M. P. S. Girard on Navigable Canals. 29 1 



"'""B"" B (B+S)"~ B+S^^"'')* 

 We have supposed D greater than x ; consequently this 

 quantity u' will always be positive. Thus, then, the primi- 

 tive depth of the water in the upper level, which was=^ 

 before thp introduction of the boat DS, has become h-\-u' 

 after it is withdrawn. 



The fall of the lock, which was x, has also become 



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 A+m' will be augmented 

 by this second passage, by a quantity 



M" = -g:^(D-(a; + M')). 



A third passage will cause a third increase j 



S 

 "*" - B+S'(^-(^+'*'+'*")) 5 



And, in general, the augmentation occasioned by the pas- 

 sage of the nth boat will be 

 S 

 «(J=-gq:g-(D-(a;-|-w'+M"+M"'4- .\ . w„ ^)). 



In examining this latter expression, we perceive that th^ 

 augmentation u („) diminishes in proportion as the number 

 n of passages increases, and that it is null when 



D — (a; + M'4-w"+w"'+ . . . M(„_ j) = o. 



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. 



After a number n of passages, the height of the level B, 

 which was = A at the commencement, being represented by 



the series h-{-u' -\-u" -{-u'" -\-u '^4- u in), their sum 



may be found by substituting for the successive augmenta- 

 tions u', m", m"', &;c. their values in functions of the known 

 quantities D, B, S and x. 



We have already found, above, 

 SiD-x) 



u'= 



B+S 



