496 PROCEEDINGS OF THE AMERICAN ACADEMY 



Again, in [4], putting n = -, q and p being positive integers, 



p C 1 = C q = q ; 

 p 



whence O n = -. [6~\ 



p x 

 Again, in [3], making m =±= o, 



C = o, 



and, making m = — n, C n = — C n , 



or, making n = -, 0„ = — -• l~71 



From [6] and [7] we have generally 



O m = m, 

 and substituting this value of C m in [1], 



D (x m ) =' m x" 1 - 1 D x. [c] 



The Differential of the Logarithmic Function. 



Let z = mx, then will Dz = m Dx, [1] 



and log z = log m -\- log x, and D (log z) =Z> (log x). . [2] 



Dividing [2] by [1], 



D (log z) D (log x) x D (log. z) 



D z m . D x z Dx ' 



since !■= *, ... • J ^ z) = x * ( 7 \°g *> = A [3] 



m z Dz Dx L J 



B denotes a constant depending upon the base of the system of 

 logarithms. Denoting by b this base, and by log 6 a corresponding 

 logarithm, we have 



D (log b x) = ^, [4] 



and by similar notation D (log a x) = . 



A relation between A and i? is found by differentiating the identical 



equation log a x = log a b log b x, 



., , , . . ^4 Z>rr , r BDx 

 thus obtaining = log a o • , 



whence A = B • log a b = log a S B ; 



a- 1 = J». 



