40 
MR. E. B. ELLIOTT ON THE INTERCHANGE OF THE 
Jl 
cb:r. 
d 
dx,, dz 
10 
I 
' dx 
A 
dx, 
+ 
01 
dXr. 
20 
A 
dz, 
20 
j ^~ 1 I 
dxr. 
dz- 
d _^dz^ 
d 
11 
dXrs dz. 
+ 
each dzpjdxrs is the coefficient of djdzpq. 
It follows tliat in the expansion in terms of ^ and t) of the right hand member 
of (80) the substitution for each ^^ 17 ? of the corresponding d-ldzp,^ exactly produces 
the expression for djdxrs. In other words, for each r and s, the 
z transform of .(81) 
dxrs 'd^ ' 
where ^ and its partial differential coefficient are to be replaced by their equivalents 
in terms of ^ and rj by (65), where the product is to be expanded in terms of ^ and 17 , 
and where in the expanded result each product ^^ 77 '? is to be replaced by the corre¬ 
sponding operative symbol d/dz^^. 
By (77) we see that djdx^s is thus replaced by a linear ^-operator of the form under 
consideration ; in fact that 
_ .0. 1, 0; « +1, - 1, )•}..(82) 
Since the fx and the / of this operator are zero, the fact that is — 1 gives no 
difficulty as to the presence or absence of coefficients on the right like 
In precisely the same way, by giving variations to ^ and Xys in (63) and (64) instead 
of (63) and (65), we might have obtained 
d 
fO, 0, 1 ; r + Ij s, — 
of which y-operator the symbolical form is 
(83) 
- cv 
drj 
dTf 
18. The rules for transforming any linear ai-operator to its equivalent forms in y 
and in 2 ; dependent, are now very simply expressed just as was the analogous rule in 
in Art. 4. Since the ^-operator 
A 
dXfs 
or 
has for its equivalent y-operator 
and for its equivalent z-operator 
-r)% 
dr 
-Vi 
dr 
