292 M. P. S. Girard on Navigable Canals. 
This value of uw’ substituted in the second equation 
S 
w= pa 6(D-(e'+u)) 
gives * 
ne Sh B ) 
~. Spe (¢50-») 
Substituting, in the same manner. the equivalent of a’ and 
u’ in the equation 
Ss 
We Bys D-(@+u'+u’)), 
it becomes 
Pah Ga) 
* =B+S \(B+S)2 
We shall find successively : 
uw =ars(gpssP—y) 
4 PS 
+ ae (p5;r-=)) ‘ 
Therefore the sum of the successive augmentations on 
the upper level B, that is to say Be 
, ” a iv S pons 
wee ha es uD BiS (D—x) (Cex) 
‘ B 1 B 2 B 3 B n~4 
(szs)* (ess )+(ar5) + me ee 
Or. in other words, the whole elevation of the surface, 
occasioned by the passage of a number n of boats, is eX 
ssed by a decreasing geometrical progression, the num- 
ber of whose terms is n, = whose ratio is 
Whence we conclude that whenever the superficies of the 
besin S is not so small, in proportion to the superficies B of 
the upper level, as to be neglected, the rise of water occa 
sioned by the descent of a boat S, will always be propo! 
tlonate to a certain power of the traction ‘ 
B 
B+S ? : 
power whose value will always be so much the less as the 
boat approaches the numerical close of the series of descend 
ing boats, 
