1848. 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



81 



ON THE MOTION OF WATER.— By Guido Grandi. 



Tran.shited bij E. Cresy, Esq., in his Evidence before the Metrnpo- 

 Utan Sanitary Commi.inioiwrs. 



Our author has taken considerable pains to construct a paraliolic 

 table, siven in his work (Book 2, cap. 5) ; by a reference to which 

 much labour will be saved by those who desire to make similar in- 

 vestigations ; he thus describes it :• — 



" This table is divided into three columns. The first containing: 

 a natural series of luimbers from 1 to 1800, representing: equal 

 parts, as inches or other measures. These numbers are the heights 

 from which tlie water falls. The second column contains the roots 

 of the opposite numbers in the first, and expresses tlie velocity of 

 the water, corresponding to the height in the first column, in inte- 

 gers and decimals : when the root is somewhat greater than the 

 truth, the sign -|- is prefixed, and when less -. The third column 

 contains the product of the first and second, and must be read off 

 as exceeding or falling short of tlie truth, accoi'ding as tlifi sign -|- 

 or — is jirefixed to its second factor. 



It is clear that if the numbers of the first column express the 

 height of a jiarabola, the numbers in the second will be its ordi- 

 nates wlien its latux rectum, or parameter, is 1 ; or at least, they 

 will be proportional to the ordinates in subduplicate ratio of unity 

 to the hittis rectum of a given parabola, and the numbers in the 

 third column will be the rectangles circumscribing the parabola 

 wMch has unity for its latus rectum, and will be moreover propor- 

 tional to the area of the parabola, H'hich is always |rds of the cir- 

 cumscribing rectangle. 



If the parabola lias 2j for its latus rectum in terms of the first 

 column, all its ordinates are to the ordinates of the parabola of 

 the same height, having 1 for its latus rectum, in subduplicate 

 ratio of 2^ to 1, that is, as li to 1, or as the circumscribed rect- 

 ann:le to the parabola, it is clear that the jiarabola whose latus 

 rectum is 2^- wiU be equal to the rectangle which circumscribes the 

 parabola wliose latus rectum is unity ; but such a rectangle is equal 

 to the product of the base by the height, which is the number 

 opposite in the third column, therefore the numbers in the third 

 column express the area of a parabola whose latus rectum is 2j, and 

 is proportional thereto when the latus rectum is any other quantity. 



Moreover, since the numbers in the first column express the 

 height of water standing in a vessel, or the distance of each par- 

 ticle of running water above its base, and the numbers in the 

 second column representing the velocity caused by such a height, 

 the numbers in the third column express the quantity of water 

 which will issue through such a width in a given time, through a 

 hole or section whose height would be equal to the whole distance 

 from the surface of the water or origin of the river, and the base 

 of such a section as the number in the first column. 



The difference of numbers of the third column will be the 

 quantity of water which escapes in an equal time through a hole 

 or section of equal breadth, and of a height equal to the difference 

 of the corresponding numbers of the first column. 



By adding two or more numbers together of the third column 

 we shall have the sum of the quantity of water carried in a given 

 time through several canals of the same width, whose sections corre- 

 spond to the numbers of the first column ; and in the aggregate 

 of such numbers, or the nearest thereto, in the third column will 

 correspond to that number in the first, which indicates a height 

 capable of comprising the channels united, as will be better under- 

 stood by the following examples : — 



1st. Given two streams, the breadth of the first of which is L =r 

 760 feet. The velocity of the surface B E corresponding to the 

 fall A B of 1 foot (which, according to Guglielmini's table is equiva- 



M 



lent to 216 feet 5 inches per minute, that is, 3J feet in a second, 

 or 2J miles per hour), the height of the surface B C = 30 feet, 

 whence A C 31 feet ; then the whole parabola A E D C, according 



to the third column of our table opposite 31 feet, will be found 

 7175'88, from which subtracting the parabola A E B, wliich is 

 found in our third column to be 4.1 '52, tlie parabolic trapezium 

 B E D C will be 7134'36, and this will be the scale of the velocity 

 of the section B C, wliich multiplied by the breadth L gives a 

 quantity of water = 512211360. 



