l58 Proceedings of the Royal Irish Academy. 



But a distinction has to be made in the case when f{z) = has two or more 

 equal roots. If, instead of this, we multiply the equation by e'^'^, where a is 

 a root of f{z) = 0, and integrate, we shall find that the case of equal roots 

 does not require separate treatment, as the above example shows. 



Let us now apply the proposed method to the solution of Legendre's 

 equation. The equation is 



X»7/ - x-D'y - 2xDy + n{n + 1)^ = 0, (1) 



which, but for the first term, would be a homogeneous equation. Multiply 

 by x^, and choose h so as to make the last three terms a perfect differential 

 coefficient. We have 



sd'Dhj -o^^^Dhj - 20*^ 'Dy + n{n+l):>^y = 0. 



The expression - cc^^'By + hc^^^y, when differentiated, gives the second and 

 third term, and will also give the last term if 



k {k +1) = n {n + 1) : (2) 



that is, if k = n or - {n + 1). Putting k ■-= n, we get a first integral of the 

 equation in the form 



D-bfB-y - x''^-Dy + nx"^'y = A', (3) 



where A' is an arbitrary constant. It will be noticed that the integration of 

 the first term is expressed merely. 



Multiplying by «'-("+^), we can integrate once more; and we get 



D- hr * (" + 'W' hf'Dhj - of "y = - Ax~ C-" ^^) - B, 



when A is written for - A'l2n + 1, and B is another arbitrary constant. 

 Multiplying again by re"^ and changing signs, we have 



{1 - oS'D-^x-'^''*^)iyhfI)-']y = Ax-^""^') + B9f\ 



If we write this 



(1 - (^)y = Av-C") + Bx'\ 



we arri^•e at the complete solution of the differential equation in the form 

 ^ = (1 - 0)- 1 { ^:>r (« + 1) + Bjf^ ] , (4) 



where r/. is the operator x"'I)'hv~^ ^^'^^W'^x"^!)-. In order to get the solution in 

 the ordinary form, expand {1 - (py^ in the form 1 + + ^^ + . . ., and perform 

 the necessary operations. Thus the term Aof^"^^'> gives rise to a particular 

 solution, the product of A into a series. The first term of the series is ro"(""^'). 

 The second term is got by operating on the first by ^ ; that is, we must differ- 

 entiate twice, multiply by x'\ integrate, multiply by x~^^"*^\ integrate again 

 and multiply by ;/;". The third term is got by operating on the second by (^, 



