86 The Rev. Robert Harlev on 



into (a). Here it will be convenient to write (|3) and (/3') 

 as under : — 



Y*-»Y*- I + (»-l)X-0 .... (/3o) 



m m - l [(m - lJD'T"-^ - (m - l)(mD' - m - 1) 



[mD'-2]'"- 2 XY = [m]'»X . . . (/3' ) 

 where 



Now (j3 ) is changed into (a) by substituting 



-<-).(-r-'.- ! : 



for m, X, Y respectively. Making the same substitutions 

 in O' ), and remembering that in this case 



D'= » . 



n — 1 



we obtain the differential equation 



<-^>1-^J"(-?) + »< :d+ »- »> 



[_(D + 2)]-' n+1 /-^T^ n - 2 y 



/ 1V ,+1 

 -[-<»- l)]- fB " 1J (--) ■<"- 1 > 



which, by the process employed in Art. 8, reduces to 



<D]»+V - (n - 1)(D - n + 1)(^ - *^) V* - °- 



Operating on both sides of this equation with 

 (D-n + l^'P-n)- 1 , 

 we have 



= c x a;" + Co«" _l . . . (a' ) 

 c lt c % being arbitrary constants. In order to determine these 



