle 8^ inches 
 
 from the end of the pipe . . . "00556 
 
 4. Twenty-four right angles . . . -00253 
 
 From these data we find consecutively, the theoretical dis- 
 charge being -021885 cubic feet per second, and the theoretical 
 
 v 2 
 head H = ^ , that 
 
 1. v = -319 v2^ii, and therefore H = 9'82 x -- ; 
 
 2. v - -282 \/2~7H, H = 12-58 x -; 
 
 3. v - -254 v'sTn, H = 15-50 x ~ 
 
 4. v = -116 \/2ffH, ,, H = 74-34 x -j- ; 
 
 The loss of head, therefore, by the introduction of 15 semi- 
 circular bends, is 2*76 ^- ; by the introduction of one right 
 
 angle, 5'68 g ; and by the introduction of 24 right angles, 
 
 64-52 g , or about 12 times the loss due to one right angle. 
 
 This shows that the resistance does not increase as the number 
 of bends, as we before remarked, p. 138, when they are close to 
 
 i? 
 each other. The loss of head from one right angle, 5-68-2~> 
 
 is more than double the loss from 15 semicircular bends, or 
 * Philosophical Transactions for 1831, p. 438.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 157 
 
 2-76 n The loss of head for a right angular bend, deter- 
 
 2 
 
 mined from Venturi's experiment, is 1'876 JT ; formula (145) 
 makes it o ; and Weisbach's empirical formula, ('9457 sin. <p 
 
 + 2-047 sin. 4 ?) ~, makes it only -984 The formulae 
 
 now in use give, therefore, results considerably under the truth. 
 It appears to us, that perhaps the quantity of water moving 
 directly towards the bend must also be taken into considera- 
 tion, and certainly the loss of mechanical effect from con- 
 traction, and eddies at the bend, as well as the loss arising 
 from change of direction. 
 
 BRANCH PIPES. 
 
 When a pipe is joined to another, the quantity of water 
 flowing below the junction B, diagram 1, Fig. 42, must be 
 
 equal to the sum of the quantities flowing in the upper 
 branches in the case of supply; and when the branch pipe 
 draws off a portion of the water, as in diagram 2, the quantity 
 flowing above the junction must be equal to the quantities 
 flowing in the lower branches. Both cases differ only in the 
 motion being from or to the branches, which, in pipes, are 
 generally grafted at right angles to the main, for practical 
 convenience, as shown at b b, and then carried on in any 
 given direction. The loss of head arising from change 
 
 of direction, equation (145), is sin. 2 2 <p <r-> in which 2 <p 
 
 = angle ABO; but as in general 2 <p is a right angle for 
 branches to mains, this source of loss becomes then simply
 
 158 THE DISCHARGE OF WATER FROM 
 
 V 3 
 
 z In addition to this, a loss of head is sustained at the 
 
 . 9 
 
 junction, from a certain amount of force required to unite or 
 
 separate the water in the new channel. In the case of drawing 
 off, diagram 2, this loss was estimated by D'Aubuisson, from 
 experiments by Genieys, to be about twice the theoretical 
 
 9 ,* 
 head due to the velocity in the branch, or -g , so that the 
 
 . v 2 2v* 30 2 
 whole loss or head arising from the junction is ^ + Q~ == "o ' 
 
 or three times the theoretical head due to the velocity. In the 
 case of supply, the loss is probably nearly the same. The 
 actual loss is, however, very uncertain ; but, as was before 
 observed when discussing the loss of head occasioned by 
 
 bends, two or three times -Q is in general so comparatively 
 
 small, that its omission does not materially effect the final re- 
 sults. A loss also arises from contraction, &c. See pp. 97, 98. 
 The calculations for mains and branches become often very 
 troublesome, but they may always be simplified by rejecting 
 at first any minor corrections for contraction at orifice of 
 entry, bends, junctions, or curves. If, in diagram 2, Fig. 42, 
 we put h for the head at B, or height of the surface of the 
 reservoir over it ; A a for the fall from B to A ; h d for the fall 
 from B to D; / equal the length of pipe from B to the re- 
 servoir ; l & equal the length B A ; l d equal the length B D ; 
 r equal the mean radius of the pipe B c ; r & the mean radius 
 of the pipe BA; r d the mean radius of BD; v the mean 
 velocity in B c ; # a the velocity in B A ; and v d the velocity in 
 B D, we then find, by means of equation (73), the fall from the 
 reservoir to A equal to 
 
 the fall from the reservoir to D equal to 
 
 and, as the quantity of water passing from c to B is equal to
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 159 
 
 the sum of the quantities passing from B to A and from 
 
 B to D, 
 
 (148.) vr 2 = v a r| + v d r\. 
 
 By means of these three equations we can find any three of 
 
 the quantities h, A a , h d , r, r a , r d , b, J a , b d , the others being 
 
 given. Equations (146) and (147) may be simplified by neg- 
 
 lecting c t , the coefficient due to the orifice of entry from the 
 
 reservoir, and 1, the coefficient of velocity. They will then 
 
 become 
 
 (148.) ^ + ^ = * 
 
 and 
 
 (149.) A + A<, = c, 
 
 The mean value of c t for a velocity of 4 feet per second is 
 
 005741, and of ^-, -0000891. The values for any other ve- 
 
 locities may be had from the table of coefficients of friction 
 given at p. 120. When /, h, and r are given, the velocity v 
 
 /2g rA\* 
 
 can be had from the equation, v = ( X 7- ) , or more im- 
 
 \ Cf * / 
 
 mediately from TABLE VIII. 
 
 GENERAL EQUATION FOR MEAN VELOCITY. 
 
 We are now enabled to give a general equation for finding 
 the whole head H, and the mean velocity v, in any channel ; 
 and to extend equations (73) and (74)* so as to comprehend 
 the corrections due to bends, curves, &c. Designating, as 
 before, the height due to the resistance at the orifice of entry by 
 h r , and the corresponding coefficient by c r ; 
 h f the head due to friction, and c { the coefficient of friction ; 
 h b the head due to bends, and c b the coefficient of bends ; 
 h c the head due to curves, and c c the coefficient of curves ; 
 h e the head due to erogation, and c 6 the coefficient of erogation ; 
 and 
 
 h f the head due to other resistances, and c x their mean co- 
 efficient ; then we get 
 
 * See Note B.
 
 160 THE DISCHARGE OF WATER FROM 
 
 V* 
 
 (150.) H = 7i r + h t + h b + h c + h e + 7^ + g-; 
 
 that is to say, by substituting for 7t r , 7t f , &c., their values as 
 previously found, 
 
 c^O.^. 
 
 or, more briefly, 
 
 (151.) H = (l + c r + c t x - + c b + c c + c e + C *)^L > 
 
 from which we find 
 
 <!.) .2 
 
 c t X + c b 
 
 It is to be observed here, that for very long uniform 
 channels, the value of the mean velocity will be found in 
 
 general equal to j --M- [ , as the other resistances and the 
 
 ' Cfl ) 
 
 head due to the velocity are all trifling compared with the 
 friction, and may be rejected without error; but, as we before 
 observed, it is advisable in practice, when determining the 
 diameter of pipes, p. 123, to increase the value of c f , TABLE, 
 p. 120, or to increase the diameter found from the formula by 
 one-sixth, which will increase the discharging power by one- 
 half. (See TABLE XIII.) 
 
 HYDROSTATIC AND HYDRAULIC PRESSURE. PIEZOMETER. 
 
 When water is at rest in any vessel or channel, the pres- 
 sure on a unit of surface is proportionate to the head at its 
 centre*, measured to the surface, and is expressed in Ibs. for 
 
 * This is only correct when the surface is small in depth compared 
 with the head. If H be the depth of a rectangular surface in feet, and 
 also the head of water measured to the lower horizontal edge, then the 
 pressure in Ibs. is expressed by 31 H 3 ; and the centre of pressure is at 
 rds of the depth.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 161 
 
 measures in feet, by 62 H s, in which H is the head, and s 
 the surface exposed to the pressure, both in feet measures. 
 This is the hydrostatic pressure. In the pipe A B c D F E, 
 Fig. 43, the pressures at the points B, c, D, F, and E, on the 
 sides of the tube will be respectively as the heads B b, cc, Dd, 
 F f t and E e, if all motion in the tube be prevented by stopping 
 
 the discharging orifice at E. In this case the pressure is a 
 maximum and hydrostatic ; but if the discharging orifice at 
 E be partially or entirely open, a portion of each pressure at 
 B, c, D, F, &c., is absorbed in overcoming the different resist- 
 ances of friction, bends, &c., between it and the orifice of entry 
 at A, and also by the velocity in the tube, and the difference 
 is the hydraulic pressure. 
 
 Bernoulli first showed that the head due to the pressure 
 at any point, in any lube, is equal to the effective head at 
 that point, minus the head due to the velocity. When the 
 resistances in a tube are nothing, the effective head becomes 
 the hydrostatic head, and by representing the former by h^ we 
 shall have, adopting the notation in equation (150), 
 
 A rf =H-(A r + A, + A e + &cO, 
 
 and consequently the head due to the hydraulic pressure equal 
 
 (153.) h p = h et -f= u-(h r + h f + h b + &c.) -. 
 
 Z 9 ^g 
 
 If small tubes be inserted, as shown in Fig. 43, at the points 
 B, c, D, and F, the heights B b 1 , c c 1 , D d l , F/ 1 , which the water 
 rises to in these, will be represented by the corresponding 
 values of 7* p in the preceding equation ; and the difference 
 between the heights c c 1 , F/ 1 , at c and F, for instance, added
 
 162 THE DISCHARGE OF WATER FROM 
 
 to the fall from c to F, will evidently express the head due 
 to all the resistances between c and F. When H = ve, 
 and the orifice at E is open, we have, from equation (150) 
 
 H = h r + h { + 7/. b + h c + &c. + ^ , and therefore h p = 0, 
 
 that is, the pressure at the discharging orifice is nothing. 
 
 The vertical tubes at B, c, D, F, when properly graduated, 
 are termed piezometers or pressure gauges; they not only 
 show the actual pressure at the points where placed, but also 
 the difference between any two, D d 1 B b 1 , for instance, 
 added to the difference of head between B and D, or D d' 2 will 
 give vd l B b 1 + od 2 for the head or pressure due to the 
 resistances between B and D. This instrument affords, per- 
 haps, the very best means of determining the loss of head 
 due to bends, curves, diaphragms, &c. The loss of head due 
 to friction, bend, diaphragms, &c., between K and L, Fig. 43, 
 is equal to K A; L/ + K v. If M be the same distance from 
 L as K is, L / M m will be the height due to the friction 
 (L and M being on the same level) ; therefore K& L + KV 
 1,1 + um = K k + K# + Mm 2 L is the head due to 
 the diaphragm and bend both together. If the diaphragm be 
 absent, we get the head due to the bend, and if the bend be 
 absent, the head due to the diaphragm in like manner. 
 
 When the discharging orifice, as at E, is quite open, we have 
 seen that the pressure there is zero ; but when, as at G, it is 
 only partly open, this is no longer the case, and the hydraulic 
 pressure increases from zero to hydrostatic pressure, as the 
 orifice decreases from the full section to one indefinitely small 
 compared with it. A piezometer, placed a short distance in- 
 side G, will give this pressure ; and the difference between it 
 and the whole head will be the head due to the resistances 
 and velocity in the pipe ; from which, and also the length 
 and diameter, the discharge may be calculated as before 
 shown. Again, by means of the head M m 1 , and that due to 
 the velocity of approach, we can also find the discharge 
 through the diaphragm G; see equation (45) and the remarks 
 following it. This result must be equal to the other, and we
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 163 
 
 may hi this way test the formulae anew or correct them by 
 the practical results. 
 
 The velocity of discharge of the tube A c D E, may be cal- 
 culated by means of any piezometric height c c 1 ; for, by put- 
 ting the whole fall from c 1 to E equal to H C I, we get, disregard- 
 
 ing bends, v = j ' I , in which l c i =. c E. This is evi- 
 
 dent from equation (152), as we have supposed that no part 
 of the head is absorbed in generating velocity, or in over- 
 coming the resistance of bends. If the bend at D were taken 
 
 f 2?H c i H 
 into consideration, then v = 
 
 CATCHMENT BASINS. FALL OF RAIN. 
 
 A catchment basin is a district which drains itself into a 
 river and its tributaries. It is bounded, generally, by the 
 summits of the neighbouring hills, ridges, or high lands ; and 
 may vary in extent from a few square miles to many thou- 
 sands ; that of the Shannon is over 4500 square miles. The 
 average quantity of water which discharges itself into a river 
 will, cceteris paribus, depend on the extent of its catchment 
 basin, and the whole quantity of rain discharged on the area 
 of the catchment basin, including lakes and rivers. The 
 quantity of rain which falls annually varies with the district 
 and the year ; and it also varies at different parts of the same 
 district. The average quantity in Ireland may be taken at 
 about 34 inches deep, that which falls in Dublin being 27 
 inches*, and that in Cork 41 inches nearly. The quantity 
 
 * The average yearly fall in Dublin for seven years, ending with 1849, 
 was 26'407 inches ; and the maximum fall in any month took place in 
 April, 1846, being 5'082 inches. The average fall in inches per month 
 for seven years, ending with 1849, was as follows : October, 3'060 ; 
 August, 2-936 ; January, 2-544 ; April, 2*503 ; November, 2-300 ; July, 
 2-116; June, 2-005 ; December, 1-938; September, 1-860 ; May, 1-814 ; 
 March, 1-739 ; February, 1-534. Proceedings of the Royal Irish Academy, 
 vol. v. p. 18. 
 
 M 2
 
 164 THE DISCHARGE OF WATER FROM 
 
 varies a good deal with the altitude of the district, [n parts 
 of Westmoreland it rises sometimes to 140 inches ; in London, 
 an average of 20 years' observations gives a fall of nearly 25 
 inches. In the district surrounding the Bann reservoirs in 
 the County Down, the average fall has been found so high 
 as 72 inches. Indeed, it is requisite to obtain the fall from 
 observation for any particular district, when it is necessary to 
 apply the results to scientific purposes. 
 
 The proportion between the quantity which falls, and that 
 which passes from a catchment basin into its river, also varies 
 very considerably. When the sides of a catchment basin are 
 steep, and the water passes off rapidly into the adjacent river 
 or tributaries, there is less loss by evaporation and perco- 
 lation than when they are nearly flat. The soil, subsoil, and 
 stratification have also considerable effect on the proportion. 
 Reservoirs being generally constructed adjacent to steep side 
 falls, give a much larger proportion of the quantity fallen 
 than can be obtained from rivers in flatter districts ; besides, 
 the quantity of rain which falls on the high summits, near 
 reservoirs, almost always exceeds the average fall, consider- 
 ably. As 640 acres is equal to one square mile, and one 
 acre is equal to 43,560 square feet, a fall of one inch of rain 
 is equal to 3630 cubic feet per acre, and to 3630 x 640 
 = 2323200 cubic feet per square mile : the proportion of 
 this fall, for each acre, or square mile of the catchment basin, 
 which enters the river, must depend entirely on the district 
 and local circumstances, the full or maximum quantity being 
 retained on lakes. 
 
 It is too often taken for granted that the discharge from a 
 catchment basin takes place, into the conveying channels, in 
 nearly the same time that a given quantity of rain falls. Per- 
 haps the largest registry on record in Great Britain is a fall 
 of 2 inches in an hour. The maximum fall in any hour of 
 any year seldom exceeds half of this amount, and then per- 
 haps only once in several years. The quantity which falls 
 will not be discharged into the channels in the same time. 
 The quantity discharged, and time, will depend a good deal
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 165 
 
 QUANTITY PER ACRE FOR A GIVEN DEPTH OF FALL. 
 
 Fall in 
 inches 
 per hour. 
 
 Cubic feet 
 per acre. 
 
 Fall in 
 inches 
 rer hour. 
 
 Cubic feet 
 per acre. 
 
 Fall in 
 inches 
 per hour. 
 
 Cubit feet 
 per acre. 
 
 Fall in 
 inches 
 per hour. 
 
 Cubic feet 
 per acre. 
 
 2 
 
 7260 
 
 | 
 
 1815 
 
 i 
 
 454 
 
 A 
 
 181 
 
 l! 
 
 6352 
 
 1 
 
 1361 
 
 1 
 
 403 
 
 & 
 
 121 
 
 1* 
 
 5445 
 
 1 
 
 907 
 
 ^ 
 
 363 
 
 * 
 
 91 
 
 H 
 
 4537 
 
 * 
 
 726 
 
 A 
 
 302 
 
 4 
 
 73 
 
 1 
 
 3630 
 
 i 
 
 605 
 
 A 
 
 259 
 
 A 
 
 61 
 
 1 
 
 2723 
 
 1 
 
 519 
 
 i 
 
 227 
 
 A 
 
 52 
 
 on the season and district. The arterial channel receives the 
 supply at different places and from different distances, and 
 the water in passing into and from it does not encounter the 
 same amount of resistance as if it all passed first into the 
 upper end. Less sectional area is, therefore, necessary than 
 if the whole discharge had to pass through the whole length 
 of the channel. The relation of the quantity of rainfall to 
 the portion which flows into the main channel, as well as the 
 time which it takes to arrive at it, and the places of arrival, 
 must be known before the proper size of a new channel can 
 be determined, particularly sewers in urban districts. A 
 pipe sufficient to discharge the water from 200 acres need not 
 be 20 times the discharging power of one exactly suited to 
 10 acres of the same district, for the discharge from the out- 
 lying 190 acres will not arrive at the main in the same time 
 as that from the adjacent 10 acres. 
 
 In the Holborn and Finsbuiy divisions Mr. Roe calculated 
 that an 18-inch cylindrical pipe, laid at an inclination of 1 in 
 80, is sufficient for 20 acres of house-sewage, while a 5-inch 
 pipe laid at an inclination of 1 in 20 is necessary for 1 acre, 
 and a 3-inch pipe, laid also at 1 in 20, for acre. In each 
 of these cases, however, the discharge must depend on the 
 head and length of the pipe as well as the inclination at 
 which it is laid. Assuming the inclination of those pipes to 
 correspond with the hydraulic inclination, we have calculated
 
 166 THE DISCHARGE OF WATER FROM 
 
 their discharging powers to be respectively 807, 72, arid 20 
 cubic feet per minute, the areas to be drained being 20, 1, 
 and acres. In all calculations of this kind it is necessary, 
 for accuracy, to ascertain not only the maximum rainfall 
 per hour, but also the proportions discharged per hour, ac- 
 cording to the season and district, into the main channel, 
 and also the junctions or places of arrival. In urban dis- 
 tricts 1500, 2100, and sometimes 3600 cubic feet per hour 
 per acre, have to be discharged after extraordinary rainfalls. 
 These may be taken as maximum results. 
 
 In urban districts, however, a much larger quantity of 
 water is conveyed more rapidly, cateris paribus, to the 
 mains, than in suburban districts and catchment basins gene- 
 rally, in which the maximum discharge per acre per hour, 
 even in the steeper and higher districts, seldom exceeds 
 700 cubic feet, and varies from about 20 cubic feet for the 
 larger and natter districts upwards. This arises from the 
 impervious nature of the surfaces it falls upon in towns, and 
 the lesser waste in passing to the drains, as well as a portion 
 of the supply being often artificial. From 70 to 90 cubic 
 feet* per acre per hour, is generally taken for the maximum 
 discharge from the average number of catchment basins; 
 this is equal to a supply of from T ^th to ; Lth of an inch 
 in depth from the whole area. Thorough drainage increases 
 the supply and discharge. Every catchment basin has, 
 however, its own peculiar data, and a knowledge of these 
 is necessary before we can draw any correct conclusions for 
 new waterworks in connection with it. It may be re- 
 marked, however, that any conclusions drawn from experi- 
 ments on the supply of tributaries, particularly in high 
 districts, are wholly inapplicable to the main channel into 
 which they flow. The flow into tributaries and mountain 
 streams, or rivers, is always more rapid than into main 
 channels and rivers in flat districts, and the supply from 
 
 * Some interesting observations on rainfall and flood discharges are 
 given in the Transactions of the Institution of Civil Engineers, Ireland, 
 for 1851, pp. 19-33 and pp. 44-52.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 167 
 
 springs often forms a large portion of the water flowing 
 in them. 
 
 The effects of evaporation are very variable; sometimes 
 58 or 60 per cent, of the annual fall is carried off in this way 
 from ordinary flat tillage soils, and other estimates are much 
 higher; much, however, depends on the soil, subsoil, in- 
 clination, stratification, and season. The evaporation from 
 water surfaces exceeds the annual fall in these countries by 
 about one-third ; and that from flat marsh and callow lands 
 exceeds the evaporation from ordinary tillage, porous, and 
 high lands. When the flat lands along the banks of rivers 
 extend considerably on both sides, an extra fall is necessary 
 into the main channel, along the normal drains, otherwise 
 such lands must suffer from excessive eveporation as well 
 as floods. Evaporation also varies with the climate. 
 
 HORSE POWER. 
 
 Taking the value of a horse's power at 33,000 Ibs.* raised 
 1 foot high in one minute, the theoretical horse power of an 
 overfall, or a given quantity of water, is expressed by the 
 fall in feet, multiplied by the discharge in cubic feet per 
 minute, multiplied by 62 and divided by 33,000. The 
 effective power depends on the nature, circumstances, and 
 construction of the machine to which the theoretical power 
 is applied. For an overshot wheel, the ratio of the power to 
 the effect may be taken as 3 : 2, and, therefore, the effective 
 horse power will be 49,500 Ibs. weight of water falling 1 foot 
 in one minute -f. The maximum effect of overshot wheels 
 is found to vary : Smeaton found it '76 times the theoretical 
 effect; D'Aubuisson also *76 times; and Weisbach found 
 78 times the theoretical effect in the wheel of a stamp mill, 
 at Frieberg, which was 23 feet high, 3 feet wide, and con- 
 tained 48 buckets. To find the effective horse power we 
 
 * 22,000 Ibs. is much nearer the average power of horses as their 
 work for twelve hours is ordinarily performed through the country. 
 
 t Small wheels are not so effective as larger ones under similar 
 circumstances.
 
 168 THE DISCHARGE OF WATER FROM 
 
 must divide these decimals, viz. '76 and '78, into 33,000, 
 which will give 43,300 and 43,600 Ibs nearly, falling 1 foot 
 for a horse power. 
 
 For breast wheels, the ratio of the theoretical to the 
 effective power must vary very considerably ; the mean value 
 may be taken at 1 : *5 ; and, therefore, the value of a horse 
 power would be 66,000 Ibs. falling 1 foot in a minute. M. 
 Morin found the efficiency to vary from 1 : *7 to 1 : *52 j 
 and Egen, with a wheel 23 feet in diameter, 4 feet wide, 
 having 69 ventilated buckets exceedingly well constructed, 
 found the ratio at best but as 1 : *52, and under ordinary 
 circumstances as 1 : '48, the mean being as 1 : '5. Very 
 wide wheels give a larger effect, sometimes as high as *73 
 in one, but a good deal depends on the manner of bringing 
 on the water and the construction of the buckets. 
 
 For undershot wheels, the mean effect may be taken -at 
 33 in one, or 100,000 Ibs. falling 1 foot high in one minute, 
 for the effective horse power. Here, however, the height 
 must be (in general) determined from the velocity. A 
 maximum effect of -5 in 1 is sometimes obtained, and a 
 minimum effect of -16 in 1. 
 
 Turbines may be said to give a useful effect of f in 1, or 
 49,500 Ibs. falling 1 foot in a minute for the effective horse 
 power, the same as for an overshot wheel. A maximum 
 effect of -821 in 1 has been obtained*, and the efficiency 
 varies from '57 to '80, or less. 
 
 Poncelet wheels give a useful effect of from '5 to *6 in 1 ; 
 floating wheels "38 in 1; impact wheels from '16 to '4; and 
 Barker's mill from '15 to '35 in 1. 
 
 Corn mills will grind about a bushel of corn per horse 
 power per hour, but a good deal depends on the state of the 
 stones, and on the grain. The value of the work done in 
 an hour being once known, the value of a horse power may 
 be determined accordingly. 
 
 * Proceedings Royal Irish Academy, vol. v. p. 214.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 169 
 
 NOTE A. 
 
 DISCHARGE FROM ONE VESSEL OR CHAMBER INTO ANOTHER. 
 LOCK-CHAMBERS. 
 
 WE have given, at pages 98 and 99, formulas for the time 
 water in a prismatic vessel takes to fall a given depth, when 
 
 discharged from an orifice at the side or bottom. The time 
 the surface ST, Diagram 1, Fig. 44, takes to rise to st, when 
 supplied through an orifice or tube OR, from an upper large 
 chamber or canal, whose surface s't' remains always at the 
 
 2A/"* 
 
 same level, is *^=* and we thence get the time of 
 
 *dv/2/ 
 rising from R to s for measures in feet. 
 
 (A.) 
 
 i = 
 
 and for measures in inches 
 (B.) i : 
 
 in which A is the area of the horizontal section at ST; a the 
 sectional area of the communicating channel or orifice OR; 
 
 * The time of rising from 8 to s is exactly double the time it would 
 take if the pressure f remained uniform to fill the same depth below E.
 
 170 THE DISCHARGE OF WATER FROM 
 
 c d the coefficient of discharge suited to it, and h L and f as 
 shown in the diagram. 
 
 In order to find the time of filling the lower vessel to the 
 level ST, supposing it at first empty, we have the contents 
 of the portion below OR equal to Ah 2 , and the time of filling 
 it equal to 
 
 8-025 c d a h\' 
 
 then the time of filling up to any level ST, for measures 
 in feet, is equal to the sum of (A) and (c) ; that is, 
 
 (D.) ,= ^ 
 
 8-025 c d a 
 
 8-025 c d a h\ 
 and for measures in inches 
 
 (E.) T =_ 
 
 27-8 c d a h\ 
 
 When ST coincides with st 
 
 ( V ) T _A(2* 1 + A.) 
 
 8-025 c d ah\' 
 
 for measures in feet, and 
 (G.) 
 
 27 
 
 for measures in inches. These equations are exactly suited 
 to the case of a closed lock-chamber filled from an adjacent 
 canal. 
 
 When the upper level sV is also variable, as in Diagram 2, 
 the time which the water in both vessels takes to come to the 
 same uniform level s't' st, is
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 171 
 
 c d (A + c) 
 
 in which ^ +/i h =/+/! is the difference of levels at 
 the beginning of the flow ; c the horizontal section of the 
 upper chamber; and the other quantities as in Diagram 1. 
 As c/! = A/, we find 
 
 Now, in order to find the time of falling a given depth d 
 below the first level sV, we have the head above s't'st equal 
 to J[ d in the upper vessel, and the depth below it in the 
 
 lower vessel equal to ^"~ -; whence the difference of 
 levels in the two vessels at the end of the fall d, is 
 
 A A. 
 
 The time of falling through d is, therefore, from equation (H), 
 
 f . 
 
 c^a (A + c) >2^ c d a (A + c) 
 
 t 
 
 C d (A + C) 
 
 in which \/2g 8'025 for measures in feet and equal 27*8 
 for measures in inches. The whole time of filling to a level 
 the lower empty vessel, is found by adding the time of filling 
 the portion below R, determined in a manner similar to 
 equations (68) and (69) to be 
 
 (K.) 
 
 c d a
 
 172 THE DISCHARGE OF WATER FROM 
 
 to the time of filling above R, given in equation (H), when 
 h is taken equal to zero. Equations (H), (i), and (K) are 
 applicable to the case of the upper and lower chambers of 
 a double lock, after making the necessary change in the 
 diagrams. 
 
 The above equations require further extensions when water 
 flows into the upper vessel while also flowing from it into the 
 lower; such extensions are, however, of little practical value, 
 and we therefore omit them. For sluices in flood-gates with 
 square arrises, c d may be taken at about *545, but with 
 rounded arrises the coefficient will rise much higher. See 
 SECTIONS III. and VII. 
 
 NOTE B. 
 
 In equations (74) and (151), the coefficient of friction c { 
 depends on the velocity v, and its value can be found from 
 an approximate value of that velocity from the small table, 
 page 120. If, however, we use both powers of the velocity, 
 as in equation (83), we shall get, when H is the whole head, 
 and h the head from the surface to the orifice of entry 
 
 (av + # v ) + (1 + O ~ + h H, a quadratic equation 
 
 from which we find 
 
 / gal 
 
 _ 
 ~ 
 
 (1 + Cl)r 
 
 gal 
 
 or a more general value of the velocity than that given in 
 equation (74). If now we put c s = c r + c b -f c c + c e + c x in 
 equation (151) we shall find 
 
 gal 
 
 (1 +c s }r 
 
 _ gal
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 173 
 
 for a more general expression of equation (152), when 
 the simple power of the velocity, as in equation (83), is 
 taken into consideration. For measures in English feet, we 
 may take a = '0000223 and b = -0000854, which correspond 
 to those of Eytelwein, in equation (97). The value of a is 
 the same in English as in French measures, but the value 
 of b in equation (83), for measures in metres, must be divided 
 by 3*2809 to find its corresponding value for measures in 
 
 English feet. In considering the head ^ c r , due to con- 
 traction at the orifice of entry as not implicitly comprised 
 in the primary values of a and b, equation (83), Eytel- 
 wein is certainly more correct than D'Aubuisson, Traite 
 d'Hydraulique, pp. 223 et 224, as this head varies with the 
 nature of the junction, and should be considered in con- 
 nection with the head due to the velocity, or separately. It 
 can never be correctly considered as a portion of the head 
 due to friction. In all Du Buat's experiments, this head was 
 considered as a portion of that due to the velocity, and the 
 
 whole head (1 + c r ) deducted to find the hydraulic 
 inclination. 
 
 VALUES OF a AND b FOR MEASURES IN ENGLISH FEET. 
 
 a. b. 
 
 Equation (88.) '0000445 -0000944 
 
 (90.) -0000173 -0001061 
 
 (94.) -0000243 -0001114 
 
 (98.) -0000223 -0000854 
 
 (109.) -0000189 -0001044 
 
 (111.) -0000241 -0001114 
 
 (114.) -0000035 -000115 
 
 Mean values for all straight! . 0000221 .Q000892 
 channels, pipes, or rivers J
 
 174 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE I. COEFFICIENTS OF DISCHARGE FROM SQUARE AND DIF- 
 FERENTLY PROPORTIONED RECTANGULAR LATERAL ORIFICES IN 
 THIN VERTICAL PLATES. 
 
 
 
 Square orifice 
 
 Rectangular 
 
 Rectangular 
 
 S f 
 
 
 8 /y X 8". 
 
 orifice 8" x 4". 
 
 orifice 8' X 2". 
 
 lyi 
 
 S ? 
 
 Ratio of the sides 
 
 Ratio of the sides 
 
 Ratio of the sides 
 
 IfiJ 
 
 o <u 
 
 1 to 1. 
 
 2tol. 
 
 4to 1. 
 
 Sf|| 
 
 5.S-S 
 
 1*1 
 
 t! "& 
 
 Ill 
 
 ^ g 
 
 g 
 
 ||| 
 
 ll| 
 
 III 
 
 Ill 
 
 i*j 
 
 Pi" 
 
 S | 
 
 
 !*' 
 
 111 
 
 | 
 
 Jll 
 
 1 = 
 
 x 
 
 o-ooo 
 
 
 
 619 
 
 
 667 
 
 
 713 
 
 0-197 
 
 025 
 
 
 597 
 
 
 630 
 
 
 668 
 
 0-394 
 
 050 
 
 
 595 
 
 
 618 
 
 607 
 
 642 
 
 0-591 
 
 075 
 
 
 594 
 
 593 
 
 615 
 
 612 
 
 639 
 
 0-787 
 
 100 
 
 572 
 
 594* 
 
 596 
 
 614 
 
 615 
 
 638 
 
 1-181 
 
 150 
 
 578 
 
 593 
 
 600 
 
 613 
 
 6-20 
 
 637 
 
 1-575 
 
 200 
 
 582 
 
 593* 
 
 603 
 
 612 
 
 623 
 
 636 
 
 1-969 
 
 250 
 
 585 
 
 593 
 
 605 
 
 612* 
 
 625 
 
 636 
 
 2362 
 
 300 
 
 587 
 
 594 
 
 607 
 
 613 
 
 627 
 
 635 
 
 2-756 
 
 350 
 
 588 
 
 594 
 
 609 
 
 613 
 
 628 
 
 635 
 
 3-150 
 
 400 
 
 589 
 
 594 
 
 610 
 
 613 
 
 629 
 
 635 
 
 3-545 
 
 450 
 
 591 
 
 595 
 
 610 
 
 614 
 
 629 
 
 634 
 
 3937 
 
 500 
 
 592 
 
 595 
 
 611 
 
 614 
 
 630 
 
 634 
 
 4-724 
 
 600 
 
 593 
 
 596 
 
 612 
 
 614 
 
 630 
 
 633 
 
 5-512 
 
 700 
 
 595 
 
 597 
 
 613 
 
 614 
 
 630 
 
 632 
 
 6-299 
 
 800 
 
 596 
 
 597 
 
 614 
 
 615 
 
 631* 
 
 631 
 
 7-087 
 
 900 
 
 597 
 
 598 
 
 615 
 
 615 
 
 630 
 
 631 
 
 7-874 
 
 I -000 
 
 598 
 
 599 
 
 615 
 
 615 
 
 630 
 
 630 
 
 9-843 
 
 1 250 
 
 599 
 
 600 
 
 616 
 
 616 
 
 630 
 
 630 
 
 11-811 
 
 1-500 
 
 600 
 
 601 
 
 616 
 
 616 
 
 629 
 
 629 
 
 15-748 
 
 2-000 
 
 602 
 
 602 
 
 617 
 
 617 
 
 628 
 
 629 
 
 19-685 
 
 2-500 
 
 603 
 
 603 
 
 617* 
 
 617* 
 
 628 
 
 628 
 
 23622 
 
 3-000 
 
 604 
 
 604 
 
 617 
 
 617 
 
 627 
 
 627 
 
 27-560 
 
 3-500 
 
 604 
 
 604 
 
 616 
 
 616 
 
 627 
 
 627 
 
 31-497 
 
 4-000 
 
 605 
 
 605 
 
 616 
 
 616 
 
 627 
 
 627 
 
 35-434 
 
 4-500 
 
 605* 
 
 605* 
 
 615 
 
 6|5 
 
 626 
 
 626 
 
 39-371 
 
 5000 
 
 605 
 
 605 
 
 615 
 
 615 
 
 626 
 
 626 
 
 43-307 
 
 5-500 
 
 604 
 
 604 
 
 614 
 
 614 
 
 625 
 
 625 
 
 47-245 
 
 6-000 
 
 604 
 
 604 
 
 614 
 
 614 
 
 624 
 
 624 
 
 51-182 
 
 6-500 
 
 603 
 
 603 
 
 613 
 
 613 
 
 622 
 
 622 
 
 55-119 
 
 7-000 
 
 603 
 
 603 
 
 612 
 
 612 
 
 62 L 
 
 621 
 
 59-056 
 
 7-500 
 
 602 
 
 602 
 
 611 
 
 611 
 
 620 
 
 620 
 
 62993 
 
 8-000 
 
 f?02 
 
 602 
 
 611 
 
 611 
 
 618 
 
 618 
 
 66-930 
 
 8-500 
 
 602 
 
 602 
 
 610 
 
 610 
 
 617 
 
 617 
 
 70-867 
 
 9-000 
 
 601 
 
 601 
 
 609 
 
 609 
 
 615 
 
 615 
 
 74-805 
 
 9-500 
 
 601 
 
 601 
 
 608 
 
 608 
 
 614 
 
 614 
 
 78-742 
 
 10-000 
 
 601 
 
 601 
 
 607 
 
 607 
 
 613 
 
 614 
 
 118-112 
 
 15-000 
 
 601 
 
 601 
 
 .603 
 
 603 
 
 606 
 
 606
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 175 
 
 TABLE I. COEFFICIENTS OF DISCHARGE FROM SQUARE AND DIF- 
 FERENTLY PROPORTIONED RECTANGULAR LATERAL ORIFICES IN 
 THIN VERTICAL PLATES. 
 
 Rectangular 
 orifice 8" x 1-18". 
 Ratio of the sides 
 7 to 1 nearly. 
 
 Rectangular 
 orifice 8" x 0-8". 
 atio of the sides 
 lOtol. 
 
 Rectangular 
 orifices' x 0-4'. 
 Ratio of the sides 
 20 to 1. 
 
 Ratio of the head 
 to the 
 length of the orifice. 
 
 Heads of water mea- 
 sured to the upper 
 sides of the orifices, 
 in English inches. 
 
 Heads taken 
 back from 
 the orifice. 
 
 g 
 
 !. 
 |*1 
 
 Heads taken 
 back from 
 the orifice. 
 
 Heads taken 
 at the 
 orifice. 
 
 Heads taken 
 back from 
 the orifice. 
 
 Heads taken 
 at the 
 orifice. 
 
 
 766 
 
 
 783 
 
 
 795 
 
 
 
 
 725 
 
 
 750 
 
 705 
 
 778 
 
 025 
 
 0-197 
 
 630 
 
 687 
 
 660 
 
 720 
 
 701 
 
 762 
 
 050 
 
 0-394 
 
 632 
 
 674 
 
 660 
 
 707 
 
 697 
 
 745 
 
 075 
 
 0-591 
 
 634 
 
 '668 
 
 659 
 
 697 
 
 694 
 
 729 
 
 100 
 
 0-787 
 
 638 
 
 659 
 
 659 
 
 685 
 
 688 
 
 708 
 
 150 
 
 1-181 
 
 640 
 
 654 
 
 658 
 
 678 
 
 683 
 
 695 
 
 200 
 
 1-576 
 
 640* 
 
 651 
 
 658 
 
 672 
 
 679 
 
 686 
 
 250 
 
 1-969 
 
 640 
 
 647 
 
 657 
 
 668 
 
 676 
 
 681 
 
 300 
 
 2-362 
 
 631) 
 
 645 
 
 656 
 
 665 
 
 673 
 
 677 
 
 350 
 
 2-756 
 
 638 
 
 643 
 
 656 
 
 662 
 
 670 
 
 675 
 
 400 
 
 3-150 
 
 637 
 
 641 
 
 655 
 
 659 
 
 668 
 
 672 
 
 450 
 
 3-543 
 
 637 
 
 640 
 
 654 
 
 657 
 
 666 
 
 669 
 
 500 
 
 3-937 
 
 636 
 
 637 
 
 653 
 
 655 
 
 663 
 
 665 
 
 600 
 
 4-724 
 
 635 
 
 636 
 
 651 
 
 653 
 
 660 
 
 661 
 
 700 
 
 5-512 
 
 634 
 
 635 
 
 650 
 
 651 
 
 658 
 
 659 
 
 800 
 
 6-299 
 
 634 
 
 634 
 
 649 
 
 650 
 
 657 
 
 657 
 
 900 
 
 7-087 
 
 633 
 
 633 
 
 648 
 
 649 
 
 655 
 
 656 
 
 1-000 
 
 7-874 
 
 632 
 
 632 
 
 646 
 
 646 
 
 653 
 
 653 
 
 1 -250 
 
 9-843 
 
 632 
 
 632 
 
 644 
 
 644 
 
 650 
 
 651 
 
 1-500 
 
 11-811 
 
 631 
 
 631 
 
 642 
 
 642 
 
 647 
 
 647 
 
 2-000 
 
 15-748 
 
 630 
 
 630 
 
 640 
 
 640 
 
 644 
 
 645 
 
 2-500 
 
 19-685 
 
 630 
 
 630 
 
 638 
 
 638 
 
 642 
 
 643 
 
 3-000 
 
 23-622 
 
 629 
 
 629 
 
 637 
 
 637 
 
 640 
 
 640 
 
 3-500 
 
 27-560 
 
 629 
 
 6-29 
 
 636 
 
 636 
 
 637 
 
 637 
 
 4-000 
 
 31-497 
 
 628 
 
 628 
 
 634 
 
 634 
 
 635 
 
 635 
 
 4-500 
 
 35-434 
 
 628 
 
 628 
 
 633 
 
 633 
 
 632 
 
 632 
 
 5-000 
 
 39-371 
 
 627 
 
 627 
 
 631 
 
 631 
 
 629 
 
 629 
 
 5-500 
 
 43-307 
 
 626 
 
 626 
 
 628 
 
 628 
 
 626 
 
 626 
 
 6-000 
 
 47-245 
 
 624 
 
 621 
 
 625 
 
 6-25 
 
 6'22 
 
 622 
 
 6-500 
 
 51-182 
 
 622 
 
 622 
 
 622 
 
 622 
 
 618 
 
 618 
 
 7-000 
 
 55-119 
 
 620 
 
 620 
 
 619 
 
 619 
 
 615 
 
 615 
 
 7-500 
 
 59-056 
 
 618 
 
 618 
 
 617 
 
 617 
 
 613 
 
 613 
 
 8-000 
 
 6-2-993 
 
 616 
 
 616 
 
 615 
 
 615 
 
 612 
 
 612 
 
 8-500 
 
 66-930 
 
 615 
 
 615 
 
 614 
 
 614 
 
 612 
 
 612 
 
 9-000 
 
 70-867 
 
 613 
 
 613 
 
 -612 
 
 612 
 
 611 
 
 611 
 
 9-500 
 
 74-805 
 
 612 
 
 612 
 
 612 
 
 612 
 
 611 
 
 611 
 
 10-000 
 
 78-742 
 
 608 
 
 608 
 
 610 
 
 610 
 
 609 
 
 609 
 
 15-000 
 
 118-112
 
 176 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE II. FOR FINDING THE VELOCITIES FROM THE ALTITUDES 
 
 AND THE ALTITUDES FROM THE VELOCITIES. 
 
 Altitudes feet T J 5 inch to feet 3| inches. 
 
 1 
 
 Coefficients of velocity, and the corresponding velocities of 
 
 1 
 
 discharge in inches per second. 
 
 .3 J 
 
 Iff* 
 
 *l* 
 
 fSi Jill . 
 
 *!i 
 
 " ^ -g 
 
 
 op 5-3 
 
 <3 o ^ 
 
 3 ip '3 o I a 01 "S P 
 
 
 1 
 
 II fl> ^ 
 
 a&l* 
 
 ^ II 
 
 
 2|? 
 
 ^5|? 
 
 ;i*|f 
 
 < 
 
 ?> * 
 
 
 CO 5, 
 
 "* s> 
 
 "* 5> 
 
 to 
 
 OJL 
 
 2-78 
 
 2-71 
 
 2-66 
 
 2-39 
 
 2-27 
 
 2-22 
 
 ' 
 
 3-48 
 
 3-38 
 
 3-32 
 
 2-99 
 
 2-83 
 
 2-78 
 
 o SJ 
 
 6-95 
 
 6-77 
 
 6-64 
 
 5-98 
 
 5-66 
 
 5-56 
 
 0? 
 
