8 
[419 
420] 
419. 
A THEOREM ON DIFFERENTIAL OPERATORS. 
[From a paper by Prof. Sylvester, “Note on the Test Operators which occur in the 
Calculus of Invariants, &c.” Philosophical Magazine, vol. xxxn. (1866), pp. 461—472, 
see p. 471.] 
The paper concludes with an Observation from Professor Cayley as follows: 
“ In the case of two variables, if 
pi = ( ax + h) j- x +(c®+ %) Jr, 
then in the notation of matrices, 
Pi = 
a, b\ . .id d\ 
c, d\ \dx' Ty)' 
*>■ 
_ (a, b 
whence also 
.id d 
‘»le, dj {X> y) \fa' df) i 
P*P 2 = P„*P 1 
_ JL (a, b 
■ dF ( *’ y) (&’ 
d 
dy 
= 3 P 3 , 
which accords with your theorem, 
E x *E 2 * = E. 2 * Ex* — E X E.,* + 3E 3 *.” 
d d 
I have taken the liberty of writing in the above ^^ for 8 X , 8 y , and P for 8 
in the original. It will be useful to bear in mind that in any operator such as 
Ex* or E 2 *, the asterisk forms an integral part of the symbol. Thus E X *E 2 *, if we 
choose, may be written under the form of Ex* multiplied by E 2 *, i.e. (E x *) x (E 2 *), 
where the cross is the sign of ordinary algebraical multiplication. 
The 
solution 0 
The 
which is 
or a posit 
The peci 
been ex] 
y contaii 
of the < 
F, Q' tf 
showing t 
To compk 
C. V