IHiS. 1 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



117 



( 



and u = rf™, the equation will be reduced to the following -,0 = 



s ; . in which a and a are the unknown 



ftp+ 1 n + 1 ' 



quantities, and f,/, b, z, the ^iven quantities ; or else, if a certain 

 breadth be given, and the heiglit remains unknown, we shall have 6 = 



e 9-+lrf-P+I-/»^-lrf»' + ^ 1 _ Now in the case of hori- 



zontal, or nearly horizontal streams, the canal of derivation being 

 open, whose bottom regulates also the height of the water of the 

 river ; that is to say, the portion which acts to produce the greater 

 or lesser quantity which it deducts, the other remaining inactive 

 in regard to such a canal of derivation, the formula will be d = 



IX. — Corollary 1. In the second formula of the preceding para- 

 graph let m = re = 1 we change it to d = 6 X a/ 



-in which 



el — c p 



if / = 4000, p = 3500, numbers expressing the quantity of water 

 which passes through a given section of the river both before and 

 after the subtraction of the water, 6 = 10 feet, a = 200 feet, c = 300 

 feet, performing the proper operation, the logarithm of rf = 

 1-334.503-t answering to 21ig?sj{. The value of the lirst height be- 



fore the diminution will he z = 





-, where d being known, all 



feet. 



X. — Corollary 2. Making m = jt = ^, which is the case of Torri- 

 celli, Mariotti, and others, transmuting the aforesaid second for- 

 aa pp J d'i\/ 1 1 



mula into rf = 6 'j/ 



and z : 



substituting the 



cc{l-py "Wpp" 



proper quantities and placing the values of the quantities I, p, b, a, c, 

 as above, the logarithm of z = 1-4846658, the number to which is 

 30"i"j^, from wliich it appears that if the river be lowered by the 

 water, diminished by the effluent so that the first height shaU be 

 to the second after the diminution as235"fj^,"j to 21.]j?§3l, the quantity 

 which passes through a given section below the point of diminu- 

 tion, before the water is subtracted, will be to the quantity which 

 passes through the same section after the water is subtracted as 

 40 to 35 in the first case, and the height in the second case as 



QA 75416 +n O7Mfl407 



XI. Corollary 3. Taking the third formula of the preceding 

 paragraph, in which we have supposed e,/, d, 6, c given, making 

 n, (p, m = 1, by the hypothesis of Castelli, let us seek the width of 



■//). 



the canal of derivation. 



We shall have o =: ^ „ x ^ 



// bb 



Suppose e :/: : 9 : 8, rf = 20, 6 = 18, e = 300, the logarithm of a = 

 1'9929051, corresponding approximatively to the number 98, of so 

 many feet will be the widtli of the canal of derivation, that the 

 first height may be to the second, after the water is diminished, as 

 9 to 8 ; but on the supposition that m, n,tp = ^, will be the formula 



, ,^ cd^d (et/e — f's/f) , , , 



changed to o = x ^ *^ . — , , \ and the logarithm nearest 



fV/ a/ b 



to a will be 1-8900925, whose nearest number, 78, will be the width 

 required. 



XII. Corollary 4. Using the general formula, s = ' 



to obtain the residual height of a river, after a certain quantity of 



c q z — a b r 



water has been subtracted, we shall have d = 



Now 



by substituting for g, r, m, their respective values «"'', b , d , we 



shall have d — l'^ 



-(• 



~'P+ 1 



■ab' 



S + l\ 1 



1 ; if then 9, S, m will be 



equal each to 5, we shall have the equation, 

 2 (}■ i' rf^ „ „ _,, _ , 2 a' 6' 



'do 



-2d3; 



^ a* 



6« 



= 0. 



C - c' ■ c^ 



Or if, for greater simplicity, we reduce it to the following expres- 



2 zc m- \/ b z 



abb 



'): 



X a^' j ; and since x = 



b^ y/ aa, 

 ' \/ c 



sion, d =: 'V ( ~' 



we shall have d = ( W'^^-'-^o^^bj^^^bz + aab^y 

 XIII. Corollary 5. — Using the preceding formula, in which we 



have constructed two parabolas, according to what has there been 

 laid down, 



abPjtl' 



p + m 



<p+;) 



p + n 



p + m 



<t>+p 



c d p 

 now d 



= c z p — ab p , and thence d p =z p c P 



I ip + P ubp + n\p , ,. <J> + B m 



p + m — I — 1 and making ;; — - r=b-l/ 



we have, making the necessary substitution d P'*'"' = 



( y -) „ / X "■. Now let B C be the curve whose equa- 



„ ^p + n-m 

 Wonis z'^*P=:b"' yp. TakeBA^ — p , and from the point 



A describe another curve A E, expressed by the equation d^ + '" = 



/ nbP'^"~'^\P 



[y — 'J< — / X 6*", we shall have D E = d, C B =: ^, and 



C E the required difference of height. 



XIV. Scholium 1. — We shall give some examples of the fall of 

 the surface of rivers, produced by derivative canals, as tliey liave 

 been called, and these examples will be taken from the Adige, 

 which, as is known, affords many such, and on which I had cause, 

 at various times, to make several observations for its general regu- 

 lation. It was found 



1st. That the Bova della Badia, in flood time, measures 10.7.4 

 Venetian feet, or 1528 lines, above the sill; a its breadth is 12^ 

 feet, or 1800 lines. The reduced height of the Adige, opposite it, 

 at flood time, was 11.3.1, or 1621 lines, being 402 feet wide, or 

 57888 lines; now l)y a preceding rule, § VI., in wliich we supposed 



1 • b \/ a 



the velocity as the height, having x ;r: — - — = 269, and conse- 

 quently d = V (^ *"" •'^'') ^ I'^^^i which being subtracted from 

 the height of the Adige before the diminution, there remains 

 23 lines, that is 1 inch 11 lines for the required diminution. 



2nd. At the mouth or sluice of the Sabbadina we found that 

 »=r 19.1.11, or 2759 lines; 6 = 9.2.11, or 1231 lines ; a = 27ifeet, 

 or 3960 lines ; c = 2280 feet, or 30240 lines, whence x =r 554 and 

 d^ i\/ {z z — xx) = 2703, which, deducted from 2759, the first 

 height, gives 56 lines or 4^ inches. 



3rd. At the sluice of the New River, when it was of wood, it 

 was found that z ~ 10.8.4 = 1480 lines, b — 4.10.8 = 704 



