VARIABLES IN CERTAIN LINEAR DIFFERENTIAL OPERATORS. 
51 
Now, from (91) and its correlative obtained by a cyclical interchange, the second 
parts of these three equal operators are themselves equal. Consequently 
. ( 120 ) 
•Do yoi ^10 ■^01 
which, and its correlatives, are the formulge for the transformation of and Vg. 
It is easy and very instructive to prove (120) directly from the symbolical 
expressions in (117) and (118) by the method of Art. 19 or 23. 
Of other important operators Mr. Forsyth’s and Ag ('Phil. Trans.,’ A., 
vol. 180, p. 74) should have their formulae of transformation noted. They are the 
complete operators (0, 1, 0 ; 1, — 1, 1 j and (0, 0, 1 ; 1, 1, — 1| of which and 
are all but the first terms. Thus their formul£e of transformation are merely (114) 
and (114a) themselves, i.e., cyclically interchanging the variables once, 
A,(a;, y^) = - = fO, 0, 1 ; 2, 0, - 1},, .... (121) 
A,{x, yz) = [0, I, 0 ; 2, - 1, 0],= - .(122) 
' ^01 
