MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
185 
/(, + + *) ^ ^ + eqp . T^s +.*«. 
d 2 f(x,y) F d 3 f{x,y) hlc 1 
dy 2 ' 1.2 “l - dx dy 2 ' 1.2 -,-,<XC ‘ 
, £/Ky) _J!_ , &c 
^ dy 3 * 1 . 2 . 3 
If a third variable z becomes z + l by a third operation -4/", then the actual com- 
Z 2 cZ 2 z 3 d 3 
position of the equivalent polynomial 1 + l d z -f y~r ? + y t r 3 +, &c. gives 
'P'l' 1 •4' ,, = l + (^^ x +& ^) + y~ 2^ d x -\-k dy+ld z ) 2 +Y^%^(hd x -\-kd 1/ -\-ld z ) ? '-\-,&i.c.; 
and so this method may be extended, whatever be the number of variables. 
9. Let 0 denote any linear operation, and make 
0 2 P 
0 = 1 + 0 + — 4- ytoTs +, &c., 
then 0 is itself a linear operation ; and if in (7.) we put 0 = 0' = 0 " =,& c., and sup- 
pose the number of these operations to be n, we have by that article 
Yi % A- A3 
0» =1+w0 + _ + _J_ +5&c . 
0 0 2 6 s 
V ~ 1 + m + 172^ + 1.2.3m 3 + ’ &C ’’ 
m being any positive integer, we have by the same principles 
fl 3 
Again, suppose 
f l =1 + 04. — + Y7273 +, &c. — 0 
<P 
1 H — . 0 -j- 
1 m 1 m* 
6 2 
+ 
Q 3 
+,&c. 
* 2 ' 1.2^ m 3 1 . 2 . 3 
Hence if (p denote an operation which, repeated m times, gives 0, and in which we 
1 n 
shall employ the notation 0 m , then <p n denotes the same as 0”*, the latter being the 
operation which, repeated m times, is the same as 0”. With this meaning understood, 
it follows that 
0 m = 1 — j— — ~ . 0 — {— — 2 
1 Yil ! ™ z 
Lastly, if we put 
ri 3 0 2 
m 2 ' 1.2 
fl 3 
1 _ 
' m 3 
S 3 
1.2.3 
+ , &C. 
Cl — 1 — n 0 + yyyj — j 0.3 +? 
compound this by the formula of (7.) with the operation 
w 2 fi 2 n 3 fi 3 
^=1 +W 0 + — + r -^ 1 +,&c., 
