1889.] Variables in certain Linear Differential Operators. 361 



II. Ternary Operators. 



Let x, y, z be variables connected by a relation of any form known 

 or unknown. Let x rs , y r s, denote respectively 



1 d r+s x 1 d r+s y 1 d r+s z 



r\s \ dy r dz s rlsl dz r dcc s r\s \ dx r dy s 



Let s" = *tfrtr\ a = 2 x'ryr 



LP + 2<i j r+s<m 



be the expansion of the rath power of an increment of x in terms of 

 the corresponding increments of y and z ; and define as 



m{fJL, v, v' ; ra, n, ri} x 

 the operator 2(> + i/r + i>V)X; m) 



dx n -\- r n'- 



and as m{jn, i>, v ; ra, w, m{/A, f, f ' ; ra, n'} z , the operators 



obtained from this by cyclically interchanging x, y, z once and twice 

 respectively. 



Attention is confined to positive integral values of ra, except that 

 the value zero of ra is admitted, in so far as its admission requires the 

 introduction of no new idea. By n and ri are denoted positive 

 integers or zeroes, or in certain special cases — 1. Thus the field of 

 investigation is narrower than the analogue of that covered in dealing 

 with binary operators. 



The comprehensive theorem for the transformation of these ternary 

 operators is that 



{fi, v, v ; ra, n, ri} x = <^ — v(n + l), v, — — ; ri, ra— 1 > 



I ra J y 



— | —v'(ri + l), v; ri + 1, ra— 1, nj^. 



There are three classes of cyclically persistent operators, of different 

 characters, each corresponding to a cube root of unity, viz. : — 



{— ra, 1,1; ra, ra — 1, ra — 1} X — { — ra, 1, 1 ; ra, ra — 1, ra — l} y 



= {— ra, 1, 1 ; ra, ra— 1, m—l} z , 

 {— ra, w, to 2 ; ra, ra— 1, ra— 1}^ == w{— ra, u> 2 ; ra, ra— 1, ra— 1}^ 



= a> 2 { — m, w, a; 2 ; ra, ra — 1, ra— 1} Z , 

 { — ra, to 2 , uj; ra, m— 1, ra— 1}^ = w 2 {—m, a> 2 , to; ra, ra— 1, ra — l} y 



= to\—m, io 2 } iv', ra, ra — 1, m — 1}.. 