 9-829 
 
 9-57 
 
 9-40 
 
 8-45 
 
 8-01 
 
 7-86 
 
 Ojg 
 
 12-038 
 
 11-72 
 
 11-51 
 
 10-35 
 
 9-81 
 
 9-63 
 
 OJ 
 
 13-900 
 
 13-54 
 
 13-29 
 
 11-95 
 
 11-33 
 
 11-12 
 
 Of), 
 
 15-541 
 
 15-14 
 
 14-86 
 
 13-36 
 
 12-67 
 
 12-43 
 
 Of 
 
 17-024 
 
 16-58 
 
 16-27 
 
 14-64 
 
 13-87 
 
 13-62 
 
 o 07, 
 
 18-388 
 
 17-91 
 
 17-58 
 
 15-81 
 
 14-99 
 
 14-71 
 
 0? 
 
 19-658 
 
 19-15 
 
 18-79 
 
 16-91 
 
 16-02 
 
 15-73 
 
 o 4 
 
 20-850 
 
 20-31 
 
 19-93 
 
 17-93 
 
 16-99 
 
 16-68 
 
 Of 
 
 21-978 
 
 21-41 
 
 21-01 
 
 18-90 
 
 17-91 
 
 17-58 
 
 0{' 
 
 23-051 
 
 22-45 
 
 22-04 
 
 19-82 
 
 18-79 
 
 18-44 
 
 Of 
 
 24-076 
 
 23-45 
 
 23-02 
 
 20-70 
 
 19-62 
 
 19-26 
 
 Oli 
 
 25-059 
 
 24-41 
 
 24-00 
 
 21-55 
 
 20-42 
 
 20-05 
 
 Of 
 
 26-005 
 
 25-33 
 
 24-86 
 
 22-36 
 
 21-19 
 
 20-80 
 
 0>$ 
 
 26-917 
 
 26-22 
 
 25-73 
 
 23-15 
 
 21-94 
 
 21-53 
 
 1 " 
 
 27-800 
 
 27-08 
 
 26-58 
 
 23-91 
 
 22-66 
 
 22-24 
 
 11 
 
 29-486 
 
 28-72 
 
 28-19 
 
 25-36 
 
 24-03 
 
 23-59 
 
 o if 
 
 31-081 
 
 30-27 
 
 29-71 
 
 26-73 
 
 25-33 
 
 24-87 
 
 o if 
 
 32-598 
 
 31-75 
 
 31-16 
 
 28-03 
 
 26-57 
 
 26-08 
 
 o i! 
 
 34-048 
 
 33-19 
 
 32-58 
 
 29-30 
 
 27-75 
 
 27-26 
 
 o if 
 
 35-438 
 
 34-52 
 
 33-88 
 
 30-48 
 
 28-88 
 
 28-35 
 
 If 
 
 36-776 
 
 35-82 
 
 35-16 
 
 31-63 
 
 29-97 
 
 29-42 
 
 IJ 
 
 38-067 
 
 37-08 
 
 36-39 
 
 32-74 
 
 31-02 
 
 30-45 
 
 2 
 
 39-315 
 
 38-29 
 
 37-59 
 
 33-81 
 
 32-04 
 
 31-45 
 
 2J 
 
 40-525 
 
 39-47 
 
 38-74 
 
 34-85 
 
 33-03 
 
 32-42 
 
 o 2! 
 
 41-700 
 
 40-62 
 
 39-87 
 
 35-86 
 
 33-99 
 
 33-36 
 
 2j- 
 
 42-843 
 
 41-73 
 
 40-96 
 
 36-84 
 
 34-92 
 
 34-27 
 
 2, 
 
 43-956 
 
 42-81 
 
 42-02 
 
 37-80 
 
 35-82 
 
 35-16 
 
 2j 
 
 45-041 
 
 4387 
 
 43-06 
 
 38-74 
 
 36-71 
 
 36-03 
 
 2J- 
 
 46-101 
 
 44-90 
 
 44-07 
 
 39-65 
 
 37-57 
 
 36-88 
 
 2i 
 
 47-137 
 
 45-90 
 
 45-06 
 
 40-54 
 
 38-42 
 
 37-71 
 
 3 
 
 48-151 
 
 46-90 
 
 46-03 
 
 41-41 
 
 39-24 
 
 38-52 
 
 3J 
 
 49-144 
 
 47-87 
 
 46-98 
 
 4226 
 
 40-05 
 
 39-32 
 
 3 L 
 
 50-117 
 
 48-81 
 
 47-91 
 
 43-10 
 
 40-85 
 
 40-09 
 
 3| 
 
 51-072 
 
 49-74 
 
 48-82 
 
 43-92 
 
 41-62 
 
 40-86 
 
 31 
 
 52-009 
 
 50-66 
 
 49-72 
 
 44-73 
 
 42-39 
 
 41-61 
 
 3J 
 
 52-930 
 
 51-55 
 
 50-60 
 
 45-52 
 
 43-14 
 
 42-34 
 
 3J 
 
 53-834 
 
 52-43 
 
 51-47 
 
 46-30 
 
 43-88 
 
 43-07 
 
 3J 
 
 54-725 
 
 53-30 
 
 52-32 
 
 47-06 
 
 44-60 
 
 43-78
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 177 
 
 TABLE II. FOB FINDING THE VELOCITIES FBOM THE ALTITUDES, 
 
 AND THE ALTITUDES FBOM THE VELOCITIES. 
 
 Altitudes feet Jj inch to Qfeet 3| inches. 
 
 Coefficients of velocity, and the corresponding velocities of 
 discharge in inches per second. 
 
 1 
 1 . 
 
 "2^ 
 
 83 
 
 8^ 
 
 S3- I-S3- 
 
 S3 
 
 .5 J 
 
 
 sSl . 
 
 S? 1 
 
 S 1 . "*~ 1 
 
 n lO 13 
 
 00 -~H 
 
 5 * '3 c? 
 
 Js 10-3 5 
 
 S *& *o 06 
 
 3 i 1 'G *~ 3 OO *o ^ 
 
 3 Oi "o "*** 
 
 
 ^lil ' 
 
 lw. 
 
 CS " 
 
 d?*' 
 
 pH-ii'S 
 
 c.-'ii" 
 
 .1 
 
 
 ^ 
 
 
 i-H S> 
 
 i-l 5> 
 
 r-l S> 
 
 
 1-95 
 
 1-85 
 
 1-75 
 
 1-72 
 
 1-68 
 
 1-62 
 
 6 o,fo 
 
 2-43 
 
 2-31 
 
 2-18 
 
 2-15 
 
 2-11 
 
 2-03 
 
 0^} 
 
 4-87 
 
 4-63 
 
 4-36 
 
 4-29 
 
 4-21 
 
 4-06 
 
 T ' g 
 
 6-88 
 
 6-55 
 
 6-17 
 
 6-06 
 
 5-96 
 
 5-74 
 
 
 
 8-43 
 9-73 
 
 8-02 
 9-26 
 
 7-56 
 8-73 
 
 7-43 
 
 8-58 
 
 7-29 
 8-42 
 
 7-03 
 8-12 
 
 0^ 
 0* 
 
 10-88 
 
 10-35 
 
 9-76 
 
 9-59 
 
 9-42 
 
 9-08 
 
 0^ 
 
 11-92 
 
 11-24 
 
 10-69 
 
 10-50 
 
 10-32 
 
 9-94 
 
 Of 
 
 12-87 
 
 12-25 
 
 11-55 
 
 1135 
 
 11-14 
 
 10-74 
 
 OT% 
 
 13-76 
 
 12-97 
 
 12-34 
 
 12-13 
 
 11-91 
 
 11-48 
 
 
 
 14-60 
 
 13-89 
 
 13-09 
 
 12-86 
 
 12-64 
 
 12-18 
 
 T 9 g 
 
 15-38 
 
 14-64 
 
 13-80 
 
 13-56 
 
 13-32 
 
 12-84 
 
 Of 
 
 16-14 
 
 15-35 
 
 14-48 
 
 14-22 
 
 13-97 
 
 13-46 
 
 OU 
 
 16-85 
 
 16-03 
 
 15-12 
 
 14-85 
 
 14-59 
 
 14-06 
 
 Of 
 
 17-54 
 
 16-69 
 
 15-74 
 
 15-46 
 
 15-19 
 
 14-63 
 
 Ojj! 
 
 18-20 
 
 17-32 
 
 16-33 
 
 16-04 
 
 15-76 
 
 15-09 
 
 Of 
 
 18-84 
 
 17-93 
 
 16-90 
 
 16-61 
 
 16-31 
 
 15-72 
 
 0{| 
 
 19-46 
 
 18-51 
 
 17-46 
 
 17-15 
 
 16-85 
 
 16-24 
 
 1 
 
 20-64 
 
 19-64 
 
 18-52 
 
 18-19 
 
 17-87 
 
 17-22 
 
 1J 
 
 21-76 
 
 20-70 
 
 19-52 
 
 19-18 
 
 18-84 
 
 18-15 
 
 ]J 
 
 22-82 
 
 21-71 
 
 20-47 
 
 20-11 
 
 19-75 
 
 19-04 
 
 If 
 
 23-85 
 
 22-69 
 
 21-38 
 
 21-01 
 
 20-63 
 
 19-88 
 
 o 14 
 
 24-81 
 
 23-60 
 
 22-26 
 
 21-87 
 
 21-48 
 
 20-70 
 
 If 
 
 25-74 
 
 24-49 
 
 23-10 
 
 22-69 
 
 22-29 
 
 21-48 
 
 If 
 
 26-65 
 
 25-35 
 
 23-91 
 
 23-49 
 
 23-07 
 
 22-23 
 
 1 
 
 27-52 
 
 26-18 
 
 24-69 
 
 24-26 
 
 23-82 
 
 22-96 
 
 2 
 
 2837 
 
 26-99 
 
 25-45 
 
 25-00 
 
 24-56 
 
 23-67 
 
 2* 
 
 29-19 
 
 27-77 
 
 26-19 
 
 25-73 
 
 25-27 
 
 24-35 
 
 2J 
 
 29-99 
 
 28-53 
 
 26-91 
 
 26-43 
 
 25-96 
 
 25-02 
 
 2| 
 
 30-77 
 
 29-27 
 
 27-60 
 
 27-12 
 
 26-64 
 
 25-67 
 
 2 
 
 31-53 
 
 30-00 
 
 28-29 
 
 27-79 
 
 27-29 
 
 26-30 
 
 2| 
 
 32-27 
 
 30-70 
 
 28-95 
 
 28-44 
 
 27-94 
 
 26-92 
 
 2| 
 
 33-00 
 
 31-39 
 
 29-60 
 
 29-08 
 
 28-57 
 
 27-53 
 
 2Z 
 
 33-71 
 
 32-07 
 
 30-24 
 
 29-71 
 
 29-18 
 
 28-12 
 
 3 
 
 34-40 
 
 32-73 
 
 30-86 
 
 30-32 
 
 29-78 
 
 28-70 
 
 3 
 
 35-08 
 
 33-38 
 
 31-47 
 
 30-92 
 
 30-37 
 
 29-27 
 
 3 
 
 35-75 
 
 34-01 
 
 32-07 
 
 31-51 
 
 30-95 
 
 29-83 
 
 3j 
 
 36-41 
 
 34-64 
 
 32-66 
 
 32-09 
 
 31-52 
 
 30-37 
 
 3 
 
 37-05 
 
 35-25 
 
 33-24 
 
 32-66 
 
 32-08 
 
 30-91 
 
 3i 
 
 37-68 
 
 35-85 
 
 33-81 
 
 33-22 
 
 3262 
 
 31-44 
 
 3; 
 
 38-31 
 
 36-45 
 
 34-37 
 
 33-77 
 
 3316 
 
 31-96 
 
 3,
 
 178 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE II. FOR FINDING THE VELOCITIES FROM THE ALTITUDES, 
 
 AND THE ALTITUDES FROM THE VELOCITIES. 
 
 Altitudes Ofeet 4 inches to 1 foot. 
 
 
 Coefficients of velocity and the corresponding velocities of 
 discharge in inches per second. 
 
 
 1. 
 
 
 
 
 
 
 Altitudes h 
 
 Values of 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 in feet and 
 
 v = 27-8 -J~h, 
 
 Coeffic 1 . 
 
 Coeffic*. 
 
 Coeffit 1 . 
 
 Coeffic 1 . 
 
 Coeffic 1 . 
 
 inches. 
 
 the theoretical 
 
 974. 
 
 956. 
 
 86. 
 
 815. 
 
 8. 
 
 
 velocity. 
 
 
 
 
 
 
 6 4 
 
 55-60 
 
 54-15 
 
 53-15 
 
 47-82 
 
 45-31 
 
 44-48 
 
 4 
 
 56-462 
 
 54-99 
 
 53-98 
 
 48-56 
 
 46-02 
 
 45-17 
 
 4 
 
 57-311 
 
 55-82 
 
 54-79 
 
 49-29 
 
 46-71 
 
 45-85 
 
 4| 
 
 58-148 
 
 5664 
 
 55-59 
 
 5001 
 
 47-39 
 
 46-52 
 
 o 44 
 
 58-973 
 
 57-44 
 
 56-38 
 
 50-72 
 
 48-06 
 
 47-18 
 
 4f 
 
 59-786 
 
 58-23 
 
 57-16 
 
 51-42 
 
 48-73 
 
 47-83 
 
 4* 
 
 60-589 
 
 59-01 
 
 57-92 
 
 52-11 
 
 49-38 
 
 48-47 
 
 4J 
 
 61-368 
 
 59-77 
 
 5867 
 
 52-78 
 
 50-02 
 
 49-09 
 
 5 
 
 62-163 
 
 60-55 
 
 59-43 
 
 53-46 
 
 50-66 
 
 49-73 
 
 5 
 
 62-935 
 
 61-30 
 
 60-17 
 
 54-12 
 
 51-29 
 
 50-35 
 
 5| 
 
 63-698 
 
 62-04 
 
 60-90 
 
 54-78 
 
 51-91 
 
 50-96 
 
 5f 
 
 64-452 
 
 62-78 
 
 61-62 
 
 55-43 
 
 52-53 
 
 51-56 
 
 54 
 
 65-197 
 
 63-50 
 
 62-33 
 
 56-07 
 
 53-14 
 
 52-16 
 
 5f 
 
 65-933 
 
 64-22 
 
 63-03 
 
 56-70 
 
 53-74 
 
 52-75 
 
 5| 
 
 66662 
 
 64-93 
 
 63-73 
 
 57-33 
 
 54-33 
 
 5333 
 
 5f 
 
 67-383 
 
 65-63 
 
 64-42 
 
 57-95 
 
 5492 
 
 53-91 
 
 6 
 
 68-096 
 
 66-33 
 
 65-10 
 
 58-56 
 
 55-50 
 
 54-48 
 
 61 
 
 69-50 
 
 67-69 
 
 66-44 
 
 59-77 
 
 56-64 
 
 55-60 
 
 6) 
 
 70-876 
 
 69-03 
 
 67-76 
 
 60-95 
 
 57-24 
 
 56-70 
 
 6| 
 
 72-227 
 
 70-35 
 
 69-05 
 
 62-11 
 
 58-86 
 
 57-78 
 
 7 
 
 73-552 
 
 71-64 
 
 70-32 
 
 63-25 
 
 59-95 
 
 58-84 
 
 7J 
 
 74-854 
 
 72-91 
 
 71-56 
 
 64-37 
 
 61-01 
 
 59-88 
 
 74 
 
 76-133 
 
 74-15 
 
 72-78 
 
 65-47 
 
 62-05 
 
 60-91 
 
 7* 
 
 77-392 
 
 75-38 
 
 73-99 
 
 66-56 
 
 63-07 
 
 61-91 
 
 8 
 
 78-630 
 
 76-59 
 
 75-17 
 
 67-62 
 
 64-08 
 
 62-90 
 
 8J 
 
 79-849 
 
 77-77 
 
 76-34 
 
 68-67 
 
 65-08 
 
 63-88 
 
 8J 
 
 81-050 
 
 78-94 
 
 77-48 
 
 69-70 
 
 6606 
 
 64-84 
 
 8| 
 
 82-234 
 
 80-10 
 
 78-62 
 
 7072 
 
 67-02 
 
 65-79 
 
 9 
 
 83-40 
 
 81-23 
 
 7973 
 
 7172 
 
 67-97 
 
 66-72 
 
 9} 
 
 84-550 
 
 82-35 
 
 80-83 
 
 72-71 
 
 68-91 
 
 67-64 
 
 9} 
 
 85-685 
 
 83-46 
 
 81-92 
 
 73-69 
 
 69-83 
 
 68-55 
 
 9} 
 
 86-805 
 
 84-55 
 
 82-99 
 
 74-65 
 
 70-75 
 
 6944 
 
 10 
 
 87-911 
 
 85-63 
 
 84-04 
 
 75-60 
 
 71-65 
 
 70-33 
 
 o 104 
 
 89-004 
 
 86-69 
 
 85-09 
 
 76-54 
 
 72-54 
 
 71-20 
 
 10.J 
 
 90-082 
 
 87-74 
 
 86-12 
 
 77-47 
 
 73-42 
 
 72-07 
 
 o io| 
 
 91-148 
 
 88-79 
 
 87-14 
 
 7839 
 
 74-29 
 
 72-92 
 
 11 
 
 92-202 
 
 89-80 
 
 88-15 
 
 79-29 
 
 75-14 
 
 73-76 
 
 o IH 
 
 93-244 
 
 90-82 
 
 89-14 
 
 80-19 
 
 75-99 
 
 74-59 
 
 11 \ 
 
 94-27-1 
 
 91-82 
 
 90-13 
 
 81-08 
 
 76-83 
 
 75-42 
 
 111 
 
 95-294 
 
 9282 
 
 91-10 
 
 81-95 
 
 7766 
 
 76-23 
 
 1 
 
 96302 
 
 9380 
 
 92-06 
 
 82-82 
 
 78-49 
 
 7704
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 179 
 
 TABLE II. FOE FINDING THE VELOCITIES FROM THE ALTITUDES, 
 
 AND THE ALTITUDES FROM THE VELOCITIES. 
 
 Altitudes Qfeet 4 inches to I foot. 
 
 Coefficients of velocity and the corresponding velocities of 
 
 
 discharge in inches per second. 
 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 
 11. 
 
 12. 
 
 Altitudes h 
 
 Coeffic'. 
 
 Coeffic 1 . 
 
 Coeffic 1 . 
 
 Coeffic 1 . 
 
 Coeffic'. 
 
 Coeffic'. 
 
 in feet and 
 
 7 
 
 666. 
 
 628. 
 
 617. 
 
 606. 
 
 584. 
 
 inches. 
 
 38-92 
 
 37-03 
 
 34-92 
 
 34-31 
 
 33-69 
 
 32-47 
 
 6 4 
 
 39-52 
 
 37-60 
 
 35-46 
 
 34-84 
 
 34-22 
 
 32-97 
 
 4| 
 
 40-12 
 
 38-17 
 
 35-99 
 
 35-36 
 
 34-73 
 
 33-47 
 
 4\ 
 
 40-70 
 
 38-73 
 
 36-52 
 
 35-88 
 
 35-24 
 
 33-96 
 
 4| 
 
 41-28 
 
 39-28 
 
 37-03 
 
 36-39 
 
 35-74 
 
 34-44 
 
 4* 
 
 41-85 
 
 39-82 
 
 37-55 
 
 36-89 
 
 36-23 
 
 34-92 
 
 4| 
 
 42-41 
 
 40-35 
 
 38-05 
 
 37-38 
 
 36-72 
 
 35-38 
 
 4J 
 
 42-96 
 
 40-87 
 
 38-54 
 
 37-86 
 
 37-19 
 
 35-84 
 
 4 
 
 43-51 
 
 41-40 
 
 39-04 
 
 38-35 
 
 37-67 
 
 36-30 
 
 5 
 
 44-05 
 
 41-91 
 
 39-52 
 
 38-83 
 
 38-14 
 
 36-75 
 
 5 
 
 44-59 
 
 42-42 
 
 40-00 
 
 39-30 
 
 38-60 
 
 37-20 
 
 5i 
 
 45-12 
 
 42-92 
 
 40-48 
 
 39-77 
 
 39-06 
 
 37-64 
 
 5| 
 
 45-64 
 
 43-42 
 
 40-94 
 
 40-23 
 
 39-51 
 
 38-07 
 
 54 
 
 46-15 
 
 43-91 
 
 41-41 
 
 40-68 
 
 39-96 
 
 38-51 
 
 5? 
 
 46-66 
 
 44-40 
 
 41-86 
 
 41-13 
 
 40-40 
 
 38-93 
 
 5| 
 
 47-17 
 
 44-88 
 
 42-32 
 
 41-58 
 
 40-83 
 
 39-35 
 
 6J 
 
 47-67 
 
 45-35 
 
 42-76 
 
 42-02 
 
 41-27 
 
 39-77 
 
 6 
 
 48-65 
 
 46-29 
 
 43-65 
 
 42-88 
 
 42-12 
 
 40-59 
 
 6} 
 
 49-61 
 
 47-20 
 
 44-51 
 
 43-73 
 
 42-95 
 
 41-39 
 
 6.1 
 
 50-56 
 
 48-10 
 
 45-36 
 
 44-56 
 
 43-77 
 
 42-18 
 
 6| 
 
 51-49 
 
 48-99 
 
 46-19 
 
 45-38 
 
 44-57 
 
 42-95 
 
 7 
 
 52-40 
 
 49-85 
 
 47-01 
 
 46-18 
 
 45-36 
 
 43-71 
 
 7i 
 
 53-29 
 
 50-70 
 
 47-81 
 
 4697 
 
 46-14 
 
 44-46 
 
 7i 
 
 54-17 
 
 51-54 
 
 48-60 
 
 47-75 
 
 46-90 
 
 45-20 
 
 7| 
 
 55-04 
 
 52-37 
 
 49-38 
 
 48-51 
 
 47-65 
 
 45-92 
 
 8 
 
 55-89 
 
 53-18 
 
 50-15 
 
 49-27 
 
 48-39 
 
 46-63 
 
 8i 
 
 56-74 
 
 53-98 
 
 50-90 
 
 50-01 
 
 49-12 
 
 47-33 
 
 8i 
 
 57-56 
 
 54-77 
 
 51-64 
 
 50-74 
 
 49-83 
 
 48-02 
 
 8| 
 
 58-38 
 
 55-54 
 
 52-38 
 
 51-46 
 
 50-54 
 
 48-71 
 
 9 
 
 59-19 
 
 56-31 
 
 53-10 
 
 52-17 
 
 51-24 
 
 49-38 
 
 9J 
 
 ' 59-98 
 
 57-07 
 
 53-81 
 
 52-87 
 
 51-93 
 
 50-04 
 
 9J 
 
 j 60-76 
 
 57-81 
 
 54-51 
 
 53-56 
 
 52-60 
 
 50-69 
 
 9| 
 
 61-54 
 
 58-55 
 
 55-22 
 
 54-24 
 
 53-27 
 
 51-34 
 
 10 
 
 62-30 
 
 59-28 
 
 55-89 
 
 54-92 
 
 5394 
 
 51-98 
 
 10J 
 
 63-06 
 
 60-00 
 
 5657 
 
 55-58 
 
 54-59 
 
 52-61 
 
 o 104 
 
 63-80 
 
 60-70 
 
 57-24 
 
 56-24 
 
 55-24 
 
 5323 
 
 10| 
 
 64-54 
 
 61-41 
 
 57-90 
 
 56-89 
 
 55-87 
 
 53-85 
 
 11 
 
 65-27 
 
 62-10 
 
 58-56 
 
 57-53 
 
 56-51 
 
 54-45 
 
 11J 
 
 65-99 
 
 62-79 
 
 59-70 
 
 58-17 
 
 57-13 
 
 55-06 
 
 llj 
 
 66-71 
 
 63-47 
 
 59-84 
 
 58-80 
 
 57-75 
 
 55-65 
 
 llf 
 
 67-41 
 
 64-14 
 
 60-48 
 
 59-42 
 
 5836 
 
 56-24 
 
 1
 
 180 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE II. FOR FINDING THE VELOCITIES FROM THE ALTITUDES, 
 
 AND THE ALTITUDES FROM THE VELOCITIES. 
 
 Altitudes I foot 0| inch to 5 feet 3 inches. 
 
 
 Coefficients of velocity and the corresponding velocities of 
 
 
 discharge in inches per second. 
 
 
 1. 
 
 
 
 
 
 
 Altitudes h 
 
 Values of 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 in feet and 
 
 v = 27-8 v/I 
 
 Coeffic'. 
 
 Coeffic*. 
 
 CoeffiV. 
 
 Coeffic*. 
 
 Coeffic 1 . 
 
 inches. 
 
 the theoretica 
 
 -974. 
 
 956. 
 
 860. 
 
 815. 
 
 8. 
 
 
 velocity. 
 
 
 
 
 
 
 i 6'i 
 
 98-288 
 
 95-73 
 
 93-96 
 
 84-53 
 
 80-10 
 
 78-63 
 
 1 1 
 
 100-234 
 
 97-63 
 
 95-82 
 
 86-20 
 
 81-69 
 
 80-19 
 
 1 14 
 
 102-144 
 
 99-49 
 
 97-65 
 
 87-84 
 
 83-25 
 
 81-71 
 
 2 
 
 104-018 
 
 101-31 
 
 99-44 
 
 89-46 
 
 84-77 
 
 83-21 
 
 1 
 
 105-859 
 
 103-11 
 
 101-20 
 
 91-04 
 
 86-28 
 
 84-69 
 
 3 
 
 107-669 
 
 104-87 
 
 102-93 
 
 92-60 
 
 87-75 
 
 86-14 
 
 3^ 
 
 109-449 
 
 106-60 
 
 104-63 
 
 94-13 
 
 89-20 
 
 87-56 
 
 4 
 
 111-200 
 
 108-31 
 
 106-31 
 
 95-63 
 
 90-63 
 
 88-96 
 
 4| 
 
 112-924 
 
 109-99 
 
 107-96 
 
 97-11 
 
 92-03 
 
 90-34 
 
 5 
 
 114-622 
 
 111-42 
 
 109-58 
 
 98-58 
 
 93-42 
 
 91-70 
 
 5i 
 
 116-296 
 
 113-27 
 
 111-18 
 
 100-01 
 
 94-78 
 
 93-04 
 
 6 
 
 117-945 
 
 114-78 
 
 112-76 
 
 101-43 
 
 96-13 
 
 94-36 
 
 7 
 
 121-177 
 
 118-03 
 
 115-85 
 
 104-21 
 
 98-76 
 
 9694 
 
 8 
 
 124-325 
 
 121-09 
 
 118-86 
 
 106-92 
 
 101-33 
 
 99-46 
 
 9 
 
 127-396 
 
 124-08 
 
 121-79 
 
 109-56 
 
 103-83 
 
 101-92 
 
 10 
 
 130-394 
 
 127-00 
 
 124-66 
 
 112-14 
 
 106-27 
 
 104-31 
 
 11 
 
 133-324 
 
 129-86 
 
 127-46 
 
 114-66 
 
 108-66 
 
 106-66 
 
 2 
 
 136-192 
 
 132-65 
 
 130-20 
 
 117-12 
 
 111-00 
 
 108-95 
 
 2 H 
 
 140-383 
 
 136-73 
 
 134-21 
 
 120-73 
 
 114-41 
 
 112-31 
 
 2 3 
 
 144-453 
 
 140-70 
 
 138-10 
 
 124-23 
 
 117-73 
 
 115-56 
 
 2 4 
 
 148-411 
 
 144-55 
 
 141-88 
 
 127-64 
 
 120-96 
 
 118-73 
 
 2 6 
 
 152-267 
 
 148-31 
 
 145-57 
 
 130-95 
 
 124-10 
 
 121-81 
 
 2 7 
 
 156-027 
 
 151-97 
 
 149-16 
 
 134-18 
 
 127-16 
 
 124-82 
 
 2 9 
 
 159-699 
 
 155-55 
 
 152-67 
 
 137-34 
 
 130-15 
 
 127-76 
 
 2 10| 
 
 163-288 
 
 159-04 
 
 156-10 
 
 140-43 
 
 133 80 
 
 130-63 
 
 3 
 
 166-800 
 
 162-46 
 
 159-46 
 
 143-45 
 
 135-94 
 
 133-44 
 
 3 1J 
 
 170-240 
 
 165-81 
 
 162-75 
 
 146-41 
 
 138-75 
 
 136-19 
 
 3 3 
 
 178-611 
 
 169-10 
 
 165-97 
 
 149-31 
 
 141-49 
 
 138-89 
 
 3 4i 
 
 176-918 
 
 172-32 
 
 169-13 
 
 152-15 
 
 144-19 
 
 141-53 
 
 3 6 
 
 180-165 
 
 175-48 
 
 172-24 
 
 154-94 
 
 146-83 
 
 144-13 
 
 3 74 
 
 183-354 
 
 178-59 
 
 175-29 
 
 157-68 
 
 149-43 
 
 146-68 
 
 3 9 
 
 186-488 
 
 181-64 
 
 178-28 
 
 160-38 
 
 151-99 
 
 149-19 
 
 3 10J 
 
 189-571 
 
 184-64 
 
 181-23 
 
 163-03 
 
 154-50 
 
 151-66 
 
 4 
 
 192-604 
 
 187-60 
 
 184-13 
 
 165-64 
 
 156-97 
 
 154-08 
 
 4 2 
 
 196-576 
 
 191-46 
 
 187-93 
 
 169-06 
 
 160-21 
 
 157-26 
 
 4 4 
 
 200-469 
 
 195-26 
 
 191-65 
 
 172-40 
 
 163-38 
 
 160-37 
 
 4 6 
 
 204-287 
 
 198-98 
 
 195-30 
 
 175-69 
 
 166-49 
 
 16343 
 
 4 8 
 
 208-036 
 
 20263 
 
 198-88 
 
 178-91 
 
 169-55 
 
 166-43 
 
 4 10 
 
 211-718 
 
 206-21 
 
 202-40 
 
 182-08 
 
 172-55 
 
 169-37 
 
 5 
 
 215-338 
 
 209-74 
 
 205-86 
 
 18509 
 
 175-50 
 
 172-27 
 
 5 3 
 
 220-656 
 
 214-92 
 
 210-95 
 
 189-76 
 
 179-83 
 
 17652
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 181 
 
 TABLE II. FOR FINDING THE VELOCITIES FROM THE ALTITUDES, 
 
 AND THE ALTITUDES FROM THE VELOCITIES. 
 
 Altitudes I foot inch to 5 feet 3 inches. 
 
 Coefficients of velocity and the corresponding velocities of 
 
 
 discharge in inches per second. 
 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 
 11. 
 
 12. 
 
 Altitudes h 
 
 Coeffic'. 
 
 Coeffic 1 . 
 
 Cocffic 1 . 
 
 Coeffic 1 . 
 
 Coeffic 1 . 
 
 Coeffic*. 
 
 in feet and 
 
 7. 
 
 666. 
 
 628. 
 
 617. 
 
 606. 
 
 584. 
 
 inches. 
 
 68-80 
 
 65-46 
 
 6172 
 
 60-64 
 
 59-56 
 
 57-40 
 
 i e'i 
 
 70-16 
 
 66-76 
 
 62-95 
 
 61-84 
 
 60-74 
 
 58-54 
 
 1 1 
 
 71-50 
 
 68-03 
 
 64-15 
 
 63-02 
 
 61-90 
 
 59-65 
 
 1 11 
 
 72-81 
 
 69-28 
 
 65-32 
 
 64-18 
 
 63-03 
 
 60-75 
 
 1 2 
 
 74-10 
 
 70-50 
 
 66-48 
 
 65-32 
 
 64-15 
 
 61-82 
 
 1 2i 
 
 75-37 
 
 71-71 
 
 67-62 
 
 66-43 
 
 65-25 
 
 62-88 
 
 1 3 
 
 76-61 
 
 72-89 
 
 68-73 
 
 67-53 
 
 66-33 
 
 63-92 
 
 1 31 
 
 77-84 
 
 74-06 
 
 69-83 
 
 68-61 
 
 67-34 
 
 64-94 
 
 1 4 
 
 79-05 
 
 75-21 
 
 70-92 
 
 69-67 
 
 68-43 
 
 65-95 
 
 1 4J 
 
 80-24 
 
 76-34 
 
 71-98 
 
 70-72 
 
 69-46 
 
 66-94 
 
 1 5 
 
 81-41 
 
 77-45 
 
 73-03 
 
 71-75 
 
 70-48 
 
 67-92 
 
 1 51 
 
 82-56 
 
 78-55 
 
 74-07 
 
 72-77 
 
 71-47 
 
 68-88 
 
 1 6 
 
 84-82 
 
 80-70 
 
 76-10 
 
 74-77 
 
 73-43 
 
 70-77 
 
 1 7 
 
 87-03 
 
 82-80 
 
 78-08 
 
 76-71 
 
 75-34 
 
 72-61 
 
 1 8 
 
 89-18 
 
 84-85 
 
 80-00 
 
 78-60 
 
 77-20 
 
 74-40 
 
 1 9 
 
 91-28 
 
 86-84 
 
 81-89 
 
 80-45 
 
 79-02 
 
 76-15 
 
 1 10 
 
 93-33 
 
 88-79 
 
 8373 
 
 82-26 
 
 80-79 
 
 77-86 
 
 1 11 
 
 95-33 
 
 90-70 
 
 85-53 
 
 84-03 
 
 82-53 
 
 79-54 
 
 2 
 
 98-27 
 
 93-50 
 
 88-16 
 
 86-62 
 
 85-07 
 
 81-98 
 
 2 1* 
 
 101-12 
 
 96-21 
 
 90-72 
 
 89-13 
 
 87-54 
 
 84-36 
 
 2 3 
 
 103-89 
 
 98-84 
 
 93-20 
 
 91-57 
 
 89-94 
 
 86-67 
 
 2 41 
 
 106-59 
 
 101-41 
 
 95-62 
 
 93-95 
 
 92-27 
 
 88-92 
 
 2 6 
 
 109-22 
 
 103-91 
 
 97-99 
 
 96-27 
 
 94-55 
 
 91-12 
 
 2 71 
 
 111-79 
 
 106-36 
 
 100-29 
 
 98-53 
 
 96-78 
 
 93-26 
 
 2 9 
 
 114-30 
 
 108-75 
 
 102-54 
 
 100-75 
 
 9895 
 
 95-36 
 
 2 104 
 
 116-76 
 
 111-09 
 
 104-75 
 
 102-92 
 
 101-08 
 
 97-41 
 
 3 
 
 119-17 
 
 11338 
 
 106-91 
 
 105-04 
 
 103-17 
 
 99-42 
 
 3 11 
 
 121-53 
 
 115-62 
 
 109-03 
 
 10712 
 
 105-21 
 
 101-39 
 
 3 3 
 
 1213-84 
 
 117-83 
 
 111-10 
 
 109-16 
 
 107-21 
 
 103-32 
 
 3 44 
 
 126-12 
 
 119-99 
 
 113-14 
 
 111-16 
 
 109-18 
 
 105-22 
 
 3 6 
 
 i 128-35 
 
 122-11 
 
 115-15 
 
 113-13 
 
 111-11 
 
 107-08 
 
 3 71 
 
 ' 130-54 
 
 124-20 
 
 117-11 
 
 115-06 
 
 113-01 
 
 108-91 
 
 3 9 
 
 132-70 
 
 126-25 
 
 119-05 
 
 116-97 
 
 114-88 
 
 110-71 
 
 3 101 
 
 i 134-82 
 
 128-27 
 
 120-96 
 
 118-84 
 
 116-72 
 
 112-48 
 
 4 
 
 137-60 
 
 130-92 
 
 123-45 
 
 121-29 
 
 119-12 
 
 114'80 
 
 4 2 
 
 140-33 
 
 133-51 
 
 125-89 
 
 123-69 
 
 121-48 
 
 117-07 
 
 4 4 
 
 143-00 
 
 136-06 
 
 128-29 
 
 126-05 
 
 12380 
 
 119-30 
 
 4 6 
 
 \ 145-63 
 
 138-55 
 
 130-65 
 
 128-36 
 
 126-07 
 
 121-49 
 
 4 8 
 
 ! 148-20 
 
 141-00 
 
 132-96 
 
 130-63 
 
 128-30 
 
 123-64 
 
 4 10 
 
 150-74 
 
 143-42 
 
 135-23 
 
 132-86 
 
 130-49 
 
 125-76 
 
 5 
 
 154-46 
 
 146-96 
 
 13857 
 
 136-14 
 
 133-72 
 
 128-86 
 
 5 3
 
 182 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE II. FOB FINDING THE VELOCITIES FROM THE ALTITUDES, 
 
 AND THE ALTITUDES FROM THE VELOCITIES. 
 
 Altitudes 5 feet 6 inches to 17 feet. 
 
 
 Coefficients of velocity, and the corresponding velocities of 
 
 
 discharge in inches per second. 
 
 
 1. 
 
 
 
 
 
 
 Altitudes h 
 
 Values of 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 in feet and 
 
 v = 27-8 N/A, 
 
 Coeffic 1 . 
 
 Coeffic 1 . 
 
 Coeffic 1 . 
 
 Coeffic'. 
 
 Coeffic*. 
 
 inches. 
 
 the theoretical 
 
 974. 
 
 956. 
 
 86. 
 
 815. 
 
 8. 
 
 
 velocity. 
 
 
 
 
 
 
 5 6 
 
 225-848 
 
 219-98 
 
 215-91 
 
 194-23 
 
 184-07 
 
 180-68 
 
 5 9 
 
 230-924 
 
 224-92 
 
 220-76 
 
 198-59 
 
 188-20 
 
 184-74 
 
 6 
 
 235-891 
 
 229-76 
 
 225-51 
 
 202-87 
 
 192-25 
 
 188-71 
 
 6 3 
 
 240-755 
 
 234-50 
 
 230-16 
 
 207-05 
 
 196-22 
 
 192-60 
 
 6 6 
 
 245-524 
 
 239-14 
 
 234-72 
 
 211-15 
 
 200-10 
 
 196-42 
 
 6 9 
 
 250-200 
 
 243-69 
 
 23919 
 
 215-17 
 
 203-91 
 
 200-16 
 
 7 
 
 254-791 
 
 248-17 
 
 243-58 
 
 219-12 
 
 207-65 
 
 203-83 
 
 7 3 
 
 259-301 
 
 252-56 
 
 247-89 
 
 222-99 
 
 211-33 
 
 207-44 
 
 7 6 
 
 263-734 
 
 256-88 
 
 252-13 
 
 226-81 
 
 214-94 
 
 210-99 
 
 7 9 
 
 268-093 
 
 261-12 
 
 256-30 
 
 230-56 
 
 218-50 
 
 214-47 
 
 8 
 
 272-383 
 
 265-30 
 
 260-40 
 
 234-25 
 
 221-99 
 
 217-91 
 
 8 3 
 
 276-607 
 
 269-41 
 
 264-44 
 
 237-88 
 
 225-43 
 
 221-29 
 
 8 6 
 
 280-766 
 
 273-47 
 
 268-41 
 
 241-46 
 
 228-82 
 
 224-61 
 
 8 9 
 
 284-865 
 
 277-46 
 
 272-33 
 
 244-98 
 
 232-17 
 
 227-89 
 
 9 
 
 288-906 
 
 281-39 
 
 276-19 
 
 248-46 
 
 235-46 
 
 231-12 
 
 9 3 
 
 292-891 
 
 285-28 
 
 280-00 
 
 251-89 
 
 238-71 
 
 234-31 
 
 9 6 
 
 296-823 
 
 289-11 
 
 283-76 
 
 255-27 
 
 241-91 
 
 237-46 
 
 9 9 
 
 300703 
 
 292-88 
 
 287-47 
 
 258-60 
 
 245-07 
 
 240-56 
 
 10 
 
 304-534 
 
 296-62 
 
 291-13 
 
 261-90 
 
 248-19 
 
 243-63 
 
 10 3 
 
 308-317 
 
 300-30 
 
 294-75 
 
 265-15 
 
 251-28 
 
 246-65 
 
 10 6 
 
 312-054 
 
 303-94 
 
 297-32 
 
 268-37 
 
 254-32 
 
 249-64 
 
 10 9 
 
 315-747 
 
 307-54 
 
 303-85 
 
 271-54 
 
 257-33 
 
 252-60 
 
 11 
 
 319-398 
 
 311-09 
 
 305-34 
 
 274-68 
 
 260-31 
 
 255-52 
 
 11 3 
 
 323-007 
 
 314-61 
 
 308-79 
 
 277-79 
 
 262-25 
 
 258-41 
 
 11 6 
 
 326-576 
 
 318-09 
 
 312-21 
 
 280-86 
 
 266-16 
 
 261-26 
 
 11 9 
 
 330-107 
 
 321-52 
 
 315-58 
 
 283-89 
 
 269-04 
 
 264-09 
 
 12 
 
 333-600 
 
 324-93 
 
 318-92 
 
 286-90 
 
 271-88 
 
 266-88 
 
 12 3 
 
 337-057 
 
 328-29 
 
 322-23 
 
 289-87 
 
 274-70 
 
 269-65 
 
 12 6 
 
 340-479 
 
 331-63 
 
 325-50 
 
 292-81 
 
 277-49 
 
 272-38 
 
 12 9 
 
 343-867 
 
 334-93 
 
 328-74 
 
 295-73 
 
 280-25 
 
 275-09 
 
 13 
 
 347-222 
 
 338-19 
 
 331-94 
 
 298-61 
 
 282-99 
 
 277-78 
 
 13 3 
 
 350-545 
 
 341-43 
 
 335-12 
 
 301-47 
 
 285-69 
 
 280-44 
 
 13 6 
 
 353-836 
 
 344-64 
 
 338-27 
 
 304-30 
 
 288-38 
 
 283-07 
 
 13 9 
 
 357-097 
 
 347-81 
 
 341-39 
 
 307-10 
 
 291-03 
 
 285-68 
 
 14 
 
 360-329 
 
 350-96 
 
 344-47 
 
 309-88 
 
 293-67 
 
 288-26 
 
 14 6 
 
 366-707 
 
 357-17 
 
 350-57 
 
 315-37 
 
 298-87 
 
 293-37 
 
 15 
 
 372-976 
 
 363-28 
 
 356-57 
 
 320-76 
 
 303-98 
 
 298-38 
 
 15 6 
 
 379-141 
 
 369-28 
 
 362-46 
 
 326-06 
 
 309-00 
 
 303-31 
 
 16 
 
 385-208 
 
 375-19 
 
 368-26 
 
 331-28 
 
 313-94 
 
 308-17 
 
 16 6 
 
 391-181 
 
 381-01 
 
 373-97 
 
 336-42 
 
 318-81 
 
 312-94 
 
 17 
 
 397-063 
 
 386-74 
 
 379-59 
 
 341-47 
 
 323-61 
 
 317-65
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 183 
 
 TABLE II. FOR FINDING THE VELOCITIES FROM THE ALTITUDES, 
 
 AND THE ALTITUDES FROM THE VELOCITIES. 
 
