286 M. P. S. Girard on Navigable Canals. 
evident that a _ a — r=o, this equa- 
tion becomes a™!°=a". means of any convenient 
Multiplier. the Base as well as the Siimenct may oe chang- 
ed at pleasure. The following a gna contain the prin- 
ciple of this transformation. 
a\* a'r (r\" (ar 
attzs (¢) P ether (¢ )-"-The equation 
ALeT Ta . (a +(m— 8.) (a'mr). . “st +(m+n—1) 
£ Hee. 2d Ein = Ayr) $2 Emr): 
(n~ Ir) >3 according to notation above, will become 
aimtnir a gmlr yc x(a + Side aking p=m-+n, whence n=P 
—m, and.a’-+mr=a, whence a’=a—mr, and reversing the 
equation we shall have (d—mr)™"". gp" = (a — mr)"; agait 
mak gs p=m, and reducing, &c. we have a°?”=1. 
ry faculty the amet of which is equal to Faro, i 
ue a) io unity. If in the same equation we make p=®, 
apts ihee ihe preceing seeing that (a —mr)or= 1, it will 
bee 
cor Fe BCE Corona | 
4g 35 % 
aie ¢ ves the “expression. for the numerical pig 
when ihe. exponent is negative. It follows in the same 
manner that, @ spre | 
1 
7 (am mr), (abr. > See. 
_ Kraunp inhi eat Arithmetic, Breposes also this n0- 
tation 1. wie 
To aid tis short aheeoh. ~ the theory of the facul- 
ties of numbers any further, and show its application 
would trespass. too much on the useful pages of the 
pee’ ; as it can interest but very, few it might perhaps be 
as well to omit it entirely. 
Se 
Arr. XI.—Second Memoir on Nt avigable Canals considered 
~ relation tite the lift and distribution of thetr Locks : by 
IRARD. 
[Translated from the French, Annales de Chimie et de Physique BA: 
—by Isaac Heepennis, 
othe first, in a memoir which I had the honor to 
coiieasiicnis to the acadamy a few months since, the rig- 
