36 
MR. E. B. ELLIOTT ON THE INTEROHANOE OF THE 
in verification of which we may notice that it is unaltered by interchange of x and y, 
in virtue of (48). Another way of writing the result is to say that 
2(0, 1; 0, - l},-x{-^^x^[0, 1; 1, - 1}, 
= 2 { 0 , 1 ; 0 , - 1 ; 1 , - 1 }, . ( 62 ) 
is a self reciprocal operator of positive character. 
We are now enabled to write (61) 
m[0, 1; m, — m — l}^ 
= m(l, 0; — m, m — 1; 0, — 1}^—1; 1,-1}^, 
which becomes (60) upon interchanging x and y, replacing m by ~ m, and using (48) 
and the values for x^ and x^, in terms of y^ and y^. Thus we have another verification 
of the consistency of our results. 
II. Ternary Operators. 
16. Let X, y, z be three variables connected by a relation of any form known or 
unknown. Let Xrs, y^ z,-s denote respectively 
1 d’- + ^x 1 d’'+H/ 1 d’' + ^z 
r !s ! dy^ dz^ ’ v’!s ! dz’’ d'jf ?’! s ! dr^ dy^ 
Let rj, i be any set of corresponding increments of x, y, z. They are connected 
by a single relation, which may be written in either of the forms 
^ = (a^lO 1 + 0 + (a?30 + ^11 T) 
+ (^30’y® + ^21 + .To3 D + • • • , • • ((13) 
'n = (2/10 1 + .1/01 + (2/20 + 2/11+ 2/02 
+ (2/30 + 2/21 + 2/12 + 2/03 ^^)+••• > • ■ ( 34 ) 
C = (^10 ^ + ^01 1) + (^20 P + ^11 + % 1^) 
+ (^30 + ^12 + 2:03 >?^) +.( 65 ) 
Let m be a positive integer, and let denote the coefficient of rf' in wlien 
expanded in ascending products of positive integral powmrs of y and so that 