 Altitudes 5 feet 6 inches to 17 feet. 
 
 Coefficients of velocity, and the corresponding velocities of 
 
 
 discharge in inches per second. 
 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 
 11. 
 
 12. 
 
 Altitudes h 
 
 Coeffic'. 
 
 Coeffic'. 
 
 Coeffic'. 
 
 Coeffic 1 . 
 
 Coeffic'. 
 
 Coeffic'. 
 
 in feet and 
 
 7. 
 
 666. 
 
 628. 
 
 617. 
 
 606. 
 
 584. 
 
 inches. 
 
 158-09 
 
 150-41 
 
 141-83 
 
 139-35 
 
 136-86 
 
 131-90 
 
 5 6 
 
 161-65 
 
 153-80 
 
 145-02 
 
 142-48 
 
 139-94 
 
 134-86 
 
 5 9 
 
 165-12 
 
 157-10 
 
 148-14 
 
 145-55 
 
 142-95 
 
 137-76 
 
 6 
 
 168-53 
 
 160-34 
 
 151-19 
 
 148-55 
 
 145-90 
 
 140-60 
 
 6 3 
 
 171-87 
 
 163-52 
 
 154-19 
 
 151-49 
 
 148-79 
 
 143-39 
 
 6 6 
 
 175-14 
 
 166-63 
 
 157-13 
 
 154-37 
 
 151-62 
 
 146-12 
 
 6 9 
 
 178-35 
 
 169-69 
 
 160-01 
 
 157-21 
 
 154-40 
 
 148-80 
 
 7 
 
 181-51 
 
 172-69 
 
 162-84 
 
 159-99 
 
 157-14 
 
 151-43 
 
 7 3 
 
 184-61 
 
 175-65 
 
 165-62 
 
 162-72 
 
 159-82 
 
 154-02 
 
 7 6 
 
 187-67 
 
 178-55 
 
 168-36 
 
 165-41 
 
 162-46 
 
 156-57 
 
 7 9 
 
 190-67 
 
 ]81-41 
 
 171-06 
 
 168-06 
 
 165-06 
 
 159-07 
 
 8 
 
 193-62 
 
 184-22 
 
 173-71 
 
 170-67 
 
 167-62 
 
 161-54 
 
 8 3 
 
 196-54 
 
 186-99 
 
 176-32 
 
 173-23 
 
 170-14 
 
 163-97 
 
 8 6 
 
 199-41 
 
 189-72 
 
 178-90 
 
 175-76 
 
 172-63 
 
 166-36 
 
 8 9 
 
 202-23 
 
 192-41 
 
 181-43 
 
 178-26 
 
 175-08 
 
 168-72 
 
 9 
 
 205-02 
 
 195-07 
 
 183-94 
 
 180-71 
 
 177-49 
 
 171-05 
 
 9 3 
 
 207-78 
 
 197-68 
 
 186-40 ! 183-14 
 
 179-87 
 
 173-34 
 
 9 6 
 
 210-49 
 
 200-27 
 
 188-84 
 
 185-53 
 
 182-23 
 
 175-61 
 
 9 9 
 
 213-17 
 
 202-82 
 
 191-25 
 
 187-90 
 
 184-55 
 
 177-85 
 
 10 
 
 215-82 
 
 205-34 
 
 193-62 
 
 190-23 
 
 186-84 
 
 180-06 
 
 10 3 
 
 218-44 
 
 207-83 
 
 195-97 
 
 192-54 
 
 189-10 
 
 182-24 
 
 10 6 
 
 221-02 
 
 210-29 
 
 198-29 
 
 194-82 
 
 191-34 
 
 184-40 
 
 10 9 
 
 223-58 
 
 212-72 
 
 200-58 
 
 197-07 
 
 193-55 
 
 186-53 
 
 11 
 
 226-10 
 
 215-12 
 
 202-85 
 
 199-30 
 
 195-74 
 
 188-64 
 
 11 3 
 
 228-60 
 
 217-50 
 
 205-09 
 
 201-50 
 
 197-91 
 
 190-72 
 
 11 6 
 
 231-07 
 
 219-85 
 
 207-31 
 
 203-68 
 
 200-04 
 
 192-78 
 
 11 9 
 
 233-52 
 
 222-18 
 
 209-50 
 
 205-83 
 
 202-16 
 
 194-82 
 
 12 
 
 235-94 
 
 224-48 
 
 211-67 
 
 207-96 
 
 204-26 
 
 196-84 
 
 12 3 
 
 238-34 
 
 226-76 
 
 213-82 
 
 210-08 
 
 206-33 
 
 198-84 
 
 12 6 
 
 240-71 
 
 229-02 
 
 215-95 
 
 212-17 
 
 208-38 
 
 200-82 
 
 12 9 
 
 243-06 
 
 231-25 
 
 218-06 
 
 214-24 
 
 210-42 
 
 202-78 
 
 13 
 
 245-38 
 
 233-46 
 
 220-14 
 
 216-29 
 
 212-43 
 
 204-72 
 
 13 3 
 
 247-69 
 
 235-65 
 
 222-21 
 
 218-32 
 
 214-42 
 
 206-64 
 
 13 6 
 
 249-97 
 
 237-83 
 
 224-26 
 
 220-33 
 
 216-40 
 
 208-54 
 
 13 9 
 
 252-23 
 
 239-98 
 
 226-29 
 
 222-32 
 
 218-36 
 
 210-43 
 
 14 
 
 256-70 
 
 244-23 
 
 230-29 
 
 226-26 
 
 222-22 
 
 214-16 
 
 14 6 
 
 261-08 
 
 248-40 
 
 234-23 
 
 230-13 
 
 226-02 
 
 217-82 
 
 15 
 
 265-40 
 
 252-51 
 
 238-10 
 
 233-93 
 
 229-76 
 
 221-42 
 
 15 6 
 
 269-65 
 
 256-55 
 
 241-91 
 
 237-67 
 
 233-44 
 
 224-96 
 
 16 
 
 273-83 
 
 260-53 
 
 245-66 
 
 241-36 
 
 237-06 
 
 228-45 
 
 16 6 
 
 277-94 
 
 264-44 
 
 249-36 
 
 244-99 
 
 240-62 
 
 231-89 
 
 17
 
 184 
 
 THE DISCHARGE OP WATER FROM 
 
 TABLE II. FOR FINDING THE VELOCITIES FROM THE ALTITUDES, 
 
 AND THE ALTITUDES FROM THE VELOCITIES. 
 
 Altitudes 17 feet 6 inches to 40 feet. 
 
 
 Coefficients of velocity, and the corresponding velocities of 
 discharge in inches per second. 
 
 
 1. 
 
 
 
 
 
 
 Altitudes A 
 
 Values of 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 in feet and 
 
 v = 27-8 v/A, 
 
 Coeffic 1 . 
 
 Coeffic 4 . 
 
 Coeffic*. 
 
 Coeffic*. 
 
 Coeffic*. 
 
 inches. 
 
 the theoretical 
 
 974. 
 
 956. 
 
 86. 
 
 815. 
 
 8. 
 
 
 velocity. 
 
 
 
 
 
 
 17 6 
 
 402-860 
 
 392-39 
 
 385-13 
 
 346-46 
 
 328-33 
 
 322-29 
 
 18 
 
 408-575 
 
 397-95 
 
 390-60 
 
 351-37 
 
 332-99 
 
 326-86 
 
 18 6 
 
 414-211 
 
 403-44 
 
 395-99 
 
 356-22 
 
 337-58 
 
 331-37 
 
 19 
 
 419-772 
 
 408-86 
 
 401-30 
 
 361-00 
 
 342-11 
 
 335-82 
 
 19 6 
 
 425-258 
 
 414-20 
 
 406-55 
 
 365-72 
 
 346-59 
 
 340-21 
 
 20 
 
 430-676 
 
 419-48 
 
 411-73 
 
 370-38 
 
 351-00 
 
 344-54 
 
 20 6 
 
 436-026 
 
 424-69 
 
 416-84 
 
 374-98 
 
 355-36 
 
 348-82 
 
 21 
 
 441-311 
 
 429-84 
 
 421-89 
 
 379-53 
 
 359-59 
 
 353-05 
 
 21 6 
 
 446-534 
 
 434-92 
 
 426-89 
 
 384-02 
 
 363-93 
 
 357-23 
 
 22 
 
 451-697 
 
 439-95 
 
 431-82 
 
 388-46 
 
 368-13 
 
 361-36 
 
 22 6 
 
 456-801 
 
 444-92 
 
 436-70 
 
 392-85 
 
 372-29 
 
 365-44 
 
 23 
 
 461-848 
 
 449-84 
 
 441-53 
 
 397-19 
 
 376-41 
 
 369-48 
 
 23 6 
 
 466-841 
 
 450-70 
 
 446-30 
 
 401-48 
 
 380-48 
 
 373-47 
 
 24 
 
 471-782 
 
 459-52 
 
 451-02 
 
 405-73 
 
 384-50 
 
 377-43 
 
 24 6 
 
 476-671 
 
 464-28 
 
 455-70 
 
 409-94 
 
 388-49 
 
 381-34 
 
 25 
 
 481-510 
 
 468-99 
 
 460-32 
 
 414-10 
 
 392-43 
 
 385-21 
 
 25 6 
 
 486-301 
 
 473-66 
 
 464-90 
 
 418-22 
 
 396-34 
 
 389-04 
 
 26 
 
 491-046 
 
 478-28 
 
 469-44 
 
 422-30 
 
 400-20 
 
 392-84 
 
 26 6 
 
 495-745 
 
 482-86 
 
 473-93 
 
 426-34 
 
 404-03 
 
 396-60 
 
 27 
 
 500-40 
 
 487-39 
 
 478-38 
 
 430-34 
 
 407-83 
 
 400-32 
 
 27 6 
 
 505-012 
 
 491-88 
 
 482-79 
 
 434-31 
 
 411-58 
 
 404-01 
 
 28 
 
 509-582 
 
 496-33 
 
 487-16 
 
 438-24 
 
 415-31 
 
 407-67 
 
 28 6 
 
 514-112 
 
 500-75 
 
 491-49 
 
 442-14 
 
 419-00 
 
 411-29 
 
 29 
 
 518-602 
 
 505-12 
 
 495-78 
 
 446-00 
 
 422-66 
 
 414-88 
 
 29 6 
 
 523-054 
 
 509-45 
 
 500-04 
 
 449-83 
 
 426-29 
 
 418-44 
 
 30 
 
 527-468 
 
 513-75 
 
 504-26 
 
 453-62 
 
 429-89 
 
 421-97 
 
 30 6 
 
 531-845 
 
 518-02 
 
 508-44 
 
 457-39 
 
 433-45 
 
 425-48 
 
 31 .0 
 
 536-187 
 
 622-25 
 
 512-59 
 
 461-12 
 
 436-99 
 
 428-95 
 
 31 6 
 
 540-494 
 
 526-44 
 
 516-71 
 
 464-82 
 
 440-50 
 
 432-40 
 
 32 
 
 644-767 
 
 530-60 
 
 520-80 
 
 468-50 
 
 443-98 
 
 435-81 
 
 32 6 
 
 549-006 
 
 534-73 
 
 524-85 
 
 472-15 
 
 447-44 
 
 439-20 
 
 33 
 
 553-213 
 
 538-83 
 
 528-87 
 
 475-76 
 
 450-87 
 
 442-57 
 
 33 6 
 
 557-388 
 
 542-90 
 
 532-86 
 
 479-35 
 
 454-27 
 
 445-91 
 
 34 
 
 561-532 
 
 546-93 
 
 536-83 
 
 482-92 
 
 457-65 
 
 449-23 
 
 34 6 
 
 565-646 
 
 550-94 
 
 540-76 
 
 486-46 
 
 461-00 
 
 452-52 
 
 35 
 
 569-730 
 
 554-92 
 
 544-66 
 
 489-97 
 
 464-33 
 
 455-78 
 
 36 
 
 677-812 
 
 562-79 
 
 552-39 
 
 496-92 
 
 470-92 
 
 462-25 
 
 37 
 
 585-782 
 
 570-55 
 
 560-01 
 
 503-77 
 
 477-41 
 
 468-63 
 
 38 
 
 593-646 
 
 578-21 
 
 567-53 
 
 510-54 
 
 483-82 
 
 474-92 
 
 39 
 
 601-406 
 
 585-77 
 
 574-94 
 
 517-21 
 
 490-15 
 
 481-12 
 
 40 
 
 609-067 
 
 593-23 
 
 582-27 
 
 523-80 
 
 496-39 
 
 487-25
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 185 
 
 TABLE II. FOB FINDING THE VELOCITIES FBOM THE ALTITUDES, 
 
 AND THE ALTITUDES FROM THE VELOCITIES. 
 
 Altitudes 17 feet 6 inches to AQfeet. 
 
 Coefficients of velocity, and the corresponding velocities of 
 
 
 discharge in inches per second. 
 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 
 11. 
 
 12. 
 
 Altitudes h 
 
 Coeffic'. 
 
 Coeffic'. 
 
 Coeffic*. 
 
 Coeffic'. 
 
 Coeffic*. 
 
 Coeffic 1 . 
 
 in feet and 
 
 7. 
 
 666. 
 
 628. 
 
 617. 
 
 606. 
 
 584. 
 
 inches. 
 
 282-00 
 
 268-30 
 
 253-00 
 
 248-56 
 
 244-13 
 
 235-27 
 
 17 6 
 
 286-00 
 
 272-11 
 
 256-59 
 
 252-09 
 
 247-60 
 
 238-61 
 
 18 
 
 289-95 
 
 275-86 
 
 260-12 
 
 255-57 
 
 251-01 
 
 241-90 
 
 18 6 
 
 293-84 
 
 279-57 
 
 263-32 
 
 259-00 
 
 254-38 
 
 245-14 
 
 19 
 
 297-68 
 
 283-22 
 
 267-06 26238 
 
 257-71 
 
 248-35 
 
 19 6 
 
 301-47 
 
 286-83 
 
 270-46 265-73 
 
 260-99 
 
 251-51 
 
 20 
 
 305-22 
 
 290-39 
 
 273-82 
 
 269-03 
 
 264-23 
 
 254-64 
 
 20 6 
 
 308-92 
 
 293-91 
 
 277-08 
 
 272-23 
 
 267-37 
 
 257-67 
 
 21 
 
 312-57 
 
 297-39 
 
 280-42 
 
 275-51 
 
 270-60 
 
 260-78 
 
 21 6 
 
 316-19 
 
 300-83 
 
 283-67 
 
 278-70 
 
 273-73 
 
 263-79 
 
 22 
 
 319-76 
 
 304-23 
 
 286-87 
 
 281-85 
 
 276-82 
 
 266-77 
 
 22 6 
 
 323-29 
 
 307-59 
 
 290-04 
 
 284-96 
 
 279-88 
 
 269-72 
 
 23 
 
 326-79 
 
 310-92 
 
 293-18 
 
 288-04 
 
 282-91 
 
 272-64 
 
 23 6 
 
 330-25 
 
 314-21 
 
 296-28 
 
 291-09 
 
 285-90 
 
 275-52 
 
 24 
 
 333-67 
 
 317-46 
 
 299-35 
 
 294-11 
 
 288-86 
 
 278-38 
 
 24 6 
 
 337-06 
 
 320-69 
 
 302-39 
 
 297-09 
 
 291-80 
 
 281-20 
 
 25 
 
 340-41 
 
 323-88 
 
 305-40 
 
 300-05 
 
 294-70 
 
 284-00 
 
 25 6 
 
 343-73 
 
 327-04 
 
 308-38 
 
 302-98 
 
 297-57 
 
 286-77 
 
 26 
 
 347-02 
 
 330-17 
 
 311-33 
 
 305-87 
 
 300-42 
 
 289-52 
 
 26 6 
 
 350-28 
 
 333-13 
 
 314-25 
 
 308-75 
 
 303-24 
 
 292-23 
 
 27 
 
 353-51 
 
 336-34 
 
 317-15 
 
 311-59 
 
 306-04 
 
 294-93 
 
 27 6 
 
 356-71 
 
 339-38 
 
 320-02 
 
 314-41 
 
 308-81 
 
 297-60 
 
 28 
 
 359-88 
 
 342-40 
 
 322-86 
 
 317-20 
 
 311-55 
 
 300-24 
 
 28 6 
 
 363-02 
 
 345-39 
 
 325-68 
 
 319-98 
 
 314-27 
 
 302-86 
 
 29 
 
 366-14 
 
 348-35 
 
 328-48 
 
 322-72 
 
 316-97 
 
 305-46 
 
 29 6 
 
 369-23 
 
 351-29 
 
 331-25 
 
 325-45 
 
 319-65 
 
 308-04 
 
 30 
 
 372-29 
 
 354-21 
 
 334-00 
 
 328-15 
 
 322-30 
 
 310-60 
 
 30 6 
 
 375-33 
 
 357-10 
 
 336-73 
 
 330-83 
 
 324-93 
 
 313-13 
 
 31 
 
 378-35 
 
 359-97 
 
 339-43 
 
 333-48 
 
 327-54 
 
 315-60 
 
 31 6 
 
 381-34 
 
 362-81 
 
 342-11 
 
 336-12 
 
 330-13 
 
 318-14 
 
 32 
 
 384-30 
 
 365-64 
 
 344-78 
 
 338-74 
 
 332-70 
 
 320-62 
 
 32 6 
 
 387-25 
 
 368-44 
 
 347-42 
 
 341-33 
 
 335-25 
 
 323-08 
 
 33 
 
 390-17 
 
 371-22 
 
 350-04 
 
 343-01 
 
 337-78 
 
 325-51 
 
 33 6 
 
 393-07 
 
 373-98 
 
 352-64 
 
 346-47 
 
 340-29 
 
 327-93 
 
 34 
 
 395-95 
 
 376-72 
 
 355-23 
 
 349-00 
 
 342-78 
 
 330-34 
 
 34 6 
 
 398-81 
 
 379-44 
 
 357-79 
 
 351-52 
 
 345-26 
 
 332-72 
 
 35 
 
 404-47 
 
 384-82 
 
 362-87 
 
 356-51 
 
 350-15 
 
 337-44 
 
 36 
 
 410-05 
 
 390-13 
 
 367-87 
 
 361-43 
 
 354-98 
 
 342-10 
 
 37 
 
 415-55 
 
 395-37 
 
 372-81 
 
 366-28 
 
 359-75 
 
 346-69 
 
 38 
 
 420-98 
 
 400-54 
 
 377-68 
 
 371-11 
 
 364-45 
 
 351-22 
 
 39 
 
 426-35 
 
 405-64 
 
 382-49 
 
 375-79 
 
 369-09 
 
 355-70 
 
 40
 
 186 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE III. SQUARE BOOTS FOR FINDING THE EFFECTS OF THE 
 
 VELOCITY OF APPROACH WHEN THE ORIFICE IS SMALL IN PRO- 
 PORTION TO THE HEAD. ALSO FOR FINDING THE INCREASE IN 
 THE DISCHARGE FROM AN INCREASE OF HEAD. (See p. 5 5 
 
 No. 
 
 Square 
 root. 
 
 No. 
 
 Square 
 root. 
 
 No. 
 
 Square 
 root. 
 
 No. 
 
 Square 
 root. 
 
 i-ooo 
 
 oooo 
 
 1-115 
 
 1-0559 
 
 475 
 
 1-2141 
 
 1975 
 
 1-4053 
 
 1-001 
 
 0005 
 
 1-120 
 
 1 -05H3 
 
 49 
 
 1-2207 
 
 1-99 
 
 i-4io7 : 
 
 1-002 
 
 0010 
 
 1-125 
 
 1-0607 
 
 5 
 
 1-2247 
 
 2-00 
 
 1-4142 
 
 1-004 
 
 0020 
 
 1-13 
 
 10630 
 
 51 
 
 1-2288 
 
 2-01 
 
 4177 
 
 1-005 
 
 1-0025 
 
 1 135 
 
 1 -0654 
 
 525 
 
 1-2349 
 
 2-025 
 
 4230 
 
 1-006 
 
 0030 
 
 1-14 
 
 1-0677 
 
 54 
 
 1-2410 
 
 2-04 
 
 1-4283 
 
 1-008 
 
 1-0040 
 
 1-145 
 
 1 -0700 
 
 155 
 
 1-2450 
 
 2-05 
 
 4318 
 
 1-009 
 
 1-0044 
 
 1-15 
 
 10723 
 
 1-56 
 
 1-2490 
 
 206 
 
 4353 
 
 1-010 
 
 1-0050 
 
 1 155 
 
 1-0747 
 
 1-575 
 
 2550 
 
 2075 
 
 4405 
 
 1-011 
 
 1-0055 
 
 1-16 
 
 1-0770 
 
 1-58 
 
 2570 
 
 2-09 
 
 4457 
 
 1012 
 
 1-0060 
 
 1-165 
 
 1-0794 
 
 159 
 
 2610 
 
 2-10 
 
 4491 
 
 1-014 
 
 0070 
 
 1-17 
 
 1-0817 
 
 1-6 
 
 2649 
 
 211 
 
 4526 
 
 1015 
 
 1-0075 
 
 1-175 
 
 1 -0840 
 
 1 61 
 
 2689 
 
 2-125 
 
 4577 
 
 1-016 
 
 1-0080 
 
 1-18 
 
 1-0863 
 
 1-625 
 
 2748 
 
 2-14 
 
 4629 
 
 1-018 
 
 1-0090 
 
 1-185 
 
 1-0886 
 
 1-64 
 
 2806 
 
 2 15 
 
 4663 
 
 1-019 
 
 1-0095 
 
 1-19 
 
 1-0909 
 
 1-65 
 
 2845 
 
 2-16 
 
 4697 
 
 1-020 
 
 1-0100 
 
 1-195 
 
 1-0932 
 
 1-66 
 
 2884 
 
 2-175 
 
 4748 
 
 1-0225 
 
 1-0112 
 
 1-2 
 
 0954 
 
 1-675 
 
 12942 
 
 2-19 
 
 4799 
 
 1-025 
 
 1-0124 
 
 1-21 
 
 1000 
 
 1-69 
 
 1-3000 
 
 22 
 
 4832 
 
 1-0275 
 
 1-0137 
 
 1-22 
 
 1045 
 
 1-7 
 
 1-3038 
 
 2-21 
 
 4866 
 
 1-03 
 
 1-0149 
 
 1-23 
 
 lorn 
 
 1-71 
 
 1-3077 
 
 2225 
 
 49 1 6 
 
 1-0325 
 
 10161 
 
 1-24 
 
 1136 
 
 1-725 
 
 1-3134 
 
 224 
 
 -49J7 
 
 1-035 
 
 1-0174 
 
 125 
 
 1180 
 
 1-74 
 
 1-3191 
 
 2-25 
 
 5000 
 
 1-0375 
 
 1-0186 
 
 1-26 
 
 1225 
 
 1-75 
 
 3229 
 
 2-26 
 
 5033 
 
 1-04 
 
 1-0198 
 
 1-27 
 
 1269 
 
 1-76 
 
 3267 
 
 2275 
 
 1-5083 
 
 1-0425 
 
 1-0210 
 
 1-28 
 
 1314 
 
 1-775 
 
 3323 
 
 2-29 
 
 15133 
 
 1-045 
 
 1-0223 
 
 1-29 
 
 1358 
 
 1-79 
 
 3379 
 
 2-3 
 
 15166 
 
 1-0475 
 
 1-0235 
 
 1-30 
 
 1402 
 
 1-80 
 
 L-3416 
 
 2-31 
 
 5199 
 
 1-05 
 
 1-0247 
 
 1-31 
 
 1446 
 
 1-81 
 
 3454 
 
 2325 
 
 5248 
 
 1-055 
 
 1-0271 
 
 1 325 
 
 1511 
 
 1-825 
 
 13509 
 
 2-34 
 
 5297 
 
 1-06 
 
 1-0296 
 
 1-34 
 
 1576 
 
 1-84 
 
 3565 
 
 235 
 
 5330 
 
 1-065 
 
 1-0320 
 
 1-35 
 
 1619 
 
 1-85 
 
 3601 
 
 2-36 
 
 5362 
 
 1-07 
 
 1-0344 
 
 1-36 
 
 1662 
 
 1-86 
 
 3638 
 
 2-375 
 
 5411 
 
 075 
 
 1-0368 
 
 1-375 
 
 1726 
 
 1 -875 
 
 3693 
 
 239 
 
 5460 
 
 08 
 
 1 -0392 
 
 1-39 
 
 1790 
 
 1-89 
 
 3748 
 
 2-4 
 
 5492 
 
 085 
 
 1-0416 
 
 140 
 
 1-1832 
 
 1-9 
 
 3784 
 
 241 
 
 5524 
 
 09 
 
 1-0440 
 
 141 
 
 1-1874 
 
 191 
 
 3820 
 
 2-425 
 
 5572 
 
 095 
 
 1 -0464 
 
 1-425 
 
 I 1937 
 
 1-925 
 
 3875 
 
 2-44 
 
 562 1 
 
 1 
 
 1-0488 
 
 1-44 
 
 1-2000 
 
 1-91 
 
 3928 
 
 245 
 
 5652 
 
 105 
 
 1-0512 
 
 145 
 
 2042 
 
 1-95 
 
 3964 
 
 2-46 
 
 5684 
 
 110 
 
 1 0536 
 
 146 
 
 1-2083 
 
 1-96 
 
 4000 
 
 2475 
 
 5732
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 187 
 
 TABLE III. SQUARE ROOTS FOR FINDING THE EFFECTS OF THE 
 
 VELOCITY OF APPROACH WHEN THE ORIFICE IS SMALL IN PRO- 
 PORTION TO THE HEAD. ALSO FOR FINDING THE INCREASE IN 
 THE DISCHARGE FROM AN INCREASE OF HEAD. (See p. 55.) 
 
 No. 
 
 Square 
 root. 
 
 No. 
 
 Square I 
 root. 
 
 No. 
 
 Square 
 root. 
 
 No. 
 
 Square 
 root. 
 
 2-49 
 
 1-5780 
 
 3-0000 
 
 1-7321 
 
 45 
 
 2-1213 
 
 26 
 
 5-0990 
 
 25 
 
 1-5811 
 
 3-025 
 
 1-7393 
 
 50 
 
 2-5361 
 
 27 
 
 5-1962 
 
 251 
 
 1-5843 ' 
 
 3-05 
 
 1-7^64 
 
 5-5 
 
 2-3452 
 
 28 
 
 5-2915 
 
 2525 
 
 1-5890 
 
 3075 
 
 1-7536 
 
 6-0 
 
 2-4495 
 
 29 
 
 5-3852 
 
 254 
 
 1-5937 
 
 31 
 
 1-7607 
 
 65 
 
 2-5495 
 
 30 
 
 5-4772 
 
 2-55 
 
 15969 
 
 3125 
 
 1-7678 
 
 7-0 
 
 2-6458 
 
 31 
 
 5-5678 
 
 256 
 
 1-6000 
 
 315 
 
 1-7748 
 
 75 
 
 2-7386 
 
 32 
 
 5-6569 
 
 2575 
 
 1-6047 
 
 3 175 
 
 1-7819 
 
 8-0 
 
 2-8284 
 
 33 
 
 5-7446 
 
 259 
 
 1-6093 
 
 3-2 
 
 1-7889 
 
 8-5 
 
 2-9155 
 
 34 
 
 5-8310 
 
 2-6 
 
 1-6125 
 
 3225 
 
 1-7958 
 
 9-0 
 
 3-0000 
 
 35 
 
 5-9161 
 
 2-61 
 
 1-6155 
 
 325 
 
 1-8028 
 
 95 
 
 3-0822 
 
 36 
 
 6-0000 
 
 2625 
 
 1-6202 
 
 3275 
 
 1-8097 
 
 10-0 
 
 3-1623 
 
 37 
 
 60828 
 
 2-64 
 
 1-6248 
 
 33 
 
 1-8166 
 
 10-5 
 
 2-2404 
 
 38 
 
 6-1644 
 
 2-65 
 
 1-6279 
 
 3325 
 
 1-8235 
 
 11-0 
 
 3-3166 
 
 39 
 
 6-2450 
 
 266 
 
 16310 
 
 3-35 
 
 1-8303 
 
 11-5 
 
 3-3912 
 
 40 
 
 6-3246 
 
 2675 
 
 1-6355 
 
 3375 
 
 1-8371 
 
 12-0 
 
 3-4641 
 
 41 
 
 6-4031 
 
 269 
 
 1-6401 
 
 31 
 
 1-8439 
 
 12-5 
 
 35355 
 
 42 
 
 6-4807 
 
 2-7 
 
 1-6432 
 
 3425 
 
 1-8507 
 
 13-0 
 
 3-6056 
 
 43 
 
 6-5574 
 
 271 
 
 1-6462 
 
 345 
 
 1-8574 
 
 13-5 
 
 3-6742 
 
 44 
 
 66332 
 
 2725 
 
 1-6508 
 
 3475 
 
 1-8641 
 
 14-0 
 
 3-7417 
 
 45 
 
 6-7082 
 
 274 
 
 1-6553 
 
 35 
 
 1-8108 
 
 14-5 
 
 3-8079 
 
 46 
 
 6-7823 
 
 275 
 
 1-6583 
 
 3525 
 
 1-8775 
 
 15-0 
 
 3-8730 
 
 47 
 
 6-8557 
 
 276 
 
 1-6613 
 
 355 
 
 1-8841 
 
 15-5 
 
 3-9370 
 
 48 
 
 6-9282 
 
 2775 
 
 1-6658 
 
 3575 
 
 1-8908 
 
 160 
 
 4-0000 
 
 49 
 
 7-0000 
 
 279 
 
 1-6703 
 
 36 
 
 8974 
 
 16-5 
 
 4-0620 
 
 50 
 
 70711 
 
 2-8 
 
 1-6733 
 
 3625 
 
 1-9039 
 
 17-0 
 
 4 1-231 
 
 51 
 
 7 1414 
 
 2-81 
 
 1-6763 
 
 3-65 
 
 1-9105 
 
 175 
 
 4-1833 
 
 52 
 
 72111 
 
 2-825 
 
 1-6808 
 
 3-675 
 
 9170 
 
 18-0 
 
 42426 
 
 53 
 
 7-2810 
 
 2-84 
 
 1-6852 
 
 37 
 
 9235 
 
 18-5 
 
 4-3012 
 
 54 
 
 7-3485 
 
 2-85 
 
 1-6882 
 
 3725 
 
 9300 
 
 19-0 
 
 4-3589 
 
 55 
 
 7-4162 
 
 2-86 
 
 1-6912 
 
 375 
 
 1-9365 
 
 19-5 
 
 44159 
 
 56 
 
 7-4833 
 
 2-875 
 
 1-6956 
 
 3775 
 
 1-9429 
 
 20-0 
 
 4-4721 
 
 57 
 
 7-5498 
 
 2-89 
 
 1-7000 
 
 38 
 
 1-9494 
 
 2-05 
 
 4-5277 
 
 58 
 
 7-6158 
 
 29 
 
 1-7029 
 
 3-825 
 
 1-9538 
 
 21-0 
 
 4-5826 
 
 59 
 
 7-6811 
 
 291 
 
 1-7059 
 
 385 
 
 96-21 
 
 215 
 
 4-6368 
 
 60 
 
 7-7460 
 
 2925 
 
 1-7103 
 
 3-875 
 
 9685 
 
 22-0 
 
 4-6904 
 
 61 
 
 7-8102 
 
 2-94 
 
 1-7146 
 
 3-9 
 
 9748 
 
 22-5 
 
 47434 
 
 62 
 
 7-8740 
 
 295 
 
 1-7176 
 
 3925 
 
 9812 
 
 230 
 
 4-7958 
 
 63 
 
 79373 
 
 296 
 
 1-7205 
 
 395 
 
 9875 
 
 235 
 
 4-8477 
 
 64 
 
 8-0000 
 
 2975 
 
 1-7248 
 
 3 975 
 
 9938 
 
 240 
 
 4-8990 
 
 65 
 
 806-23 
 
 299 
 
 1-7292 
 
 4-0 
 
 2-0000 
 
 250 
 
 50000 
 
 66 
 
 8- . 240
 
 188 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE IV. FOB FINDING THE DISCHARGE THROUGH RECTANGULAR ORIFICES; 
 
 h 
 IN WHICH nz=-T. ALSO FOR FINDING THE EFFECTS OF THE VELOCITY 
 
 OF APPROACH TO WEIRS, AND THE DEPRESSION ON THE CREST. (See p. 55.) 
 
 l + 
 
 *i 
 
 (1 + n)* 
 
 (l+ w) 2- i 
 
 !%. 
 
 J 
 
 (! + )' 
 
 (i+*)i-i 
 
 1-000 
 
 0000 
 
 1-0000 
 
 1-0000 
 
 1-115 
 
 0390 
 
 1-1774 
 
 1384 
 
 1-001 
 
 oooo 
 
 1-0015 
 
 1-0015 
 
 1-120 
 
 0416 
 
 1-1853 
 
 1437 
 
 1-002 
 
 0001 
 
 1-0030 
 
 1-0029 
 
 1-125 
 
 0442 
 
 1-1932 
 
 1491 
 
 1-004 
 
 0003 
 
 1-0060 
 
 1-0058 
 
 1-13 
 
 0469 
 
 1-2012 
 
 1543 
 
 1-005 
 
 0004 
 
 0075 
 
 1-0072 
 
 1-135 
 
 0496 
 
 1-2092 
 
 1596 
 
 006 
 
 0005 
 
 1-0090 
 
 1-0086 
 
 1-14 
 
 0524 
 
 12172 
 
 1648 
 
 1-008 
 
 0007 
 
 0120 
 
 1-0113 
 
 1-145 
 
 0552 
 
 1-2251 
 
 1700 
 
 1-009 
 
 0009 
 
 1-0135 
 
 1-0127 
 
 1-15 
 
 0581 
 
 1-2332 
 
 1751 
 
 1-010 
 
 0010 
 
 1-0150 
 
 10140 
 
 1-155 
 
 0610 
 
 1-2413 
 
 1803 
 
 1-011 
 
 0012 
 
 1-0165 
 
 1-0154 
 
 1-16 
 
 0640 
 
 1-2494 
 
 1854 
 
 1-012 
 
 0013 
 
 1-0181 
 
 1-0167 
 
 1-165 
 
 0670 
 
 1-2574 
 
 1904 
 
 1-014 
 
 0017 
 
 1-0211 
 
 10194 
 
 1-17 
 
 0701 
 
 1-2655 
 
 1955 
 
 1-015 
 
 0018 
 
 1-0226 
 
 1-0207 
 
 1-175 
 
 0732 
 
 1-2737 
 
 2005 
 
 1-016 
 
 0020 
 
 1-0241 
 
 1-0221 
 
 1-18 
 
 0764 
 
 1-2818 
 
 2054 
 
 1-018 
 
 0024 
 
 1-0271 
 
 1-0247 
 
 1-185 
 
 0-96 
 
 1-2900 
 
 2104 
 
 1-019 
 
 0026 
 
 1-0286 
 
 1-0260 
 
 T19 
 
 0828 
 
 1-2981 
 
 V2153 
 
 020 
 
 0028 
 
 1-0301 
 
 1 0273 
 
 1-195 
 
 0861 
 
 1-3003 
 
 1-2202 
 
 1-0225 
 
 0034 
 
 1-0339 
 
 1-0306 
 
 1-2 
 
 0894 
 
 1-3145 
 
 1-2251 
 
 1 -025 
 
 0040 
 
 1-0377 
 
 1-0338 
 
 1-21 
 
 0962 
 
 1-3310 
 
 1-2348 
 
 0275 
 
 0046 
 
 1-0415 
 
 1-0370 
 
 1-22 
 
 1032 
 
 13475 
 
 1-2443 
 
 1-03 
 
 0052 
 
 1-0453 
 
 1-0401 
 
 123 
 
 1103 
 
 3641 
 
 12538 
 
 0325 
 
 0059 
 
 1-0491 
 
 1-0433 
 
 1-24 
 
 1176 
 
 1-3808 
 
 1-2632 
 
 1-035 
 
 0065 
 
 1-0530 
 
 1-0464 
 
 125 
 
 1250 
 
 3975 
 
 1-2725 
 
 1-0375 
 
 0073 
 
 1-0568 
 
 1-0495 
 
 1-26 
 
 1326 
 
 14143 
 
 1-2818 
 
 1-04 
 
 0080 
 
 1-0606 
 
 1-OJ26 
 
 1-27 
 
 1403 
 
 4312 
 
 1-2909 
 
 1-0425 
 
 0088 
 
 10644 
 
 1 055? 
 
 1-28 
 
 1482 
 
 4482 
 
 1-3000 
 
 1-045 
 
 0095 
 
 1-0683 
 
 1-0587 
 
 1-29 
 
 1562 
 
 4652 
 
 1 3090 
 
 10475 
 
 0104 
 
 1-0721 
 
 1-0617 
 
 1-30 
 
 1643 
 
 4822 
 
 1-3179 
 
 1-05 
 
 0112 
 
 1-0759 
 
 1-0648 
 
 131 
 
 1726 
 
 4994 
 
 1 -3268 
 
 1-055 
 
 0129 
 
 1-0836 
 
 1-0707 
 
 132) 
 
 1853 
 
 5252 
 
 1-3399 
 
 106 
 
 0147 
 
 1-0913 
 
 1-0766 
 
 1-34 
 
 1983 
 
 5512 
 
 1 3529 
 
 1-065 
 
 0166 
 
 1-0991 
 
 1-0825 
 
 1-35 
 
 2071 
 
 5686 
 
 1-3615 
 
 1-07 
 
 0185 
 
 1068 
 
 1-0883 
 
 1 36 
 
 2160 
 
 5860 
 
 1-3700 
 
 1-075 
 
 0205 
 
 1146 
 
 1-0940 
 
 1-375 
 
 2296 
 
 6123 
 
 1-3827 
 
 1-08 
 
 0226 
 
 1224 
 
 10997 
 
 1-39 
 
 2436 
 
 6388 
 
 1-3952 
 
 1-085 
 
 02 48 
 
 1302 
 
 1054 
 
 1-40 
 
 2530 
 
 6565 
 
 1-4035 
 
 1-09 
 
 0270 
 
 1380 
 
 mo 
 
 1-41 
 
 2625 
 
 6743 
 
 1-4118 
 
 1-095 
 
 0293 
 
 1458 
 
 1166 
 
 1-425 
 
 2771 
 
 1-7011 
 
 1-4240 
 
 1-1 
 
 0316 
 
 1537 
 
 1221 
 
 1-44 
 
 2919 
 
 1-7280 
 
 1-4361 
 
 1-105 
 
 0340 
 
 1-1616 
 
 1275 
 
 1-45 
 
 3019 
 
 1-7460 
 
 1-4442 
 
 I 110 
 
 0365 
 
 1-1695 
 
 1330 
 
 1-46 
 
 3120 
 
 1-7641 
 
 1-4521 
 
 Values of n from to "46. 
 
 [Contiiitifd ,
 
 ORIFICES, WEIKS, PIPES, AND RIVERS. 
 
 189 
 
 TABLE IV. FOE FINDING THE DISCHARGE THROUGH RECTANGULAR ORIFICES; 
 
 h 
 
 IN WHICH n = -7- ALSO FOR FINDING THE EFFECTS OF THE VELOCITY 
 a 
 
 OF APPROACH TO WEIRS, &c. (See p. 55.) 
 
 1 1 
 
 J 
 
 /i i _ \ 5 
 
 
 
 
 1 _i_ 7i\i 
 
 1 4-\2 2 
 
 i -+ n 
 
 7t 
 
 1 -r n ) 
 
 
 
 
 1 T n ) 
 
 1 T n ) * 
 
 1-475 
 
 3274 
 
 1 7914 
 
 1-4640 
 
 1975 
 
 9627 
 
 2-7756 
 
 1-8128 
 
 1-49 
 
 3430 
 
 1-8188 
 
 1-4758 
 
 1-99 
 
 9850 
 
 2-8072 
 
 1-8222 
 
 15 
 
 3536 
 
 1-8371 
 
 1-4836 
 
 2- 
 
 I -0000 
 
 2-8284 
 
 18284 
 
 l'5l 
 
 3642 
 
 1-8555 
 
 1-4913 
 
 2-01 
 
 1-0150 
 
 28497 
 
 1-8346 
 
 1-525 
 
 3P04 
 
 1-8832 
 
 1-5028 
 
 2-025 
 
 1-0377 
 
 2-8816 
 
 1-8439 
 
 1-54 
 
 3968 
 
 1-9111 
 
 1-5143 
 
 2-04 
 
 1-0606 
 
 2-9137 
 
 1-P531 
 
 1-55 
 
 4079 
 
 1-9297 
 
 1-5218 
 
 2-05 
 
 1-0759 
 
 2-9352 
 
 1-8592 
 
 1-56 
 
 4191 
 
 1 9484 
 
 1 5294 
 
 2-06 
 
 1-0913 
 
 2-9567 
 
 1 -8653 
 
 1 575 
 
 4360 
 
 1-9766 
 
 1-5406 
 
 2-075 
 
 1-1146 
 
 2-9890 
 
 1-8744 
 
 1-58 
 
 4417 
 
 1-9860 
 
 1-5443 
 
 2-09 
 
 1-1380 
 
 3-0215 
 
 1-8835 
 
 1-59 
 
 4532 
 
 2-0049 
 
 15517 
 
 2-10 
 
 1-1537 
 
 3-0432 
 
 1-8895 
 
 1-6 
 
 4648 
 
 20239 
 
 1-5591 
 
 2-U 
 
 1-1695 
 
 3-0650 
 
 1-8955 
 
 1-61 
 
 4764 
 
 2-0429 
 
 1-5664 
 
 2-125 
 
 1-1932 
 
 3-0977 
 
 1-9045 
 
 1-625 
 
 4941 
 
 20715 
 
 1-5774 
 
 2 14 
 
 1217-2 
 
 3-1306 
 
 1-9134 
 
 1-64 
 
 5120 
 
 2-1002 
 
 1-5882 
 
 2-15 
 
 1-2332 
 
 3-1525 
 
 1-9193 
 
 1-65 
 
 5240 
 
 2-1195 
 
 1-5954 
 
 2-16 
 
 1-2494 
 
 3-1745 
 
 1-9252 
 
 1-66 
 
 5362 
 
 2-1388 
 
 1 -6026 
 
 2-175 
 
 1-2737 
 
 3-2077 
 
 19340 
 
 1 "675 
 
 5546 
 
 2-1678 
 
 1-6132 
 
 2-19 
 
 1-2981 
 
 3-2409 
 
 1-9428 
 
 1-69 
 
 5732 
 
 2-1970 
 
 1-6238 
 
 22 
 
 1-3145 
 
 32631 
 
 1-9486 
 
 1-7 
 
 5857 
 
 22165 
 
 1-6309 
 
 2 21 
 
 1-3310 
 
 3-2854 
 
 1-9544 
 
 1-71 
 
 5983 
 
 2-2361 
 
 1 6379 
 
 2-225 
 
 1-3558 
 
 3-3189 
 
 1-9631 
 
 1-725 
 
 6173 
 
 2-2656 
 
 1-6483 
 
 5-24 
 
 1 3808 
 
 33525 
 
 1-9717 
 
 1-74 
 
 6366 
 
 2-295-2 
 
 1-6586 
 
 225 
 
 1-3975 
 
 3-3750 
 
 1-9775 
 
 175 
 
 '6495 
 
 2-3150 
 
 l-6f>55 
 
 2-26 
 
 1-4143 
 
 3-3975 
 
 1-9832 
 
 1-76 
 
 6626 
 
 2-3349 
 
 1-6724 
 
 2275 
 
 1-4397 
 
 3-4314 
 
 1-9917 
 
 1-775 
 
 6823 
 
 2-3648 
 
 1-6826 
 
 229 
 
 1-4652 
 
 3-4654 
 
 2-0002 
 
 179 
 
 7022 
 
 2-3949 1-6927 
 
 2-3 
 
 1-4822 
 
 3-4881 
 
 2-0059 
 
 1-80 
 
 7155 
 
 2-4150 1-6994 
 
 231 
 
 1-4994 
 
 3-5109 
 
 2-0115 
 
 1-81 
 
 7290 
 
 2-4351 
 
 1-7061 
 
 2325 
 
 1-5252 
 
 3-5451 
 
 2-0200 
 
 1 -825 
 
 7493 
 
 2465* 
 
 1-7161 
 
 234 
 
 1-5512 
 
 3-5795 
 
 2-0284 
 
 1-84 
 
 7699 
 
 2-4959 
 
 17260 
 
 235 
 
 1-5686 
 
 3-6025 
 
 2-0339 
 
 1-85 
 
 7837 
 
 2-5163 
 
 1-7326 
 
 2-36 
 
 1-5860 
 
 3-6255 
 
 2-0395 
 
 1-86 
 
 7975 
 
 2-5367 
 
 1-7392 
 
 2375 
 
 1-6123 
 
 3-6601 
 
 2-0478 
 
 1-875 
 
 8185 
 
 2-5674 
 
 1-7490 
 
 2-39 
 
 1-638} 
 
 36948 
 
 2-0561 
 
 1-89 
 
 8396 
 
 2-5983 
 
 1-7587 
 
 2-4 
 
 1-6565 
 
 3-7181 
 
 2-0616 
 
 1-9 
 
 8538 
 
 2-6190 
 
 1-7652 
 
 2-41 
 
 1-6743 
 
 3-7413 
 
 2-0670 
 
 191 
 
 8681 
 
 2-6397 
 
 1-7716 
 
 2-425 
 
 1-7011 
 
 37763 
 
 2-0752 
 
 1-925 
 
 8896 
 
 2-6709 
 
 1-7813 
 
 244 
 
 1--J280 
 
 38114 
 
 2-0834 
 
 194 
 
 9114 
 
 2-7021 
 
 1-7907 
 
 2-45 
 
 1-7460 3-8349 
 
 2-0888 
 
 1-95 
 
 9259 ; 2-7^30 
 
 17971 
 
 2-46 
 
 1-7641 
 
 3-8584 
 
 2-0942 
 
 1-96 
 
 9406 2-7440 
 
 1-8034 
 
 2-475 
 
 1-7914 
 
 38937 
 
 2-1023 
 
 Values of n from -475 to 1-475. 
 
