VARIABLES IN CERTAIN LINEAR DIFFERENTIAL OPERATORS. 
41 
these rules are merely—To find the equivalent y-operator to a given linear £c-operator, 
multiply its symbolical form by — dr)/d^; and to find its equivalent z-operator, 
multiply that symbolical form by — dljd^. The y-operator thus obtained has of 
course to be expanded in terms of ^ and ^ by (64), and the z-operator in terms of ^ 
and 7] by (65), before being intelligible except b}^ means of (76) to (78). 
In verification let it be noticed that, since by first principles of the theory of partial 
differentiation the three sets of ratios 
d7] ' ' 
drj dr) 
~ ■ ’ 
d.r] d^ ’ 
are equal, precisely the same results are obtained by cyclical interchange of x, y, 2 ; 
and i, 7), C- 
19. Now as in (76) 
m{l, 0, 0 ; m, n, 
Its forms in y and z respectively are then 
- ^ . and - ^. 
Of these by two cyclic interchanges in (78), and by (77) itself, respectively, the 
expressions are 
— (0, 0, 1; n -{■ I, n, m — l)^,, and — [0, 1, 0 ; n -{■ I, 711 — 1, 
consequently 
m(l, 0, 0 ; m, n, 71 ]^ = — {0, 0, 1 ; 7i -\- 1, n', 771 — l}y 
= — {0, 1, 0 ; 71 + 1, m. — 1, n}~. . 
In the same way the y and s transforms of 
{0, 1, 0 ; m, n, n'}^, i.e., + , 
(84) 
are 
^.e,, 
_ H + l^.p _ ^M-1 u + 1 , 
' dt] d^ ^ I '=• 
_ + and + + 
dr] 
MDCCCXC,—A. 
G 
