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



evident that a"'i'"=(a4-(m — l)r)"'i-''. When r=o, thisequa- 

 tion becomes a'"'°=a"'. By means of any convenient 

 Multiplier, the Base as well as the difference ntiay be chang- 

 ed at pleasure. The following equations contain the prin- 

 ciple of this transformation. 



(a \ " \a'r /r \ " /a/\ 



a) 'n^=\r') •i-J""^-The equation 



a'{a'-\-r) .... (^a'-[-(m— l)r) (a'-\"mr) .... {a'-{-{m-\-n — 'l) 

 r) = 



<( a'(a'+r). .. (a' + (m- l)r) )>X<( (a'+mr) . , . (a'+mr+ 

 [n—l)r) )> ; according to the notation above, will become 

 «""*"''■=«""'' X (a' +mr)"''' ; taking p=m4-'^, whence n=p 

 — m, and a'-{-mr^a, whence a'=a—mr, and reversing the 

 equation we shall have (a — m/*)'"'''. a''-~'"'''=(a — mr)^'''; again 

 making p=m, and reducing, &ic. we have a°'* == 1 . That 

 is every faculty the exponent of which is equal to Yjero., is 

 equal to unity. Ifin the same equation we make p=o, 

 and, from the preceding seeing that (a— mr)'''''=l, it will 

 become 



1 1 



■ ~(«— mr)""'- ~(a-r)™i-''. 



This gives the expression for the numerical faculties 

 when the exponent is negative. It follows in the same 

 1 1_ _ 



mannerthat,a -' ''=(«+^^)™i-=(^^^yn,o ^' 



Kramp in his Universal Arithmetic, proposes also this no- 

 tation 1.2.3.4.... m = r"''=m/ 



To extend this short sketch of the theory of the facul- 

 ties of numbers any further, and show its application 

 k.c. would trespass too much on the useful pages of the 

 journal ; as it can interest but very ie\y it might perhaps be 

 as well to omit it entirely. 



Art, XI. — Second Memoir on JVavigable Canals considered 

 in relation to the lift and distribution of their Locks : by 

 M. P. S. Girard. 



[Translated from the French, Annales de Chimie et de Physique, Nov^ 

 1821 : — by Isaac Doolittle. 



1 gave the firs*^, in a memoir which I had the honor to 

 communicate to the acadamy a few months since, the rig- 