 [Continued on next page.
 
 190 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE IV. FOB FINDING THE DISCHARGE THROUGH RECTANGULAR ORIFICES; 
 
 h 
 IN WHICH n = -. ALSO FOR FINDING THE EFFECTS OF THE VELOCITY OF 
 
 APPROACH TO WEIRS, &c. (See p. 55.) 
 
 l + . 
 
 i 
 
 (1 + n){ 
 
 (l+*)5_t& 
 
 l + n. 
 
 'A 
 
 (i + )i 
 
 (l + n )l-l 
 
 2-49 
 
 1-8188 
 
 3-9292 
 
 2-1104 
 
 3 
 
 2-8284 
 
 5-1962 
 
 23677 
 
 2-5 
 
 1-8371 
 
 3-9528 
 
 2 1157 
 
 3025 
 
 2-8816 
 
 52612 
 
 2-3796 
 
 2-51 
 
 1-8555 
 
 3-9766 
 
 21211 
 
 3-05 
 
 29352 
 
 53266 
 
 2-3914 
 
 2-525 
 
 1-8832 
 
 4'0123 
 
 2-1291 
 
 3-075 
 
 2-9890 
 
 53922 
 
 2-4032 
 
 2-54 
 
 1-9111 
 
 4-0481 
 
 2-1370 
 
 J3-1 
 
 3-0432 
 
 5-4581 
 
 2-4149 
 
 2-55 
 
 1-9297 
 
 4-0720 
 
 2-1423 
 
 3-125 
 
 3-0977 
 
 55243 
 
 2-4266 
 
 2-56 
 
 1-9484 
 
 4-0960 
 
 2-1476 
 
 3-15 
 
 3-1525 
 
 5-5907 
 
 2-4382 
 
 2-575 
 
 1 -9766 
 
 4-1321 
 
 2 1554 
 
 ! 3-175 
 
 3-2077 
 
 5-6574 
 
 24497 
 
 2-59 
 
 2-0049 
 
 4-1682 
 
 2-1633 
 
 32 
 
 3-2631 
 
 57243 
 
 2-4612 
 
 2-6 
 
 2-0239 
 
 4-1924 
 
 2-1685 
 
 3 225 
 
 3-3189 
 
 5-7915 
 
 24726 
 
 2-61 
 
 2-0429 
 
 4-2166 
 
 2-1747 
 
 I 3 25 
 
 33750 
 
 5-8590 
 
 2-4840 
 
 2-625 
 
 2-0715 
 
 4-2530 
 
 2-1815 
 
 3275 
 
 3-4314 
 
 5-9268 
 
 24953 
 
 2-64 
 
 2-1002 
 
 4-2895 
 
 2-1893 
 
 33 
 
 3-4881 
 
 5-9947 
 
 2-5066 
 
 2-65 
 
 2-1195 
 
 4-3139 
 
 2-1944 
 
 3-325 
 
 35451 
 
 6-0630 
 
 2-5179 
 
 2-66 
 
 2-1388 
 
 4-3383 
 
 2-1996 
 
 335 
 
 3-6025 
 
 6 1315 
 
 2-5290 
 
 2-675 
 
 2-1678 
 
 4-3751 
 
 2-2073 
 
 3375 
 
 3-6601 
 
 6-2003 
 
 2-5401 
 
 2-69 
 
 2-1970 
 
 4-4119 
 
 2-2149 
 
 34 
 
 3-7181 
 
 6-2693 
 
 25512 
 
 2-7 
 
 2-2165 
 
 4-4366 
 
 2-2200 
 
 3-425 
 
 3-7763 
 
 6-3386 
 
 25623 
 
 2-71 
 
 2-2361 
 
 4-4612 
 
 22251 
 
 3-45 
 
 3-8349 
 
 6-4081 
 
 25732 
 
 2-725 
 
 2-2656 
 
 4-4983 
 
 2-2327 
 
 3-475 
 
 3-8937 
 
 64779 
 
 2-5842 
 
 2-74 
 
 2-2952 
 
 4-5355 
 
 2-2403 
 
 3-5 
 
 3-9528 
 
 6-5479 
 
 2-5951 
 
 2-75 
 
 2-3150 
 
 4-5604 
 
 2-2453 
 
 3525 
 
 4-0123 
 
 6-6182 
 
 2-6059 
 
 2-76 
 
 2-3349 
 
 4-5853 
 
 2-2504 
 
 3-55 
 
 4-0720 
 
 6-6887 
 
 2-6167 
 
 2-775 
 
 2-3648 
 
 4-6227 
 
 2-2579 
 
 3-575 
 
 4-1321 
 
 67595 
 
 2-6274 
 
 2-79 
 
 2-3949 
 
 4-6602 
 
 2-2654 
 
 36 
 
 4-1924 
 
 6-8305 
 
 2-6381 
 
 2-8 
 
 2-4150 
 
 4-6853 
 
 2-2703 
 
 3-625 
 
 4-2530 
 
 6-9018 
 
 2-6488 
 
 2-81 
 
 2-1351 
 
 4-7104 
 
 22753 
 
 3-65 
 
 4-3139 
 
 69733 
 
 2-6594 
 
 2-825 
 
 2-4654 
 
 4-7482 
 
 2-2827 
 
 3 675 
 
 4-3751 
 
 7-0451 
 
 2-6700 
 
 2-84 
 
 2-4959 
 
 4-7861 
 
 2-2902 
 
 37 
 
 4-4366 
 
 71171 
 
 2-6805 
 
 2-85 
 
 2-5163 
 
 4-8114 
 
 2-2951 
 
 3-725 
 
 4-4983 
 
 7-1893 
 
 2-6910 
 
 2-86 
 
 2-5367 
 
 4-8367 
 
 2-3000 
 
 375 
 
 4-5604 
 
 7-2618 
 
 27015 
 
 2-875 
 
 2-5674 
 
 4-8748 
 
 2-3074 
 
 3-775 
 
 4-6227 
 
 73346 
 
 27119 
 
 2-89 
 
 2-5983 
 
 4-9130 
 
 2-3147 
 
 3-8 
 
 4-6853 
 
 7-4076 
 
 27223 
 
 2-9 
 
 2-6190 
 
 4-9385 
 
 23196 
 
 3-825 
 
 4-7482 
 
 7-4808 
 
 2-7326 
 
 2-91 
 
 2-6397 
 
 4-9641 
 
 23244 
 
 3-85 
 
 4-8114 
 
 75542 
 
 2-7429 
 
 2-925 
 
 2-6708 
 
 5-0025 
 
 23317 
 
 3-875 
 
 4-8748 
 
 7-6279 
 
 2-7531 
 
 2-94 
 
 2-7021 
 
 5-0411 
 
 2-3389 
 
 3-9 
 
 4-9385 
 
 7-7019 
 
 2-7634 
 
 2-95 
 
 2-7230 
 
 5-0668 
 
 2-3438 
 
 3925 
 
 5-0025 
 
 7-7761 
 
 27735 
 
 2-96 
 
 2-7440 
 
 5-0926 
 
 2-3486 
 
 3-95 
 
 5-0668 
 
 7-8505 
 
 2-7837 
 
 2-975 
 
 2-7,56 
 
 5-1313 
 
 2-3558 
 
 3-975 
 
 51313 
 
 7-9251 
 
 2-7938 
 
 2-99 
 
 2-8072 
 
 5-1702 
 
 2-3630 
 
 4- 
 
 5-1962 
 
 8- 
 
 2-8038 
 
 Values of n from T49 to 3.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 191 
 
 TABLE IV. FOR FINDING THE DISCHARGE THROUGH RECTANGULAR ORIFICES; 
 
 IN WHICH n = -,- ALSO FOR FINDING THE EFFECTS OF THE VELOCITY OF 
 
 a 
 APPROACH TO WEIRS, &c. (See p. 55.) 
 
 1 + 71. 
 
 Jl 
 
 (l + )l 
 
 (i +)!_*!. 
 
 l + n. 
 
 i. 
 
 (1 + )*. 
 
 (l+n)*-n%. 
 
 4-5 
 
 65479 
 
 9-5459 
 
 2-9980 
 
 26- 
 
 125-0000 
 
 1325745 
 
 75745 
 
 5-0 
 
 8-0000 
 
 11-1803 
 
 3-1803 
 
 27 
 
 1325745 
 
 1402961 
 
 7-7216 
 
 5-5 
 
 95459 
 
 12-8986 
 
 33527 
 
 28- 
 
 140-2961 
 
 148 1621 
 
 7-8660 
 
 6-0 
 
 11-1803 
 
 14-6969 
 
 3-5166 
 
 29- 
 
 148-1621 
 
 156-1698 
 
 8-0077 
 
 65 
 
 12-8986 
 
 16-5718 
 
 36732 
 
 30- 
 
 156-1698 
 
 164-3168 
 
 8-1470 
 
 7-0 
 
 14-6969 
 
 18-5203 
 
 3-8234 
 
 31- 
 
 164-3168 
 
 172-6007 
 
 8-2839 
 
 7'5 
 
 165718 
 
 20-5396 
 
 39678 
 
 32- 
 
 172-6007 
 
 181-0193 
 
 8-4186 
 
 8-0 
 
 18-5203 
 
 22-6274 
 
 4-1071 
 
 33- 
 
 181-0193 
 
 189-5706 
 
 8-5513 
 
 8-5 
 
 20-5396 
 
 24-7815 
 
 4-2419 
 
 34- 
 
 189-5706 
 
 198- -'524 
 
 8-6818 
 
 9-0 
 
 22-6274 
 
 27-0000 
 
 4-3726 
 
 35- 
 
 198-2524 
 
 207-0628 
 
 8-8104 
 
 9-5 
 
 24-7815 
 
 29-2810 
 
 4-4995 
 
 36- 
 
 207-0628 
 
 216-0000 
 
 89372 
 
 10-0 
 
 27-0000 
 
 31-6228 
 
 4-6228 
 
 37- 
 
 216-0000 
 
 225-0622 
 
 9-0622 
 
 10-5 
 
 29-2810 
 
 34-0239 
 
 4-7429 
 
 38- 
 
 225-0622 
 
 234-2477 
 
 9-1855 
 
 11-0 
 
 31-6228 
 
 36-4829 
 
 4-8601 
 
 39- 
 
 2342477 
 
 2 13-5549 
 
 9-307-2 
 
 115 
 
 34-0239 
 
 38-9984 
 
 4-9745 
 
 40- 
 
 243-5549 
 
 252-9822 
 
 94273 
 
 120 
 
 36-4829 
 
 41-5692 
 
 5-0863 
 
 41- 
 
 252-9822 
 
 26-2-5281 
 
 9-5459 
 
 12-5 
 
 38-9984 
 
 44-1942 
 
 5-1958 
 
 42 
 
 262-5281 
 
 272-1911 
 
 9-630 
 
 13-0 
 
 41-5692 
 
 46-8722 
 
 5-3030 
 
 43 
 
 272-1911 
 
 281-9699 
 
 9-7788 
 
 135 
 
 44-1942 
 
 49-6022 
 
 5-4080 
 
 44- 
 
 28 L -9699 
 
 291-8630 
 
 9-8931 
 
 14-0 
 
 46-8722 
 
 52-3832 
 
 5-5110 
 
 45 
 
 291-8630 
 
 301-8692 
 
 10-0062 
 
 14-5 
 
 49-6022 
 
 552144 
 
 5-6122 
 
 46- 
 
 301-8692 
 
 311-9872 
 
 10-1180 
 
 15-0 
 
 52-3832 
 
 58-0947 
 
 57115 
 
 47" 
 
 311-9872 
 
 32-2-2158 
 
 10-2286 
 
 155 
 
 55-2144 
 
 61-0236 
 
 5-8092 
 
 48- 
 
 322-2158 332-5538 
 
 10-3380 
 
 16-0 
 
 58-0947 
 
 64- 
 
 5-9053 
 
 49- 
 
 332-5538 
 
 343-0000 
 
 10-4462 
 
 165 
 
 61-0236 
 
 67-0247 
 
 6-0011 
 
 50- 
 
 343-0000 
 
 353-5534 
 
 105534 
 
 17'0 
 
 64- 
 
 70-0928 
 
 6-0928 
 
 51- 
 
 353-5534 
 
 364-21-28 
 
 10-6594 
 
 175 
 
 67-0247 
 
 73-2078 
 
 6-1831 
 
 52 
 
 364-2128 
 
 3749773 
 
 107645 
 
 18-0 
 
 70-0928 
 
 76-3675 
 
 6-2747 
 
 53 
 
 374-9'/73 
 
 385-8458 
 
 10-8685 
 
 18-5 
 
 73-2078 
 
 79-5715 
 
 6-3637 
 
 54 
 
 385-8458 
 
 396-8173 
 
 10-9715 
 
 19-0 
 
 76-3675 
 
 82-8191 
 
 6-4516 
 
 55 
 
 3968173 
 
 407-8909 
 
 110736 
 
 19-5 
 
 79-5715 
 
 86- 1097 
 
 6-5382 
 
 56- 
 
 407-8909 
 
 419-0656 
 
 111747 
 
 20-0 
 
 82-8191 
 
 89-4427 
 
 6-6236 
 
 57- 
 
 419-0656 
 
 430-3406 
 
 11-2750 
 
 20-5 
 
 86-1097 
 
 92-8177 
 
 6-7080 
 
 58- 
 
 430-3406 
 
 441-7148 
 
 11-3742 
 
 21-0 
 
 89-4427 
 
 962341 
 
 6-7914 
 
 59 
 
 441-7148 
 
 453-1876 
 
 11 4728 
 
 215 
 
 92-8177 
 
 99-6914 
 
 6-8737 
 
 60- 
 
 453-18:6 
 
 464-7580 
 
 11-5704 
 
 22-0 
 
 962341 
 
 103-1892 
 
 69551 
 
 61- 
 
 464-7580 
 
 476-4252 
 
 11-6672 
 
 225 
 
 99-6914 J106-7269 
 
 7-0355 
 
 62- 
 
 476-4252 
 
 488-1885 
 
 11-7633 
 
 23 
 
 1031892 
 
 110-3041 
 
 7-1149 
 
 63 
 
 488-1885 
 
 500-0470 
 
 11-8585 
 
 23-5 
 
 1067269 
 
 1 13-9205 
 
 7-1936 
 
 64- 
 
 500-0470 
 
 5 12 -0000 
 
 11-9530 
 
 24 
 
 1103041 
 
 1175755 
 
 7-2714 
 
 65- 
 
 512-0000 
 
 524-0488 
 
 12-0468 
 
 25- 
 
 1175755 
 
 125- 
 
 7'424> 
 
 66' 
 
 524-0468 
 
 536-1865 
 
 121397 
 
 Values of n from 3-5 to G6.
 
 192 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE V. COEFFICIENTS OF DISCHARGE FOR DIFFERENT RATIOS 
 
 OF THE CHANNEL TO THE ORIFICE. 
 
 Coefficients for heads in still water '550 and '573. 
 
 
 Coefficient -550 for heads in 
 
 Coefficient -573 for heads in 
 
 
 still water. 
 
 still water. 
 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 Ratio 
 
 the height 
 
 for orifices : 
 
 for weirs : 
 
 the height 
 
 for orifices : 
 
 for weirs : 
 
 of the 
 
 due to the 
 
 the heads 
 
 the heads 
 
 due to the 
 
 the heads 
 
 the heads 
 
 channel 
 
 velocity of 
 
 measured 
 
 measured 
 
 velocity of 
 
 measured 
 
 measured 
 
 to the 
 
 approach 
 
 to the 
 
 the full 
 
 approach 
 
 to the 
 
 the full 
 
 orifice. 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 30- 
 
 000 
 
 550 
 
 550 
 
 000 
 
 573 
 
 573 
 
 20- 
 
 001 
 
 550 
 
 551 
 
 001 
 
 573 
 
 574 
 
 15- 
 
 001 
 
 550 
 
 551 
 
 001 
 
 573 
 
 574 
 
 lo- 
 
 003 
 
 551 
 
 552 
 
 003 
 
 574 
 
 576 
 
 g- 
 
 004 
 
 551 
 
 553 
 
 004 
 
 574 
 
 576 
 
 s' 
 
 005 
 
 551 
 
 554 
 
 005 
 
 574 
 
 577 
 
 7' 
 
 006 
 
 552 
 
 555 
 
 007 
 
 575 
 
 578 
 
 f 
 
 008 
 
 552 
 
 557 
 
 009 
 
 576 
 
 580 
 
 5-5 
 
 010 
 
 553 
 
 558 
 
 Oil 
 
 576 
 
 582 
 
 5-0 
 
 012 
 
 553 
 
 559 
 
 013 
 
 577 
 
 584 
 
 4-5 
 
 015 
 
 554 
 
 562 
 
 016 
 
 578 
 
 586 
 
 4-0 
 
 019 
 
 555 
 
 565 
 
 021 
 
 579 
 
 589 
 
 3-75 
 
 022 
 
 556 
 
 566 
 
 024 
 
 580 
 
 592 
 
 3-50 
 
 025 
 
 557 
 
 569 
 
 028 
 
 581 
 
 594 
 
 3-25 
 
 029 
 
 558 
 
 572 
 
 032 
 
 582 
 
 598 
 
 30 
 
 035 
 
 559 
 
 575 
 
 038 
 
 584 
 
 602 
 
 2-75 
 
 042 
 
 561 
 
 580 
 
 045 
 
 586 
 
 607 
 
 2-50 
 
 051 
 
 564 
 
 586 
 
 055 
 
 589 
 
 614 
 
 2-25 
 
 064 
 
 567 
 
 594 
 
 069 
 
 593 
 
 623 
 
 2-00 
 
 082 
 
 572 
 
 606 
 
 089 
 
 598 
 
 636 
 
 1-95 
 
 086 
 
 573 
 
 609 
 
 094 
 
 599 
 
 639 
 
 1-90 
 
 091 
 
 575 
 
 612 
 
 100 
 
 601 
 
 643 
 
 1-85 
 
 097 
 
 576 
 
 615 
 
 106 
 
 603 
 
 647 
 
 1-80 
 
 103 
 
 578 
 
 619 
 
 113 
 
 604 
 
 651 
 
 1-75 
 
 110 
 
 579 
 
 623 
 
 120 
 
 606 
 
 655 
 
 1-70 
 
 117 
 
 581 
 
 627 
 
 128 
 
 609 
 
 660 
 
 1-65 
 
 125 
 
 583 
 
 632 
 
 137 
 
 611 
 
 666 
 
 1-60 
 
 134 
 
 586 
 
 637 
 
 147 
 
 614 
 
 671 
 
 1-55 
 
 144 
 
 588 
 
 643 
 
 158 
 
 617 
 
 678 
 
 1-50 
 
 155 
 
 591 
 
 649 
 
 171 
 
 620 
 
 685 
 
 1-45 
 
 168 
 
 594 
 
 656 
 
 185 
 
 624 
 
 694 
 
 1-40 
 
 183 
 
 598 
 
 664 
 
 201 
 
 628 
 
 703 
 
 1-35 
 
 199 
 
 602 
 
 673 
 
 220 
 
 633 
 
 713 
 
 1-30 
 
 218 
 
 607 
 
 683 
 
 241 
 
 638 
 
 724 
 
 1-25 
 
 240 
 
 612 
 
 695 
 
 266 
 
 645 
 
 737 
 
 1-20 
 
 265 
 
 619 
 
 707 
 
 295 
 
 652 
 
 753 
 
 1-15 
 
 297 
 
 626 
 
 723 
 
 330 
 
 661 
 
 770 
 
 1-10 
 
 333 
 
 635 
 
 741 
 
 372 
 
 671 
 
 791 
 
 1-05 
 
 378 
 
 646 
 
 762 
 
 424 
 
 684 
 
 816 
 
 1-00 
 
 434 
 
 659 
 
 787 
 
 489 
 
 699 
 
 845 
 
 See the auxiliary table, p. 68.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 193 
 
 TABLE V. COEFFICIENTS OF DISCHARGE FOR DIFFERENT RATIOS 
 
 OF THE CHANNEL TO THE ORIFICE. 
 
 Coefficients for heads in still water -584 and '595. 
 
 Ratio 
 
 Coefficient -584 for heads in 
 still water. 
 
 Coefficient -595 for heads in 
 still water. 
 
 of the 
 
 Ratio of 
 
 Coefficients Coefficients 
 
 Ratio of iCoefficients Coefficients 
 
 channel 
 to the 
 
 the height 
 due to the 
 
 for orifices . 
 the heads 
 
 for weirs : 
 the heads 
 
 the height for orifices : 
 due to the the heads 
 
 for weirs : 
 the heads 
 
 orifice. 
 
 velocity of 
 
 measured 
 
 measured 
 
 velocity of measured 
 
 measured 
 
 
 approach 
 
 to the 
 
 the full 
 
 approach to the 
 
 the full 
 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 30- 
 
 000 
 
 584 
 
 584 
 
 000 
 
 595 
 
 595 
 
 20- 
 
 001 
 
 584 
 
 585 
 
 001 
 
 595 
 
 596 
 
 15- 
 
 002 
 
 584 
 
 585 
 
 002 
 
 595 
 
 596 
 
 10- 
 
 003 
 
 585 
 
 587 
 
 004 
 
 596 
 
 598 
 
 9-0 
 
 004 
 
 585 
 
 588 
 
 004 
 
 596 
 
 599 
 
 8-0 
 
 005 
 
 586 
 
 588 
 
 006 
 
 597 
 
 600 
 
 7-0 
 
 007 
 
 586 
 
 590 
 
 007 
 
 597 
 
 601 
 
 6-0 
 
 010 
 
 587 
 
 592 
 
 010 
 
 598 
 
 603 
 
 5-5 
 
 Oil 
 
 587 
 
 593 
 
 012 
 
 599 
 
 605 
 
 5-0 
 
 014 
 
 588 
 
 595 
 
 014 
 
 599 
 
 607 
 
 4-5 
 
 017 
 
 589 
 
 598 
 
 018 
 
 600 
 
 610 
 
 4-0 
 
 022 
 
 590 
 
 601 
 
 023 
 
 602 
 
 613 
 
 3-75 
 
 025 
 
 591 
 
 604 
 
 026 
 
 603 
 
 616 
 
 3-50 
 
 029 
 
 592 
 
 606 
 
 030 
 
 604 
 
 619 
 
 3-25 
 
 033 
 
 594 
 
 610 
 
 035 
 
 605 
 
 622 
 
 3-0 
 
 039 
 
 595 
 
 614 
 
 041 
 
 607 
 
 627 
 
 2-75 
 
 047 
 
 598 
 
 620 
 
 049 
 
 609 
 
 633 
 
 2-50 
 
 058 
 
 601 
 
 627 
 
 060 
 
 613 
 
 641 
 
 2-25 
 
 072 
 
 605 
 
 637 
 
 075 
 
 617 
 
 651 
 
 2-0 
 
 093 
 
 611 
 
 651 
 
 097 
 
 623 
 
 666 
 
 1-95 
 
 099 
 
 612 
 
 654 
 
 103 
 
 625 
 
 669 
 
 1-90 
 
 104 
 
 614 
 
 660 
 
 109 
 
 627 
 
 673 
 
 1-85 
 
 111 
 
 615 
 
 662 
 
 115 
 
 628 
 
 678 
 
 1-80 
 
 118 
 
 617 
 
 666 
 
 123 
 
 630 
 
 682 
 
 1-75 
 
 125 
 
 620 
 
 671 
 
 131 
 
 633 
 
 687 
 
 1-70 
 
 134 
 
 622 
 
 676 
 
 140 
 
 635 
 
 693 
 
 1-65 
 
 143 
 
 624 
 
 682 
 
 149 
 
 638 
 
 699 
 
 1-60 
 
 154 
 
 627 
 
 689 
 
 160 
 
 641 
 
 706 
 
 1-55 
 
 166 
 
 631 
 
 696 
 
 173 
 
 644 
 
 713 
 
 1-50 
 
 179 
 
 634 
 
 703 
 
 187 
 
 648 
 
 721 
 
 1-45 
 
 194 
 
 638 
 
 712 
 
 202 
 
 652 
 
 730 
 
 1-40 
 
 211 
 
 643 
 
 722 
 
 220 
 
 657 
 
 741 
 
 1-35 
 
 230 
 
 648 
 
 732 
 
 241 
 
 663 
 
 752 
 
 1-30 
 
 253 
 
 654 
 
 745 
 
 265 
 
 669 
 
 765 
 
 1-25 
 
 279 
 
 661 
 
 759 
 
 293 
 
 677 
 
 780 
 
 1-20 
 
 310 
 
 669 
 
 775 
 
 325 
 
 685 
 
 797 
 
 1-15 
 
 348 
 
 678 
 
 794 
 
 366 
 
 695 
 
 818 
 
 1-10 
 
 393 
 
 689 
 
 816 
 
 414 
 
 707 
 
 842 
 
 1-05 
 
 448 
 
 703 
 
 842 
 
 473 
 
 722 
 
 870 
 
 1-00 
 
 518 
 
 719 
 
 874 
 
 548 
 
 740 
 
 905 
 
 See the auxiliary table, p. 68.
 
 194 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE V. COEFFICIENTS OF DISCHARGE FOR DIFFERENT RATIOS 
 
 OF THE CHANNEL TO THE ORIFICE. 
 
 Coefficients for heads in still water -606 and -617. 
 
 
 Coefficient '606 for heads in 
 
 Coefficient -617 for heads in 
 
 
 still water. 
 
 still water. 
 
 Ratio 
 
 
 
 of the 
 
 Ratio of 
 
 Coefficients Coefficients 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients' 
 
 channel 
 to the 
 
 the height 
 due to the 
 
 for orifices : 
 the heads 
 
 for weirs : 
 the heads 
 
 the height 
 due to the 
 
 for orifices 
 the heads 
 
 for weirs : ! 
 the heads 
 
 orifice. 
 
 velocity of 
 
 measured 
 
 measured 
 
 velocity of 
 
 measured 
 
 measured 
 
 
 approach 
 
 to the 
 
 the full 
 
 approach 
 
 to the 
 
 the full 
 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 30- 
 
 000 
 
 606 
 
 606 
 
 000 
 
 617 
 
 617 
 
 20- 
 
 001 
 
 606 
 
 607 
 
 001 
 
 617 
 
 618 
 
 15- 
 
 002 
 
 607 
 
 607 
 
 002 
 
 618 
 
 619 
 
 10- 
 
 004 
 
 607 
 
 609 
 
 004 
 
 618 
 
 620 
 
 9-0 
 
 005 
 
 607 
 
 610 
 
 005 
 
 618 
 
 621 
 
 8-0 
 
 006 
 
 608 
 
 611 
 
 006 
 
 619 
 
 622 
 
 7-0 
 
 008 
 
 608 
 
 612 
 
 008 
 
 619 
 
 624 
 
 6-0 
 
 010 
 
 609 
 
 615 
 
 Oil 
 
 620 
 
 626 
 
 5-5 
 
 012 
 
 610 
 
 616 
 
 013 
 
 621 
 
 628 
 
 5-0 
 
 015 
 
 611 
 
 619 
 
 015 
 
 622 
 
 630 
 
 4-5 
 
 018 
 
 612 
 
 621 
 
 019 
 
 623 
 
 633 
 
 4-0 
 
 023 
 
 613 
 
 625 
 
 024 
 
 624 
 
 637 
 
 3-75 
 
 027 
 
 614 
 
 628 
 
 028 
 
 626 
 
 640 
 
 3-50 
 
 031 
 
 615 
 
 631 
 
 032 
 
 627 
 
 643 
 
 3-25 
 
 036 
 
 617 
 
 635 
 
 037 
 
 628 
 
 647 
 
 3-00 
 
 043 
 
 619 
 
 640 
 
 044 
 
 630 
 
 653 
 
 2-75 
 
 051 
 
 621 
 
 646 
 
 053 
 
 633 
 
 660 
 
 2-50 
 
 062 
 
 625 
 
 654 
 
 065 
 
 637 
 
 668 
 
 2-25 
 
 078 
 
 629 
 
 665 
 
 081 
 
 642 
 
 679 
 
 2-00 
 
 101 
 
 636 
 
 681 
 
 105 
 
 649 
 
 696 
 
 1-95 
 
 107 
 
 638 
 
 685 
 
 111 
 
 650 
 
 700 
 
 1-90 
 
 113 
 
 639 
 
 689 
 
 118 
 
 652 
 
 704 
 
 1-85 
 
 119 
 
 641 
 
 693 
 
 125 
 
 654 
 
 709 
 
 1-80 
 
 128 
 
 644 
 
 698 
 
 133 
 
 657 
 
 714 
 
 1-75 
 
 136 
 
 646 
 
 703 
 
 142 
 
 659 
 
 720 
 
 1-70 
 
 146 
 
 649 
 
 709 
 
 152 
 
 662 
 
 726 
 
 1-65 
 
 156 
 
 652 
 
 716 
 
 163 
 
 665 
 
 733 
 
 1-60 
 
 167 
 
 655 
 
 723 
 
 175 
 
 669 
 
 741 
 
 1-55 
 
 180 
 
 658 
 
 731 
 
 188 
 
 673 
 
 749 
 
 1-50 
 
 195 
 
 662 
 
 739 
 
 204 
 
 677 
 
 759 
 
 1-45 
 
 212 
 
 667 
 
 749 
 
 221 
 
 681 
 
 768 
 
 1-40 
 
 231 
 
 672 
 
 760 
 
 241 
 
 687 
 
 780 
 
 1-35 
 
 252 
 
 678 
 
 772 
 
 264 
 
 694 
 
 793 
 
 1-30 
 
 278 
 
 685 
 
 786 
 
 291 
 
 701 
 
 808 
 
 1-25 
 
 307 
 
 693 
 
 803 
 
 322 
 
 709 
 
 825 
 
 1-20 
 
 342 
 
 702 
 
 821 
 
 359 
 
 719 
 
 845 
 
 1-16 
 
 384 
 
 713 
 
 843 
 
 404 
 
 731 
 
 868 
 
 1-10 
 
 436 
 
 726 
 
 868 
 
 459 
 
 745 
 
 895 
 
 1-05 
 
 499 
 
 742 
 
 898 
 
 527 
 
 763 
 
 928 
 
 1-00 
 
 680 
 
 762 
 
 936 
 
 615 
 
 784 
 
 969 
 
 See the auxiliary table, p. 68.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 195 
 
 TABLE V. COEFFICIENTS OF DISCHARGE FOR DIFFERENT RATIOS 
 
 OF THE CHANNEL TO THE ORIFICE. 
 
 Mean Coefficient '628. 
 Coefficients for heads in still water -628 and '639. 
 
 
 Coefficient -628 for heads in 
 
 Coefficient '639 for heads in 
 
 Ratio 
 
 still water. 
 
 still water. 
 
 of the 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 channel 
 to the 
 
 ;he height 
 due to the 
 
 or orifices : 
 the heads 
 
 for weirs : 
 the heads 
 
 the height 
 due to the 
 
 or orifices : 
 the heads 
 
 for weirs : 
 the heads 
 
 orifice. 
 
 velocity of 
 
 measured 
 
 measured 
 
 velocity of 
 
 measured 
 
 measured 
 
 
 approach 
 
 to the 
 
 the full 
 
 approach 
 
 to the 
 
 the full 
 
 
 o the head. 
 
 centres. 
 
 depth. 
 
 o the head. 
 
 centres. 
 
 depth. 
 
 30- 
 
 000 
 
 628 
 
 628 
 
 000 
 
 639 
 
 639 
 
 20- 
 
 001 
 
 628 
 
 629 
 
 001 
 
 639 
 
 640 
 
 15- 
 
 002 
 
 629 
 
 630 
 
 002 
 
 640 
 
 641 
 
 10- 
 
 004 
 
 629 
 
 632 
 
 004 
 
 640 
 
 643 
 
 9-0 
 
 005 
 
 630 
 
 632 
 
 005 
 
 641 
 
 644 
 
 8-0 
 
 006 
 
 630 
 
 634 
 
 006 
 
 641 
 
 645 
 
 7-0 
 
 008 
 
 631 
 
 635 
 
 008 
 
 642 
 
 647 
 
 6-0 
 
 Oil 
 
 631 
 
 638 
 
 Oil 
 
 643 
 
 649 
 
 5-5 
 
 013 
 
 632 
 
 640 
 
 014 
 
 643 
 
 651 
 
 5-0 
 
 016 
 
 633 
 
 642 
 
 017 
 
 644 
 
 654 
 
 4-5 
 
 020 
 
 634 
 
 645 
 
 021 
 
 646 
 
 657 
 
 4-0 
 
 025 
 
 636 
 
 649 
 
 026 
 
 647 
 
 662 
 
 3-75 
 
 029 
 
 637 
 
 652 
 
 030 
 
 648 
 
 665 
 
 3-50 
 
 033 
 
 638 
 
 656 
 
 034 
 
 650 
 
 668 
 
 3-25 
 
 039 
 
 639 
 
 659 
 
 040 
 
 652 
 
 673 
 
 3-0 
 
 046 
 
 642 
 
 666 
 
 048 
 
 654 
 
 678 
 
 2-75 
 
 055 
 
 645 
 
 672 
 
 057 
 
 657 
 
 686 
 
 2-50 
 
 067 
 
 649 
 
 682 
 
 070 
 
 661 
 
 695 
 
 2-25 
 
 084 
 
 654 
 
 694 
 
 088 
 
 666 
 
 708 
 
 2-0 
 
 109 
 
 661 
 
 711 
 
 114 
 
 674 
 
 727 
 
 1-95 
 
 116 
 
 663 
 
 715 
 
 120 
 
 676 
 
 731 
 
 1-90 
 
 123 
 
 665 
 
 720 
 
 128 
 
 679 
 
 736 
 
 1-85 
 
 130 
 
 668 
 
 725 
 
 135 
 
 681 
 
 741 
 
 1-80 
 
 139 
 
 670 
 
 731 
 
 144 
 
 684 
 
 747 
 
 1-75 
 
 148 
 
 673 
 
 737 
 
 154 
 
 686 
 
 753 
 
 1-70 
 
 158 
 
 676 
 
 743 
 
 165 
 
 690 
 
 760 
 
 1-65 
 
 169 
 
 679 
 
 750 
 
 176 
 
 693 
 
 768 
 
 1-60 
 
 182 
 
 683 
 
 758 
 
 190 
 
 697 
 
 776 
 
 1-55 
 
 196 
 
 687 
 
 767 
 
 205 
 
 701 
 
 786 
 
 1-50 
 
 213 
 
 692 
 
 777 
 
 222 
 
 706 
 
 796 
 
 1-45 
 
 231 
 
 697 
 
 788 
 
 241 
 
 712 
 
 808 
 
 1-40 
 
 252 
 
 703 
 
 800 
 
 262 
 
 718 
 
 820 
 
 1-35 
 
 276 
 
 709 
 
 814 
 
 289 
 
 725 
 
 836 
 
 1-30 
 
 304 
 
 717 
 
 830 
 
 319 
 
 734 
 
 853 
 
 1-25 
 
 338 
 
 726 
 
 846 
 
 354 
 
 743 
 
 872 
 
 1-20 
 
 377 
 
 734 
 
 866 
 
 396 
 
 755 
 
 895 
 
 MB 
 
 425 
 
 750 
 
 894 
 
 447 
 
 769 
 
 921 
 
 1-10 
 
 484 
 
 765 
 
 924 
 
 509 
 
 785 
 
 953 
 
 1-05 
 
 557 
 
 784 
 
 959 
 
 588 
 
 805 
 
 991 
 
 1-00 
 
 651 
 
 807 
 
 1-002 
 
 690 
 
 830 
 
 1-038 
 
 See the auxiliary table, p. 68. 
 
 o 2
 
 196 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE V. COEFFICIENTS OF DISCHARGE FOR DIFFERENT RATIOS 
 
 OF THE CHANNEL TO THE ORIFICE. 
 
 Coefficients for heads in still water -650 and -667. 
 
 
 Coefficient -650 for heads in 
 
 Coefficient -667 for heads in 
 
 Ratio 
 
 still water. 
 
 still water. 
 
 of the 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 channel 
 to the 
 
 the height 
 due to the 
 
 for orifices : 
 the heads 
 
 for weirs : 
 the heads 
 
 the height 
 due to the 
 
 'or orifices : 
 the heads 
 
 for weirs : 
 the heads 
 
 orifice. 
 
 velocity of 
 
 measured 
 
 measured 
 
 velocity of 
 
 measured 
 
 measured 
 
 
 approach 
 
 to the 
 
 the full 
 
 approach 
 
 to the 
 
 the full 
 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 30- 
 
 000 
 
 650 
 
 650 
 
 000 
 
 667 
 
 667 
 
 20- 
 
 001 
 
 650 
 
 651 
 
 001 
 
 667 
 
 668 
 
 15- 
 
 002 
 
 651 
 
 652 
 
 002 
 
 667 
 
 669 
 
 lo- 
 
 004 
 
 651 
 
 654 
 
 004 
 
 668 
 
 671 
 
 g- 
 
 005 
 
 652 
 
 655 
 
 006 
 
 669 
 
 672 
 
 s' 
 
 007 
 
 652 
 
 656 
 
 007 
 
 669 
 
 673 
 
 7-0 
 
 009 
 
 653 
 
 658 
 
 009 
 
 670 
 
 675 
 
 6-0 
 
 012 
 
 654 
 
 661 
 
 012 
 
 671 
 
 678 
 
 5-5 
 
 014 
 
 655 
 
 663 
 
 015 
 
 672 
 
 680 
 
 5-0 
 
 017 
 
 656 
 
 665 
 
 018 
 
 673 
 
 682 
 
 4-5 
 
 021 
 
 657 
 
 669 
 
 022 
 
 674 
 
 687 
 
 4-0 
 
 027 
 
 659 
 
 674 
 
 029 
 
 676 
 
 692 
 
 3-75 
 
 031 
 
 660 
 
 677 
 
 033 
 
 678 
 
 696 
 
 3-50 
 
 036 
 
 662 
 
 681 
 
 038 
 
 679 
 
 700 
 
 3-25 
 
 042 
 
 663 
 
 686 
 
 044 
 
 681 
 
 705 
 
 3-0 
 
 049 
 
 666 
 
 692 
 
 052 
 
 684 
 
 711 
 
 2-75 
 
 059 
 
 669 
 
 699 
 
 062 
 
 687 
 
 720 
 
 2-50 
 
 073 
 
 673 
 
 709 
 
 077 
 
 692 
 
 731 
 
 2-25 
 
 091 
 
 679 
 
 723 
 
 096 
 
 698 
 
 745 
 
 2-0 
 
 118 
 
 687 
 
 742 
 
 125 
 
 707 
 
 766 
 
 1-95 
 
 125 
 
 689 
 
 747 
 
 132 
 
 709 
 
 771 
 
 1-90 
 
 133 
 
 692 
 
 752 
 
 140 
 
 712 
 
 777 
 
 1-85 
 
 141 
 
 694 
 
 758 
 
 149 
 
 715 
 
 783 
 
 1-80 
 
 150 
 
 697 
 
 764 
 
 159 
 
 718 
 
 790 
 
 1-75 
 
 160 
 
 700 
 
 771 
 
 170 
 
 721 
 
 797 
 
 1-70 
 
 172 
 
 704 
 
 779 
 
 182 
 
 725 
 
 805 
 
 1-65 
 
 184 
 
 707 
 
 786 
 
 195 
 
 729 
 
 814 
 
 1-60 
 
 198 
 
 711 
 
 795 
 
 210 
 
 733 
 
 823 
 
 1-55 
 
 213- 
 
 716 
 
 805 
 
 227 
 
 738 
 
 833 
 
 1-50 
 
 231 
 
 721 
 
 816 
 
 246 
 
 744 
 
 846 
 
 1-45 
 
 251 
 
 727 
 
 828 
 
 268 
 
 751 
 
 859 
 
 1-40 
 
 275 
 
 734 
 
 842 
 
 293 
 
 - -758 
 
 874 
 
 1-35 
 
 302 
 
 742 
 
 858 
 
 322 
 
 764 
 
 888 
 
 1-30 
 
 333 
 
 751 
 
 876 
 
 356 
 
 776 
 
 911 
 
 1-25 
 
 371 
 
 761 
 
 896 
 
 398 
 
 788 
 
 934 
 
 1-20 
 
 415 
 
 773 
 
 920 
 
 446 
 
 802 
 
 961 
 
 1-15 
 
 469 
 
 788 
 
 949 
 
 506 
 
 818 
 
 992 
 
 MO 
 
 537 
 
 806 
 
 983 
 
 580 
 
 838 
 
 1-030 
 
 1-05 
 
 621 
 
 828 
 
 1-024 
 
 675 
 
 863 
 
 1-076 
 
 1-00 
 
 732 
 
 855. 
 
