182 MR. GRAVES ON A RECTIFICATION OF THE 



§ 7. On the orders and ranks of logarithms. 



In [22] let 3/ = R + J — 1 S and a = A + J - 1 B ; 

 then, by [28], will 



- 1 



R 



fZrTior^ = — — — ^^ 



[32] 



2fT4c6s ^^, ^ g^ - >/ - 1 1 v^ A" + B» 



When I have thus separated respectively the real and imaginary parts of the 

 numerator and denominator of [22], upon assigning particular values, \ and i, 

 to i and i in [32] , I would indicate the order of a logarithm by the \ in the 

 denominator, and the rank it bears in that order by the i in the numerator ; e. g. 

 I would say of the resulting x that, in the base a, it was the ith logarithm of y 

 of the zth order. 



By [20], all the values of (A + J'^^H. B)' ^''^ comprised in the formula 



f^a-f-'(A+ A/^=^B)j- 

 or, (see [28] ) 



f{:r (2.V + COS-' ^J'_^ 3, - >/-:r-ll A/-AJTB-)} 



When, in this formula, i assumes the particular value ^, I would denominate 



f{x (2J. + c6s-'^^^^^ - >/^l ^A^TB^)} 



the Ith value of (A + J'^^l B)' 



When, with respect to the base a, x is any logarithm oiy of the \i\\ order, the 

 ith value of a' will = y. 



Employing the mode of expression above explained, I conceive that the chief 

 novelty of my system consists, not in showing that any assigned quantity, 

 relatively to a given base, has an infinite number of logarithms (which was 

 known before), but in showing that it has an infinite number of orders of 

 logarithms, and an infinite number of logarithms in each order. 



Thus, all the Neperian logarithms of 1 have been hitherto supposed to be 

 comprised in the formula 



^/^=~l 2 i T 



