[2 
2] 
SYMBOLICAL EXPRESSION. 
7 
Now U representing any homogeneous function of the order 2s, it is easily seen that 
A ? = 7 + 2i(2i+2_4s_n) ^ ; 
and repeating continually the operation A, observing that A U, A 2 £7, &c. are of the 
orders 2 (s — 1), 2(s — 2), &c. we at length arrive at 
A«.-? = A* U. 
P l P 
+ | 2i (2i + 2q - 4s - n) A« -1 U. -Jpj 
+ q( j~ 2 V) 2 i (2 i + 2) (2 i + 2g - 4s - n) (2 % + 2q - 4s - n - 2) A«“ 2 U.~- a 
this is 
>r the 
+ 2i (2i + 2) ... (2i + 2q) (2i + 2q — 4s - n) ... (2i + 2 — 4s — u) U. -j. 
i+q ' 
Changing i into s + i + i', we have an equation which we may represent by 
. U A A* U , . A?" 1 U . U 
A q — A l. l A -4-9/1 •/ —— 
^ pS+i+i’ Xl 9> 1 pS+i+i’ ' 9» ? pS+i+i'+i ~ 9’ pS+i+i'+q *** 
where in general 
r a - g (g — 1 ) ••• (g-y + 1) 
1 . 2 ... r 
x (2s + 2i + 2i v ) (2s + 2% + 2» + 2) ... (2s + 2% + 2i + 2r-2) 
x (2i + 2t' + 2q - 2s - n) ... (2i + 2% +2q-2s-n-2r + 2). 
Now the value of S, written at full length, is 
ZsP s+1 (a. 
(«)» 
(~l) s f 
1.2 ...s 
A s -^- — - Æ A s_1 - 
s pS+i J pZ+l+l 
+ Çs-i P*+ l ~ l ( ««A 
+ &c. 
17 s-1 
->s+l 
&A S 
IT 
+... 
and substituting for the several terms of this expansion the values given by the 
equation (a), we have 
S=, 4~ 1 -'- (kA'U+i,-A-'U +k,-, u) 
1.2 s V p P ' 
where in general 
fc x — a s (*A, o £s + X 1 -A s —i, o Çs—i • + -d. s _ x , 0 Ç g —x) 
— fis ^ X-1 -d«_i, i £s • • • + “ 1 ~ l £s-x+i) 
+ A (8(8-1)... (S-X + 1) \ 
± A s 1 V~ 2 ^ X<ssj ) 
\ s being the (x + l) th of the series a s ,fi s ...