 1-074 
 
 800 
 
 894 
 
 1-133 
 
 See the auxiliary table, p. 68.
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 197 
 
 TABLE V. COEFFICIENTS OF DISCHARGE FOB DIFFERENT RATIOS 
 
 OF THE CHANNEL TO THE ORIFICE. 
 
 Coefficients for heads in still water *s/'5 = '7071 and 1. 
 
 
 Coefficient -7071 for heads in 
 
 Coefficient TOGO for heads in 
 
 
 still water. 
 
 still water. 
 
 Ratio 
 
 
 
 of the 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 channel 
 to the 
 
 the height 
 due to the 
 
 for orifices : 
 the heads 
 
 for weirs : 
 the heads 
 
 the height 
 due to the 
 
 for orifices : 
 the heads 
 
 for weirs : 
 the heads 
 
 orifice. 
 
 velocity of 
 
 measured 
 
 measured 
 
 velocity of 
 
 measured 
 
 measured 
 
 
 approach 
 
 to the 
 
 the full 
 
 approach 
 
 to the 
 
 the full 
 
 
 ;o the head. 
 
 centres. 
 
 depth. 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 30- 
 
 001 
 
 707 
 
 708 
 
 001 
 
 1-001 
 
 1-002 
 
 20- 
 
 001 
 
 708 
 
 708 
 
 003 
 
 1-001 
 
 1-004 
 
 15- 
 
 001 
 
 708 
 
 709 
 
 005 
 
 1-002 
 
 1-006 
 
 lo- 
 
 005 
 
 709 
 
 712 
 
 010 
 
 1-005 
 
 1-014 
 
 g- 
 
 006 
 
 709 
 
 . -713 
 
 013 
 
 1-006 
 
 1-017 
 
 s' 
 
 008 
 
 710 
 
 714 
 
 016 
 
 1-008 
 
 1-021 
 
 7- 
 
 010 
 
 711 
 
 717 
 
 021 
 
 1-010 
 
 1-028 
 
 6- 
 
 014 
 
 712 
 
 721 
 
 029 
 
 1-014 
 
 1-038 
 
 5-5 
 
 017 
 
 713 
 
 723 
 
 034 
 
 1-017 
 
 1-045 
 
 5-0 
 
 020 
 
 714 
 
 727 
 
 041 
 
 1-021 
 
 1-055 
 
 4-5 
 
 025 
 
 716 
 
 731 
 
 052 
 
 1-026 
 
 1-067 
 
 4-0 
 
 032 
 
 718 
 
 737 
 
 067 
 
 1-033 
 
 1-084 
 
 3-75 
 
 037 
 
 720 
 
 742 
 
 077 
 
 1-038 
 
 1-096 
 
 3-50 
 
 043 
 
 722 
 
 747 
 
 089 
 
 1-044 
 
 1-110 
 
 3-25 
 
 050 
 
 724 
 
 753 
 
 105 
 
 1-051 
 
 1-127 
 
 3-00 
 
 059 
 
 728 
 
 760 
 
 125 
 
 1-061 
 
 1-149 
 
 2-75 
 
 071 
 
 732 
 
 770 
 
 152 
 
 1-073 
 
 1-178 
 
 2-50 
 
 087 
 
 737 
 
 783 
 
 190 
 
 1-091 
 
 1-216 
 
 2-25 
 
 110 
 
 745 
 
 801 
 
 246 
 
 1-116 
 
 1-269 
 
 2-00 
 
 143 
 
 756 
 
 826 
 
 333 
 
 1-155 
 
 347 
 
 1-95 
 
 151 
 
 759 
 
 832 
 
 356 
 
 1-165 
 
 1-367 
 
 1-90 
 
 161 
 
 762 
 
 839 
 
 383 
 
 1-176 
 
 389 
 
 1-85 
 
 171 
 
 765 
 
 846 
 
 412 
 
 1-188 
 
 1-413 
 
 1-80 
 
 182 
 
 769 
 
 854 
 
 446 
 
 1-203 
 
 1-441 
 
 1-75 
 
 195 
 
 773 
 
 863 
 
 484 
 
 1-218 
 
 1-471 
 
 1-70 
 
 209 
 
 778 
 
 873 
 
 529 
 
 1-237 
 
 1-505 
 
 1-65 
 
 225 
 
 783 
 
 883 
 
 579 
 
 1-257 
 
 1-543 
 
 1-60 
 
 243 
 
 788 
 
 895 
 
 641 
 
 1-281 
 
 1-589 
 
 1-55 
 
 263 
 
 795 
 
 908 
 
 711 
 
 1-308 
 
 1-638 
 
 1-50 
 
 286 
 
 802 
 
 923 
 
 800 
 
 1-342 
 
 1-699 
 
 1-45 
 
 312 
 
 810 
 
 939 
 
 903 
 
 1-379 
 
 1-767 
 
 1-40 
 
 342 
 
 819 
 
 958 
 
 1-042 
 
 1-429 
 
 1-854 
 
 1-35 
 
 378 
 
 830 
 
 980 
 
 1-216 
 
 1-489 
 
 1-958 
 
 1-30 
 
 421 
 
 842 
 
 1-003 
 
 1-449 
 
 1-565 
 
 2-088 
 
 1-25 
 
 471 
 
 857 
 
 1-033 
 
 1-778 
 
 1-667 
 
 2-259 
 
 1-20 
 
 532 
 
 875 
 
 1-066 
 
 2-273 
 
 1-810 
 
 2-499 
 
 1-15 
 
 608 
 
 897 
 
 1-107 
 
 3-100 
 
 2-025 
 
 2844 
 
 1-10 
 
 704 
 
 923 
 
 1-155 
 
 4-762 
 
 2-400 
 
 3-440 
 
 1-05 
 
 830 
 
 957 
 
 1-216 
 
 9-756 
 
 3-280 
 
 4-803 
 
 J-00 
 
 1-000 
 
 1-000 
 
 1-293 
 
 infinite. 
 
 infinite. 
 
 infinite.
 
 198 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VI THE DISCHARGE OVER WEIRS OR NOTCHES OF ONE 
 
 FOOT IN LENGTH, IN CUBIC FEET PER MINUTE. 
 
 Depths J inch to 10 inches. Coefficients -667 to -617. 
 
 GREATER COEFFICIENTS. 
 
 Heads in 
 inches. 
 
 Theoretical 
 discharge. 
 
 Coefficient 
 667. 
 
 Coefficient 
 650. 
 
 Coefficient 
 639. 
 
 Coefficient 
 628. 
 
 Coefficient 
 617. 
 
 25 
 
 965 
 
 644 
 
 627 
 
 617 
 
 606 
 
 596 
 
 5 
 
 2-730 
 
 1-821 
 
 1-775 
 
 1-744 
 
 1-714 
 
 1-684 
 
 75 
 
 5-016 
 
 3345 
 
 3260 
 
 3-205 
 
 3-150 
 
 3-095 
 
 I 
 
 7722 
 
 5-151 
 
 5-019 
 
 4934 
 
 4-849 
 
 4-764 
 
 1-25 
 
 10-792 
 
 7-198 
 
 7015 
 
 6-896 
 
 6-777 
 
 6-659 
 
 1-5 
 
 14-186 
 
 9-462 
 
 9-221 
 
 9-065 
 
 8-909 
 
 8-753 
 
 1-75 
 
 17-877 
 
 11924 
 
 11-620 
 
 11-423 
 
 11227 
 
 11-030 
 
 2- 
 
 21-842 
 
 14-569 
 
 14-197 
 
 13-957 
 
 13717 
 
 13477 
 
 225 
 
 26-062 
 
 17-383 
 
 16-940 
 
 16-654 
 
 16-367 
 
 16-080 
 
 2-5 
 
 30-524 
 
 20-360 
 
 19-841 
 
 19-505 
 
 19-169 
 
 18-833 
 
 275 
 
 35215 
 
 23-489 
 
 22-890 
 
 22-503 
 
 22-115 
 
 21-728 
 
 3- 
 
 40-125 
 
 26-763 
 
 26-081 
 
 25-640 
 
 25-199 
 
 24-757 
 
 3-25 
 
 45244 
 
 30-178 
 
 29-408 
 
 28-911 
 
 28-413 
 
 27-915 
 
 35 
 
 50-563 
 
 33726 
 
 32-866 
 
 32-310 
 
 31-754 
 
 31-197 
 
 3-75 
 
 56-077 
 
 37-403 
 
 36-450 
 
 35-833 
 
 35-216 
 
 34-599 
 
 4- 
 
 61-777 
 
 41-205 
 
 40-155 
 
 39-476 
 
 38-796 
 
 38-116 
 
 4-25 
 
 67-658 
 
 45-128 
 
 43-978 
 
 43233 
 
 42-489 
 
 41-745 
 
 4-5 
 
 73-714 
 
 49 167 
 
 47-914 
 
 47-103 
 
 46-292 
 
 45-482 
 
 4-75 
 
 79-942 
 
 53-321 
 
 51-962 
 
 51-083 
 
 50-203 
 
 49-324 
 
 5 
 
 86-335 
 
 57-585 
 
 56-118 
 
 55-168 
 
 54-218 
 
 53-269 
 
 525 
 
 92891 
 
 61-958 
 
 60379 
 
 59-357 
 
 58-335 
 
 57-314 
 
 55 
 
 99-604 
 
 66-436 
 
 64-743 
 
 63-647 
 
 62-551 
 
 61-456 
 
 5-75 
 
 106-472 
 
 71-017 
 
 69-207 
 
 68-036 
 
 66-864 
 
 65-693 
 
 6- 
 
 113-491 
 
 75-698 
 
 73-769 
 
 72-521 
 
 71-272 
 
 70-024' 
 
 625 
 
 120-657 
 
 80-478 
 
 78-427 
 
 77-100 
 
 75772 
 
 74-445 
 
 6-5 
 
 127-969 
 
 85-355 
 
 83-180 
 
 81-772 
 
 80-365 
 
 78-957 
 
 6-75 
 
 135-422 
 
 90-326 
 
 88-024 
 
 86-535 
 
 85-045 
 
 83555 
 
 7- 
 
 143-015 
 
 95'391 
 
 92-960 
 
 91-387 
 
 89-813 
 
 88-240 
 
 7-25 
 
 150-744 
 
 100-546 
 
 97-983 
 
 96325 
 
 94-667 
 
 93-009 
 
 75 
 
 158-608 
 
 105-792 
 
 103-095 
 
 101-350 
 
 99-606 
 
 97-861 
 
 7'75 
 
 166-604 
 
 111-125 
 
 108-292 
 
 106-460 
 
 104-627 
 
 102-795 
 
 8- 
 
 174731 
 
 116-546 
 
 113-575 
 
 111-653 
 
 109-731 
 
 107-809 
 
 8-25 
 
 182-984 
 
 122-051 
 
 118-940 
 
 116-927 
 
 114-914 
 
 112901 
 
 8-5 
 
 191-365 
 
 127-640 
 
 124-387 
 
 122-282 
 
 120-177 
 
 ] 18-072 
 
 8-75 
 
 199-869 
 
 133-313 
 
 129-915 
 
 127-716 
 
 125-518 
 
 123-319 
 
 9' 
 
 208-496 
 
 139-067 
 
 135522 
 
 133229 
 
 130-935 
 
 128-642 
 
 9-25 
 
 217-243 
 
 144-901 
 
 141-207 
 
 138-818 
 
 136-428 
 
 134039 
 
 9-5 
 
 226-111 
 
 150-816 
 
 146-972 
 
 144485 
 
 141 -997 
 
 139510 
 
 1 9-75 
 
 235-093 
 
 156-807 
 
 152-810 
 
 150-225 
 
 147- (539 
 
 145-053 
 
 10- 
 
 244-193 
 
 162-877 
 
 158-725 
 
 156-039 
 
 153353 
 
 150-666
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 199 
 
 TABLE VI. THE DISCHARGE OVER WEIRS OR NOTCHES OF ONE 
 
 FOOT IN LENGTH, IN CUBIC FEET PER MINUTE. 
 
 Depths 10-25 inches to 32 inches. Coefficients -667 to -617. 
 
 GREATER COEFFICIENTS. 
 
 Heads in Theoretical 
 inches. discharge. 
 
 Coefficient 
 667. 
 
 Coefficient 
 650. 
 
 Coefficient Coefficient 
 639. -628. 
 
 Coefficient 
 617. 
 
 10-25 
 
 253-407 
 
 169-023 
 
 164-715 
 
 161 927 159-140 
 
 156-352 
 
 10-5 
 
 262-734 
 
 175-244 
 
 170777 
 
 167-887 164-997 
 
 162-107 
 
 10-75 
 
 272173 
 
 181-540 
 
 176-913 
 
 173 919 170 925 
 
 167-931 
 
 11- 
 
 281-723 
 
 187-909 
 
 183-120 
 
 180021 176-922 
 
 173-823 
 
 1125 
 
 291-382 
 
 194-352 
 
 189-398 
 
 186-193 182-988 
 
 179-782 
 
 11-5 
 
 301-148 
 
 200-866 
 
 195-746 
 
 192-434; 189-121 
 
 185-808 
 
 11-75 
 
 311-024 
 
 207-451 
 
 202-164 
 
 198-743 195-321 
 
 191-900 
 
 12- 
 
 321- 
 
 214-107 
 
 208-650 
 
 205-119 201-588 
 
 198-057 
 
 12-5 
 
 341-275 
 
 227628 
 
 221-826 
 
 218-072 214-318 210-564 
 
 13- 
 
 361-950 
 
 241-421 
 
 235-268 
 
 231-286 227-305 223-323 
 
 135 
 14- 
 
 383-031 
 404-507 
 
 255-482 
 269-806 
 
 248-970 
 262-930 
 
 244-757 240-543 236-330 
 258-480 254-030 1 249-581 
 
 14-5 
 
 426-368 
 
 284-387 
 
 277-139 
 
 272 449! 267-759 
 
 263-069 
 
 15 
 
 448-611 
 
 299-223 
 
 291-597 
 
 286-662^ 281-728 
 
 276-793 
 
 155 
 
 471-228 
 
 314-309 
 
 306-298 
 
 301-115 295-931 
 
 290-748 
 
 16- 
 
 494-212 
 
 329-639 
 
 321-238 
 
 315-801 310-365 
 
 304-929 
 
 165 
 
 517*558 
 
 345-211 
 
 336-413 
 
 330-720 325 026 
 
 319-333 
 
 17 
 
 541-261 
 
 361-021 
 
 351-820 
 
 345-866! 339-912 
 
 333958 
 
 175 
 
 565-315 
 
 377-065 
 
 367-455 
 
 361-236 355 018 
 
 348-799 
 
 18- 
 
 589-715 
 
 393340 
 
 383315 
 
 376-828 370 341 
 
 363-854 
 
 18-5 
 
 614-443 
 
 409-833 
 
 399-388 
 
 392-629 
 
 385-870 
 
 379-111 
 
 19- 
 
 639-533 
 
 426-569 
 
 415696 
 
 408-62 
 
 401-627 
 
 394-592 
 
 195 
 
 664-944 
 
 443-518 
 
 432-214 
 
 424-899 
 
 417-585 
 
 410-270 
 
 20- 
 
 690-682 
 
 460-685 
 
 448-943 
 
 441-346 
 
 433-748 
 
 426-151 
 
 20-5 
 
 716737 
 
 478-064 
 
 465-879 
 
 457-995 
 
 450-111 
 
 442-227 
 
 21- 
 
 743-125 
 
 495-664 
 
 483-031 
 
 4-4-857 
 
 466-683 
 
 458-508 
 
 21-5 
 
 769-823 
 
 513-472 
 
 500-385 
 
 491-917 
 
 483-449 
 
 474-981 
 
 22- 
 
 796-832 
 
 531-487 
 
 517-941 
 
 509-176 
 
 500-410 
 
 491-645 
 
 i 22 5 
 
 824-151 
 
 549709 
 
 535-698 
 
 526-632 517-567 
 
 508-501; 
 
 23- 
 
 851-775 
 
 568-134 
 
 553-654 
 
 544-2841 534-915 
 
 525-545 
 
 235 
 
 879*700 
 
 586-760 
 
 571-805 
 
 562-128 552-452 
 
 542-775 
 
 24- 
 
 907-925 
 
 605-586 
 
 590151 
 
 580-164 570-177 
 
 560- 190l 
 
 25- 
 
 965253 
 
 643-824 
 
 627-414 
 
 616-797 
 
 606-179 
 
 595-561 
 
 26- 
 
 1023-748 
 
 682-840 
 
 665-436 
 
 654-175 
 
 642914 
 
 631-653 
 
 27- 
 
 1083-375 
 
 722-611 
 
 704-194 
 
 692-277 
 
 680-360 
 
 668-442' 
 
 28- 
 
 1144-116 
 
 763-125 
 
 743-675 
 
 731-090 
 
 718-5051 705-920 
 
 29- 
 
 1205-950 
 
 804-369 
 
 783-868 
 
 770-602 
 
 757-337 744-071: 
 
 30- 
 
 1268-864 
 
 846332 
 
 824-762 
 
 810-804 
 
 796-847 782-889 
 
 31- 
 
 1332833 
 
 889-000 
 
 866-341 
 
 851-680 
 
 837-019 822-358 
 
 32- 
 
 1397-842 
 
 932-361 
 
 908-597 
 
 193-221 
 
 877-845 862-469
 
 200 
 
 THE DISCHARGE OP WATER FROM 
 
 TABLE VI. THE DISCHARGE OVER WEIRS OR NOTCHES OF ONE 
 
 FOOT IN LENGTH, IN CUBIC FEET PER MINOTE. 
 
 Depths 33 inches to 72 inches. Coefficients -667 to -017. 
 
 GREATER COEFFICIENTS. 
 
 Heads in 
 inches. 
 
 Theoretical 
 discharge. 
 
 Coefficient 
 667. 
 
 Coefficient 
 650. 
 
 Coefficient 
 639. 
 
 Coefficient 
 628. 
 
 Coefficient 
 617. 
 
 33- 
 
 1463-875 
 
 976-405 
 
 951-519 
 
 935416 
 
 919314 
 
 903211 
 
 34- 
 
 1530-917 
 
 1021-122 
 
 995-096 
 
 978-256 
 
 961-416 
 
 944-576 
 
 35 
 
 1598-951 
 
 1066500 
 
 1039318 
 
 1021-730 
 
 1004-141 
 
 986-553 
 
 36- 
 
 1667-964 
 
 1112-532 
 
 1084-177 
 
 1065-829 1047-481 
 
 1029-134 
 
 37 
 
 1737-943 1159-208 
 
 1129-663 
 
 1110546 
 
 1091-428 
 
 1072-311 
 
 38- 
 
 1808-8751 1200-520 
 
 1175-T6[) 
 
 1155871 
 
 1135974 
 
 1116076 
 
 39- 
 
 1880-746 
 
 1254-458 
 
 1222485 
 
 1201 797 
 
 1181-108 
 
 1160-420 
 
 40- 
 
 1953544 
 
 1303014 
 
 1269-804 
 
 1248315 
 
 1226-826 
 
 1205337 
 
 41- 
 
 2027-258 
 
 1352-181 
 
 1317718 
 
 1295-418 
 
 1273118 
 
 1250-818 
 
 42 
 
 2101-876 
 
 1401-951 
 
 1366-219 
 
 1343-099 
 
 1319-978 
 
 1296-857 
 
 43- 
 
 2177-387 
 
 1452-317 
 
 1415-302 
 
 1391-350 
 
 1367-399 
 
 1343-448 
 
 44. 
 
 2253783 
 
 1503273 
 
 1464-959 
 
 1440- 167 
 
 1415-376 1390584 
 
 45 
 
 2331052 
 
 1554-812 
 
 1515-184 
 
 1489-542 1463-901 1438259 
 
 46 
 
 2409-183 
 
 1606-925 
 
 1565969 
 
 1539-468 
 
 1512-967 
 
 1486-466 
 
 47- 
 
 2488-170 
 
 1659609 
 
 1617-311 
 
 1589-941 
 
 1562-571 
 
 1535-201 
 
 48- 
 
 2568- 
 
 1712-856 
 
 1669-200 
 
 1640-952 
 
 1612-704 
 
 1584-456 
 
 49- 
 
 2648-666 
 
 1766-660 
 
 1721-633 
 
 1692-498 
 
 1663362 
 
 1634-227 
 
 50- 
 
 2730-160 
 
 1821-021 
 
 1774-604 
 
 1744572 
 
 1714-540 
 
 1684-509 
 
 51- 
 
 2812-474 
 
 1875-920 
 
 1828 108 
 
 1797-17ll 1766-234 
 
 1735296 
 
 52- 
 
 2895597 
 
 193T363 
 
 1882-138 
 
 1850-286 1818-435 
 
 1786-583 
 
 53- 
 
 2979-525 
 
 1987-343 
 
 1936-691 
 
 1903-916 1871-142 
 
 1838-367 
 
 54- 
 
 3064-253 
 
 2043-857 
 
 1991 764 
 
 1958-058 1924-351 
 
 1890-644 
 
 55- 
 
 3149-755 
 
 2100-887 
 
 2047-341 
 
 2012-693 1978-046 
 
 1943399 
 
 56- 
 
 3236-050 
 
 2158-445 
 
 2103-433 
 
 2067-836 2032-239 
 
 1996-643 
 
 57- 
 
 3323-117! 2216-519 
 
 2160-026 
 
 2123-472! 2086-917 
 
 2050-363 
 
 58- 
 
 3410-946 2275-101 
 
 2217-115 
 
 2179-594 
 
 2142-074 
 
 2104-554 
 
 59- 
 
 3499 542 2334-195 
 
 2274702 
 
 2236-207 
 
 2197-712 
 
 2159-217 
 
 60- 
 
 3588-889 2393-789 
 
 2332778 
 
 2293-300 
 
 2253-822 
 
 2214-344 
 
 61- 
 
 3678-984 2453-882 
 
 2391-340 
 
 2350-871 
 
 2310-402 
 
 2269-933 
 
 62- 
 
 3769 825 2514-473 
 
 2450-386 
 
 2408-918 
 
 2367-450 
 
 23-25-982 
 
 63- 
 
 3861-393 2575-549 
 
 2509-905 
 
 2467-430 
 
 2424955 
 
 2382-479 
 
 64- 
 
 3953694 2637 114 
 
 2569-901 
 
 2526-4 10 ! 2482-920 
 
 2439429 
 
 65- 
 
 4046-720; 2699-162 
 
 2630-368 
 
 2585-854' 2541 -340 i 2496'826| 
 
 66- 
 
 4140-465 
 
 2761-690 
 
 2691-302 
 
 2645-757 2600-212 
 
 2554667 
 
 67- 
 
 4234922 
 
 2824-693 
 
 2752699 
 
 2706-115 
 
 2659531 
 
 2612-947 
 
 68- 
 
 4330-086 
 
 2888-167 
 
 2814-556 
 
 2766-925 
 
 2719-294 
 
 2671-068 
 
 69- 
 
 4425-954 
 
 2952-111 
 
 2876-870 
 
 2828-185 2779-499 2730-814 
 
 70- 
 
 4522-516 
 
 3016-518 
 
 2939635 
 
 2889-888 2840-140 2790-392! 
 
 71- 
 
 4619-774 
 
 3081-389 3002-853 
 
 2952-036 2901-218 
 
 2850-401 
 
 72- 
 
 4717-718 
 
 3146-718 3066-518 
 
 3014-622 2962727 
 
 | 
 
 2910-832
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 201 
 
 TABLE VI. THE DISCHARGE OVER WEIRS OR NOTCHES OF ONE 
 
 FOOT IN LENGTH, IN CUBIC FEET PER MINUTE. 
 
 Depths i inch to 10 inches. Coefficients -606 to -518. 
 
 LESSER COEFFICIENTS. 
 
 Heads in 
 
 1 inches. 
 
 Coefficient 
 606. 
 
 Coefficient 
 595. 
 
 Coefficient 
 584. 
 
 Coefficient 
 562. 
 
 Coefficient 
 540. 
 
 Coefficient 
 518. 
 
 25 
 
 585 
 
 574 
 
 564 
 
 542 
 
 521 
 
 500 
 
 5 
 
 1 654 
 
 1624 
 
 1 594 
 
 1534 
 
 1-474 
 
 1-414 
 
 75 
 
 3-039 
 
 2-985 
 
 2929 
 
 2-819 
 
 2-708 
 
 2-598 
 
 1- 
 
 4-680 
 
 4-595 
 
 4-510 
 
 4340 
 
 4-170 
 
 4-000 
 
 1-25 
 
 6540 
 
 6-42 L 
 
 6303 
 
 6-065 
 
 5-828 
 
 5-590 
 
 15 
 
 8-597 
 
 8-441 
 
 8-284 
 
 7-973 
 
 7-660 
 
 7-348 
 
 1-75 
 
 10-833 
 
 10-637 
 
 10-440 
 
 10-047 
 
 9-653 
 
 9-260 
 
 2- 
 
 12236 
 
 12-996 
 
 12-756 
 
 12-275 
 
 11-795 
 
 11-314 
 
 2 25 
 
 15794 
 
 15-507 
 
 15-220 
 
 14647 
 
 14-073 
 
 13-500 
 
 25 
 
 18-498 
 
 18-162 
 
 17-826 
 
 17 155 
 
 16-483 
 
 15-811 
 
 275 
 
 21340 
 
 20953 
 
 20-566 
 
 19-791 
 
 19-016 
 
 18-241 
 
 3- 
 
 24316 
 
 23-874 
 
 23433 
 
 22-550 
 
 21-668 
 
 20-785 
 
 325 
 
 27-418 
 
 26920 
 
 26-422 , 
 
 25-427 
 
 24432 
 
 23-436 
 
 3-5 
 
 30641 
 
 30-085 
 
 29-529 
 
 28-416 
 
 27-304 
 
 26-192 
 
 3-75 
 
 33-982 
 
 33366 
 
 32-749 
 
 31-515 
 
 30-281 
 
 29-048 
 
 4- 
 
 37437 
 
 36-757 
 
 36-078 
 
 34-719 
 
 33-360 
 
 32-000 
 
 4-25 
 
 41-001 
 
 40-256 
 
 39-512 
 
 38-024 
 
 36-535 
 
 35-047 
 
 4-5 
 
 44671 
 
 43-860 
 
 43-049 
 
 41-427 
 
 39-806 
 
 38-184 
 
 4-75 
 
 48-445 
 
 47565 
 
 46-686 
 
 44-927 
 
 43-169 
 
 41-410 
 
 5- 
 
 52-319 
 
 51-369 
 
 50-420 
 
 48-520 
 
 46621 
 
 44-722 
 
 5-25 
 
 56-292 
 
 55-270 
 
 54-248 
 
 52-205 
 
 50-161 
 
 48-117 
 
 55 
 
 60-360 
 
 59-264 
 
 58-169 
 
 55-977 
 
 53-786 
 
 51-595 
 
 575 
 
 64-522 
 
 63351 
 
 62-180 
 
 59-837 
 
 57-495 
 
 55-153 
 
 6- 
 
 68-776 
 
 67527 
 
 66-279 
 
 63-782 
 
 61-285 
 
 58-788 
 
 625 
 
 73-118 
 
 71-791 
 
 70-464 
 
 67-809 
 
 65-155 
 
 62-500 
 
 6'5 
 
 77549 
 
 76142 
 
 74-734 
 
 71-919 
 
 69-103 
 
 66-288 
 
 1 6-75 
 
 82-066 
 
 80-576 
 
 79-086 
 
 76-107 
 
 73-128 
 
 70-149 
 
 7- 
 
 86-667 
 
 85-094 
 
 83521 
 
 80-374 
 
 77-228 
 
 74-082 
 
 725 
 
 91-351 
 
 89-693 
 
 88-034 
 
 84-718 
 
 81-402 
 
 78-085 
 
 75 
 
 96-116 
 
 94-372 
 
 92-627 
 
 89-138 
 
 85-648 
 
 82-159 
 
 775 
 
 lOu-962 
 
 99-129 
 
 97-297 
 
 93-631 
 
 89-966 
 
 86-301 
 
 8- 
 
 105-887 
 
 103-965 
 
 10-2-043 
 
 98-199 
 
 94-355 
 
 90-511 
 
 1 8-25 
 
 110-889 
 
 108-816 
 
 106-863 
 
 102-837 
 
 98-812 
 
 94-786 
 
 8-5 
 
 115-967 
 
 113-862 
 
 111-757 
 
 107-547 
 
 103-337 
 
 99-127 
 
 8-75 
 
 121 121 
 
 118-922 
 
 116-723 
 
 112-326 
 
 107-929 
 
 103-532 
 
 9- 
 
 126-349 
 
 124-055 
 
 121-762 
 
 117-175 
 
 112-588 
 
 108-001 
 
 925 
 
 131-649 
 
 129-259 
 
 126-870 
 
 122-090 
 
 117-311 
 
 112-532 
 
 9-5 
 
 137023 
 
 134535 
 
 132-048 
 
 127-074 
 
 122-100 
 
 117-125 
 
 975 
 
 142-467 
 
 139-881 
 
 137-294 
 
 132-122 
 
 126-950 
 
 121-778 
 
 10- 
 
 147-981 
 
 145295 
 
 142-609 
 
 137237 
 
 131-864 
 
 126-492
 
 202 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VI. THE DISCHARGE OVER WEIRS OR NOTCHES OF ONE 
 
 FOOT IN LENGTH, IN CUBIC FEET PER MINUTE. 
 
 Depths 10-25 inches to 32 inches. Coefficients -606 to '518. 
 
 LESSER COEFFICIENTS. 
 
 Heads 
 in 
 inches 
 
 Coeffic'. 
 606. 
 
 Coeffic 1 . 
 595. 
 
 Coefficient 
 584. 
 
 Coefficient 
 562. 
 
 Coefficient 
 540. 
 
 Coefficient 
 518. 
 
 10-25 
 
 153-565 
 
 150-777 
 
 147-990 
 
 142415 
 
 136-840 
 
 131 265 
 
 10-5 
 
 159-217 
 
 156-327 
 
 153-437 
 
 147-657 
 
 141-876 
 
 136-096 
 
 10-75 
 
 164-937 
 
 161-943 
 
 158-949 
 
 152-961 
 
 146-974 
 
 140-986 
 
 11' 
 
 170-724 
 
 167-625 
 
 164-526 
 
 158-328 
 
 152-130 
 
 145933 
 
 11-25 
 
 176577 
 
 173-372 
 
 170-167 
 
 163-756 
 
 157346 
 
 150-936 
 
 11-5 
 
 182-496 
 
 179- 183 
 
 175-870 
 
 169-245 
 
 162-620 
 
 155-995 
 
 1 1 -75 
 
 188-479 
 
 185-058 
 
 181-636 
 
 174-794 
 
 167-952 
 
 161-109 
 
 12- 
 
 194-526 
 
 190-995 
 
 187-464 
 
 180-402 
 
 173340 
 
 166-278 
 
 12-5 
 
 206-810 
 
 203-056 
 
 199302 
 
 191-794 
 
 184-286 
 
 176-778 
 
 13- 
 
 219-342 
 
 215-360 
 
 211-379 
 
 203-415 
 
 195453 
 
 187-490 
 
 13-5 
 
 232-117 
 
 227-903 
 
 223-690 
 
 215-263 
 
 206-837 
 
 198-410 
 
 14- 
 
 245-131 
 
 240-682 
 
 2SJ0-232 
 
 227-333 
 
 218-434 
 
 209-535 
 
 14-5 
 
 258-379 
 
 253-689 
 
 248-999 
 
 239-619 
 
 230-239 
 
 220-859 
 
 15- 
 
 271-858 
 
 266-924 
 
 261-989 
 
 252-119 
 
 242-250 
 
 232-380 
 
 15-5 
 
 285-564 
 
 280-381 
 
 275-197 
 
 264-830 
 
 254-463 
 
 244-096 
 
 16- 
 
 299-492 
 
 294-056 
 
 288-620 
 
 277-747 
 
 266-875 
 
 256-001 
 
 16-5 
 
 313-640 
 
 307-947 
 
 302-253 
 
 290-868 
 
 279-481 
 
 268-095 
 
 17- 
 
 328-004 
 
 322-050 
 
 316-096 
 
 304-189 
 
 292-281 
 
 280-373 
 
 17-5 
 
 342-581 
 
 336-362 
 
 330-144 
 
 317-707 
 
 305-270 
 
 292-833 
 
 18- 
 
 357-367 
 
 350-880 
 
 344-394 
 
 331-420 
 
 318-446 
 
 305-472 
 
 18-5 
 
 372-352 
 
 365-594 
 
 358-835 
 
 345-317 
 
 331-799 
 
 318-281 
 
 19- 
 
 387-557 
 
 380-522 
 
 373-487 
 
 359-418 
 
 345-348 
 
 331-278 
 
 19-5 
 
 402-956 
 
 395-642 
 
 388-327 
 
 373-699 
 
 359-070 
 
 344-441 
 
 20- 
 
 418-553 
 
 410-956 
 
 403-358 
 
 388-163 
 
 372-968 
 
 357-773 
 
 20-5 
 
 434-343 
 
 426-458 
 
 418-574 
 
 402-806 
 
 387-038 
 
 371-270 
 
 21- 
 
 450-334 
 
 442-159 
 
 433-985 
 
 417-636 
 
 401-288 
 
 384-939 
 
 21-5 
 
 466-513 
 
 458-045 
 
 449-577 
 
 432-641 
 
 415-704 
 
 398-768 
 
 22- 
 
 482-880 
 
 474115 
 
 465-350 
 
 447-819 
 
 430-289 
 
 412-759 
 
 22-5 
 
 499-436 
 
 490-370 
 
 481-304 
 
 463-173 
 
 445-042 
 
 426-910 
 
 23- 
 
 516-176 
 
 506-806 
 
 497-437 
 
 478-698 
 
 4 <9 959 
 
 441-219 
 
 23-5 
 
 533-098 
 
 523-421 
 
 513-745 
 
 494-391 
 
 475-038 
 
 455-685 
 
 24- 
 
 550-203 
 
 540-215 
 
 530-228 
 
 510-254 
 
 490-280 
 
 470-305 
 
 25- 
 
 584-943 
 
 574326 
 
 563-708 
 
 542472 
 
 521-237 
 
 500-001 
 
 26- 
 
 620-391 
 
 609- 1 30 
 
 597-869 
 
 575-346 
 
 552-824 
 
 530-301 
 
 27- 
 
 656-525 
 
 644608 
 
 632-691 
 
 608-857 
 
 585-023 
 
 561-188 
 
 28- 
 
 693-334 
 
 680-749 
 
 668-164 
 
 642-993 
 
 617-823 
 
 592-652 
 
 29- 
 
 730-806 
 
 717-540 
 
 704-275 
 
 677-744 
 
 651-213 
 
 624-682 
 
 30- 
 
 768-932 
 
 754974 
 
 741-017 
 
 713-102 
 
 685-187 
 
 657-272 
 
 31- 
 
 807-697 
 
 793-036 
 
 778-374 
 
 749-052 
 
 719-730 
 
 690-407 
 
 32- 
 
 847-092 
 
 831-716 
 
 816-340 
 
 785-587 
 
 754-835 
 
 724-082
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 203 
 
 TABLE VI. THE DISCHARGE OVEB WEIRS OB NOTCHES OF ONE 
 
 FOOT IN LENGTH, IN CUBIC FEET PEB MINUTE. 
 
 Depths 33 inches to 1% inches. Coefficients -606 to -518. 
 
 LESSER COEFFICIENTS. 
 
 ' 
 
 11 
 
 Coefficient 
 606. 
 
 Coefficient 
 595. 
 
 Coefficient 
 584. 
 
 Coefficient 
 562. 
 
 Coefficient 
 540. 
 
 Coefficient 
 518. 
 
 33 
 
 887-108 
 
 871-006 
 
 854-903 
 
 822-698 
 
 790-493 
 
 758-287 
 
 34 
 
 927-736 
 
 910-896 
 
 894-056 
 
 860-375 
 
 826-695 
 
 793-015 
 
 35 
 
 968-964 
 
 951-37 
 
 933-787 
 
 898-610 
 
 863-434 
 
 828-257 
 
 36 
 
 1010-786 
 
 992-439 
 
 974-091 
 
 937-396 
 
 900-701 
 
 864-005 
 
 37 
 
 1053-193 
 
 1034-076 
 
 1014-959 
 
 976724 
 
 938-489 
 
 900-254 
 
 38 
 
 1096-178 1076-281 
 
 1056-383 
 
 1016-588 
 
 976-793 
 
 936-997 
 
 39 
 
 1139-732 1119-044 
 
 1098-356 
 
 1056-979 
 
 1015-603 
 
 974226 
 
 40 
 
 1183-848, 1162-359 
 
 1140-870 
 
 1097-892 
 
 1054-914 
 
 1011936 
 
 41 
 
 1228-518; 1206-219 
 
 1183-919 
 
 1139-319 
 
 1094-719 
 
 1050-120 
 
 42 
 
 1273-737 1250-616 
 
 1227496 
 
 1181-254 
 
 1135-013 
 
 1088-772 
 
 43 
 
 1319497 
 
 1295545 
 
 1271-594 
 
 1223-691 
 
 1175-789 
 
 1127-886 
 
 44 
 
 1365-792 
 
 1341-001 
 
 1316-209 
 
 1266-626 
 
 1217-043 
 
 1167-460 
 
 45 
 
 1412-618 
 
 1386-976 
 
 1361-334 
 
 1310-051 
 
 1258-7613 
 
 1207-485 
 
 46 
 
 1459-965 
 
 1433-464 
 
 1406-963 
 
 1353-961 
 
 1300-959 
 
 1247-957 
 
 47 
 
 1507-831 
 
 1480-461 
 
 1453-091 
 
 1398-352 
 
 1343612 
 
 1288-872 
 
 48 
 
 1556-208 1527-960 
 
 1499-712 
 
 1443216 
 
 1386-720 
 
 1330224 
 
 49 
 
 1605-092 
 
 1575-956 
 
 1546-821 
 
 1488-550 
 
 1430-280 
 
 1372-009 
 
 50 
 
 1654-477 
 
 1624445 
 
 1594-413 
 
 1534-350 
 
 1474-286 
 
 1414-223 
 
 51 
 
 1704-359 
 
 1673-422 
 
 1642-485 
 
 1580-610 
 
 1518-736 
 
 1456-862 
 
 52 
 
 1754-732 
 
 1722-880 
 
 1691-029 
 
 1627-326 
 
 1563622 
 
 1499-919 
 
 53 
 
 1805-592 
 
 1772-817 
 
 1740043 
 
 1674-493 
 
 1608-944 
 
 1543394 
 
 54 
 
 1856-937 1823-231 
 
 1789-524 
 
 1722-110 
 
 1654697 
 
 1587-283 
 
 55 
 
 1908-751 1874-104 
 
 1839-457 
 
 1770-162 
 
 1700-868 
 
 1631-573 
 
 56 
 
 1961-046 
 
 1925450 
 
 1889-853 
 
 1818-660 
 
 1747-467 
 
 1676-274 
 
 57 
 
 2013-809 
 
 1977-255 
 
 1940-700 
 
 1867-592 
 
 1794-483 
 
 1721375 
 
 58 
 
 2067-033 
 
 202P-513 
 
 1991-992 
 
 1916952 
 
 1841-911 
 
 1766-870 
 
 59 
 
 2120722 
 
 2082-227 
 
 2043-733 
 
 1966-743 
 
 1889-753 
 
 1812-763 
 
 60 
 
 2174-867 
 
 2135-389 
 
 2095-911 
 
 2016-956 
 
 1938-000 
 
 1859045 
 
 61 
 
 2229-464 
 
 2188-995 
 
 2148-527 
 
 2067-589 
 
 1986-651 
 
 1905-714 
 
 62 
 
 2284-514 
 
 2243-046 
 
 2201-578 
 
 2118-642 
 
 2035-706 
 
 1952-769 
 
 63 
 
 2340-004 
 
 2297-529 
 
 2255-054 
 
 2170-103 
 
 2085-152 
 
 2000-202 
 
 64 
 
 2395-939 
 
 235-2-448 
 
 2308-957 
 
 2221-976 
 
 2134-995 
 
 2048-013 
 
 65 
 
 2452-312 
 
 2407-798 
 
 2363-284 
 
 2274-2 i7 
 
 2185-229 
 
 2096-201 
 
 66 
 
 2509-122 
 
 2463577 
 
 24 18-032 
 
 2326-941 
 
 2235-851 
 
 2144-761 
 
 67 
 
 2566-363 
 
 2519-779 
 
 2473-194 
 
 2380-026 
 
 2286-858 
 
 2193-690 
 
 68 
 
 2624-032 
 
 2576-401 
 
 2528-770 
 
 2433-508 
 
 2338-246 
 
 2242-985 
 
 69 
 
 2682-128 
 
 2633-443 
 
 2584-757 
 
 2487-386 
 
 2390-015 
 
 2292-644 
 
 70 
 
 2740-645 
 
 2690-897 
 
 2611-149 
 
 2541-654 
 
 2442-159 
 
 2342-663 
 
 71 
 
 2799-583 
 
 2748-766 
 
 2697-948 
 
 2596-313 
 
 2494-678 
 
 2393043 
 
 72 
 
 2858937 
 
 2807-042 
 
 2755 147 
 
 2651-3-38 
 
 2547-568 2443 778
 
 204 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VII. FOR FINDING THE MEAN VELOCITY FROM THE MAXI- 
 MUM VELOCITY AT THE SURFACE, IN MILL RACES, STREAMS, AND 
 RIVERS WITH UNIFORM CHANNELS; AND THE MAXIMUM VELOCITY 
 FROM THE MEAN VELOCITY. (See p. 101.) 
 
