M. P. S. Girard on Navigable Canals. 297 



which, at the commencement, were a;', x\, x' ,„ a?'iv , Sec. 

 will have become x",x",iX'\„ x''^^, &tc., and bj substitu- 

 ting these quantities in heu of the original falls, we may, 

 in the same manner, determine the augmentations u'\ tjl' ,, 

 u"i„ u"\^ occasioned on the levels B< B„ B„; Biv , &£c. By the 

 passage of a second boat, we shall therefore have 



n _ S ,^ B, U"^ 



'^""~ B,,+s(^ ^"")-(B,,+S) 



"'"~B,,,+S^^ "" "'>~ B,,4-8 



«' ^—m-x") iM'£tB,.!f^±B^.J_ 



"''-Biv+S^ •^^~ (Biv+S) 



equations which will give the augmentations u" , u"„ v!' „,^ 

 M"iv, Ssc. in functions of the quantities S, B, B„ B^„ Bj^ . 

 &c. and in functions of the primitive fall? x\ x\, x\,^ x'i^ , 

 &;c. of the locks E, E^, E,,, Ej^, Sec, in paying attention to 

 the new falls of these locks 



x'\=x' ^-\-u\—u' j^ 

 X i=x^.-{-u^i ic 1^1 

 x",^=x',^^-{-u\,,-u'i^ &;c. 



The levels B B,, B,;, Biv, &c. will be respectively raised 

 by the quantities 

 u',-\-vf', . . . u'^+u"^, . . . u'„,+u",„ . . . w'iv+?^"iv, &:c. 



After the passage of the second boat, the falls of locks 

 E. E ^ E,^ Eiv, <^c. will become x"\ x'" „ x"\,^ x'",^, Sec. 

 In the same manner may be determined the elevations u"'j 

 '^" „ w'"/,/ w'"iv, ^'c. of water by the passage of a third boat, 

 and the heights assigned in functions of the primitive quan- 

 tities S, V>, B,. B /, Biv, &ic. x' , x\i «' ^ a;'iv> &c. in consider- 

 ing that we have 



X nt^^^ tit'ir'^ III "T"W ),, W iv W iv OiC. 



x'" =x' -\-v( +w" —v! —u" 



(n) (n) (n) (n) (n+1) (n+l) 



Adding together all these falls of the locks from the first 

 to the last, we have 



X:"^x"', + x"',,+ x"',,-{., &C. + X'" = 



x'i+v'i+u",+x',+x\„+, &:c.+x' /[-u' -u'' 



(n) (i+l) '.71+1) 



Vol. VII.— No. 2. 38 



