184 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
and putting for these symbols their expansions found in the preceding article, we get 
c.}{ 
m d -2 m d s 
1 + h ' d * + \%+Hks+>* us 
•} 
— 1 + (A + h') d x -j- 
{h + hJf dj 2 (h + h') 3 d 3 
1 . 2 
+ 
1 . 2.3 
-j-, &C. 
a relation which may be verified by actually compounding the two polynomials of 
the first member. 
Now in this act of verification the operations h d x h! d x have only such properties 
as are common to any two linear operations which are relatively free : hence if 0, 0 r 
represent any such operations, we have generally 
f 6 2 6 3 „ 1 f , S' 2 
| 1 + 0 + J~o “f“ 1.2.3 + > & c * j • | 1 4- 0 + + j o ,5 +5 &C. 
— 1 + (0 + 0 ) i fTo - + 1.2.3* 
and it is easy to extend an identity exactly similar to any number of operations which 
QII2 QII3 
are all relatively free ; for in introducing a new polynomial, I + 0" + — -J- y-^-|--,&c\, 
we have only to regard 0 + 0' as itself a free linear operation, and therefore the result 
would be 
1 + (#+#+ ^ + «±^ + a±j^ +J& e. 
8. If the subject be a function of two variables, x and y, then using to denote 
that x must be changed into x + h, and \py that y must become y k, these opera- 
tions are relatively free, it being of no consequence which operation is first performed ; 
therefore the operations A^, A y of taking the corresponding finite differences are also 
free ; from whence, lastly, the differentiations relative to x and y, represented by 
d x , d y , must be of the same character. 
Now since 
and 
7,2 d 2 h 3 d 3 
, , , , , , l‘‘>V , EAl , . 
therefore, by the general identity of (7-), we have 
W,= \ + (hd, + k d,) + ( AsL+lM + y + k -M +> &c. 
And now introducing the subject f(x,y ), and expanding the terms in the right 
member of this identity by the formula given in the preceding section, it will be- 
come, in the common notation, 
