360 Mr. E. B. Elliott. On the Interchange of the [June 20, 



be either positive or negative. The value zero of n is admitted, and 

 that of m, though somewhat special, is not excluded. 



What is sought and effected is the expression of an y such operator 

 {jtt, v ; m, n} x in terms of operators of the same form {//, v ; ra', n'} y , 

 in y dependent. The process depends on the use of a certain sym- 

 bolical form for { v; m, n } x , and on the proof that a simple factor 

 produces from that symbolical form the symbolical form of the equi- 

 valent y operator. v 



If m + n >1, so that none of the coefficients of powers of rj in p 1 is 

 wanting from {fi, v\ m, n} x , the inclusive formula of transformation 

 is found to be — 



{ju, v; m,n} x = — j v(n + l), ^ ; n + 1, ra— 1 1 ; 



and the conclusion is deduced, among others, that there are two 

 classes of self reciprocal operators, a class of positive and one of 

 negative character, viz. : — 



{— m, 1 ; m, m—l\ x — {— m, 1 ; ra, ra— 1}^, 



and {m, 1 ; m, m—l} x = — {m, 1 ; ra, m—Y} y . 



Particular attention is devoted to the special cases of m = and 

 n == — 1 ; also to the transformation of Q and V, the annihilators of in- 

 variants aud of pure reciprocants. V of course, not involving the first 

 derivative, is not an operator of the class {/x, v ; ra, n} itself, but is 

 linear in such operators. 



If m = — n the formula of transformation is found to be 



m{/x, v- ra, —m} x 



= —{vm(l—m'),[i; 1 — ra, ra— + (/* + ^)2/i~ m {0, 1; 1, — 



In particular 



ra 



If m + n<0, = — r say, it is 



m{/x, v j m, — m— rja; = — (i'm(l — ?w— r), jm; 1— ra — r, ra — l} y 

 + (^m)Xi»>{0 r l; 1-r, 

 + ( /l +m+,)XW 1 {0,1; 2-r, -1} 



+ . . 



+ ( /i + m+w _,)XW. 1 {0 ) l; 0, -1} 

 + (^ + VTO + V r)X£S P {0, 1; 1, -1}*,. 



