M. P. S. Girard on Mivigahle Canals, 299 



DS, must be equal to each other ; making this commot 

 quantity =a, we shall have 

 S 



''-B^+S ^^~^'">~(B +S) 



--1—m > ^ (B..+B,+B)a 

 «-Bi,+S^^-^i^>'" Bi,+S 

 And generally : 



^- B „,+s ^^-^ <""'-'' (Bf„7FS) 



Now the position of the level B(„), in relation to the 

 summit level of the canal, or, which amounts to the same 

 thing, the sura of all the levels B -fB, -f-B,^^+. . . B:,. ..), 

 which precede it, being given and represented by (B;, we 

 have : 



„- S .(D-x„)- " C^) 



B(,.) + S' ;"" B(„)+S' 

 or, to abridge, by making 



B(„)4-S=?/; D-x(„)=^, 

 we have 



a ((B)+i/)-Sz=o 

 equation of the right line, easy of construction, and which 

 expresses the relation which should exist between the ex- 

 tent of any one of the levels and the fall of the sluice 

 which terminates it, in order that this level, and each of 

 those which precede it, shall acquire an uniform rise of 

 level at each passage of a boat, whatever may be the num- 

 ber of passages. 



For after the passage of the first boat, the falls of the 

 sluices on the whole length of the canal have become : 

 x i-]ra—a=x' , 

 x' „-\-a—a=x ,^ 



Since nothing is changed, either in the extent of the 

 levels, or in the primitive fall of its sluices, it is evident 

 that the passage of a second boat will cause the same rise «, 

 of water ifi each level, and so on indefinitely. 



