300 M. P. S' Girard on Navigable Canals. 



We have supposed that each level was exposed to an 

 equal chance of deperdition of water ; but if, from the ef- 

 fect of some particular circumstances, any one of the lev- 

 els B(,.). was exposed to a greater loss than the others, it 

 would be necessary that its level should be raised by a 

 quantity A, proportionate to its greater losses ; whilst the 

 levels which precede it should only be raised by the quan- 

 tity a, and we shall have 



Bc»)4-S _ ^ ^ (B('0+S) 



whence we immediately conclude that the length of the 

 level B(n) being given, the fall of the sluice x(n) should be 

 so much the smaller as the rise of water A, destined to re- 

 place the loss attributed to this level, becomes more con- 

 siderable. 



If, on the contrary, the level B(n) were to receive a 

 quantity of water, from some other channel, it might gain 

 instead of losing water, and consequently preserve water 

 sufficient for the purposes of navigation, even after being 

 depressed as much as it had been elevated by the auxiliary 

 stream. The quantity A changes its sign in this hypothesis 

 and the equation 

 -A==-A_(D -. Q)_, (B,+B.+B..+. . . . fee.) 



B(n)4-S^ ^^' _ B(«)+S 



shews that the fall x(n) of the sluice, which terminates thfr 

 level, may be so much the greater as the volume of watei' 

 introduced into this level, is increased. 



This level, receiving an additional supply, may in its 

 turn be considered as a culminant point of a new canal, to 

 which all that we have heretofore said of the other may be 

 applied without restriction ; the same thing will again occur 

 after a second, and a third supply of water, and we are thus 

 brought back to the propositions laid down in our first me- 

 moir. 



If, in the general equation, 



we suppose the rise of water in all the levels, except the 

 summit level w' to be null, the equation will then become 



DB u 

 — X{n) — ^ — = 0, 