 For the velocity in feet per minute, multiply by 5. 
 
 Maximum velocity 
 at the surface in 
 inches per second. 
 
 Mean velocity in 
 large channels in 
 incnes per second. 
 
 Mean velocity in 
 smaller channels in 
 inches per second. 
 
 Maximum velocity 
 at the surface in 
 inches per second. 
 
 Mean velocity in 
 large channels in 
 incnes per second. 
 
 Mean velocity in 
 smaller channels in 
 inches per second. 
 
 Maximum velocity 
 at the surface in 
 inches per second. 
 
 Mean velocity in 
 large channels in 
 inches per second. 
 
 Mean velocity in 
 smaller channels in 
 inches per second. 
 
 1 
 
 84 
 
 75 
 
 41 
 
 34-24 
 
 33-37 
 
 81 
 
 67-64 
 
 68-86 
 
 2 
 
 1-67 
 
 1-51 
 
 42 
 
 35-07 
 
 34-23 
 
 82 
 
 68-47 
 
 69-77 
 
 3 
 
 2-51 
 
 2-27 
 
 43 
 
 35-91 
 
 35-09 
 
 83 
 
 69-31 
 
 70-68 
 
 4 
 
 3-34 
 
 3-04 
 
 44 
 
 36-74 
 
 35-95 
 
 84 
 
 70-14 
 
 71-59 
 
 5 
 
 4-18 
 
 3-81 
 
 45 
 
 37-58 
 
 36-82 
 
 85 
 
 70-98 
 
 72-50 
 
 6 
 
 5-01 
 
 4-58 
 
 46 
 
 38-41 
 
 37-69 
 
 86 
 
 71-81 
 
 73-42 
 
 7 
 
 5-85 
 
 5-36 
 
 47 
 
 39-25 
 
 38-56 
 
 87 
 
 72-65 
 
 74-33 
 
 8 
 
 6-68 
 
 6-14 
 
 48 
 
 40-08 
 
 39-43 
 
 88 
 
 73-48 
 
 75-24 
 
 9 
 
 7-52 
 
 6-92 
 
 49 
 
 40-92 
 
 40-30 
 
 89 
 
 74-32 
 
 76-16 
 
 10 
 
 8-35 
 
 7-71 
 
 50 
 
 41-75 
 
 41-17 
 
 90 
 
 75-15 
 
 77-08 
 
 11 
 
 9-19 
 
 8-50 
 
 51 
 
 42-59 
 
 42-05 
 
 91 
 
 75-99 
 
 77-99 
 
 12 
 
 10-02 
 
 9-29 
 
 52 
 
 43-42 
 
 42-92 
 
 92 
 
 76-82 
 
 78-91 
 
 13 
 
 10-86 
 
 10-09 
 
 53 
 
 44-26 
 
 43-80 
 
 93 
 
 77-66 
 
 79-83 
 
 14 
 
 11-69 
 
 10-88 
 
 54 
 
 45-09 
 
 44-68 
 
 94 
 
 78-49 
 
 80-75 
 
 15 
 
 12-53 
 
 11-69 
 
 55 
 
 45-93 
 
 45-56 
 
 95 
 
 79-33 
 
 81-67 
 
 16 
 
 13-36 
 
 12-49 
 
 56 
 
 46-76 
 
 46-44 
 
 96 
 
 80-16 
 
 82-59 
 
 17 
 
 14-20 
 
 13-30 
 
 57 
 
 47-60 
 
 47-32 
 
 97 
 
 81-00 
 
 83-51 
 
 18 
 
 15-03 
 
 14-11 
 
 58 
 
 48-43 
 
 48-21 
 
 98 
 
 81-83 
 
 84-43 
 
 19 
 
 15-87 
 
 14-92 
 
 59 
 
 4927 
 
 49-09 
 
 99 
 
 82-67 
 
 85-36 
 
 20 
 
 16-70 
 
 15-73 
 
 60 
 
 50-10 
 
 49-98 
 
 100 
 
 83-50 
 
 86-28 
 
 21 
 
 17-54 
 
 16-55 
 
 61 
 
 50-94 
 
 50-87 
 
 101 
 
 84-34 
 
 87-20 
 
 22 
 
 18-37 
 
 17-37 
 
 62 
 
 51-77 
 
 51-76 
 
 102 
 
 85-17 
 
 88-13 
 
 23 
 
 19-21 
 
 18-19 
 
 63 
 
 52-61 
 
 52-65 
 
 103 
 
 86-01 
 
 89-06 
 
 24 
 
 20-04 
 
 1902 
 
 64 
 
 53-44 
 
 53-54 
 
 104 
 
 86-84 
 
 89-98 
 
 25 
 
 20-88 
 
 19-85 
 
 65 
 
 54-28 
 
 54-43 
 
 105 
 
 87-68 
 
 90-91 
 
 26 
 
 21-71 
 
 20-68 
 
 66 
 
 55-11 
 
 55-33 
 
 106 
 
 88-51 
 
 91-84 
 
 27 
 
 22-55 
 
 21-51 
 
 67 
 
 55-95 
 
 56-H2 
 
 107 
 
 89-35 
 
 92-77 
 
 28 
 
 23-38 
 
 22-34 
 
 68 
 
 56-78 
 
 57-12 
 
 108 
 
 90-18 
 
 93-69 
 
 29 
 
 24-22 
 
 23-18 
 
 69 
 
 57-62 
 
 58-02 
 
 109 
 
 91-02 
 
 94-62 
 
 30 
 
 25-05 
 
 24-02 
 
 70 
 
 58-45 
 
 58-91 
 
 110 
 
 91-85 
 
 95-55 
 
 31 
 
 25-89 
 
 24-86 
 
 71 
 
 59-29 
 
 59-81 
 
 111 
 
 92-69 
 
 96-49 
 
 32 
 
 26-72 
 
 25-70 
 
 72 
 
 60-12 
 
 60-71 
 
 112 
 
 93-52 
 
 97-42 
 
 33 
 
 27-56 
 
 26-54 
 
 73 
 
 60-96 
 
 61-61 
 
 113 
 
 94-36 
 
 98-35 
 
 34 
 
 28-39 
 
 27-39 
 
 74 
 
 61-79 
 
 62-52 
 
 114 
 
 95-19 
 
 99-28 
 
 35 
 
 29-23 
 
 28-24 
 
 75 
 
 62-63 
 
 63-42 
 
 115 
 
 96-03 
 
 100-21 
 
 36 
 
 30-06 
 
 29-09 
 
 76 
 
 63-46 
 
 64-32 
 
 116 
 
 96-86 
 
 101-15 
 
 37 
 
 30-90 
 
 29-94 
 
 77 
 
 64-30 
 
 65-23 
 
 117 
 
 97-70 
 
 102-08 
 
 38 
 
 31-73 
 
 30-79 
 
 78 
 
 65-13 
 
 66-13 
 
 118 
 
 98-53 
 
 103-02 
 
 39 
 
 32-57 
 
 31-65 
 
 79 
 
 65-97 
 
 67-04 
 
 119 
 
 99-37 
 
 103-95 
 
 40 
 
 33-40 
 
 32-51 
 
 80 
 
 66-80 
 
 67-95 
 
 120 
 
 100-20 
 
 104-89
 
 ORIFICES, WEIRS, PIPES, AND RITERS. 
 
 205 
 
 TABLE VIII. FOB FINDING THE MEAN VELOCITIES OF WATER 
 
 FLOWING IN PIPES, DRAINS, STREAMS, AND RIVERS. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 hydraulic mean depth. 
 
 Diameters of pipes inch to % inches. Falls per mile 1 inch to 
 12 jeet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations 
 one in 
 
 T ' 6 inch. 
 
 J inch. 
 
 iinch. 
 
 | inch. 
 
 \ inch. 
 
 F. I. 
 
 
 
 
 
 
 
 1 
 
 63360 
 
 14 
 
 24 
 
 38 
 
 49 
 
 57 
 
 2 
 
 31680 
 
 22 
 
 37 
 
 59 
 
 76 
 
 90 
 
 3 
 
 21120 
 
 28 
 
 48 
 
 75 
 
 97 
 
 1-15 
 
 4 
 
 15840 
 
 34 
 
 57 
 
 89 
 
 1-15 
 
 1-36 
 
 5 
 
 12672 
 
 38 
 
 65 
 
 1-02 
 
 1-30 
 
 1-55 
 
 6 
 
 10560 
 
 42 
 
 72 
 
 1-13 
 
 1-45 
 
 1-72 
 
 7 
 
 9051 
 
 46 
 
 78 
 
 1-24 
 
 1-58 
 
 1-88 
 
 8 
 
 7920 
 
 50 
 
 85 
 
 1-33 
 
 1-71 
 
 2-02 
 
 9 
 
 7040 
 
 53 
 
 90 
 
 1-43 
 
 1-83 
 
 2-16 
 
 10 
 
 6336 
 
 57 
 
 96 
 
 1-51 
 
 1-94 
 
 2-30 
 
 11 
 
 5760 
 
 60 
 
 1-01 
 
 1-60 
 
 1-96 
 
 2-42 
 
 1 
 
 5280 
 
 63 
 
 1-06 
 
 1-68 
 
 2-15 
 
 2-54 
 
 1 3 
 
 4224 
 
 71 
 
 1-20 
 
 1-90 
 
 2-43 
 
 2-88 
 
 1 6 
 
 3520 
 
 79 
 
 33 
 
 2-10 
 
 2-69 
 
 3-19 
 
 1 9 
 
 3017 
 
 87 
 
 45 
 
 2-29 
 
 2-94 
 
 3-48 
 
 2 
 
 2640 
 
 93 
 
 56 
 
 2-47 
 
 3-16 
 
 3-75 
 
 2 3 
 
 Interpolated. 
 
 99 
 
 67 
 
 2-63 
 
 3-37 
 
 3-99 
 
 2 6 
 
 2112 
 
 1-05 
 
 1-77 
 
 2-79 
 
 3-58 
 
 4-24 
 
 2 9 
 
 Interpolated. 
 
 1-11 
 
 87 
 
 2-94 
 
 3-77 
 
 4-47 
 
 3 
 
 1760 
 
 1-16 
 
 1-96 
 
 3-09 
 
 3-96 
 
 4-69 
 
 3 3 
 
 Interpolated. 
 
 1-21 
 
 2-05 
 
 3-23 
 
 4-14 
 
 4-91 
 
 3 6 
 
 1508 
 
 1-26 
 
 2-14 
 
 3-37 
 
 4-32 
 
 5-12 
 
 3 9 
 
 Interpolated. 
 
 1-31 
 
 2-22 
 
 3-50 
 
 4-48 
 
 5-31 
 
 4 
 
 1320 
 
 1-36 
 
 2-30 
 
 3-63 
 
 4-65 
 
 5-51 
 
 4 6 
 5 
 
 Interpolated. 
 1056 
 
 1-45 
 1-54 
 
 2-45 
 2-61 
 
 3-87 
 4-11 
 
 4-96 
 5-27 
 
 5-88 
 6-24 
 
 5 6 
 
 Interpolated. 
 
 1-62 
 
 2-75 
 
 4-33 
 
 5-55 
 
 6-58 
 
 6 
 
 880 
 
 1-71 
 
 2-89 
 
 4-55 
 
 5-83 
 
 6-91 
 
 6 6 
 
 Interpolated. 
 
 1-78 
 
 3-02 
 
 4-76 
 
 6-10 
 
 7-22 
 
 7 
 
 754 
 
 1-86 
 
 3-15 
 
 4-97 
 
 6-36 
 
 7-54 
 
 7 6 
 
 Interpolated. 
 
 1-93 
 
 3-27 
 
 6-16 
 
 6-61 
 
 7-83 
 
 8 
 
 660 
 
 2-01 
 
 3-39 
 
 5-35 
 
 6-86 
 
 8-12 
 
 8 6 
 
 Interpolated. 
 
 2-07 
 
 3-51 
 
 5-53 
 
 7-09 
 
 8-40 
 
 9 
 
 587 
 
 2-14 
 
 3-62 
 
 5-72 
 
 7-32 
 
 8-68 
 
 9 6 
 
 Interpolated. 
 
 2-20 
 
 3-74 
 
 5-89 
 
 755 
 
 8-94 
 
 10 
 
 528 
 
 2-28 
 
 3-85 
 
 6-07 
 
 7-77 
 
 9-21 
 
 10 6 
 
 Interpolated. 
 
 2-33 
 
 3-95 
 
 6-24 
 
 7-99 
 
 9-47 
 
 11 
 
 480 
 
 2-40 
 
 4-06 
 
 6-40 
 
 8-20 
 
 9-72 
 
 11 6 
 
 Interpolated. 
 
 2-46 
 
 4-16 
 
 6-57 
 
 8-41 
 
 9-97 
 
 12 
 
 440 
 
 2-52 
 
 4-27 
 
 6-73 
 
 8-62 
 
 10-21
 
 206 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VIII. FOR FINDING THE MEAN VELOCITIES OF WATER 
 
 FLOWING IN PIPES, DRAINS, STREAMS, AND RIVERS. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 
 hydraulic mean depth. 
 
 Diameters of pipes ^ inch to 2 inches. Falls per mile 13 feet to 
 5280 feet. 
 
 Falls per mile in feet, and 
 the hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations 
 one in 
 
 T ' s inch. 
 
 4 inch. 
 
 inch. 
 
 | inch. 
 
 4 inch. 
 
 F. 
 
 13-2 
 
 400 
 
 2-66 
 
 4-50 
 
 7-10 
 
 9-10 
 
 10-78 
 
 13-6 
 
 Interpolated. 
 
 2-71 
 
 4-59 
 
 7-24 
 
 9-27 
 
 10-98 
 
 14-1 
 
 375 
 
 2-76 
 
 4-67 
 
 7-37 
 
 9-44 
 
 11-18 
 
 14-6 
 
 Interpolated. 
 
 2-82 
 
 4-76 
 
 7-52 
 
 9-63 
 
 11-41 
 
 15-1 
 
 350 
 
 2-87 
 
 4-85 
 
 7-66 
 
 9-82 
 
 11-63 
 
 15-6 
 
 Interpolated. 
 
 2-94 
 
 4-96 
 
 7-83 
 
 10-03 
 
 11-88 
 
 16-2 
 
 325 
 
 3-00 
 
 5-07 
 
 7-99 
 
 10-24 
 
 12-13 
 
 17-6 
 
 300 
 
 3-14 
 
 5-30 
 
 8-37 
 
 10-72 
 
 12-70 
 
 19-2 
 
 275 
 
 3-30 
 
 5-58 
 
 8-80 
 
 11-27 
 
 13-35 
 
 21-1 
 
 250 
 
 3-48 
 
 5-89 
 
 9-39 
 
 11-90 
 
 14-10 
 
 23-5 
 
 225 
 
 3-70 
 
 6-26 
 
 9-87 
 
 12-65 
 
 14-99 
 
 26-4 
 
 200 
 
 3-96 
 
 6-70 
 
 10-57 
 
 13-54 
 
 16-04 
 
 30-2 
 
 175 
 
 4-28 
 
 7-24 
 
 11-42 
 
 14-63 
 
 17-33 
 
 35-2 
 
 150 
 
 4-68 
 
 7-92 
 
 12-49 
 
 16-00 
 
 18-96 
 
 377 
 
 140 
 
 4-88 
 
 8-24 
 
 13-00 
 
 16-66 
 
 19-74 
 
 42-2 
 
 125 
 
 5-21 
 
 8-81 
 
 13-90 
 
 17-80 
 
 21-09 
 
 48- 
 
 110 
 
 5-62 
 
 9-50 
 
 14-98 
 
 19-19 
 
 22-74 
 
 52-8 
 
 100 
 
 5-94 
 
 10-05 
 
 15-85 
 
 20-30 
 
 24-06 
 
 58-7 
 
 90 
 
 6-33 
 
 10-69 
 
 16-87 
 
 21-61 
 
 25-60 
 
 66- 
 
 80 
 
 6-78 
 
 11-47 
 
 18-10 
 
 23-17 
 
 27-46 
 
 75-4 
 
 70 
 
 7-35 
 
 12-42 
 
 19-59 
 
 25-09 
 
 29-73 
 
 88- 
 
 60 
 
 8-05 
 
 13-61 
 
 21-48 
 
 27-51 
 
 32-60 
 
 105-6 
 
 50 
 
 8-99 
 
 15-19 
 
 23-96 
 
 30-69 
 
 36-37 
 
 117-3 
 
 45 
 
 9-57 
 
 16-18 
 
 25-53 
 
 32-70 
 
 38-75 
 
 132-0 
 
 40 
 
 10-28 
 
 17-37 
 
 27-41 
 
 35-11 
 
 41-60 
 
 150-8 
 
 35 
 
 11-14 
 
 18-84 
 
 29-71 
 
 38-06 
 
 45-10 
 
 176- 
 
 30 
 
 12-23 
 
 20'68 
 
 32-62 
 
 41-78 
 
 49-51 
 
 212-2 
 
 25 
 
 13-66 
 
 23-09 
 
 36-43 
 
 46-67 
 
 55-30 
 
 264- 
 
 20 
 
 15-64 
 
 26-44 
 
 41-71 
 
 53-43 
 
 63-30 
 
 352- 
 
 15 
 
 18-61 
 
 31-46 
 
 49-63 
 
 63-57 
 
 75-33 
 
 528- 
 
 10 
 
 23-73 
 
 40-11 
 
 63-28 
 
 81-06 
 
 96-05 
 
 586-7 
 
 9 
 
 25-26 
 
 42-70 
 
 67-37 
 
 86-29 
 
 102-25 
 
 660- 
 
 8 
 
 27-08 
 
 45-78 
 
 72-22 
 
 92-51 
 
 109-61 
 
 754-3 
 
 7 
 
 29-29 
 
 49-51 
 
 78-10 
 
 100-04 
 
 118-54 
 
 880-0 
 
 6 
 
 32-05 
 
 54-15 
 
 85-43 
 
 109-43 
 
 129-66 
 
 1056- 
 
 5 
 
 35-08 
 
 60-15 
 
 94-89 
 
 121-54 
 
 144-02 
 
 mo- 
 
 4 
 
 40-40 
 
 68-29 
 
 107-73 
 
 137-99 
 
 163-51 
 
 mo- 
 
 3 
 
 47-48 
 
 80-25 
 
 126-61 
 
 162-17 
 
 192-16 
 
 2640- 
 
 2 
 
 59-47 
 
 100-53 
 
 158-59 
 
 203-14 
 
 240-70 
 
 5280- , 
 
 1 
 
 88-13 
 
 148-97 
 
 235-02 
 
 301-04 
 
 356-70
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 207 
 
 TABLE VIII. FOB FINDING THE MEAN VELOCITIES OF WATER 
 
 FLOW1NO IN PIPES, DRAINS, STREAMS, AND RIVERS. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 
 hydraulic mean depth. 
 
 Diameters of pipes 2^ inches to 5 inches. Falls per mile I inch to 
 12 feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations 
 one in 
 
 | inch. 
 
 | inch. 
 
 J inch. 
 
 1 inch. 
 
 1J in. in- 
 terpolated 
 
 F. I. 
 
 
 
 
 
 
 
 1 
 
 63360 
 
 65 
 
 73 
 
 79 
 
 85 
 
 96 
 
 2 
 
 31680 
 
 1-02 
 
 1-13 
 
 1-23 
 
 1-33 
 
 1-49 
 
 3 
 
 21120 
 
 1-30 
 
 1-45 
 
 1-58 
 
 1-70 
 
 1-91 
 
 4 
 
 15840 
 
 1-54 
 
 1-71 
 
 1-87 
 
 2-01 
 
 2-26 
 
 5 
 
 12672 
 
 1-76 
 
 1-95 
 
 2-13 
 
 2-29 
 
 2-58 
 
 6 
 
 10560 
 
 1-95 
 
 2-17 
 
 2-36 
 
 2-55 
 
 2-86 
 
 7 
 
 9051 
 
 2-13 
 
 2-37 
 
 2-58 
 
 2-78 
 
 3-13 
 
 8 
 
 7920 
 
 2-30 
 
 2-55 
 
 2-78 
 
 3-00 
 
 3-37 
 
 9 
 
 7040 
 
 2-46 
 
 2-73 
 
 2-98 
 
 3-21 
 
 3-61 
 
 10 
 
 6336 
 
 2-61 
 
 2-90 
 
 3-16 
 
 3-40 
 
 3-83 
 
 11 
 
 5760 
 
 2-76 
 
 3-06 
 
 3-33 
 
 3-59 
 
 4-04 
 
 1 
 
 5280 
 
 2-89 
 
 3-21 
 
 3-50 
 
 3-77 
 
 4-24 
 
 1 3 
 
 4224 
 
 3-28 
 
 3-64 
 
 3-97 
 
 4-27 
 
 4-81 
 
 1 6 
 
 3520 
 
 3-63 
 
 4-03 
 
 4-39 
 
 4-73 
 
 5-32 
 
 1 9 
 
 3017 
 
 3-96 
 
 4-39 
 
 4-79 
 
 5-16 
 
 5-80 
 
 2 
 
 2640 
 
 4-26 
 
 4-73 
 
 5-16 
 
 5-55 
 
 6-25 
 
 2 3 
 
 Interpolated. 
 
 4-55 
 
 5-04 
 
 5-50 
 
 5-92 
 
 6-66 
 
 2 6 
 
 2112 
 
 4-83 
 
 5-35 
 
 5-84 
 
 6-29 
 
 7-07 
 
 2 9 
 
 3 
 
 Interpolated. 
 1760 
 
 5-09 
 5-34 
 
 5-64 
 5-92 
 
 6-15 
 6-46 
 
 6-12 
 6-96 
 
 7-46 
 7-83 
 
 3 3 
 
 Interpolated. 
 
 5-58 
 
 6-19 
 
 6-75 
 
 7-27 
 
 8-18 
 
 3 6 
 
 1508 
 
 5-82 
 
 6-46 
 
 7-04 
 
 7-59 
 
 8-53 
 
 3 9 
 
 Interpolated. 
 
 6-05 
 
 6-71 
 
 7-31 
 
 7-88 
 
 8-86 
 
 4 
 
 1320 
 
 6-27 
 
 6-95 
 
 7-58 
 
 8-17 
 
 9-19 
 
 4 6 
 
 Interpolated. 
 
 6-69 
 
 7-42 
 
 8-09 
 
 8-71 
 
 9-80 
 
 5 
 
 1056 
 
 7-10 
 
 7-88 
 
 8-59 
 
 9-25 
 
 10-41 
 
 5 6 
 
 Interpolated. 
 
 7-48 
 
 8-30 
 
 9-05 
 
 9-76 
 
 10-97 
 
 6 
 
 880 
 
 7-86 
 
 8-72 
 
 9-51 
 
 10-25 
 
 11-53 
 
 6 6 
 
 Interpolated. 
 
 8-22 
 
 9-12 
 
 9-94 
 
 10-71 
 
 12-05 
 
 7 
 
 754 
 
 8-57 
 
 9-51 
 
 10-37 
 
 11-17 
 
 12-57 
 
 7 6 
 
 Interpolated. 
 
 8-92 
 
 9-89 
 
 10-78 
 
 11-62 
 
 13-06 
 
 8 
 
 660 
 
 9-24 
 
 10-25 
 
 11-18 
 
 12-04 
 
 13-54 
 
 8 6 
 
 Interpolated. 
 
 9-55 
 
 10-60 
 
 11-56 
 
 12-45 
 
 14-01 
 
 9 
 
 587 
 
 9-87 
 
 10-95 
 
 11-94 
 
 12-86 
 
 14-47 
 
 9 6 
 
 Interpolated. 
 
 10-18 
 
 11-28 
 
 12-31 
 
 13-26 
 
 14-91 
 
 10 
 
 528 
 
 10-48 
 
 11-62 
 
 12-67 
 
 13-65 
 
 15-36 
 
 10 6 
 
 Interpolated. 
 
 10-77 
 
 11-95 
 
 13-03 
 
 14-03 
 
 15-78 
 
 11 
 
 480 
 
 11-06 
 
 12-27 
 
 13-38 
 
 14-41 
 
 16-21 
 
 11 6 
 
 Interpolated. 
 
 11-34 
 
 12-58 
 
 13-72 
 
 14-82 
 
 16-64 
 
 12 
 
 440 
 
 11-62 
 
 12-89 
 
 14-06 
 
 15-22 
 
 17-07
 
 208 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VIII. FOB FINDING THE MEAN VELOCITIES OF WATER 
 
 FLOWING IN PIPES, DRAINS, STREAMS, AND RIVERS. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 
 hydraulic mean depth. 
 
 Diameters of pipes 2 inches to 5 inches. Falls per mile 13 feet to 
 5280 feet. 
 
 Falls per mile in feet, and 
 the hydraulic inclinations. 
 
 " Hydraulic mean depths," or "mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations 
 one in 
 
 I inch. 
 
 J inch. 
 
 inch. 
 
 1 inch. 
 
 1J in. in- 
 
 erpolated 
 
 F. 
 
 
 
 
 
 
 
 13-2 
 
 400 
 
 12-26 
 
 13-60 
 
 14-83 
 
 15-98 
 
 17-98 ' 
 
 13-6 
 
 Interpolated. 
 
 12-49 
 
 13-86 
 
 15-11 
 
 16-28 
 
 18-31 i 
 
 14-1 
 
 375 
 
 12-72 
 
 14-11 
 
 15-39 
 
 16-58 
 
 18-65 
 
 14-6 
 
 Interpolated. 
 
 12-98 
 
 14-39 
 
 15-70 
 
 16-91 
 
 19-02 
 
 15-1 
 
 350 
 
 13-23 
 
 14-68 
 
 16-00 
 
 17-24 
 
 19-40 
 
 15-6 
 
 Interpolated. 
 
 13-52 
 
 14-99 
 
 16-35 
 
 17-62 
 
 19-81 
 
 16-2 
 
 325 
 
 13-80 
 
 15-31 
 
 16-79 
 
 17-99 
 
 20-23 
 
 17-6 
 
 300 
 
 14-45 
 
 16-02 
 
 17-48 
 
 18-83 
 
 21-18 
 
 19-2 
 
 275 
 
 15-19 
 
 16-85 
 
 18-37 
 
 19-79 
 
 22-26 i 
 
 21 -1 
 
 250 
 
 16-04 
 
 17-80 
 
 19-40 
 
 20-91 
 
 23-52 
 
 23-5 
 
 225 
 
 17-05 
 
 18-91 
 
 20-62 
 
 22-21 
 
 24-99 
 
 26-4 
 
 200 
 
 18-25 
 
 20-24 
 
 22-07 
 
 23-78 
 
 26-75 
 
 30-2 
 
 175 
 
 19-71 
 
 21-87 
 
 23-85 
 
 25-69 
 
 28-90 
 
 35-2 
 
 150 
 
 21-57 
 
 23-92 
 
 26-09 
 
 28-11 
 
 31-62 
 
 37-7 
 
 140 
 
 22-45 
 
 24-91 
 
 27-16 
 
 29-26 
 
 32-92 
 
 42-2 
 
 125 
 
 23-99 
 
 26-62 
 
 29-03 
 
 31-27 
 
 35-18 
 
 48- 
 
 110 
 
 25-87 
 
 28-69 
 
 31-29 
 
 33-71 
 
 37-92 
 
 52-8 
 
 100 
 
 27-36 
 
 30-35 
 
 33-10 
 
 35-66 
 
 40-11 
 
 58-7 
 
 90 
 
 29-12 
 
 32-31 
 
 35-23 
 
 37-96 
 
 42-69 
 
 66- 
 
 80 
 
 31-23 
 
 34-64 
 
 37-78 
 
 40-70 
 
 4579 
 
 75-4 
 
 70 
 
 33-82 
 
 37-51 
 
 40-91 
 
 44-07 
 
 49-58 
 
 88-0 
 
 60 
 
 37-08 
 
 41-13 
 
 44-86 
 
 48-33 
 
 54-36 
 
 105-6 
 
 50 
 
 41-37 
 
 45-78 
 
 50-04 
 
 53-91 
 
 60-65 
 
 117-3 
 
 45 
 
 44-08 
 
 48-89 
 
 53-32 
 
 57-44 
 
 64-62 
 
 132- 
 
 40 
 
 47-32 
 
 52-49 
 
 57-25 
 
 61-67 
 
 69-37 
 
 150-8 
 
 35 
 
 51-30 
 
 56-90 
 
 62-06 
 
 66-86 
 
 75-20 
 
 176- 
 
 30 
 
 56-32 
 
 62-47 
 
 68-13 
 
 73-40 
 
 82-56 
 
 211-2 
 
 25 
 
 62-90 
 
 69-77 
 
 76-09 
 
 81-97 
 
 92-21 
 
 264- 
 
 20 
 
 72-01 
 
 79-87 
 
 87-11 
 
 93-84 
 
 105-56 
 
 352- 
 
 15 
 
 85-68 
 
 95-05 
 
 103-66 
 
 111-67 
 
 125-61 
 
 528- 
 
 10 
 
 109-26 
 
 121-19 
 
 132-17 
 
 142-39 
 
 160-17 
 
 586-7 
 
 9 
 
 116-31 
 
 129-01 
 
 140-70 
 
 151-58 
 
 170-50 
 
 660- 
 
 8 
 
 , 124-68 
 
 138-30 
 
 150-83 
 
 162-49 
 
 182-78 
 
 754-3 
 
 7 
 
 134-84 
 
 149-57 
 
 163-12 
 
 175-73 
 
 197-67 
 
 880- 
 
 6 
 
 147-69 
 
 163-60 
 
 178-42 
 
 192-22 
 
 216-22 
 
 1056- 
 
 5 
 
 163-82 
 
 181-71 
 
 198-17 
 
 213-50 
 
 240-15 
 
 1320- 
 
 4 
 
 185-99 
 
 206-31 
 
 225-00 
 
 242-39 
 
 272-66 
 
 1760- 
 
 3 
 
 218-58 
 
 242-46 
 
 264-42 
 
 284-86 
 
 320-43 
 
 2640- 
 
 2 
 
 1 273-79 
 
 303-70 
 
 331-22 
 
 356-82 
 
 401-37 
 
 5280- 
 
 1 
 
 405-74 
 
 450-07 
 
 490-84 
 
 528-79 
 
 594-82
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 209 
 
 TABLE VIII. FOB FINDING THE MEAN VELOCITIES OF WATER 
 
 FLOWING IN PIPES, DRAINS, STREAMS, AND RIVERS. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 hydraulic mean depth. 
 
 OPEN DRAINS AND PIPES. 
 
 Diameters of pipes 6 inches to 12 inches. Falls per mile 1 inch 
 to 1 2 feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 "Hydraulic mean depths," or " mean radii," and 
 velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 1 
 1 inch. 
 
 If in. in- 
 
 erpolated 
 
 2 inches. 
 
 4 inches. 
 
 3 inches. 
 
 p. i. 
 
 
 
 
 
 
 
 1 
 
 63360 
 
 1-07 
 
 1-15 
 
 1-24 
 
 1-40 
 
 1-55 
 
 2 
 
 31680 
 
 1-66 
 
 1-80 
 
 1-94 
 
 2-19 
 
 2-41 
 
 3 
 
 21120 
 
 2-12 
 
 2-30 
 
 2-48 
 
 2-80 
 
 3-08 
 
 4 
 
 15840 
 
 2-52 
 
 2-73 
 
 2-94 
 
 3-34 
 
 3-65 
 
 5 
 
 12672 
 
 2-86 
 
 3-11 
 
 3-35 
 
 3-77 
 
 4-16 
 
 6 
 
 10560 
 
 3-18 
 
 3-45 
 
 3-72 
 
 4-19 
 
 4-62 ! 
 
 7 
 
 9051 
 
 3-47 
 
 3-77 
 
 4-06 
 
 4-58 
 
 5-04 
 
 8 
 
 7920 
 
 3-75 
 
 4-06 
 
 4-38 
 
 4-94 
 
 5-44 
 
 9 
 
 7040 
 
 4-01 
 
 4-34 
 
 4-68 
 
 5-28 
 
 5-81 
 
 10 
 
 6336 
 
 4-25 
 
 4-61 
 
 4-97 
 
 5-60 
 
 6-17 
 
 11 
 
 5760 
 
 4-49 
 
 4-86 
 
 5-24 
 
 5-91 
 
 6-51 
 
 1 
 
 5280 
 
 4-71 
 
 5-11 
 
 5-51 
 
 6-21 
 
 6-84 
 
 1 3 
 
 4224 
 
 5-34 
 
 5-79 
 
 6-24 
 
 7-03 
 
 7-75 
 
 1 6 
 
 3520 
 
 5-91 
 
 6-41 
 
 6-91 
 
 7-79 
 
 8-58 
 
 1 9 
 
 3017 
 
 6-44 
 
 6-99 
 
 7-53 
 
 8-49 
 
 9-35 
 
 2 
 
 2640 
 
 6-94 
 
 7-53 
 
 8-11 
 
 9-14 
 
 10-07 
 
 2 3 
 
 Interpolated. 
 
 7-40 
 
 8-03 
 
 8-65 
 
 9-74 
 
 10-74 
 
 2 6 
 
 2112 
 
 7-86 
 
 8-52 
 
 9-18 
 
 10-35 
 
 11-40 
 
 2 9 
 
 Interpolated. 
 
 8-28 
 
 8-98 
 
 9-67 
 
 10-90 
 
 12-01 
 
 3 
 
 1760 
 
 8-70 
 
 9-43 
 
 10-16 
 
 11-45 
 
 12-62 i 
 
 3 3 
 
 Interpolated. 
 
 9-09 
 
 9-85 
 
 10-62 
 
 11-97 
 
 13-19 
 
 3 6 
 
 1508 
 
 9-48 
 
 10-28 
 
 11-08 
 
 12-48 
 
 13-76 
 
 3 9 
 
 Interpolated. 
 
 9-84 
 
 10-67 
 
 11-50 
 
 12-96 
 
 14-29 
 
 4 
 
 1320 
 
 10-21 
 
 11-07 
 
 11-93 
 
 13-44 
 
 14-81 
 
 4 6 
 
 Interpolated. 
 
 10-89 
 
 11-80 
 
 12-72 
 
 14-34 
 
 15-80 
 
 5 
 
 1056 
 
 11-56 
 
 12-54 
 
 13-51 
 
 15-23 
 
 16-78 
 
 5 6 
 
 Interpolated. 
 
 12-18 
 
 13-21 
 
 14-24 
 
 16-04 
 
 17-68 ! 
 
 6 
 
 880 
 
 12-80 
 
 13-88 
 
 14-96 
 
 16-86 
 
 18-58 ! 
 
 6 6 
 
 Interpolated. 
 
 13-38 
 
 14-51 
 
 15-64 
 
 17-62 
 
 19-42 
 
 7 
 
 754 
 
 13-96 
 
 15-14 
 
 16-32 
 
 18-&9 
 
 20-26 
 
 7 6 
 
 Interpolated. 
 
 14-51 
 
 15-73 
 
 16-95 
 
 19-10 
 
 21-05 
 
 8 
 
 660 
 
 15-05 
 
 16-32 
 
 17-58 
 
 19-82 
 
 21-84 : 
 
 8 6 
 
 Interpolated. 
 
 15-56 
 
 1687 
 
 18-18 
 
 20-49 
 
 22-58 
 
 9 
 
 587 
 
 16-07 
 
 17-43 
 
 18-78 
 
 21-17 
 
 23-32 
 
 9 6 
 
 Interpolated. 
 
 16-57 
 
 17-97 
 
 19-36 
 
 21-82 
 
 24-04 
 
 10 
 
 528 
 
 17-06 
 
 18-50 
 
 19-94 
 
 22-47 
 
 24-76 
 
 10 6 
 
 Interpolated. 
 
 17-54 
 
 19-01 
 
 20-49 
 
 23-09 
 
 25-45 
 
 11 
 
 . 480 
 
 18-01 
 
 19-53 
 
 21-04 
 
 2372 
 
 26-13 
 
 11 6 
 
 Interpolated. 
 
 18-47 
 
 20-02 
 
 21-57 
 
 24-32 
 
 26-79 
 
 12 
 
 440 
 
 18-92 
 
 20-51 
 
 22-11 
 
 24-91 
 
 27-45
 
 210 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VIII. FOR FINDING THE MEAN VELOCITIES OF WATER 
 
 FLOWING IN PIPES, DRAINS, STREAMS, AND R1VKRS. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 hydraulic mean depth. 
 
 PIPES. 
 Diameters of piyes 6 inches to 14 inches. Falls per mile 13 feet to 
 
 Falls per mile in feet, 
 and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or "mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 1J inch. 
 
 2 inches. 
 
 2 inches. 
 
 3 inches. 
 
 3^ inches. 
 
 F. 
 
 
 
 
 
 
 
 13-2 
 
 400 
 
 19-97 
 
 23-34 
 
 26-30 
 
 28-98 
 
 31-44 
 
 13-6 
 
 
 20-34 
 
 23-77 
 
 26-79 
 
 29-52 
 
 32-H3 
 
 14-1 
 
 375 
 
 20-72 
 
 24-21 
 
 27-28 
 
 30-06 
 
 32-62 
 
 14-6 
 
 
 21-13 
 
 24-69 
 
 27-83 
 
 30-67 
 
 33-27 
 
 15-1 
 
 350 
 
 21-55 
 
 25-18 
 
 28-38 
 
 31-27 
 
 33-93 
 
 15-6 
 
 
 22-01 
 
 25-72 
 
 28-99 
 
 31-94 
 
 34-66 
 
 16-2 
 
 325 
 
 22-48 
 
 26-27 
 
 29-60 
 
 32-62 
 
 35-39 
 
 17-6 
 
 300 
 
 23-53 
 
 27-50' 
 
 30-99 
 
 34-15 
 
 37-05 
 
 19-2 
 
 275 
 
 24-74 
 
 28-90 
 
 32-57 
 
 35-89 
 
 38-94 
 
 21-1 
 
 250 
 
 26-13 
 
 30-53 
 
 34-41 
 
 37-91 
 
 41-14 
 
 23-5 
 
 225 
 
 27-76 
 
 32-44 
 
 36-56 
 
 40-28 
 
 43-71 
 
 26-4 
 
 200 
 
 29-72 
 
 34-72 
 
 39-13 
 
 43-12 
 
 46-79 
 
 30-2 
 
 175 
 
 32-11 
 
 37-52 
 
 42-28 
 
 46-59 
 
 50-55 
 
 35-2 
 
 150 
 
 35-13 
 
 41-04 
 
 46-26 
 
 50-97 
 
 55-30 
 
 377 
 
 140 
 
 36-57 
 
 42-73 
 
 48-16 
 
 53-07 
 
 57-58 
 
 42-2 
 
 125 
 
 39-08 
 
 45-66 
 
 51-46 
 
 56-71 
 
 61-53 
 
 48- 
 
 110 
 
 42-13 
 
 49-23 
 
 55-48 
 
 61-13 
 
 66-33 
 
 52-8 
 
 100 
 
 44-57 
 
 52-07 
 
 58-69 
 
 64-67 
 
 70-17 
 
 58-7 
 
 90 
 
 47-43 
 
 55-42 
 
 62-46 
 
 68-83 
 
 74-68 
 
 66- 
 
 80 
 
 50-87 
 
 59-44 
 
 66-99 
 
 73-81 
 
 80-09 
 
 75-4 
 
 70 
 
 55-08 
 
 64-36 
 
 72-50 
 
 79-92 
 
 86-72 
 
 88- 
 
 60 
 
 60-39 
 
 70-57 
 
 79-53 
 
 87-63 
 
 95-09 
 
 105-6 
 
 50 
 
 67-38 
 
 78-73 
 
 88-73 
 
 97-77 
 
 106-08 
 
 117-3 
 
 45 
 
 71-79 
 
 83-88 
 
 94-54 
 
 104-17 
 
 113-03 
 
 132- 
 
 40 
 
 77-07 
 
 90-06 
 
 101-50 
 
 118-84 
 
 121-35 
 
 150-8 
 
 35 
 
 83-55 
 
 97-63 
 
 110-03 
 
 121-24 
 
 131-55 
 
 176- 
 
 30 
 
 91-72 
 
 107-18 
 
 120-79 
 
 133-10 
 
 144-41 
 
 211-2 
 
 25 
 
 102-44 
 
 119-70 
 
 134-90 
 
 148-65 
 
 161-29 
 
 264- 
 
 20 
 
 117-28 
 
 137-03 
 
 154-44 
 
 170-18 
 
 184-65 
 
 352- 
 
 15 
 
 139-56 
 
 163-06 
 
 183-78 
 
 202-50 
 
 219-72 
 
 528- 
 
 10 
 
 177-95 
 
 207-92 
 
 234-33 
 
 258-21 
 
 280-16 
 
 586-7 
 
 9 
 
 189-43 
 
 221-34 
 
 249-45 
 
 274-87 
 
 298-24 
 
 660- 
 
 8 
 
 203-07 
 
 237-28 
 
 267-42 
 
 294-67 
 
 319-72 
 
 754-3 
 
 7 
 
 219-61 
 
 256-61 
 
 289-20 
 
 318-67 
 
 345-77 
 
 880- 
 
 6 
 
 240-22 
 
 281-36 
 
 316-33 
 
 348-57 
 
 378-20 
 
 1056- 
 
 5 
 
 266-81 
 
 311-75 
 
 351-35 
 
 387-15 
 
 420-07 
 
 1320- 
 
 4 
 
 302-92 
 
 353-95 
 
 398-91 
 
 439-55 
 
 476-93 
 
 1760- 
 
 3 
 
 356-00 
 
 415-96 
 
 468-80 
 
 516-57 
 
 560-49 
 
 2640- 
 
 2 
 
 445-93 
 
 521-04 
 
 687-22 
 
 647-06 
 
 702-08 
 
 5280- 
 
 1 
 
 660-84 
 
 772-16 
 
 870-23 
 
 958-91 
 
 1040-44
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 211 
 
 TABLE VIII. FOB FINDING THE MEAN VELOCITIES OF WATER 
 
 FLOWING IN PIPES, DRAINS, STREAMS, AND RIVERS. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 
 hydraulic mean depth. 
 
 Diameters of pipes 14 inches to 22 inches. Falls per mile 1 inch 
 to IZfeet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 3i inches. 
 
 4 inches. 
 
 4 inches. 
 
 5 inches. 
 
 5 inches. 
 
 p. I. 
 
