M. Girard on Navigable Canals. 107 
it follows from this equation that the expense of water, 
y, will be positive, null or negative, according as we have: 
c> t,, raat, t, 
c= t,, —t, 
x St, —t,. 
Thus it appears that the expense of water from any lev- 
el may not only be diminished at pleasure, but that it may 
be rendered null, and that a certain quantity of water may 
even be raised from a lower to an upper contiguous basin. 
If two other boats successively pass the same lock, and if 
‘their respective drafts of water be represented as follows. G 
for the ascending boat, and ¢, for the descending one, the 
expense of water occasioned by this double passage will be 
represented by 
ae Tees (co bi) i 
In the same manner we shall have for the expense of a 
third double passage 
} xs oo — (ty 7) 
The total expense of the upper level of a lock, for any 
number n of double alternate passages will therefore be 
y ty" 4+ Be. ne ( (6, tit tn) — (646 
4ty-+ stmt) ) 
designating by odd numbers the drafts of water of the as- 
cending boats, and by even numbers those of the boats 
which descend. Therefore, if we make the sum of the 
drafts of water of the first ='T’, the sum of those of the lat- 
ter ='T’”, andthe total expense of water ie +y"+&e.=¥ 
we shall hav e, 
Y=ne — (T’ — 
The expense of waters for any eee of double passa- 
ges through the same lock, will therefore be positive, null or 
negative, Recor. as we have 
Tr T > 
ee 
n 
; oh ECGS GTN 
a ea rt 
Te Uy ple 
<a 
i 
And as the drafts of water of the boats always represent 
their weight and that of their cargoes, it follows that, in or- 