The second stream ha\ing a width M = 139 feet, its superficial 

 velocity will be G K, depending on the height F G, 8 inches 

 (which gives, by Guglielmini's table, a velocity of 176 feet in a 

 minute, rather less than 3 feet in a second, and 2 miles 56 perches 

 in an hour). The height of its surface G H is 11 feet, and con- 

 sequently F H 1 1 feet 8 inches, corresponding in our third column 

 to the value of 1656'20 for the parabola F K I H, from wliich sub- 

 tracting the parabola F K G, which our table gives opposite 

 8 inches as 22-64, there remains the trapezium GK I H 1633"56, 

 which is the scale of the velocity of the second stream, which, 

 multiplied by the width M, gives the quantity of water passing in 

 a given time tlirough this river ^ 227064'84 ; whence the two 

 quanties carried by both tlie rivers will be 5649178'44. Supposing 

 they flow together, without increase of velocity, B E ^ O R ; and 

 let the height O P, at whicii the united water runs, be the unknown 

 quantity, then since O N =; B A through R, and with the axis 

 N P, describe the parabola N R Q P, the truncated parabola 

 O R Q P will be the scale of the velocity of the united rivers, 

 which multiplied by L ^= the sum of the two quantities =z 

 5649178-44, which divided by L gives a quotient 7433-13 = the pa- 

 rabolic trapezium O R Q P, and adding the parabola N R O = 

 41-52, we shall have the parabola N R Q P = 7474-65, the nearest 

 number to which in the table is 7464-28, corresponding to a height 

 of 31 feet 10 inches. This number sought being rather more than 

 the tabular value, it will be found bv proportional parts that ^ 

 must be added. Therefore N P = 31 feet lOi inches and O P = 

 30 feet 10^ inches ; therefore the union of the streams raises the 

 level B C 10^ inches. 



But if, at the conflux of the rivers, the velocity B E augments, 

 becoming O R, so that the height N O depending on it exceeds 

 A B by 1 inch, the parabola N O R, corresponding to a height of 

 13 inches, will equal 46-93, which, added to the trapezium R OPQ, 

 found previously to be 743313, we shall have the total parabola 

 N R Q P = 7480-06, the nearest number to which, 7464-28, corre- 

 sponding to 31 feet 10 inches ; but since this is rather too little, we 

 must add ^ for the proportional part of the difference, wlience N P 

 = 31 feet 10^ inches ; from which N O = 1 foot 1 inch being sub- 

 tracted, there remains O P = 30 feet 9i inches, making the total 

 increase in this case 9^ inches. 



But if we suppose "with Guglielmini, and which is not impro- 

 bable according to actual observation, that the scale of a velocity in 

 a given section is an entire parabola and not a truncated one, the 

 velocity, as in the case of vessels depending only on pressure, 

 whence the surface alone acquires velocity when it is communicated 

 by the lower water which transports it, tlie calculation will then be 

 more quickly effected. Wherefore A C = 30 feet, the height of 

 the first river, and F H = 11 feet, height of the second. The 

 parabola A E D C = 6829-20, in our table, which, multiplied by 

 the width L 760 feet, gives for the quantity of water 5190192-00, 

 and the parabola F I H = 1516-68, which multiplied by the width 

 M = 139 feet — 210818-52, whence the sum = 5401010-52, which, 

 divided by the width L, gives, when the velocity of the surface is 

 not increased, the parabola N Q P = 7106-59, corresponding to a 

 height of 30 feet 10 inches, corresponding in the table to the number 

 7118-80, which is rather more than the preceding; wherefore 

 the rise will be 10 inches. 



Then if the velocity of the two rivers increases at their con- 

 fluence, the height will be reduced in the reciprocal ratio of that 

 velocity ; so that if the velocity be increased t^, the height will 

 be reduced to 30^ feet, that is, the increase will only be about 

 6 inches ; if the velocity increases Wr, the height will be 29 feet 

 8 inches ; so that the height, in place of increasing, will actually 

 be reduced about 4 inches by the union of the two streams ; so 

 likewise the height 30 feet, will remain precisely the same when 

 the velocity is increased by ^, since 37 : 36 ; ; 30 feet 10 inches ; 

 30 feet. 



Example 2. — The influent C B D R in a given point of its bed 

 has the height O H, having a free influx into the recipient R M, 

 when it is low, and its superficial velocity in H is what would 

 correspond to a height A H of 4 feet. Then, raising the level N S 

 of the recipient, regurgitation follows through the level of the 

 influent. It is required to find the increase in the height O H = 7 

 feet } Suppose it to increase as far as Q, draw the parabola A K R, 

 with its ordinates H Y, Q K ; let O S, cut off by the prolongation 

 of the level of the recipient, = 3 feet ; the whole height A O will 



12 