 
 
 
 
 
 
 1 
 
 63360 
 
 1-68 
 
 1-80 
 
 1-91 
 
 2-02 
 
 2-13 
 
 2 
 
 31680 
 
 2-61 
 
 2-81 
 
 2-98 
 
 3-15 
 
 3-32 
 
 3 
 
 21120 
 
 3-34 
 
 3-59 
 
 3-82 
 
 4-03 
 
 4-24 
 
 4 
 
 15840 
 
 3-96 
 
 4-25 
 
 4-52 
 
 4-78 
 
 5-02 
 
 5 
 
 12672 
 
 4-51 
 
 4-84 
 
 5-15 
 
 5-44 
 
 5-72 
 
 6 
 
 10560 
 
 5-01 
 
 5-37 
 
 5-72 
 
 6-04 
 
 6-35 
 
 7 
 
 9051 
 
 5-47 
 
 5-87 
 
 6-24 
 
 6-60 
 
 6-94 
 
 8 
 
 7920 
 
 5-90 
 
 6-33 
 
 6-74 
 
 7-12 
 
 7-48 
 
 9 
 
 7040 
 
 6-31 
 
 6-77 
 
 7-20 
 
 7-61 
 
 8-00 
 
 10 
 
 6336 
 
 6-70 
 
 7-18 
 
 7-64 
 
 8-08 
 
 8-49 
 
 11 
 
 5760 
 
 7'06 
 
 7-58 
 
 8-06 
 
 8-52 
 
 8-96 
 
 1 
 
 5280 
 
 7-42 
 
 7-96 
 
 8-47 
 
 8-95 
 
 9-41 
 
 1 3 
 
 4224 
 
 8-41 
 
 9-02 
 
 9-60 
 
 10-14 
 
 10-66 
 
 1 6 
 
 3520 
 
 9-31 
 
 9-99 
 
 10-63 
 
 11-23 
 
 11-80 
 
 1 9 
 
 3017 
 
 10-15 
 
 10-89 
 
 11-58 
 
 12-24 
 
 12-86 
 
 2 
 
 2640 
 
 10-93 
 
 11-73 
 
 12-47 
 
 13-18 
 
 13-86 
 
 2 3 
 
 Interpolated. 
 
 11-65 
 
 12-50 
 
 13-30 
 
 14-05 
 
 14-77 
 
 2 6 
 
 2112 
 
 12-37 
 
 13-28 
 
 14-12 
 
 14-93 
 
 15-69 
 
 2 9 
 
 Interpolated. 
 
 13-03 
 
 13-68 
 
 14-88 
 
 15-72 
 
 16-53 
 
 3 
 
 1760 
 
 13-69 
 
 14-69 
 
 15-63 
 
 16-52 
 
 17-36 
 
 3 3 
 
 Interpolated. 
 
 14-31 
 
 15-35 
 
 16-33 
 
 17-26 
 
 18-14 
 
 3 6 
 
 1508 
 
 14-92 
 
 16-01 
 
 17-03 
 
 18-00 
 
 18-92 
 
 3 9 
 
 Interpolated. 
 
 15-50 
 
 16-63 
 
 17-69 
 
 18-70 
 
 19-65 
 
 4 
 
 1320 
 
 16-07 
 
 17-25 
 
 18-35 
 
 19-39 
 
 20-38 
 
 4 6 
 
 Interpolated. 
 
 17-14 
 
 18-39 
 
 19-56 
 
 20-68 
 
 21-73 
 
 5 
 
 1056 
 
 18-21 
 
 19-53 
 
 20-78 
 
 21-96 
 
 23-08 
 
 5 6 
 
 Interpolated. 
 
 19-18 
 
 20-58 
 
 21-90 
 
 23-14 
 
 24-32 
 
 6 
 
 880 
 
 20-16 
 
 21-63 
 
 23-01 
 
 24-32 
 
 25-56 
 
 6 6 
 
 Interpolated. 
 
 21-07 
 
 22-61 
 
 24-05 
 
 25-42 
 
 26-72 
 
 7 
 
 754 
 
 21-98 
 
 23-59 
 
 25-09 
 
 26-52 
 
 27-87 
 
 7 6 
 
 Interpolated. 
 
 22-84 
 
 24-50 
 
 26-07 
 
 27-55 
 
 28-96 
 
 8 
 
 660 
 
 23-69 
 
 25-42 
 
 27-04 
 
 28-58 
 
 30-04 
 
 8 6 
 
 Interpolated. 
 
 24-50 
 
 26-29 
 
 27-97 
 
 29-55 
 
 31-06 
 
 9 
 
 587 
 
 25-31 
 
 27-54 
 
 28-89 
 
 30-53 
 
 32-09 
 
 9 6 
 
 Interpolated. 
 
 26-09 
 
 27-99 
 
 29-78 
 
 31-47 
 
 33-08 
 
 10 
 
 528 
 
 26-87 
 
 28-83 
 
 30-67 
 
 32-41 
 
 34-06 
 
 10 6 
 
 Interpolated. 
 
 27-61 
 
 29-62 
 
 31-52 
 
 33-31 
 
 35-01 
 
 11 
 
 480 
 
 28-35 
 
 30-42 
 
 32-37 
 
 34-20 
 
 35-95 
 
 11 6 
 
 Interpolated. 
 
 29-07 
 
 31-19 
 
 33-18 
 
 35-07 
 
 36-86 
 
 12 
 
 440 
 
 29-79 
 
 31-96 
 
 34-00 
 
 35-93 
 
 37-77 
 
 p 2
 
 212 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VIII. FOB FINDING THE MEAN VELOCITIES OF WATER 
 
 FLOWING IN PIPES, DRAINS, STREAMS, AND RIVERS. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 
 hydraulic mean depth. 
 Diameters of pipes 16 inches to 2 feet. Falls per mile 13 feet to 
 
 Falls per mile in feet, 
 and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 4 inches. 
 
 4^ inches. 
 
 5 inches. 
 
 5i inches. 
 
 6 inches. 
 
 F. 
 
 
 
 
 
 
 
 13-2 
 
 400 
 
 33-74 
 
 35-89 
 
 37-93 
 
 39-87 
 
 41-72 
 
 13-6 
 
 Interpolated. 
 
 34-37 
 
 36-56 
 
 38-64 
 
 40-61 
 
 42-50 
 
 14-1 
 
 375 
 
 35-00 
 
 37-23 
 
 39-35 
 
 41-36 
 
 43-28 
 
 14-6 
 
 Interpolated. 
 
 35-70 
 
 37-98 
 
 40-14 
 
 42-19 
 
 44-15 
 
 15-1 
 
 350 
 
 36-40 
 
 38-73 
 
 40-92 
 
 43-02 
 
 45-01 
 
 15-6 
 
 Interpolated. 
 
 37-19 
 
 39-56 
 
 41-81 
 
 43-94 
 
 45-99 
 
 16-2 
 
 325 
 
 37-97 
 
 40-40 
 
 42-69 
 
 44-87 
 
 46-96 
 
 17-6 
 
 800 
 
 39-75 
 
 42-29 
 
 44-69 
 
 46-97 
 
 49-16 
 
 19-2 
 
 275 
 
 41-78 
 
 44-45 
 
 46-97 
 
 49-38 
 
 51-67 
 
 811 
 
 250 
 
 44-14 
 
 46-95 
 
 49-62 
 
 52-16 
 
 54-58 
 
 23-5 
 
 225 
 
 46-90 
 
 49-90 
 
 5272 
 
 55-42 
 
 58-00 
 
 26-4 
 
 200 
 
 50-20 
 
 53-41 
 
 56-44 
 
 59-32 
 
 62-08 
 
 30-2 
 
 175 
 
 54-24 
 
 57-71 
 
 60-98 
 
 64-10 
 
 67-07 
 
 35-2 
 
 150 
 
 59-34 
 
 63-13 
 
 66-71 
 
 70-12 
 
 73-37 
 
 377 
 
 140 
 
 61-78 
 
 65-72 
 
 69-45 
 
 73-00 
 
 76-39 
 
 42'2 
 
 125 
 
 66-02 
 
 70-23 
 
 74-22 
 
 78-01 
 
 81-64 
 
 48- 
 
 110 
 
 71-17 
 
 75-72 
 
 80-01 
 
 84-10 
 
 88-00 
 
 52-8 
 
 100 
 
 75-29 
 
 80-09 
 
 84-64 
 
 88-97 
 
 93-10 
 
 587 
 
 90 
 
 80-13 
 
 85-25 
 
 90-08 
 
 94-69 
 
 99-09 
 
 66- 
 
 80 
 
 85-93 
 
 91-42 
 
 96-61 
 
 101-54 
 
 106-26 
 
 75-4 
 
 70 
 
 93-04 
 
 98-98 
 
 104-60 
 
 109-95 
 
 115-05 
 
 88- 
 
 60 
 
 102-02 
 
 ] 08-54 
 
 114-70 
 
 120-56 
 
 126-16 
 
 105-6 
 
 50 
 
 113-82 
 
 121-09 
 
 127-96 
 
 134-50 
 
 14074 
 
 117-3 
 
 45 
 
 121-2T 
 
 129-01 
 
 136-34 
 
 143-30 
 
 149-96 
 
 132- 
 
 40 
 
 130-20 
 
 138-51 
 
 146-38 
 
 153-86 
 
 161-00 
 
 150-8 
 
 35 
 
 141-14 
 
 150-16 
 
 158-68 
 
 16679 
 
 174-53 
 
 176- 
 
 30 
 
 154-95 
 
 164-84 
 
 174-20 
 
 183-10 
 
 191-61 
 
 211-2 
 
 25 
 
 173-05 
 
 184-10 
 
 194-56 
 
 204-50 
 
 214-00 
 
 264- 
 
 20 
 
 198-12 
 
 21077 
 
 222-73 
 
 234-11 
 
 244-98 
 
 352- 
 
 15 
 
 235-75 
 
 250-80 
 
 265-04 
 
 278-58 
 
 291-52 
 
 528- 
 
 10 
 
 300-60 
 
 319-80 
 
 337-95 
 
 355-22 
 
 &7171 
 
 5867 
 
 9 
 
 320-00 
 
 340-43 
 
 359-76 
 
 378-14 
 
 395-70 
 
 660- 
 
 8 
 
 343-04 
 
 359-65 
 
 385-67 
 
 405-37 
 
 424-20 
 
 754-3 
 
 7 
 
 370-99 
 
 394-68 
 
 417-08 
 
 438-39 
 
 458-76 
 
 880- 
 
 6 
 
 40579 
 
 431-70 
 
 456-21 
 
 479-52 
 
 501-79 
 
 1056- 
 
 5 
 
 450-71 
 
 479-49 
 
 506-71 
 
 532-60 
 
 557-34 
 
 1320- 
 
 4 
 
 511-72 
 
 544-39 
 
 575-30 
 
 604-69 
 
 632-78 
 
 1760- 
 
 3 
 
 601-38 
 
 639-78 
 
 676-10 
 
 710-64 
 
 743-65 j 
 
 2640- 
 
 2 
 
 753-29 
 
 801-39 
 
 846-89 
 
 890-16 
 
 931-50 
 
 5280- 
 
 1 
 
 1116-35 
 
 1187-62 
 
 1255-04 
 
 1319-17 
 
 1380-44
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 213 
 
 TABLE VIII. FOR FINDING THE MEAN VELOCITIES OF WATER 
 
 FLOWING IN PIPES, DRAINS, STREAMS, AND RIVERS. 
 
 The hydraulic mean depth is found for all channels, by dividing 
 
 the wetted perimeter into the area. 
 
 Hydraulic mean depths 6 inches to 10 inches. Falls per mile ] inch 
 to 1 2 feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or "mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 6 inches. 
 
 7 inches. 
 
 8 inches. 
 
 9 inches. 
 
 LO inches. 
 
 F. I. 
 
 
 
 
 
 
 
 1 
 
 63360 
 
 2-23 
 
 2-41 
 
 2-58 
 
 2-75 
 
 2-90 
 
 2 
 
 31680 
 
 3-47 
 
 3-76 
 
 4-03 
 
 4-28 
 
 4-52 
 
 3 
 
 21120 
 
 4-43 
 
 4-80 
 
 5-15 
 
 5-47 
 
 5-78 
 
 4 
 
 15840 
 
 5-26 
 
 5-69 
 
 6-10 
 
 6-49 
 
 6-85 
 
 5 
 
 12672 
 
 5-98 
 
 6-48 
 
 6-95 
 
 7-39 
 
 7-80 
 
 6 
 
 10560 
 
 6-65 
 
 7-20 
 
 7-72 
 
 8-20 
 
 8-66 
 
 7 
 
 9051 
 
 7-26 
 
 7-86 
 
 8-43 
 
 8-96 
 
 9-46 
 
 8 
 
 7920 
 
 7-83 
 
 8-48 
 
 9-09 
 
 9-67 
 
 10-21 
 
 9 
 
 7040 
 
 8-37 
 
 9-07 
 
 9-72 
 
 10-33 
 
 10-91 
 
 10 
 
 6336 
 
 8-88 
 
 9-63 
 
 10-32 
 
 10-97 
 
 11-58 
 
 11 
 
 5760 
 
 9-37 
 
 10-16 
 
 10-89 
 
 11-57 
 
 12-22 
 
 1 
 
 5280 
 
 9-84 
 
 10-67 
 
 11-43 
 
 12-15 
 
 12-83 
 
 1 3 
 
 4224 
 
 11-16 
 
 12-09 
 
 12-95 
 
 13-77 
 
 14-54 
 
 1 6 
 
 3520 
 
 12-35 
 
 13-38 
 
 14-34 
 
 15-25 
 
 16-10 
 
 1 9 
 
 3017 
 
 13-46 
 
 14-58 
 
 15-63 
 
 16-61 
 
 17-54 
 
 2 
 
 2640 
 
 14-50 
 
 15-71 
 
 16-84 
 
 17-90 
 
 18-90 
 
 2 3 
 
 Interpolated. 
 
 15-45 
 
 16-75 
 
 18-24 
 
 19-08 
 
 20-15 
 
 2 6 
 
 2112 
 
 16-42 
 
 17-79 
 
 19-64 
 
 20-26 
 
 21-40 
 
 2 9 
 
 Interpolated. 
 
 17-29 
 
 18-74 
 
 20-37 
 
 21-34 
 
 22-54 
 
 3 
 
 1760 
 
 18-17 
 
 19-69 
 
 21-10 
 
 22-42 
 
 23-68 
 
 3 3 
 
 Interpolated. 
 
 18-99 
 
 20-57 
 
 22-05 
 
 23-43 
 
 24-75 
 
 3 6 
 
 1508 
 
 19-80 
 
 21-46 
 
 23-00 
 
 24-44 
 
 25-81 
 
 3 9 
 
 Interpolated. 
 
 20-56 
 
 22-28 
 
 23-88 
 
 25-38 . 
 
 26-80 
 
 4 
 
 1320 
 
 21-33 
 
 23-11 
 
 24-77 
 
 26-82 
 
 27-80 
 
 4 6 
 
 Interpolated. 
 
 22-74 
 
 24-64 
 
 26-41 
 
 28-07 
 
 29-64 
 
 5 
 
 1056 
 
 24-16 
 
 26-17 
 
 28-05 
 
 29-81 
 
 31-48 
 
 5 6 
 
 Interpolated. 
 
 25-45 
 
 27-58 
 
 29-56 
 
 31-42 
 
 33-17 
 
 6 
 
 880 
 
 26-75 
 
 28-98 
 
 31-06 
 
 33-02 
 
 34-86 
 
 6 6 
 
 Interpolated. 
 
 27-96 
 
 30-29 
 
 32-47 
 
 34-51 
 
 36-44 
 
 7 
 
 754 
 
 29-17 
 
 31-60 
 
 33-87 
 
 36-00 
 
 38-02 
 
 7 6 
 
 Interpolated. 
 
 30-30 
 
 32-83 
 
 35-19 
 
 37-40 
 
 39-50 
 
 8 
 
 660 
 
 31-43 
 
 34-06 
 
 36-50 
 
 38-80 
 
 40-97 
 
 8 6 
 
 Interpolated. 
 
 32-51 
 
 35-22 
 
 37-75 
 
 40-12 
 
 42-37 
 
 9 
 
 587 
 
 33-58 
 
 36-39 
 
 38-99 
 
 41-45 
 
 43-77 
 
 9 6 
 
 Interpolated. 
 
 34-61 
 
 37-50 
 
 40-20 
 
 42-72 
 
 45-11 
 
 10 
 
 528 
 
 35-65 
 
 38-63 
 
 41-40 
 
 44-00 
 
 46-46 
 
 10 6 
 
 Interpolated. 
 
 36-63 
 
 39-69 
 
 42-54 
 
 45-22 
 
 47-75 
 
 11 
 
 480 
 
 37-62 
 
 40-76 
 
 43-69 
 
 46-44 
 
 49-03 
 
 11 6 
 
 Interpolated. 
 
 38-57 
 
 41-79 
 
 44-79 
 
 47-61 
 
 50-27 
 
 12 
 
 440 
 
 39-52 
 
 42-82 
 
 45-90 
 
 48-78 
 
 51-51
 
 214 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VIII. FOE FINDING THE MEAN VELOCITIES OF WATER 
 
 FLOWING IN PIPES, DRAINS, STREAMS, AND RIVERS. 
 
 The hydraulic mean depth is found for all channels by dividing the 
 
 wetted perimeter into the area. 
 
 Hydraulic mean depths 1 1 inches to 2 1 inches. Falls per mile 1 inch 
 to IZfeet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 11 inches. 
 
 12 inches 
 
 15 inches. 
 
 18 inches. 
 
 21 inches. 
 
 F. I. 
 
 
 
 
 
 
 
 1 
 
 63360 
 
 3-05 
 
 3-19 
 
 3-57 
 
 3-92 
 
 4-25 
 
 2 
 
 31680 
 
 4-75 
 
 4-97 
 
 5-57 
 
 6-12 
 
 6-62 
 
 3 
 
 21120 
 
 6-07 
 
 6-35 
 
 7-12 
 
 7-82 
 
 8-46 
 
 4 
 
 15840 
 
 7-19 
 
 7-53 
 
 8-44 
 
 9-27 
 
 10-03 
 
 5 
 
 12672 
 
 8-19 
 
 8-57 
 
 9-61 
 
 10-55 
 
 11-42 
 
 6 
 
 10560 
 
 9-10 
 
 9-52 
 
 10-67 
 
 11-72 
 
 12-68 
 
 7 
 
 9051 
 
 9-94 
 
 10-39 
 
 11-66 
 
 12-80 
 
 13-85 
 
 8 
 
 7920 
 
 10-72 
 
 11-21 
 
 12-57 
 
 13-81 
 
 14-94 
 
 9 
 
 7041 
 
 11-46 
 
 11-99 
 
 13-44 
 
 14-76 
 
 15-97 
 
 10 
 
 6336 
 
 12-16 
 
 12-72 
 
 14-27 
 
 15-66 
 
 16-95 
 
 11 
 
 5760 
 
 12-83 
 
 13-42 
 
 15-05 
 
 16-53 
 
 17-88 
 
 1 
 
 5280 
 
 13-48 
 
 14-09 
 
 15-81 
 
 17-36 
 
 18-78 
 
 1 3 
 
 4224 
 
 15-27 
 
 15-97 
 
 17-91 
 
 19-67 
 
 21-28 
 
 1 6 
 
 3520 
 
 16-91 
 
 17-68 
 
 19-83 
 
 21-78 
 
 23-56 
 
 1 9 
 
 3017 
 
 18-23 
 
 19-27 
 
 21-62 
 
 23-73 
 
 25-68 
 
 2 
 
 2640 
 
 19-85 
 
 20-76 
 
 23-28 
 
 25-63 
 
 27-66 
 
 2 3 
 
 Interpolated. 
 
 21-16 
 
 22-13 
 
 24-82 
 
 27-29 
 
 29-49 
 
 2 6 
 
 2112 
 
 22-48 
 
 23-51 
 
 26-36 
 
 28-95 
 
 31-32 
 
 2 9 
 
 Interpolated. 
 
 23-68 
 
 24-76 
 
 27-77 
 
 30-49 
 
 32-99 
 
 3 
 
 1760 
 
 24-88 
 
 26-02 
 
 29-18 
 
 32-04 
 
 34-67 
 
 3 3 
 
 Interpolated. 
 
 25-99 
 
 27-18 
 
 30-47 
 
 33-48 
 
 36-22 
 
 3 6 
 
 1508 
 
 27-11 
 
 28-35 
 
 31-77 
 
 34-92 
 
 37*78 
 
 3 9 
 
 Interpolated. 
 
 28-15 
 
 29-45 
 
 33-01 
 
 36-26 
 
 39-23 
 
 4 
 
 1320 
 
 29-20 
 
 30-54 
 
 34-25 
 
 37-60 
 
 40-69 
 
 4 6 
 
 Interpolated. 
 
 31-13 
 
 32-56 
 
 36-52 
 
 40-10 
 
 43-39 
 
 5 
 
 1056 
 
 33-07 
 
 34-59 
 
 38-79 
 
 42-59 
 
 46-09 
 
 5 6 
 
 Interpolated. 
 
 34-85 
 
 36-44 
 
 40-87 
 
 44-88 
 
 48-56 
 
 6 
 
 880 
 
 36-62 
 
 3830 
 
 42-95 
 
 47-16 
 
 51-03 
 
 6 6 
 
 Interpolated. 
 
 38-28 
 
 40-03 
 
 44-90 
 
 49-30 
 
 53-34 
 
 7 
 
 754 
 
 39-93 
 
 41-76 
 
 46-84 
 
 51-43 
 
 55-65 
 
 7 6 
 
 Interpolated. 
 
 41-48 
 
 43-39 
 
 48-66 
 
 53-43 
 
 57-81 
 
 8 
 
 660 
 
 43-04 
 
 45-01 
 
 50-48 
 
 55-42 
 
 59-97 
 
 8 6 
 
 Interpolated. 
 
 44-50 
 
 46-54 
 
 52-20 
 
 57-32 
 
 62-02 
 
 9 
 
 587 
 
 45-97 
 
 48-08 
 
 53-92 
 
 59-21 
 
 64-06 
 
 9 6 
 
 Interpolated. 
 
 47-39 
 
 49-56 
 
 55-58 
 
 61-03 
 
 66-04 
 
 10 
 
 528 
 
 48-80 
 
 51-04 
 
 57-24 
 
 62-85 
 
 68-01 
 
 10 6 
 
 Interpolated. 
 
 50-15 
 
 52-45 
 
 58-83 
 
 64-59 
 
 69-89 
 
 11 
 
 480 
 
 51-51 
 
 53-87 
 
 60-41 
 
 66-33 
 
 71-78 
 
 11 6 
 
 Interpolated. 
 
 52-81 
 
 55-23 
 
 61-94 
 
 68-01 
 
 73-59 
 
 12 
 
 440 
 
 54-11 
 
 56-59 
 
 63-47 
 
 69-68 
 
 75-40
 
 ORIFICES, WEIKS, PIPES, AND RIVERS. 
 
 215 
 
 TABLE VIII. FOB FINDING THK MEAN VELOCITIES OF WATER 
 
 FLOWING IN PIPES, DRAINS, STREAMS, AND RIVERS. 
 
 The hydraulic mean depth is found for all channels by dividing the 
 
 wetted perimeter into the area. 
 
 Hydraulic mean depths 24 inches to 4 feet. Falls per mile 1 inch 
 to 12 feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 24 inches. 
 
 30 inches. 
 
 36 inches. 
 
 42 inches. 
 
 48 inches. 
 
 F. I. 
 
 
 
 
 
 
 
 1 
 
 63360 
 
 4-54 
 
 5-09 
 
 5-59 
 
 6-04 
 
 6-47 
 
 2 
 
 31680 
 
 7-09 
 
 7-94 
 
 8-71 
 
 9-42 
 
 10-08 
 
 3 
 
 21120 
 
 9-06 
 
 10-15 
 
 11-14 
 
 12-04 
 
 12-89 
 
 4 
 
 15840 
 
 10-73 
 
 12-03 
 
 13-20 
 
 14-27 
 
 15-27 
 
 5 
 
 12672 
 
 12-22 
 
 13-69 
 
 15-03 
 
 16-25 
 
 17-39 
 
 6 
 
 10560 
 
 13-57 
 
 15-21 
 
 16-69 
 
 18-05 
 
 19-31 
 
 7 
 
 9051 
 
 14-83 
 
 16-61 
 
 18-23 
 
 19-71 
 
 21-09 
 
 8 
 
 7920 
 
 15-99 
 
 17-92 
 
 19-66 
 
 21-27 
 
 22-76 
 
 9 
 
 7041 
 
 17-10 
 
 19-16 
 
 21-02 
 
 22-73 
 
 24-33 
 
 10 
 
 6336 
 
 18-15 
 
 20-33 
 
 22-31 
 
 24-13 
 
 25-82 
 
 11 
 
 5760 
 
 19-15 
 
 21-45 
 
 23-54 
 
 25-46 
 
 27-24 
 
 1 
 
 5280 
 
 20-11 
 
 22-53 
 
 24-72 
 
 26-73 
 
 28-61 
 
 1 3 
 
 4224 
 
 22-78 
 
 25-53 
 
 28-01 
 
 30-29 
 
 32-42 
 
 1 6 
 
 3520 
 
 25-23 
 
 28-27 
 
 31-02 
 
 33-54 
 
 35-90 
 
 1 9 
 
 3017 
 
 27-49 
 
 30-81 
 
 33-80 
 
 36-55 
 
 39-12 
 
 2 
 
 2640 
 
 29-62 
 
 33-18 
 
 36-41 
 
 39-38 
 
 42-14 
 
 2 3 
 
 Interpolated. 
 
 31-57 
 
 35-38 
 
 38-82 
 
 41-98 
 
 44-92 
 
 2 6 
 
 2112 
 
 33-53 
 
 37-57 
 
 41-22 
 
 44-58 
 
 47-71 
 
 2 9 
 
 Interpolated. 
 
 35-32 
 
 39-58 
 
 43-43 
 
 46-96 
 
 50-26 
 
 3 
 
 1760 
 
 37-11 
 
 41-58 
 
 45-63 
 
 49-34 
 
 52-81 
 
 3 3 
 
 Interpolated. 
 
 38-78 
 
 43-45 
 
 47-68 
 
 51-56 
 
 55-] 8 
 
 3 6 
 
 1508 
 
 40-45 
 
 45-32 
 
 49-73 
 
 53-78 
 
 57-55 
 
 3 9 
 
 Interpolated. 
 
 42-00 
 
 47-07 
 
 51-64 
 
 55-85 
 
 59-77 
 
 4 
 
 1320 
 
 43-56 
 
 48-81 
 
 53-56 
 
 57-92 
 
 61-98 
 
 4 6 
 
 Interpolated. 
 
 46-45 
 
 52-05 
 
 57-11 
 
 61-76 
 
 66-09 
 
 5 
 
 1056 
 
 49-34 
 
 55-28 
 
 60-66 
 
 65-60 
 
 70-20 
 
 5 6 
 
 Interpolated. 
 
 51-99 
 
 58-25 
 
 63-91 
 
 69-12 
 
 73-97 
 
 6 
 
 880 
 
 54-63 
 
 61-22 
 
 67-17 
 
 72-64 
 
 77-74* 
 
 6 6 
 
 Interpolated. 
 
 57-11 
 
 63-99 
 
 70-21 
 
 75-93* 
 
 81-25 
 
 7 
 
 754 
 
 59-58 
 
 6676 
 
 73-25 
 
 79-21 
 
 84-77 
 
 7 6 
 
 Interpolated. 
 
 61-89 
 
 69-35 
 
 76-09* 
 
 87-29 
 
 88-06 
 
 8 
 
 660 
 
 64-21 
 
 71-94 
 
 78-94 
 
 85-37 
 
 91-35 
 
 8 6 
 
 Interpolated. 
 
 66-40 
 
 74-40 
 
 81-63 
 
 88-26 
 
 94-47 
 
 9 
 
 587 
 
 68-59 
 
 76-85* 
 
 84-32 
 
 91-19 
 
 97-59 
 
 9 6 
 
 Interpolated. 
 
 70-60 
 
 79-22 
 
 86-92 
 
 94-00 
 
 100-59 
 
 10 
 
 528 
 
 72-81 
 
 81-58 
 
 89-52 
 
 96-81 
 
 103-60 
 
 10 6 
 
 Interpolated. 
 
 74-83 
 
 83-84 
 
 91-99 
 
 99-49 
 
 106-47 
 
 11 
 
 480 
 
 76-84* 
 
 86-10 
 
 94-47 
 
 102-17 
 
 109-33 
 
 11 6 
 
 Interpolated. 
 
 78-78 
 
 88-28 
 
 96-86 
 
 104-75 
 
 112-10 
 
 12 
 
 440 
 
 80-72 
 
 90-45 
 
 99-25 
 
 107-33 
 
 114-86 
 
 -.1
 
 216 
 
 THE DISCHARGE OF WATEK FROM 
 
 TABLE VIII. FOB FINDING THE MEAN VELOCITIES OF WATER 
 
 FLOWING IN PIPES, DRAINS, 8TBEAMS, AND KIVERS. 
 
 The hydraulic mean depth is found for all channels by dividing the 
 
 wetted perimeter into the area. 
 
 Hydraulic mean depths 4 feet 6 inches to 7 feet. Falls per mile 1 inch 
 to \%feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 54 inches. 
 
 60 inches. 
 
 66 inches. 
 
 72 inches. 
 
 84 inches. 
 
 F. I. 
 
 
 
 
 
 
 
 1 
 
 63360 
 
 6-86 
 
 7-24 
 
 7-60 
 
 7-94 
 
 8-58 
 
 2 
 
 31680 
 
 10-70 
 
 11-29 
 
 11-85 
 
 12-38 
 
 13-39 
 
 3 
 
 21120 
 
 13-68 
 
 14-62 
 
 15-14 
 
 15-83 
 
 17-11 
 
 4 
 
 15840 
 
 16-21 
 
 17-10 
 
 17-95 
 
 18-76 
 
 20-28 
 
 5 
 
 12672 
 
 18-46 
 
 19-47 
 
 20-43 
 
 21-35 
 
 23-13 
 
 6 
 
 10560 
 
 20-50 
 
 21-63 
 
 22-70 
 
 23-72 
 
 25-64 
 
 7 
 
 9051 
 
 22-39 
 
 23-62 
 
 24-79 
 
 25-90 
 
 28-00 
 
 8 
 
 7920 
 
 24-16 
 
 25-48 
 
 26-74 
 
 27-95 
 
 30-21 
 
 9 
 
 7041 
 
 25-83 
 
 27-24 
 
 28-59 
 
 29-88 
 
 32-30 
 
 10 
 
 6336 
 
 27-41 
 
 28-91 
 
 30-34 
 
 3171 
 
 34-28 
 
 11 
 
 5760 
 
 28-92 
 
 30-51 
 
 32-01 
 
 33-46 
 
 36-17 
 
 1 
 
 5280 
 
 30-37 
 
 32-03 
 
 33-62 
 
 35-13 
 
 37-98 
 
 1 3 
 
 4224 
 
 34-41 
 
 36-30 
 
 38-10 
 
 39-81 
 
 43-04 
 
 1 6 
 
 3520 
 
 38-10 
 
 40-19 
 
 42-18 
 
 44-08 
 
 47-65 
 
 1 9 
 
 3017 
 
 41-52 
 
 43-80 
 
 45-97 
 
 48-04 
 
 51-93 
 
 2 
 
 2640 
 
 44-73 
 
 47-18 
 
 49-52 
 
 51-75 
 
 55-94 
 
 2 3 
 
 Interpolated. 
 
 47-69 
 
 50-30 
 
 52-79 
 
 55-17 
 
 59-64 
 
 2 6 
 
 2112 
 
 50-65 
 
 53-42 
 
 56-07 
 
 58-59 
 
 63-34 
 
 2 9 
 
 Interpolated. 
 
 53-35 
 
 56-28 
 
 59-06 
 
 61-72 
 
 66-72 
 
 3 
 
 1760 
 
 56-06 
 
 59-13 
 
 62-05 
 
 64-85 
 
 70-10 
 
 3 3 
 
 Interpolated. 
 
 58-57 
 
 61-79 
 
 64-84 
 
 67-76 
 
 73-25 
 
 3 6 
 
 1508 
 
 61-09 
 
 64-44 
 
 67-63 
 
 70-67 
 
 76-40* 
 
 3 9 
 
 Interpolated. 
 
 63-44 
 
 66-92 
 
 70-23 
 
 73-39 
 
 79-35 
 
 4 
 
 1320 
 
 65-80 
 
 69-41 
 
 72-84 
 
 76-11* 
 
 82-29 
 
 4 6 
 
 Interpolated. 
 
 70-16 
 
 74-01 
 
 77-67* 
 
 81-16 
 
 87-74 
 
 5 
 
 1056 
 
 74-52 
 
 78-61* 
 
 82-50 
 
 86-21 
 
 93-20 
 
 5 6 
 
 Interpolated. 
 
 78-52* 
 
 82-83 
 
 86-92 
 
 90-84 
 
 98-20 
 
 6 
 
 880 
 
 82-52 
 
 87-05 
 
 91-35 
 
 95-46 
 
 103-20 
 
 6 6 
 
 Interpolated. 
 
 86-25 
 
 90-98 
 
 95-58 
 
 99-78 
 
 107-87 
 
 7 
 
 754 
 
 89-99 
 
 94-92 
 
 99-62 
 
 104-10 
 
 112-54 
 
 7 6 
 
 Interpolated. 
 
 93-48 
 
 98-61 
 
 103-48 
 
 108-14 
 
 116-91 
 
 8 
 
 660 
 
 96-98 
 
 102-30 
 
 107-35 
 
 112-19 
 
 121-28 
 
 8 6 
 
 Interpolated. 
 
 100-29 
 
 105-79 
 
 111-02 
 
 116-01 
 
 125-42 
 
 9 
 
 587 
 
 103-59 
 
 109-27 
 
 114-68 
 
 119-84 
 
 129-56 
 
 9 6 
 
 Interpolated. 
 
 106-78 
 
 112-64 
 
 118-21 
 
 123-53 
 
 133-55 
 
 10 
 
 528 
 
 109-97 
 
 116-01 
 
 121-74 
 
 127-22 
 
 137-54 
 
 10 6 
 
 Interpolated. 
 
 113-02 
 
 119-22 
 
 125-11 
 
 130-74 
 
 141-34 
 
 11 
 
 480 
 
 116-06 
 
 122-43 
 
 128-48 
 
 134-27 
 
 145-15 
 
 11 6 
 
 Interpolated. 
 
 119-00 
 
 125-52 
 
 131-73 
 
 137-66 
 
 148-82 
 
 12 
 
 440 
 
 121-93 
 
 128-61 
 
 134-97 
 
 141-05 
 
 152-49
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 217 
 
 TABLE VIII. FOR FINDING THE MEAN VELOCITIES OF WATER 
 
 FLOWING IN PIPES, DRAINS, STREAMS, AND RIVERS. 
 
 The hydraulic mean depth is found for all channels by dividing the 
 
 wetted perimeter into the area. 
 
 Hydraulic mean depths 8 feet to 12 feet. Falls per mile 1 inch 
 to ISfeet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic nclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 96 inches. 
 
 108 
 inches. 
 
 120 
 inches. 
 
 132 
 inches. 
 
 144 
 inches. 
 
 F. I. 
 
 
 
 
 
 
 
 1 
 
 63360 
 
 9-18 
 
 9-75 
 
 10-28 
 
 10-79 
 
 11-27 
 
 2 
 
 31680 
 
 14-32 
 
 15-20 
 
 16-03 
 
 16-82 
 
 17-57 
 
 3 
 
 21120 
 
 18-30 
 
 19-43 
 
 20-49 
 
 21-50 
 
 22-46 
 
 4 
 
 15840 
 
 21-69 
 
 23-02 
 
 24-28 
 
 25-47 
 
 26-62 
 
 5 
 
 12672 
 
 24-70 
 
 26-21 
 
 27-64 
 
 29-00 
 
 30-31 
 
 6 
 
 10560 
 
 27-43 
 
 29-11 
 
 30-70 
 
 32-21 
 
 33-66 
 
 7 
 
 9051 
 
 29-96 
 
 31-80 
 
 33-53 
 
 35-18 
 
 36-76 
 
 8 
 
 7920 
 
 32-32 
 
 34-30 
 
 36-18 
 
 37-96 
 
 39-66 
 
 9 
 
 7041 
 
 34-55 
 
 36-67 
 
 38-67 
 
 40-58 
 
 42-40 
 
 10 
 
 6336 
 
 36-67 
 
 38-92 
 
 41-04 
 
 43-07 
 
 45-00 
 
 11 
 
 5760 
 
 38-69 
 
 41-06 
 
 43-31 
 
 45-44 
 
 47-48 
 
 1 
 
 5280 
 
 40-63 
 
 43-12 
 
 45-48 
 
 47-72 
 
 49-86 
 
 1 3 
 
 4224 
 
 46-04 
 
 48-87 
 
 51-54 
 
 54-07 
 
 56-50 
 
 1 6 
 
 3520 
 
 50-98 
 
 54-11 
 
 57-06 
 
 59-87 
 
 62-56 
 
 1 9 
 
 3017 
 
 55-60 
 
 58-96 
 
 62-18 
 
 65-25 
 
 68-17 
 
 2 
 
 2640 
 
 59-85 
 
 63-52 
 
 66-98 
 
 70-28 
 
 73-44* 
 
 2 3 
 
 Interpolated. 
 
 6380 
 
 67-72 
 
 71-41 
 
 74-93* 
 
 78-29 
 
 2 6 
 
 2112 
 
 67-76 
 
 71-91 
 
 75-84* 
 
 79-58 
 
 83-15 
 
 2 9 
 
 Interpolated. 
 
 71-38 
 
 75-75* 
 
 79-89 
 
 83-83 
 
 87-59 
 
 3 
 
 1760 
 
 75-00* 
 
 79-59 
 
 83-94 
 
 88-08 
 
 92-03 
 
 3 3 
 
 Interpolated. 
 
 7837 
 
 83-17 
 
 87-71 
 
 92-03 
 
 96-16 
 
 3 6 
 
 1508 
 
 81-74 
 
 86-75 
 
 91-48 
 
 95-99 
 
 100-30 
 
 3 9 
 
 Interpolated. 
 
 84-88 
 
 90-09 
 
 95-01 
 
 99-69 
 
 104-16 
 
 4 
 
 1320 
 
 88-03 
 
 93-43 
 
 98-53 
 
 103-38 
 
 108-02 
 
 4 6 
 
 Interpolated. 
 
 93-87 
 
 99-62 
 
 105-06 
 
 110-24 
 
 115-18 
 
 5 
 
 1056 
 
 99-70 
 
 105-82 
 
 111-59 
 
 117-09 
 
 122-34 
 
 5 6 
 
 Interpolated. 
 
 105-06 
 
 111-49 
 
 117-58 
 
 123-38 
 
 128-91 
 
 6 
 
 880 
 
 110-41 
 
 117-17 
 
 123-57 
 
 129-66 
 
 135-48 
 
 6 6 
 
 Interpolated. 
 
 115-40 
 
 122-47 
 
 129-16 
 
 135-53 
 
 141-61 
 
 7 
 
 754 
 
 120-40 
 
 127-76 
 
 134-75 
 
 141-39 
 
 147-73 
 
 7 6 
 
 Interpolated. 
 
 125-07 
 
 132-74 
 
 139-99 
 
 146-88 
 
 153-47 
 
 8 
 
 660 
 
 129-75 
 
 137-70 
 
 145-22 
 
 152-38 
 
 159-21 
 
 8 6 
 
 Interpolated. 
 
 134-18 
 
 142-40 
 
 150-18 
 
 157-57 
 
 164-64 
 
 9 
 
 587 
 
 138-60 
 
 147-10 
 
 155-13 
 
 162-77 
 
 170-07 
 
 9 6 
 
 Interpolated. 
 
 142-87 
 
 151-63 
 
 159-91 
 
 167-78 
 
 175-31 
 
 10 
 
 528 
 
 147-14 
 
 156-16 
 
 164-68 
 
 172-80 
 
 180-55 
 
 10 6 
 
 Interpolated. 
 
 151-21 
 
 160-48 
 
 169-24 
 
 177-58 
 
 185-55 
 
 11 
 
 480 
 
 155-29 
 
 164-80 
 
 173-80 
 
 182-36 
 
 190-54 
 
 11 6 
 
 Interpolated. 
 
 159-21 
 
 168-97 
 
 178-19 
 
 186-97 
 
 195-36 
 
 12 
 
 440 
 
 163-13 
 
 173-13 
 
 182-59 
 
 191-58 
 
 200-17 
 
 18
 
 218 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE IX. FOR FINDING THE DISCHARGE IN CUBIC FEET, PER 
 
 MINUTE, WHEN THE DIAMETER OF A PIPE, OB ORIFICE, AND THE 
 VELOCITY OF DISCHARGE ARE KNOWN ; AND VICE VERSA. 
 
 ill 
 
 A** 
 
 Discharge in cubic feet per minute, for different velocities. 
 
 Velocity of 
 100 inches 
 per second. 
 
 Velocity of 
 200 inches 
 per second. 
 
 Velocity of 
 300 inches 
 per second. 
 
 Velocity of 
 400 inches 
 per second. 
 
 Velocity of 
 500 inches 
 per second. 
 
 1 
 
 4 
 
 170442 
 
 3409 
 
 5113 
 
 6818 
 
 8522 
 
 | 
 
 68177 
 
 1-3635 
 
 2-0453 
 
 2-7271 
 
 3-4089 
 
 
 1-53398 
 
 3-0679 
 
 4-6019 
 
 6-1359 
 
 7-6699 
 
 i 
 
 2-727077 
 
 5-4541 
 
 8-1812 
 
 10-9083 
 
 13-6354 
 
 H 
 
 4-26106 
 
 8-5221 
 
 12-7832 
 
 17-0442 
 
 213053 
 
 ij 
 
 6-13593 
 
 12-2718 
 
 18-4080 
 
 24-5437 
 
 30-6797 
 
 if 
 
 835167 
 
 16-7033 
 
 25-0550 
 
 33-4067 
 
 41-7584 
 
 2 
 
 10-90831 
 
 21-1817 
 
 32-7249 
 
 43-6332 
 
 545415 
 
 2| 
 
 13-80583 
 
 27-6117 
 
 41-4175 
 
 55-2233 
 
 69-0291 
 
 2f 
 
 17-04423 
 
 34-0885 
 
 51-1327 
 
 68-1769 
 
 85-2212 
 
 2f 
 
 20-62352 
 
 41-2470 
 
 61-8706 
 
 82-4941 
 
 103-1176 
 
 3 
 
 24-54369 
 
 49-0874 
 
 73-6311 
 
 98-1748 
 
 121-7185 
 
 a i 
 
 28-80475 
 
 57-6095 
 
 86-4143 
 
 115-2190 
 
 144-0238 
 
 31 
 
 33-40669 
 
 66-8134 
 
 100-2201 
 
 1336268 
 
 167-0335 
 
 3f 
 
 38-34952 
 
 76-6990 
 
 115-0486 
 
 153-3981 
 
 191-7476 
 
 4 
 
 43-63323 
 
 87-2665 
 
 130-8997 
 
 174-5329 
 
 218-1662 
 
 4i 
 
 49-25783 
 
 98-5157 
 
 147-7735 
 
 197-0313 
 
 246-2892 
 
 4i 
 
 55-22331 
 
 110-4466 
 
 165-6699 
 
 220-8932 
 
 276-1166 
 
 4| 
 
 61-52968 
 
 123-0594 
 
 184-5890 
 
 246-1187 
 
 307-6484 
 
 5 
 
 68-17692 
 
 136-3539 
 
 204-5308 
 
 272-7077 
 
 340-8846 
 
 5 4 
 
 75-16506 
 
 150-3301 
 
 225-4952 
 
 300-6603 
 
 375-8253 
 
 5? 
 
