le 



290 M. P. S, Girard on Navigable Canals. 



fore require a certain number of successive passages to aug- 

 ment the apparent height of water on the summit level ; but 

 this level would nevertheless have received a certain vol- 

 ume of water, which could replace, in whole or in part, that 

 portion which should have been lost by evaporation and fil- 

 tration, and which, without this, it would have been ne- 

 cessary to supply from the upper reservoir. 



Although the levels of a canal are generally largo enough, 

 when compared with the capacity of the locks, to render 

 the above supposition admissible, nevertheless as the most 

 general hypothesis is that of a finite relation between the 

 horizonial projections of the levels, and those of the locks, 

 we shall proceed to examine this general hypothesis, and 

 determine the law according to which the height of water 

 augments in any level by the descent, into an inferior con- 

 tiguous level, of a series of boats equally laden. 



In the first place it is evident, that by introducing into th 

 upper level of a lock a prismatic boat whose horizontal pro 

 lection is S, and whose draft of water is D, the same effect 

 will be produced, to modify the height of water in this level, 

 as by the introduction of a volume of water=DS. 



If therefore we designate by h the primitive height of wa- 

 ter, that height, after the introduction of the boat, will be- 

 come 



DS 



^+-B 



The lift X of the lock will at the same time become 

 DS B:c+DS 



a;-f- = • 



B B 



In the second place, it is evident that the water in the ba- 

 sin S, can only be raised to the height of the level B, by 

 operating in the latter a certain given depression z, as we 

 may assure ourselves by the equation 



/Bx+DS \ 

 Bz=S\^— g — -z), 



Whence 



_ S(B:c + DS) 



^= B(B + S) 



Things being brought to this state we introduce the boat 

 DS into the basin, and shut the upper gate of the lock, and 

 the primitive height of the water in the upper level will be 

 augmented by the quantity 



