y = ^^7;;;yF (m, x) when m = k 



236 The Ohio Journal of Science [Vol. XVI, No. 6, 



And thus is 



y = x'' (Ao + Bo logex) A-series +BoX'' (B-series) 

 a solution. 



And in case m has the value k three times, then in addition 

 to the solutions (1), (2), is also 



y = Cox'' -^ ^ log\x+ (B-series) logeX+ (C-series) 



a solution. This is 



dm. 

 And thus is 



Co 



y = x'^ (Ao+BologeX+ -^log\x) (A-series) 



+x'^(Bo+Cologex) (B-series) 

 -\-\^ (C-series) 



a solution. And so on. 



It is quite easy, in any particular case, to obtain the B- 



series, C-series, etc., from the A-series, far easier than by 



the method usual in the texts (compare the following with 



Johnson, p. 181 to 194). 



Illustrations. 



(a) Series starting with x°. 



d^v dv 

 dx'' dx 

 y = x"', gives m-x"^-i-|-x™ = (1) 



We may assume an ascending series, starting with x°, 

 consecutive powers diJffering by unity. 



00 



Ifj|we assume y = S ArX'"+'', 







then by (1) 



(m+r)2A.-fA,_i = () 



For> = 0, A, = - Ar-i . ^ (3) ; 



(3) Ogives the A-series, 



X , X- x^ 



y-1 12 + 12.22 12.22.32. "^^^^* 