 82-49408 
 
 164-9882 
 
 247-4822 
 
 329-9763 
 
 412-4704 
 
 5f 
 
 90-16399 
 
 180-3280 
 
 270-4920 
 
 360-6560 
 
 450-8200 
 
 G 
 
 98-17478 
 
 196-3495 
 
 294-5243 
 
 392-6991 
 
 490-8739 
 
 63 
 
 10652645 
 
 2130529 
 
 3195794 
 
 426-1058 
 
 5326323 
 
 6^ 
 
 115-2190 
 
 230-4380 
 
 345-6570 
 
 460-8760 
 
 576-0950 
 
 6| 
 
 124-25245 
 
 248-5049 
 
 372-7574 
 
 497-0098 
 
 621-2623 
 
 7 
 
 133-6268 
 
 267-2536 
 
 400-8804 
 
 534-5072 
 
 668-1340 
 
 7i 
 
 14334199 
 
 286-6840 
 
 430-0260 
 
 573-3680 
 
 716-7100 
 
 7i 
 
 153-39809 
 
 306-7962 
 
 460-1943 
 
 6135924 
 
 766-9905 
 
 7f 
 
 163-79507 
 
 327-5901 
 
 491-3852 
 
 655-1803 
 
 818-9753 
 
 8 
 
 174-53293 
 
 349-0659 
 
 523-5988 
 
 698-1317 
 
 872-6647 
 
 8| 
 
 197-03132 
 
 394-0626 
 
 591-0940 
 
 788-1253 
 
 985-1566 
 
 v2 
 
 9 
 
 220-89325 
 
 441-7865 
 
 662-6798 
 
 883-5730 
 
 1104-4663 
 
 9| 
 
 24611871 
 
 492-2374 
 
 738-3561 
 
 984-4748 
 
 1230-5936 
 
 10 
 
 272-70771 
 
 345-4154 
 
 818-1231 
 
 1090-8308 
 
 1363-5386 
 
 10| 
 
 300-66025 
 
 601-3205 
 
 901-9808 
 
 1202-6410 
 
 15033013 
 
 11 
 
 329-97633 
 
 659-9527 
 
 989-9290 
 
 1319-9053 
 
 1649-8817 
 
 iii 
 
 36065595 
 
 7213119 
 
 1081-9679 
 
 1442-6238 
 
 1803-2798 
 
 2 
 
 12 
 
 392-6991 
 
 785-3982 
 
 1178-0973 
 
 1570-7964 
 
 1963-4955
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 219 
 
 TABLE IX. FOR FINDING THE DISCHARGE IN CUBIC FEET, PER 
 
 MINUTE, WHEN THE DIAMETER OF A PIPE, OR ORIFICE, AND THE 
 VELOCITY OF DISCHARGE ARE KNOWN J AND VICE VERSA. 
 
 Discharge in cubic feet per minute, for different velocities. 
 
 III 
 IW 
 
 Velocity of 
 600 inches 
 per second. 
 
 Velocity of 
 700 inches 
 per second. 
 
 Velocity of 
 800 inches 
 per second. 
 
 Velocity of 
 900 inches 
 per second. 
 
 Velocity of 
 1000 inches 
 per second. 
 
 1-02-27 
 
 1-1931 
 
 1-3635 
 
 1 5340 
 
 1-7044 
 
 1 
 
 4-0906 
 
 47724 
 
 5-4542 
 
 6-1359 
 
 6-8177 
 
 ^ 
 
 9-2039 
 
 10-7379 
 
 12-2718 
 
 13-8058 
 
 15-3398 
 
 3 
 
 163625 
 
 19-0895 
 
 21-8166 
 
 24-5437 
 
 27-2708 
 
 1 
 
 255664 
 
 29-8274 
 
 34-0885 
 
 38-3495 
 
 42-6106 
 
 *i 
 
 36-8155 
 
 429515 
 
 49-0874 
 
 552234 
 
 61-3593 
 
 i| 
 
 50 1100 
 
 58-4617 
 
 68-8134 
 
 75-1650 
 
 83-5167 
 
 if 
 
 65-4499 
 
 76-3582 
 
 87-2665 
 
 98-1748 
 
 109-0831 
 
 2 
 
 82-8350 
 
 96-6408 
 
 110-4466 
 
 124-2525 
 
 138-0583 
 
 2-1 
 
 102-2654 
 
 1193096 
 
 136-3538 
 
 153-3981 
 
 1704423 
 
 2| 
 
 123-7411 
 
 1 44-3646 
 
 164-9882 
 
 185-6117 
 
 2062352 
 
 2| 
 
 147-2621 
 
 171-8059 
 
 196-3496 
 
 220-8933 
 
 2454369 
 
 3 
 
 172-8285 
 
 201-6333 
 
 230-4380 
 
 259-2428 
 
 288-0475 
 
 3i 
 
 2004401 
 
 233-8468 
 
 267-2535 
 
 300-6602 
 
 334-0669 
 
 3 I 
 
 230-0971 
 
 268-4467 
 
 306-7962 
 
 345-1457 
 
 3834952 
 
 B| 
 
 261-7994 
 
 305-4326 
 
 349-0659 
 
 392-6991 
 
 4363323 
 
 4 
 
 295-5470 
 
 344-8048 
 
 3940626 
 
 443-3205 
 
 4925783 
 
 4 i 
 
 331-3399 
 
 3865632 
 
 441-7865 
 
 497-0098 
 
 5522331 
 
 4 I 
 
 369-1781 
 
 430-7077 
 
 492-2374 
 
 553-7671 
 
 615-2968 
 
 4| 
 
 409-0615 
 
 477-2384 
 
 5454154 
 
 613-5923 
 
 681-7692 
 
 5 
 
 450-9904 
 
 526-1554 
 
 601-3205 
 
 676-4855 
 
 751-6506 
 
 5 4 
 
 494-9645 
 
 577-4586 
 
 659-9526 
 
 742-4467 
 
 824-9408 
 
 5* 
 
 5409839 
 
 631-1479 
 
 721 3119 
 
 811-4759 
 
 901-6399 
 
 5f 
 
 589-0486 
 
 6872235 
 
 785-3982 
 
 883-5730 
 
 981-7478 
 
 6 
 
 639-1587 
 
 745-6852 852-2116 
 
 958-7381 
 
 1065-2645 
 
 6i 
 
 691-3141 
 
 806-5330 921-7520 
 
 1036-9710 
 
 1152-1900 
 
 6 
 
 745-5147 
 
 869-7672 994-0196 1 1 18'272 1 
 
 12425245 
 
 4 
 
 801-7608 
 
 935-3876 1069-0144 
 
 1202-6412 
 
 1336-2680 
 
 7 
 
 860-051910033939 1146-7359 
 
 1290-0779 
 
 14334199 
 
 1} 
 
 920-3885 1073-7866 1227-1847 
 
 1380-5828 
 
 1533-9809 
 
 7 
 
 982-77041146-5655 13103605 
 
 1474-1556 
 
 1637-9507 
 
 7f 
 
 1047-1976 1221 7305 1396-2634 
 
 1570-7964 
 
 17453293 
 
 8 
 
 1182-1879J1379-2192 1576-2506 
 
 1773-2819 1970-3132 
 
 i 
 
 1325-3595 1546-2528 1767'1460 
 1476-7123 1722 8310 1968-9497 
 
 1988-0393 
 2215-0684 
 
 22089325 
 2461 1871 
 
 9 
 
 *H 
 
 1636-2463 1908 9540 2181-6617 
 
 2454 3694 12727 0771 
 
 10 
 
 1803-96152104-6218:2405-2820 
 
 2705-9423 
 
 3006-6025 
 
 10* 
 
 1979-85802309-8343 |2639'8I06 
 2163-9357 2524-5917 12885 2476 
 
 2969-7870 
 3245^936 
 
 32997633 
 3606-5595 
 
 11 
 iH 
 
 2356-1946 2748-8937 3141-5928 
 
 35342919 
 
 39269910 
 
 12
 
 220 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE X. FOR FINDING THE DEPTHS ON WEIRS OF DIFFERENT 
 
 LENGTHS, THE QUANTITY DISCHARGED OVER EACH BEING SUPPOSED 
 
 CONSTANT. (See pages 149 and 150.) 
 
 Ratios of 
 lengths. 
 
 Coeffi- 
 cients. 
 
 Ratios of 
 lengths. 
 
 Coeffi- 
 cients. 
 
 Ratios of 
 lengths. 
 
 Coeffi- 
 cients. 
 
 Ratios of 
 lengths. 
 
 Coeffi- 
 cients. 
 
 01 
 
 0464 
 
 405 
 
 5474 
 
 605 
 
 7153 
 
 805 
 
 8654 
 
 02 
 
 0737 
 
 410 
 
 5519 
 
 610 
 
 7193 
 
 810 
 
 8689 
 
 03 
 
 0965 
 
 415 
 
 5564 
 
 615 
 
 7232 
 
 815 
 
 8725 
 
 04 
 
 1170 
 
 420 
 
 5608 
 
 620 
 
 7271 
 
 820 
 
 8761 
 
 05 
 
 1357 
 
 425 
 
 5653 
 
 625 
 
 7310 
 
 825 
 
 8796 
 
 06 
 
 1533 
 
 430 
 
 5697 
 
 630 
 
 7349 
 
 830 
 
 8832 
 
 07 
 
 1699 
 
 435 
 
 5741 
 
 635 
 
 7388 
 
 835 
 
 8867 
 
 08 
 
 1857 
 
 440 
 
 5785 
 
 640 
 
 7427 
 
 840 
 
 8903 
 
 09 
 
 2008 
 
 445 
 
 5829 
 
 645 
 
 7465 
 
 845 
 
 8938 
 
 10 
 
 2154 
 
 450 
 
 5872 
 
 650 
 
 7504 
 
 850 
 
 8973 
 
 11 
 
 2296 
 
 455 
 
 5916 
 
 655 
 
 7542 
 
 855 
 
 9008 
 
 12 
 
 2433 
 
 460 
 
 5959 
 
 660 
 
 7580 
 
 860 
 
 9043 
 
 13 
 
 2566 
 
 465 
 
 6002 
 
 665 
 
 7619 
 
 865 
 
 9078 
 
 14 
 
 2696 
 
 470 
 
 6045 
 
 670 
 
 7657 
 
 870 
 
 9113 
 
 15 
 
 2U23 
 
 475 
 
 6088 
 
 675 
 
 76 ( >5 
 
 875 
 
 9148 
 
 16 
 
 2947 
 
 480 
 
 6130 
 
 680 
 
 7733 
 
 880 
 
 9183 
 
 17 
 
 3069 
 
 485 
 
 6173 
 
 685 
 
 7771 
 
 885 
 
 9218 
 
 18 
 
 3188 
 
 490 
 
 6215 
 
 690 
 
 7808 
 
 890 
 
 9253 
 
 19 
 
 3305 
 
 495 
 
 6258 
 
 695 
 
 7846 
 
 895 
 
 9287 
 
 20 
 
 3420 
 
 500 
 
 6300 
 
 700 
 
 7884 
 
 900 
 
 9322 
 
 21 
 
 3533 
 
 505 
 
 6342 
 
 705 
 
 7921 
 
 905 
 
 9356 
 
 22 
 
 3644 
 
 510 
 
 6383 
 
 710 
 
 7959 
 
 910 
 
 9391 
 
 23 
 
 3754 
 
 515 
 
 6425 
 
 715 
 
 7996 
 
 915 
 
 9425 
 
 24 
 
 3862 
 
 520 
 
 6466 
 
 720 
 
 8033 
 
 920 
 
 9459 
 
 25 
 
 3969 
 
 525 
 
 6508 
 
 725 
 
 80TO 
 
 925 
 
 9494 
 
 26 
 
 4074 
 
 530 
 
 6549 
 
 730 
 
 8107 
 
 930 
 
 9528 
 
 27 
 
 4177 
 
 535 
 
 6590 
 
 735 
 
 8144 
 
 935 
 
 9562 
 
 28 
 
 4280 
 
 540 
 
 6631 
 
 740 
 
 8181 
 
 940 
 
 9596 
 
 29 
 
 4381 
 
 545 
 
 6672 
 
 745 
 
 8218 
 
 945 
 
 9630 
 
 30 
 
 4481 
 
 550 
 
 6713 
 
 750 
 
 8255 
 
 950 
 
 9664 
 
 31 
 
 4580 
 
 555 
 
 6754 
 
 755 
 
 8291 
 
 955 
 
 9698 
 
 32 
 
 4678 
 
 560 
 
 6794 
 
 760 
 
 8328 
 
 960 
 
 9732 
 
 33 
 
 4775 
 
 565 
 
 6834 
 
 765 
 
 8365 
 
 965 
 
 9762 
 
 34 
 
 4871 
 
 570 
 
 6875 
 
 770 
 
 8401 
 
 970 
 
 9799 
 
 35 
 
 4966 
 
 575 
 
 6915 
 
 775 
 
 8437 
 
 975 
 
 9833 
 
 36 
 
 5061 
 
 580 
 
 6955 
 
 780 
 
 8474 
 
 980 
 
 9866 
 
 37 
 
 5154 
 
 585 
 
 6995 
 
 785 
 
 8510 
 
 985 
 
 9900 
 
 38 
 
 5246 
 
 590 
 
 7035 
 
 790 
 
 8546 
 
 990 
 
 9933 
 
 39 
 
 5338 
 
 595 
 
 7074 
 
 795 
 
 8582 
 
 995 
 
 9967 
 
 40 
 
 5429 
 
 600 
 
 7114 
 
 800 
 
 8618 
 
 1-000 
 
 1-0000
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 221 
 
 TABLE XT. RELATIVE DIMENSIONS OF EQUAL DISCHARGING TRAPE- 
 ZOIDAL CHANNELS, WITH SLOPES FROM TO 1, UP TO 2 TO I. 
 
 Half sum of the top and bottom is the mean width. The ratio of 
 the slope, multiplied by the depth, subtracted from the mean 
 width, will give the bottom; and if added, will give the top. 
 
 TABLE XII. gives the discharge in cubic feet per minute from the 
 primary channel, 70 wide, and the corresponding depths taken in 
 feet. For lesser or greater channels and discharges, see Rules, 
 pp. 123, 124, 125, 126, and 135. 
 
 The mean widths are given in the top horizontal line, and the corresponding 
 depths in the other horizontal lines. They may be taken in inches, feet, 
 yards, fathoms, or any other measures whatever. 
 
 70 
 
 60 
 
 50 
 
 40 
 
 35 
 
 30 
 
 25 
 
 20 
 
 15 
 
 10 
 
 125 
 
 13 
 
 15 
 
 17 
 
 20 
 
 23 -26 
 
 29 
 
 35 
 
 48 
 
 25 
 
 27 
 
 30 
 
 35 
 
 40 
 
 45 
 
 52 
 
 58 
 
 71 
 
 98 
 
 375 
 
 41 
 
 46 
 
 54 
 
 60 
 
 67 
 
 76 
 
 88 
 
 1-09 
 
 1-51 
 
 5 
 
 55 
 
 62 
 
 73 
 
 80 
 
 89 
 
 1-02 
 
 1-19 
 
 1-48 
 
 2-04 
 
 625 
 
 68 
 
 78 
 
 91 
 
 1-00 
 
 1-12 
 
 1-29 
 
 1-50 
 
 1-88 
 
 2-62 
 
 75 
 
 82 
 
 94 
 
 1-10 
 
 1-20 
 
 1-35 
 
 1-56 
 
 1-82 
 
 2-28 
 
 3-22 ! 
 
 875 
 
 96 
 
 1-10 
 
 1-29 
 
 1-41 
 
 1-58 
 
 1-83 
 
 2-14 
 
 2-69 
 
 386 
 
 1- 
 
 1-10 
 
 1-26 
 
 1-48 
 
 1-62 
 
 1-81 2-10 
 
 2-46 
 
 3-11 
 
 4-50 
 
 1-125 
 
 1-24 
 
 1-42 
 
 1-67 
 
 1-83 
 
 2-04 2-37 
 
 2-79 
 
 3-54 
 
 5-19* 
 
 1-25 
 
 1-39 
 
 1-58 
 
 1-86 
 
 2-04 
 
 2-28 2-65 
 
 3-12 1 3-98 
 
 5-89 
 
 1 1-375 
 
 1-53 
 
 1-74 
 
 2-05 
 
 2-25 
 
 2-51 2 92 
 
 3-46 ! 4-43 
 
 660 
 
 1-5 
 
 1-67 
 
 1-90 
 
 2-24 
 
 2-46 
 
 2-75 
 
 3-20 
 
 3-80 4-88 
 
 7-31 
 
 1 1-625 
 
 1-81 
 
 2-06 
 
 2-43 
 
 2-67 
 
 2-99 
 
 3-47 
 
 4-15 5-34 
 
 8-08 
 
 1-75 
 
 1-95 
 
 2-22 
 
 2-62 
 
 2-88 
 
 3-23 
 
 3-75 
 
 4-50 5-80 
 
 8-86 
 
 1-875 
 
 2-09 
 
 2-38 
 
 2-81 
 
 3-09 
 
 3-47 
 
 4-03 
 
 4-86 
 
 6-29 
 
 9-68 
 
 2- 
 
 2-23 
 
 2-54 
 
 3-00 
 
 3-31 
 
 3-72 
 
 4-32 
 
 5-22 6-78 
 
 10-50 
 
 2-125 
 
 2-37 
 
 2-70 
 
 319 
 
 3-52 
 
 3-96 
 
 4-61 
 
 5-58 7-29 
 
 11-37 
 
 2-25 
 
 2-51 
 
 2-86 
 
 3-38 
 
 3-73 
 
 4-21 
 
 4-91 
 
 5-95 i 7-81* 
 
 12-25 
 
 2-375 
 
 2-65 
 
 3-02 
 
 3-57 
 
 3-94 
 
 4-45 
 
 5-20 
 
 6-31 i 8 32 
 
 13-12 
 
 2-5 
 
 2-79 
 
 3-18 
 
 3-76 
 
 4-16 
 
 4-70 
 
 5-50 
 
 6-68 8-84 
 
 14-00 
 
 2-625 
 
 2-93 
 
 3-34 
 
 3-95 
 
 4-38 
 
 4-95 
 
 5-79 
 
 7-06 
 
 938 
 
 1492 
 
 2-75 
 
 3-07 
 
 3-51 
 
 4-15 
 
 4-60 
 
 5-21 
 
 6-09 
 
 7-45 
 
 9-93 
 
 1584 
 
 2-875 
 
 3-21 
 
 3-67 
 
 4-34 
 
 4-82 
 
 5-46 
 
 6-39 
 
 7-83 10-48 
 
 1676 
 
 3- 
 
 3-35 
 
 3-84 
 
 4-54 
 
 5-04 
 
 5-72 
 
 6-69 
 
 8-22 11 03 
 
 17-68 
 
 3-125 
 
 3-49 
 
 4-00 
 
 4-73 
 
 5-26 
 
 5-97 
 
 7-00 
 
 8-62 11-60 
 
 1868 
 
 3-25 
 
 3-63 
 
 4-17 
 
 4-93 
 
 5-49 
 
 6-23 7-31 
 
 9-02 1217 
 
 1968 
 
 3-375 
 
 3-77 
 
 4-33 
 
 5-13 
 
 5-72 
 
 6-49 7-62 
 
 9-42 12-74 
 
 2068 
 
 3-5 
 
 3-91 
 
 4-50 
 
 5-33 
 
 5-95 
 
 6-75 
 
 7-93 
 
 9-82 13-32 
 
 2168 
 
 3-625 
 
 4-05 
 
 4-66 
 
 5-53 
 
 6-17 
 
 7-01 8-25 
 
 10-23* 13-92 
 
 22 76 i 
 
 3-75 
 
 4-19 
 
 4-82 
 
 5-73 
 
 6-40 
 
 7-28 8-57 
 
 10 65 14-53 23-84 
 
 3-875 
 
 4-33 
 
 4-98 
 
 5-93 
 
 6-62 
 
 7-54 8-89 
 
 11 06 15-14 24 92 1 
 
 4- 
 
 4-48 
 
 5-14 
 
 6-13 
 
 6-85 
 
 7-81 
 
 9-21 
 
 11-48 15 75 ; 26 00 
 
 4-25 
 
 4-76 
 
 5-46 
 
 6-54 
 
 7-30 
 
 8-35 
 
 9-85 
 
 12-33 16 98 
 
 28-18 
 
 4-5 
 
 5-05 
 
 579 
 
 6-95 
 
 7-75 
 
 8-90 10-50 
 
 13-19 18-22 
 
 30-36 
 
 4-75 
 
 5-33 6-12 
 
 7-35 
 
 8-20 
 
 9-45 
 
 11-14 
 
 14-07 1950 
 
 32-68 
 
 5- 
 
 5-62 6-45 
 
 7-75 
 
 866 
 
 10-00 
 
 11-79 
 
 1496 2080 
 
 35-00 
 
 5-25 
 
 5-90 
 
 6-78 
 
 8-16 
 
 9-14 
 
 10-55 
 
 12-51* 
 
 1586 22-13 
 
 37-40 
 
 5-5 
 
 6-18 
 
 7-12 
 
 8-57 
 
 9-62 
 
 11-10 
 
 13-24 
 
 16-77 
 
 2347 
 
 39-81 
 
 5-75 
 
 6-46 
 
 7-46 
 
 8-98 
 
 10-11 
 
 11-66 
 
 1394 
 
 17-71 
 
 24-86 
 
 42-33 
 
 6- 
 
 6-75 
 
 7-80 
 
 9-40 
 
 10-60 
 
 12-22 
 
 14-65 
 
 18-65 
 
 2625 
 
 44-86
 
 222 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE XII. DISCHARGES FROM THE PRIMARY CHANNEL IN THE 
 
 FIRST COLUMN OF TABLE XI. 
 
 If the dimensions of the primary channel be in inches, divide 
 the discharges in this table by 500 ; if in yards, multiply by 
 15-6 ; and if in fathoms, by 88-2, &c.: see pp. 126, 127. The 
 final figures in the discharges may be rejected when they do not 
 exceed one-half per cent., or 0'5 in 100. (See pages 123 to 126.) 
 
 Depths of a 
 channel 
 whose mean 
 width is 
 70: infect. 
 
 Falls, inclinations, and discharges in cubic feet per minute. 
 Interpolate for intermodule falls; divide greater falls by 4, 
 and double the corresponding discharges. 
 
 linch 
 per mile, 
 1 in 63360 
 
 2 inches 
 per mile, 
 1 in 31680 
 
 3 inches 6 inches 
 per mile, per mile, 
 Iin211201inl0560 
 
 9 inches 
 per mile, 
 1 in 7040. 
 
 12 inches 
 per mile, 
 1 in 5280. 
 
 15 inches 
 per mile, 
 1 in 4224. 
 
 125 
 
 47 
 
 72 
 
 93 
 
 139 
 
 175 
 
 205 
 
 233 
 
 25 
 
 136 
 
 210 
 
 268 
 
 403 
 
 506 
 
 596 
 
 675 
 
 375 
 
 249 
 
 389 
 
 498 
 
 746 
 
 940 
 
 1105 
 
 1252 
 
 50 
 
 387 
 
 603 
 
 770 
 
 1155 
 
 1454 
 
 1709 
 
 1935 
 
 625 
 
 541 
 
 849 
 
 1078 
 
 1617 
 
 2036 
 
 2395 
 
 2714 
 
 75 
 
 714 
 
 1112 
 
 1420 
 
 2128 
 
 2681 
 
 3153 
 
 3573 
 
 875 
 
 900 
 
 1401 
 
 1791 
 
 2685 
 
 3382 
 
 3978 
 
 4507 
 
 1- 
 
 1100 
 
 1714 
 
 2190 
 
 3283 
 
 4134 
 
 4862 
 
 5507 
 
 1-125 
 
 1310 
 
 2042 
 
 2614 
 
 3909 
 
 4927 
 
 .5792 
 
 6577 
 
 1-25 
 
 1534 
 
 2384 
 
 3058 
 
 4581 
 
 5766 
 
 6780 
 
 7690 
 
 1-375 
 
 1767 
 
 2757 
 
 3521 
 
 5279 
 
 6661 
 
 7823 
 
 8863 
 
 1-50 
 
 2013 
 
 3142 
 
 4006 
 
 6016 
 
 7588 
 
 8915 
 
 10099 
 
 1-625 
 
 2268 
 
 3540 
 
 4525 
 
 6781 
 
 8541 
 
 10044 
 
 11381 
 
 1-75 
 
 2534 
 
 3950 
 
 5053 
 
 7570 
 
 9537 
 
 11210 
 
 12703 
 
 1-875 
 
 2812 
 
 4384 
 
 5599 
 
 8386 
 
 10570 
 
 12429 
 
 14083 
 
 2- 
 
 3090 
 
 4821 
 
 6161 
 
 9230 
 
 11628 
 
 13675 
 
 15513 
 
 2-125 
 
 3377 
 
 5273 
 
 6738 
 
 10092 
 
 12718 
 
 14956 
 
 16943 
 
 2-25 
 
 3674 
 
 5736 
 
 7331 
 
 10981 
 
 13833 
 
 16281 
 
 18435 
 
 2-375 
 
 3977 
 
 6210 
 
 7937 
 
 11889 
 
 14981 
 
 17645 
 
 19960 
 
 2-50 
 
 4293 
 
 6699 
 
 8563 
 
 12829 
 
 16161 
 
 19045 
 
 21534 
 
 2-625 
 
 4616 
 
 7203 
 
 9204 
 
 13800 
 
 17380 
 
 20434 
 
 23135 
 
 2-75 
 
 4947 
 
 7716 
 
 9865 
 
 14782 
 
 18624 
 
 21886 
 
 24800 
 
 2-875 
 
 5280 
 
 8233 
 
 10525 
 
 15773 
 
 19887 
 
 23360 
 
 26473 
 
 3- 
 
 5621 
 
 8762 
 
 11204 
 
 36788 
 
 21165 
 
 24833 
 
 28176 
 
 3-125 
 
 5972 
 
 9310 
 
 11900 
 
 17830 
 
 22454 
 
 26410 
 
 29925 
 
 3-25 
 
 6329 
 
 9862 
 
 12614 
 
 18897 
 
 23780 
 
 27994 
 
 31714 
 
 3-375 
 
 6689 
 
 10420 
 
 13320 
 
 19963 
 
 25145 
 
 29570 
 
 33507 
 
 3-50 
 
 7049 
 
 10995 
 
 14048 
 
 21052 
 
 26509 
 
 31262 
 
 35329 
 
 3-625 
 
 7418 
 
 11574 
 
 14785 
 
 22153 
 
 27906 
 
 32860 
 
 37186 
 
 3-75 
 
 7794 
 
 12163 
 
 15526 
 
 23284 
 
 29321 
 
 34479 
 
 39080 
 
 3-875 
 
 8178 
 
 12753 
 
 16283 
 
 24416 
 
 30756 
 
 36170 
 
 41013 
 
 4- 
 
 8566 
 
 13354 
 
 17070 
 
 25592 
 
 32225 
 
 37898 
 
 42954 
 
 4-25 
 
 9355 
 
 14582 
 
 18643 
 
 27936 
 
 35191 
 
 41368 
 
 46916 
 
 4-50 
 
 10173 
 
 15849 
 
 20267 
 
 30366 
 
 38254 
 
 44982 
 
 50973 
 
 4-75 
 
 11001 
 
 17140 
 
 21908 
 
 32818 
 
 41356 
 
 48630 
 
 55102 
 
 5- 
 
 11833 
 
 18454 
 
 23595 
 
 35355 
 
 44546 
 
 52378 
 
 59346 
 
 5-25 
 
 12696 
 
 19802 
 
 25362 
 
 37939 
 
 47795 
 
 56209 
 
 63688 
 
 5-50 
 
 13576 
 
 21172 
 
 27248 
 
 40564 
 
 51097 
 
 60079 
 
 68097 
 
 5-75 
 
 14478 
 
 22580 
 
 29160 
 
 43253 
 
 54478 
 
 64058 
 
 72591 
 
 6- 
 
 15393 
 
 23995 
 
 31122 
 
 45969 
 
 57897 
 
 68082 
 
 77154
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 223 
 
 TABLE XII. DISCHARGES FROM THE PRIMARY CHANNEL IN THE 
 
 FIRST COLUMN OF TABLE XI. 
 
 If the dimensions of the primary channel be in inches, divide the 
 discharges in this table by 500 ; if in yards, multiply by 15'6, and 
 if in fathoms by 88-2, &c.": see pp. 126 and 127. The final figures 
 in the discharges may be rejected when they do not exceed one- 
 half per cent., or 0-5 in 100. (See pages 123 to 126.) 
 
 Falls, inclinations, and discharges in cubic feet per minute. 
 Interpolate for intermediate falls ; divide greater falls by 4, 
 
 Depths of a 
 
 and double the corresponding discharges. 
 
 channel 
 
 ;18 inches 
 
 1 inches 24 inches 27 inches 
 
 30 inches 33 inches 36 inches 
 
 width is 
 
 per mile, 
 1 in 3520. 
 
 per mile, 
 in 3017. 
 
 per mile, ' per mile, 
 1 in 2640. 1 in 2347. 
 
 per mile, per mile, per mile, ]70: infect. 1 
 1 in 2112.1 in 1920.1 in 1760.1 
 
 258 
 
 281 
 
 303 
 
 323 
 
 343 
 
 362 380 
 
 125 
 
 748 
 
 815 
 
 877 
 
 936 
 
 993 1049 
 
 1100 
 
 25 
 
 1387 
 
 1511 
 
 1627 
 
 1736 
 
 1843 
 
 1952 
 
 2037 
 
 375 
 
 2145 
 
 2336 
 
 2515 
 
 2684 
 
 2852 
 
 3023 
 
 3155 
 
 50 
 
 3004 
 
 3274 
 
 3527 
 
 3753 
 
 4021 
 
 4207 
 
 4414 
 
 625 
 
 3957 
 
 4311 
 
 4645 
 
 4966 
 
 5287 
 
 5553 
 
 5817 
 
 75 
 
 4991 
 
 5422 
 
 5859 1 6274 
 
 6650 
 
 6992 
 
 7342 
 
 875 
 
 6097 
 
 6622 
 
 7159 7631 
 
 8107 
 
 8540 
 
 8974 
 
 1- 
 
 7266 
 
 7920 
 
 8531 
 
 9124 
 
 9660 
 
 10200 
 
 10693 
 
 1-125 
 
 8514 
 
 9284 
 
 9995 
 
 10658 
 
 11318 11923 j 12520 
 
 1-25 
 
 9816 
 
 10697 
 
 11539 
 
 12307 
 
 13045 
 
 13741 
 
 14479 
 
 1-375 
 
 11182 
 
 12185 
 
 13152 
 
 14007 
 
 14862 
 
 15656 
 
 16448 
 
 1-50 
 
 12601 
 
 13730 
 
 14821 
 
 157S6 
 
 16750 
 
 17657 
 
 18552 
 
 1-625 
 
 14069 
 
 15331 
 
 16525 
 
 17616 
 
 18700 
 
 19698 
 
 20696 
 
 1-75 
 
 15593 
 
 16997 
 
 18306 
 
 19517 
 
 20728 
 
 21840 
 
 22944 
 
 1-875 
 
 17157 
 
 18697 
 
 20141 
 
 21469 
 
 22803 
 
 24017 
 
 25242 
 
 2- 
 
 18766 
 
 20446 
 
 22030 
 
 23480 
 
 24938 
 
 26269 
 
 27601 
 
 2-125 
 
 20410 
 
 22247 
 
 23965 
 
 25547 
 
 27129 
 
 28578 
 
 30027 
 
 2-25 
 
 22104 
 
 24087 
 
 25947 
 
 27662 
 
 29395 
 
 30934 
 
 32512 
 
 2-375 
 
 23848 
 
 25988 
 
 27992 
 
 29841 
 
 31701 
 
 33381 
 
 35096 
 
 2-50 
 
 25669 
 
 27953 
 
 30100 
 
 32069 
 
 34086 
 
 35910 
 
 37725 
 
 2-625 
 
 27479 
 
 29933 
 
 32247 
 
 34384 
 
 36512 
 
 38471 
 
 40415 
 
 2-75 
 
 29318 
 
 31947 
 
 34408 
 
 36697 
 
 38958 
 
 41055 
 
 43135 
 
 2-875 
 
 31206 
 
 34002 
 
 36624 
 
 39050 
 
 41464 
 
 43880 
 
 45896 
 
 3- 
 
 33141 
 
 36112 
 
 38897 
 
 41482 
 
 44048 46398 
 
 48747 
 
 3-125 
 
 35126 
 
 38266 
 
 41223 
 
 43954 
 
 46672 
 
 49174 
 
 51664 
 
 3-25 
 
 37109 
 
 40438 
 
 43556 
 
 46438 
 
 ; 49330 
 
 51951 
 
 54586 
 
 3-375 
 
 39140 
 
 42631 
 
 45925 
 
 48963 
 
 51993 
 
 54775 
 
 57550 
 
 3-50 
 
 41184 
 
 44872 
 
 48343 
 
 51537 
 
 54728 
 
 57659 
 
 50580 
 
 3-625 
 
 43273 
 
 47158 
 
 50807 54162 
 
 57514 
 
 60585 
 
 63656 
 
 3-75 
 
 45407 
 
 49468 
 
 53300 56840 
 
 60341 
 
 63560 
 
 66784 
 
 3-875 
 
 47551 
 
 51818 
 
 55832 
 
 59514 
 
 i 63200 
 
 66576 
 
 69951 
 
 4- 
 
 51911 
 
 56586 
 
 60973 
 
 64974 
 
 69013 
 
 72694 
 
 76383 
 
 4-25 
 
 56448 
 
 61508 
 
 66176 
 
 70623 
 
 75017 
 
 79017 
 
 82994 
 
 4-50 
 
 61014 
 
 66500 
 
 71625 
 
 76408 
 
 81097 
 
 85426 
 
 89767 
 
 4-75 
 
 65713 
 
 71628 
 
 77140 
 
 82250 
 
 87351 
 
 92015 
 
 96653 
 
 5- 
 
 70509 
 
 76863 
 
 82779 
 
 88200 
 
 93731 
 
 98729 
 
 103745 
 
 5-25 
 
 75383 
 
 82159 
 
 88434 
 
 94344 
 
 100200 
 
 105550 
 
 110905 
 
 5-50 
 
 80379 
 
 87590 
 
 94348 
 
 100616 
 
 106823 
 
 112540 118254 
 
 5-75 
 
 85407 
 
 93093 
 
 100275 
 
 106911 
 
 113505 
 
 119616 
 
 125664 
 
 6-
 
 224 
 
 THE DISCHARGE OF WATER FROM PIPES. 
 
 TABLE XIII. THE SQUARE BOOTS OF THE FIFTH POWERS OF 
 
 NUMBERS FOR FINDING THE DIAMETER OF A PIPE, OR DIMENSIONS 
 OF A CHANNEL FROM THE DISCHARGE, OR THE REVERSE ; SHOWING 
 THE RELATIVE DISCHARGING POWERS OF PIPES OF DIFFERENT DIA- 
 METERS, AND OF ANY SIMILAR CHANNELS WHATEVER, CLOSED OR 
 
 OPEN. (See pages 13, 123, 124, Ac.) 
 
 If d be the diameter of a pipe, in feet, and D the discharge in 
 cubic feet per minute, then for long straight pipes we shall have, 
 for velocities of nearly 3 feet per second, D = 2400 (d 5 s) 4 , and 
 
 d = -044 f j ; or if D be the discharge per second, D = 40 (oT's) 4 , 
 and d = -228 f V. (See pages 112 to 121.) 
 
 Relative 
 dimensions 
 or 
 diameters 
 of pipes. 
 
 Relative 
 discharg- 
 ng powers. 
 
 Relative 
 dimensions 
 or 
 diameters 
 of pipes. 
 
 Relative 
 discharg- 
 ing powers. 
 
 Relative 
 dimensions 
 or 
 diameters 
 of pipes. 
 
 Relative 
 discharg- 
 ng powers. 
 
 Relative 
 dimensions 
 or 
 diameters 
 of pipes. 
 
 Relative 
 discharg- 
 ing powers. 
 
 25 
 
 031 
 
 10-5 
 
 357-2 
 
 30-5 
 
 5138- 
 
 61- 
 
 29062- 
 
 5 
 
 177 
 
 IV 
 
 401-3 
 
 31- 
 
 5351- 
 
 62- 
 
 30268- 
 
 75 
 
 485 
 
 11-5 
 
 448-5 
 
 31-5 
 
 5569- 
 
 63- 
 
 31503- 
 
 1- 
 
 1- 
 
 12- 
 
 498-8 
 
 32- 
 
 5793- 
 
 64- 
 
 32768- 
 
 1-25 
 
 1-747 
 
 12-5 
 
 552-4 
 
 32-5 
 
 6022- 
 
 65- 
 
 34063- 
 
 1-5 
 
 2-756 
 
 13- 
 
 609-3 
 
 33- 
 
 6256- 
 
 66- 
 
 35388- 
 
 1-75 
 
 4-051 
 
 13-5 
 
 669-6 
 
 33-5 
 
 6496- 
 
 67- 
 
 36744- 
 
 2- 
 
 5-657 
 
 14- 
 
 733-4 
 
 34- 
 
 6741- 
 
 68- 
 
 38131- 
 
 2-25 
 
 7-594 
 
 14-5 
 
 800-6 
 
 34-5 
 
 6991- 
 
 69- 
 
 39548- 
 
 2-5 
 
 9-882 
 
 15- 
 
 871-4 
 
 35- 
 
 7247- 
 
 70- 
 
 40996- 
 
 2-75 
 
 12-541 
 
 15-5 
 
 945-9 
 
 35-5 
 
 7509- 
 
 71- 
 
 42476- 
 
 3- 
 
 15-588 
 
 16- 
 
 1024- 
 
 36- 
 
 7776- 
 
 72- 
 
 43988- 
 
 3-25 
 
 19-042 
 
 16-5 
 
 1105-9 
 
 36-5 
 
 8049- 
 
 73- 
 
 45531- 
 
 3-5 
 
 22-918 
 
 17- 
 
 1191-6 
 
 37- 
 
 8327- 
 
 74- 
 
 47106- 
 
 3-75 
 
 27-232 
 
 17-5 
 
 1281-1 
 
 37-5 
 
 8611- 
 
 75- 
 
 48714- 
 
 4- 
 
 32- 
 
 18- 
 
 1374-6 
 
 38- 
 
 8901- 
 
 76- 
 
 50354- 
 
 4-25 
 
 37-24 
 
 18-5 
 
 1472-1 
 
 38-5 
 
 9197- 
 
 77- 
 
 52027- 
 
 4-5 
 
 42-96 
 
 19- 
 
 1573-6 
 
 39- 
 
 9498- 
 
 78- 
 
 53732- 
 
 4-75 
 
 49-17 
 
 19-5 
 
 1679-1 
 
 39-5 
 
 9806- 
 
 79- 
 
 55471- 
 
 5- 
 
 55-90 
 
 20- 
 
 1788-9 
 
 40- 
 
 10119- 
 
 80- 
 
 57243- 
 
 5-25 
 
 63-15 
 
 20-5 
 
 1902-8 
 
 41- 
 
 10764- 
 
 81- 
 
 59049- 
 
 5-5 
 
 70-94 
 
 21- 
 
 2020-9 
 
 42- 
 
 11432- 
 
 82- 
 
 60888- 
 
 5-75 
 
 79-28 
 
 21-5 
 
 2143-4 
 
 43- 
 
 12125- 
 
 83- 
 
 62762- 
 
 6- 
 
 88-18 
 
 22- 
 
 2270-2 
 
 44- 
 
 12842- 
 
 84- 
 
 64669- 
 
 6-25 
 
 97-66 
 
 22-5 
 
 2401-4 
 
 45- 
 
 13584- 
 
 85- 
 
 66611- 
 
 6-5 
 
 107-72 
 
 23- 
 
 2537- 
 
 46- 
 
 14351- 
 
 86- 
 
 68588- 
 
 6-75 
 
 118-38 
 
 23-5 
 
 2677-1 
 
 47- 
 
 15144- 
 
 87- 
 
 70599- 
 
 7- 
 
 129-64 
 
 24- 
 
 2821-8 
 
 48- 
 
 15963- 
 
 88- 
 
 72645- 
 
 7-25 
 
 141-53 
 
 24-5 
 
 2971-1 
 
 49- 
 
 16807' 
 
 89- 
 
 74727- 
 
 7-5 
 
 154-05 
 
 25- 
 
 3125- 
 
 50- 
 
 17678- 
 
 90- 
 
 76843- 
 
 7-75 
 
 167-21 
 
 25-5 
 
 3283-6 
 
 51- 
 
 18575- 
 
 91- 
 
 78996- 
 
 8- 
 
 181-02 
 
 26- 
 
 3446-9 
 
 52- 
 
 19499- 
 
 92- 
 
 81184- 
 
 8-25 
 
 195-50 
 
 26-5 
 
 3615-1 
 
 53- 
 
 20450- 
 
 93- 
 
 83408- 
 
 8-5 
 
 210-64 
 
 27- 
 
 3788- 
 
 54- 
 
 21428- 
 
 94- 
 
 85668- 
 
 8-75 
 
 226-48 
 
 27-5 
 
 3965-8 
 
 55- 
 
 22434- 
 
 95- 
 
 87965- 
 
 9- 
 
 243- 
 
 28- 
 
 4148-5 
 
 56- 
 
 23468- 
 
 96- 
 
 90298- 
 
 9-25 
 
 260-23 
 
 28-5 
 
 4336-2 
 
 57- 
 
 24529- 
 
 97- 
 
 92668- 
 
 9-5 
 
 278-17 
 
 29- 
 
 4528-9 
 
 58- 
 
 25620- 
 
 98- 
 
 95075- 
 
 9-75 
 
 296-83 
 
 29-5 
 
 4726-7 
 
 59- 
 
 26738- 
 
 99- 
 
 97519- 
 
 10- 
 
 316-23 
 
 30- 
 
 4929-5 
 
 60- 
 
 27886- 
 
 100' 
 
 100000- 
 
 Woodfall and Kinder, Printers, An^el Court, Skinner Street, London.
 

 
 SOUTHERN BRANCH, 
 
 yNIVERSlTY OF CALIFORNIA, 
 
 LIBRARY, 
 
 1-OS ANGELES, CALIF.