M. P. S. Girard on Navigable Canals. 303 



so that, naming generally X, X;, X„^ Xj^, &:c. the falls of 

 the locks E, E,, E„. Ej^ &c. and V. V, V,,, Vj, &c. the dif- 

 ference of level produced by any number of douMe passa- 

 ges in the levels B< B,^ B, , Bjv &c. whose transverse sec- 

 tion is supposed rectangulai, the general fornnulae, 



V = ^-(+D+X ) (B„V.,+ BV,) 

 "■" B,„+s'±''+ '"' (B„,+S) 

 will express either the rise or fall of the levels accordingly 

 as the quantities D and X shall be affected with the upper 

 or the lower signs ; and we may therefore, in either case 

 deduce analagous consequences from them. 



Let us nevertheless repeat what we have already stated 

 at the commencement of this Memoir: that in projecting 

 navigable canals, we shall much more frequently find it ne- 

 cessary to raise water from the lower levels, which never 

 dry, into the upper levels which are sometimes subject to 

 that inconveniency, than to take from the latter, the water 

 required to render the others navigable ; it will therefore 

 be proper, whenever circumstances will permit, to con- 

 fine ourselves to the case where the difference D — a; is a 

 positive quantity. 



To resume in a few words what we have said hitherto : 

 We have first considered two contiguous levels separated 

 by a single lock, the upper level having a determinate su- 

 perficies, and the lower one indefinite. We have deter- 

 mined, in this hypothesis, the law which would govern the 

 rise of water in the upper level, by the double passage of 

 any number of boats ascending and descending hrough this 

 lock, when the lift of the lock is less than the difference 

 of draft of water in the ascending and the descending boats. 

 We have found that, the number of double passages in- 

 creasing necessarily as the series of natural numbers, that 

 is to say, in an arithmetical progression, the successive ele- 

 vations of the upper level, produced by the working of the 

 lock, diminished in a geometrical ratio, so that the law 

 which connects the successive elevations of the upper lev- 

 el, and ihe number of passages to which these elevations 

 are due, may be graphically represented by the co-ordi- 



