22 
MR. E. R. ELLIOTT OK THE IKTERCHAKGE OF THE 
Accordingly it follows that 
(Mh 
dx,. dXr dXr 
{y, + 2y,f + 3ysP + 
■) . 
i ’ 
( 5 ) 
and consequently, this being true for all values of that if by aid of (1) this right 
hand member be expanded in ascending powers of the coefficients of . . . 
are exactly the expressions for clyyldx,-, dyjdx,-, dyjdx^, . . . 
Now 
d 
dX-r 
dry d dy„ d 
-• "T“ - • - “T” 
dXr dy-^ dXf dy„ 
d}h , 
dXr' dy^ 
and is therefore the result of replacing each power of ^ on the left, and therefore 
on the right, of (5) by the corresponding symbol djdys. It follows that the expression 
on the right of (5) may be taken as a symbolical representation of the equivalent 
operator to dldx,-, i.e., that 
~ ( 2/1 ^ +•••)' {yi + ^yd + + . • •) 
= —V 
r ^ 
dr 
( 6 ) 
where the meaning of the symbolisation on the right is that y is to be replaced by its 
equivalent in terms of f from (1), that the differentiation with regard to f is partial, 
that the product on the right is to be expanded as a sum of multiples of powers of 
and that then each power ^ is to be replaced by the coiTesponding symbol of 
differentiation djdys. 
4. The proof of (6) is the same for all positive integral values (including unity) of r. 
Thus the means of transforming any differential operator whatever is obtained. 
The rule according to which any linear operator is transformed may be very simply 
expressed. 
Exactly companion to the symbolical notation ^ for djdys in an operator linear in 
djdy^, djdy^, djdy^, ... is the notation rj^ for djdxs in an operator linear in djdx-^, 
djdx^, djdx^, . , . Now, writing a linear operator 
in the symbolical form 
A 
dx^ 
+ Bj^ + C 
dx 
dx\, 
+ . 
At}'^ + -j- Cy'^' -j- . . . , 
we learn by (6) that its equivalent in djdy^, djdy^, djy^, ... is obtained by multiplica¬ 
tion by — dyj d^, expansion in terms of ^ by (1), and substitution for each power ^ 
in the expanded result, of the corresponding djdys. 
