[2 
2] 
SYMBOLICAL EXPRESSION. 
9 
icity 
the 
rator 
or we have simply 
where 
Consider the term 
(-l)*fro_ (-1)* 
1.2...5 2 2s (1.2 ... s) 2 . (2i + 2s) ’ 
8 = 
(-1 y 
2 s » (1.2 ...s) 2 . (2*+ 2s) 
da 2 db- 
1.2 ... s 
1.2... X . 1.2 ...ya. &c. 
with respect to this, A s reduces itself to 
1 . 2 ... s 
m ix _ _ • 
d \ 2A 
1.2 ... X . 1.2 ... ^i. &c. \daj 
and the corresponding term of S is 
(-l) s 
2^[2i + 2s) (1.2 ... X. 1.2 .7. /¿7&c> 1 ■ 2 ''‘‘ 2X • 1' 2 • ■ • • &c - P• • • 
_(-l)M.3... (2X — 1). 1.3 ... (2fi — 1). &c. 
(2i + 2s) 2.4... 2X.2.4...2/*. &c. w *” 
which, omitting the factor ^, and multiplying by ¿c 2s , is the general term of the 
s th order in l, m,... of 
V{(1 + ¿# 2 ) (1 + m# 2 )...} 
The term itself is therefore the general term of 
¿c 21-1 dx 
V{(1 + lx 2 ) (1 + mx 2 ) ...} ’ 
or taking the sum of all such terms for the complete value of S, and the sum of the 
different values of S for the values 0, 1, 2... of the variable s, we have the required 
equation 
3,21-1 rf jX 
Y = 
0 V{(1 + lx 2 ) (1 + mx 2 ) 
Another and perhaps more remarkable form of this equation may be deduced 
b 2 
by writing “ 7 , ——, &c. for a 2 , b 2 , &c., and putting + —- h &c. = rj 2 , Irf = a. 2 . 
i + i L+m x I+tlfm 
mrf = f3 2 , &c.: we readily deduce 
x 2i ~ l dx 
V {(rj 2 + a 2 x 2 ) (r) 2 + /3 2 x 2 )...} 
b 2 
_ Q 00 1 
-&p 0 2 2 ^ +1 .1.2 ... ». i(i 
2 dr_ ,o,dr_ Y J 
p . i (i + 1) ... (i+p)\ da 2 db 2 "') (a 2 + b 2 ...y’ 
c. 
2