M. P. 8S. Girard on Navigable Canals. 29) 
_DS_ S (Br+Ds) 
S 
v= — B (BLS) — B+S-*)- 
We have supposed D greater than x; consequently this 
quantity w’ will always be positive. Thus, then. the primi- 
tive depth of the water in the upper level, which was=A 
before the introduction of the boat DS, has become h+w 
after it is withdrawn. 
The fail of the lock, which was zx, has also become 
a+’, 
Let us now suppose a second boat DS introduced in the 
level B, and that by repeating the manoeuvres we carry it in- 
to the contiguous level below, whose height is constant; it is 
easy to perceive that the height 4-+-w' will be augmented 
by this second passage, by a quantity 
ts) 
u'=Bg(D —(z+u')). ; | 
A third passage will cause a third increase : 
S 
w"= Big D _ (x-+u'+u")) ; 
And, in general, the augmentation occasioned by the pas- 
sage of the nth boat will be 
u(,) = 5s (D —(e--u -ul fu... ty 1) 
In examining this latter expression, we perceive that the 
augmentation & (,) diminishes in proportion the 
nm of passages increases, and that it is nol] when 
D— (aw! bul pul vo Wn) HO 
That is to say, when the sum of the successive elevations 
and of the fall of the lock is equal to the common draft of 
water of the boats, : a 
After a number n of passages, the height of the level B, 
which was = A at the cepepenrate ab being pcre? by 
the seriesh--u’ eu! ful tu +. eee Ul 
may be found be becteting for the successive augmenta- 
tions u’, wu", ul, &c. their values in iunactions of the known 
quantities D, B, S and zx. 
We have already found, above, 
$(D-—- 2) 
B+5 
a= 
