1V5 



M. P. S. Girard on Navigable Canals. 295 



«'/+M^+M"',+w'\+, &c ; 



indicates the total rise of each of the levels B^ B,^ B^^^ Bi 

 &c. after any number n of successive passages. 



Let us follow and examine the effect of a boat going 

 from the summit level B, through all the levels B,^ B^^ &c. 

 of the canal. 



We have just found that the boat DS, in passing through 

 the lock E,, , whose fall is x' , , has produced in the first lev- 

 el B^ , a rise 

 S 



«>b;+s(^~''''^- 



But the volume of water to produce this rise has been 

 borrowed from the second level B,, ; the height of water in 

 the latter is consequently diminished by a quantity 



B. ' 

 whence we see that the primitive fall x' , of the second lock 

 becomes 



making this fall =z„ ; and representing by v'„ the rise pro- 

 duced in the level B,, , by the descent of the boat DS in the 

 level B,,. 



We shall find, in applying the same reasoning as in deter^ 

 mining the rise in the first level B. , 

 S 



■ ^'"=B:a^(^-^'")- 



But it is evident that the final rise of waters' of the level 

 B,,, above its primitive state, is equal to what is gained by 

 the passage of the boat into the second lock E, minu,s the 

 quantity it lost by the passing of the same boat into the first 

 lock E/ ; that is to say, we has^e 

 , , B,u', 



or by substituting for v',, and z'„ their value 



^*'"-B,,+s(^~^'"^~B„ + S' 



