72 Proceedings of the Royal Irish Academy. 



we can invert the operation by dividing operatively B^^ that is /3, by 

 it. The quotient applied to \_<ji]x is 



Or we may take the division by descending terms, beginning 

 •with the term L'\ when, if 



For example, let 



^ + . - 



Then 



[DO + fli>]2/ = x\ and y = {p - aB a^B" - a^D"^ . . = x - a. 

 Also 



[Z) + a-^IP'\y = ; 



and 



2^ = - + . .'\{a^x) = cf^ ^ - ^ + . . . 



X 



= ae"^ + {x - a). 

 The two roots are interdependent, their operative ratio being 



- e « + /3. 



e 



(2) If the coefficients are integral functions of x^ the same pro- 

 cedure is adopted. For example, let 



IL"" + xD']y = {n+ l)x\ 



Then 



B' + xD} XB'-IB-v^B'^-^BK.. 



i)o + xB 



xB 



xB - ^x'B- 



1^2^2 ^ A ^37)3 



etc. 



