VARIABLES IN CERTAIN LINEAR DIFFERENTIAL OPERATORS. 
47 
is, but for a first derivative factor, an operator which persists in form after one and 
two cyclical interchanges of the variables. 
Symbolically we have, if square brackets indicate that in an operator first deriva¬ 
tives are thus omitted, 
[ I 5 1 ; b, 0]a — ^ m y) C^loV 
= (I - - I + x^^rj + + 77 
^loV ^ 01 ^ I 
The y transform of this operator is therefore, by Art. 18, 
which, since 
and 
ccio : Xqi : 1 — 1 ; '- yoi — ^01 ' ^ ^ 
10 ’ 
dfi dfi _ _ _ .df 
dv'dV ^ ~ ^ ' d^' d^~ drj’ ' d^’ 
may be written 
(- „«-■(, - ;| + - V 
and is consequently 
(— [— 1, 1 ; w, 0, 0]^. 
In exactly the same way the 2 transform of the same operator is 
(— [— m, 1, 1 ; m, 0, 0].-. 
Thus we have the formula of transformation 
r / 1 / 1 \»»-l 
[- w, 1, 1 ; m, 0, 0], = (^- -j [-m, 1,1 ; m, 0, 0 ],= -j [- m, 1, 1 ; m, 0, 0]„ 
which may be written in a form even more clearly expressive of the cyclically 
persistent property, viz.. 
