OF ARTS AND SCIENCES : JANUARY 14, 1873. 495 



D O m ) _ j. D (* m ) 



Dividing [2] by [1], 



Dz rDx ' 



, . z D (z m ) z mJl D (x") 



and, since r = -, — ^— - = — r — ^r-A 



a;' Z>z a;" 1 " 1 Dx 



Separating variables, by dividing by z? l ~\ 



J_ D(z") _ _ J_ Dpc") r31 



By a train of reasoning precisely similar to that employed in ob- 

 taining equation [4] of the article on D (x 2 ) 



1 D(x m ) n rA1 



we P rove ^i -J^T = °m L 4 ] 



C m being used to denote the constant, because it may be a function 

 of m. 



From [4] D (x m ) = C m x^Dx. [1] 



By similar notation D (x n ) = C n x"" 1 D x, 



and D (x m+n ) = C m+n x^' 1 Dx. 



To determine C m . 

 If m = 1 in equation [1] Dx l z= G x x° D x 



4=1. [2] 



Differentiating the identity 



O m+n x™*"- 1 Dx = x n O m x m - x Dx + x™ C n a;"" 1 D x, 

 and dividing by as" 1 ^" 1 X>x, 



c m+n = o m + q. [3] 



making m = n C^ = 2 C„, 



making m = 2n C 3n = 3 C„, &c. 



C pn = p G n ; O being a positive integer). [4] 



Making n = 1 in [4], Q, = p Ci = p. [5] 



