114 M. Girard on Navigable Canals 
vided among a certain number of locks whose lifts are res- 
pectively w,2,,2,, &c. we shall have 
BAL AL pe Pnj=Hh 5 
And for the loss of active force on the whole length of the 
canal, the sum of the squares : 
LP vy? FO? Hod? (n) 
which will diminish in proportion as the number x of the 
locks increases. 
The particular case where all the locks have an equal 
lift gives : 
h 
L SL HT 5, He oe On 
In which case the loss of active force becomes : 
nh? h? 
nr on 
{f the total rise of the canal be divided into any other 
h : 
number n’ of equal locks, that loss becomes = the losses 
of active force, in the two hypotheses are therefore : 
1 1 
Big uM sn 
that is, they are in inverse proportion to the number of 
locks employed to obtain a given height. 
Designating by y and 7 the positive expenditure of wa- 
ter in the two suppositions, we have : 
h 
y—'—(t'—t) 
yet) 
Whence we see that the expenditure of water is so much 
the less as the number of locks is greater, or as their lift is 
smaller. 
It becomes exactly proportional to the lift when the as- 
cending and the descending boats have the same draft of 
water, the only supposition which has been tacitly admitted 
hitherto ; for in this case we have : 
h 
, 
If we only consider the distribution of the locks of a navi- 
gable canal in relation to the expense of water, which is to 
be supplied by the higher levels, the preceding analysis 
shews the great advantages that will result from diminish- 
