84 Dr. James Bottomley on a • 



of which the integral is 



z^n^(t>i{y^x)^fly-x) . . . (50) 



If we substitute for ^2" the differential expression for 

 which it stands, we shall obtain a differential equation of 

 the second order, but by repeated applications of the 

 method this and all succeeding equations may be reduced 

 and integrated as equations of the first order in the manner 

 indicated in the treatment of the equation of the fourth 

 order (13), and finally we shall obtain as the solution of the 

 equation (2,7) the expression 



where the letters /i, /a . . . ■/2"+h denote arbitrary functions, 

 and the letters (>i p2 ps . . • • (>2"+i are the roots of the equation 



X-""''-l = .... (52) 



The foregoing method may also be used to integrate the 

 equation 



assume the equation 



'^ = px .... (54) 



where p denotes a root of the equation 



x2"+'-a = .... (55) 

 Then if we differentiate 2"+ times we shall obtain 



aRir . • •(-) 



hence by substitution in (53) we shall get 



2"+l / rf \ 2"+^ 



(l)RI> 



of which the mode of integrating has already been given. 



All the equations considered are of an even order, and 

 on reduction give rise to equations of even order until we 



