M. P. S. Girard on 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 in the 
lower level only tends to increase the quantity on the sum- 
mit level. 
If, in this equation, we substitute in the place of x’, its 
value f 
__(D—z’) 
B+S 
we can deduce from it 
SD+B x’, 
£(n) =—_.— 
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 
(D—2@) )—a 
ed (B,+B, +B +... Ba)) 
B (a) +5 aK Ons 8) 
because B=B,=B,=---- Ba- 
ad BB +B, +B,,+~ Boab= =(n 1B; 
will become eS 
n—t 
-( D—2m)—SE a 
(Ss 8) 
Orns 
whence a(n) =D — a —. 
We have also: S ny * | 
wo yeD gS O—DB) ; 
aB 
therefore B(n—I)— 2) =e 
that is to say, the falls of two adjoining locks, taken at hangs 
ard in sick canal, cae from each other by a con v4 
quantity, or, which come e same thing, the falls of all 
the locks of a system diminish, from the first to the last, in 
a 
an arithmetical progression, the ratio whereof is ~~ 
