M. P. S. Girardon Navigable Canals, 301 



which answers to the case where each intermediary level 

 gains precisely as much water on the one side as it loses on 

 the other ; so that the whole volume of water taken m the 

 lower level only tends to increase the quantjty on the sum- 

 mit level. 



If, in this equation, we substitute in the place of u , its 

 value 



we can deduce from it 



SD+B x', 



B+S 



whence we see that the falls of all the sluices, whatever be 

 their numerical rank, after the first, are equal to each other. 



Preserving the hypothesis of an equal rise in all the lev- 

 els of a canal, let us suppose farther that all these levels are 

 equal to each other in extent. 



The general equation 



B^r.^+S^ ^^^ (B,vo+S) 



because B^=B ,=B^^^=- ••• B(n-i)=B; 



and B^-f B,+B,,+ • • • B(n- i) = (n- 1)B ; 



will become 



S -rk \ (n— l)aB 



whence x(n)=T>—a^-^- — - 



We have also : 



(S + (n-l)B) 



therefore x(n—r) — oc(n)=-—- 



& 



that is to say, the falls of two adjoining locks, taken at haz- 

 ard in such a canal, differ from each other by a constant 

 quantity, or, which comes to the same thing, the falls of all 

 the locks of a system diminish, from the first to the last, in 



an arithmetical progression, the ratio whereof is — — , 



