M. Girard on Navigable Canals. 111 
when the number of boats ascending, is equal to the num- 
ber of boats which descend ; this supposition being more 
simple than any other, it is that to which we shall confine 
ourselves. 
Now it is evident that if any number n of boats pass the 
lock one by one, ascending and descending alternately, the 
condition of no expense of water will be expressed by 
Ly NE 
in 
If, on the contrary, the same number of boats go in two 
convoys, the same condition will be expressed by 
if Des A 
ra 
much nearer to 
T’—T’ 
an 
as the number n of boats become greater ; whence it follows 
that the two quantities 
T’—T’ T’—T 
A and PON 
are the two limits between which the lift of the locks of a 
canal should be confined, in order that the expenditure of 
water from the upper level should be null, in whatever num- 
ber of convoys the given number of boats successively as- 
cend and descend. 
If therefore, we give to the locks a lift which is an arith- 
metical mean proportional between the heights which ren- 
der the expenditure null in the two extreme cases we have 
supposed, that is to say, if we make 
en 2 
Re. an Von 4n 
this height will be most likely to fulfil the condition which 
we have proposed, of rendering the expense of water null, or 
nearly so, on the passage of any quantity of boats distri- 
buted in convoys as chance may offer. ; 
This lift is, as we see, 2 of that which would be proper if 
the boats alternated one by one at the passage of each lock : 
this last order of passing is the one generally adopted, as far 
as practicable, in navigable canals, and the preceding analy- 
