48 Proceedings of the Royal Irish Academy. 



(5) Solve x"" 4- ax''^'' + ^ = 0. 



In carrying out the division by + we shall first observe 



the general law that the coefficients in the quotient reduce to a simple 

 binomial form. Let t be the number of a term in the quotient (the 

 first term being numbered zero), and let s be the exponent of p in that 

 term, then the general expression for the term will reduce to 



— —r ^P'- 



ns 



Moreover, s is easily determined. For the first term « = -, and for 



r ^ 

 each successive term it must be increased by - ; so that 



n 



1 1 / ilr 1 / l + 2rY^^ — 



^ 1 + r n I 1 + 2r V ^ / 1 2 



It will be seen in the following section that a similar reduction 

 occurs in the general value of <j5)"\ The values of n and r are not 

 restricted. 



(6) The quadratic x"^ -\- ax h = 0. The complete solution is 

 given by 



the coefficients of the integral powers of except P\ vanishing. 

 The sum of the series is 



/5+ O 



2) 2 



The same series is given if we solve from the form 



X-^ -V ^ X-^ = - Y' 

 



The forms 



X + hx~'^ = - a and + -x'"^ = 



