116 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



LApbil, 



Calling A B = rf. AE = ^=CD. BD=z. FG = h. HI = 



n. L .M = c The velocity of the recipient l)ef()re receiviiitf tlie 

 influent II. Its velocity iifter having? received it, hut hefore it 

 cimld exercise any pressure and reduce it to e(|uililirium ; tliat is 

 the same wliidi it would h ive if the water of the influent ran in 

 the width of the recipient = t, the velocity which the recipient 



I L 



-i I— 



B 



M 



has after the union and after the waters have equilibrated in 

 their course = 7, and the velocity which the influent had in its 

 own level before the union = r. Then since the two masses of 

 water of tlie influent and recipient in a given and equal time can 

 pass separately in tlie level of the recipient, they ought to be able 

 to pass together through the aforesaid recipient. Hence the 



equation du + t j; = q z and z^ '^'"'^'^ first general formula; 



now since equal masses of water pass in equal times both through 

 the influent separately and through the aforesaid influent when re- 

 duced to the width of the recipient, we shall have ct x z= abr, 



whence .r=: ILL and z= Elji±lAl^ the second general formula 



expressing the whole height BD; wherefore AD, which is the 

 whole increment produced by the influent above the first state of 



the recipient will be '"^" + «& >" - c(^>? . 



cq 

 III. Corollary 1.— If the velocity be a mean proportional to the 

 height, we shall have A P~ ^V'^ + .^^V^ -d^z _^_^ ^,,^.^1^ 



reduces itself to z = W (d^ + 2dx ^y di: + ^■') and A D = 

 V('i-^ + 2rf^Vrf^-^')-rf,inwhichx=A!i^, as is ob- 



tained by substituting in the formula ctx =abr,the values of 

 < and r, which are v'^,V*, which value of x if substituted for 

 that of t, Will give the value of A D. 



IV Corollary^.— On the supposition of Castelli and of Barattieri 

 that the velocity will be as the height, we shaU have j; = 



'^(ji'l+ «A*) and AD= v(<i<^ + ^*)-</. 



V. Corollary 3. — And consequently if u = d"' r =b'' q = z"" 



where m, n, <j) are numbers which may be integers or fractions ex- 

 pressing any power of the height by the velocity, we shall have the 



1 

 general formula z = (d'n+ ^ + ax c-'i b^+^)'P+l^ in which x 

 being already eliminated, it only remains to substitute the values 

 of'/, «, c, b, taking the aforesaid exponents as fixed, supposino- z d 

 unknown, the aforesaid formula will give the general equation of 

 the whole curve of the increment of the river by the addition of 

 the other stream, the abscissa uf which will be ;:;, the ordinate d or 



more generaUy making u — dj,, r = bfi, q = zp, we shall have 



( 



c d 



and that 2 *'+/' = 



and we shall be able to determine the 



relation of s to d in the following manner : — 

 p +m m 



Letd f —b''y; now dP-^"'=b'"yP; weshaUhave z1'-^P= 

 n+ p — m 



( 



y + 



fi b' 



> 



X bJn Construct the curve A E expressed 



by the equation d P '*'"' =zb 

 TakeB 



y 



p + n — m 

 \ =:a b — and from 



the point B describe another curve, 

 which has for its equation zf^P — 

 n + p~m^ p 



( 



y + 



ab 



■) 



X6" 



Shall have D E = d, C D = z, and 'he intercepted portion C E will 

 he the increment required. 



V'l. Sclwlium 1. — In the simplest case of the velocity in propor- 

 tion to the height, using the first formula of the preceding corollary, 



change this into d d — z z , the equation of the equilateral 



hyperbola b A, of which as 



well the parameter b n as the 



,. , , 2 b /sj o, 

 diameter b m = — 



wherefore D B will be the 



fa " j/_ ^ ^ height after the union of the 



b B ivater, and B A the height 



wliich the recipient will 



have on first receiving it. And by the properties of the equilateral 



hyperbola, the square of B A beintr equal to the rectangular 



B m X 6 B, tliat is, to the ditference of tlie squares D B, D i, we 



shall have analytically d d= z z , which is the equation 



proposed ; whence appears the method of describing such a hyper- 

 bola, so as to contain every possible case of increment arising from 

 an addition of water. And calculating with the second formula 

 the two parabolas of the preceding corollary, we shall have 



d d — b y, B A = - ' and z z — b y-{- 



bb a 



and if for b y we sub- 



b b a 



stitute its equivalent d d, we shall have z z - d d-\ ■, the 



equation which is found and constructed above. 



VII. Scholium 2. — If the velocity is as the root of the height, 

 the equation resulting from the first formula of § V will ascend 

 to the sixth dimension of the unknown quantity, and the progres- 

 sion will be c-> z"—2 a- b' cc z" -\- a* b'^ -2 C d' z' — 2 a" c- 6^ d'^ 

 -\- c' d" = 0, which does not transcend the limits of a cubic equa- 

 tion ; but with the second formula 

 ba 



9 



= 5 we shall have 



z^ 



(»+vV 



X b; 



P P P 

 and supposing d' = b y\ A E will be the 



parabola expressing the aforesaid equation, and B C that of s' = 



( y -\ )" X 6, without otherwise embarrassing itself in the 



resolution of the aforesaid equation, already sufficient complicated. 

 VIII. The converse proposition to § II. deduced from the for- 

 mula there enunciated, which gives the height of a river from 

 which a quantity of water is deducted, to find the section of a 

 canal, such as shall dischauge the same quantity of water, and 

 whose height B D shall descend to B A. The equation cq z = 

 c d u-\- a b r is then changed into a b r — c q z — c d u, which 

 solves the problem. Let it be required to diminish it by such a 

 quantity of water as may have to the first, before the subtraction, 

 the ratio of /to/), whence we shall have the analogy cq;! '. cd u;;t ; p, 



by makingr=6»,andM=dm,and wehave6= ( -X — -Xd'"'^ J^Tl 



whence we deduce the height of the canal of deduction d = 



( -Xr^-X b Jm + l formula, which denotes the height which 



that river from which the water has been subtracted will have 

 acquired after such deduction. 



If the water of a river be diminished by a given height after the 

 canal or derivation be opened, and the height of the effluent b is 

 noted, required its breadth a. Let the first height before the de- 

 duction be to the second, after the latter has taken place, as e to/; 



d e 

 hence z : dy,e : /whence z =i—r . Therefore by substituting this 



value in the general formula, since we have already r = 6 ", 3 =« '', 



