6 
ON THE PROPERTIES OF A CERTAIN 
[2 
Considering the expression 
f 1 dr V 
if for a moment we write 
\1 +1’ da? J {(1 + L) a 2 ... } f ’ 
( p $ 
(1 + l)a 2 = a 1 2 , &c.; Al = + ¿¿F 2 ’ pi = «i 2 + W • ••, 
this becomes 
A,« 
1 ’ 
Pi** 
m V • • a- + i .I , » 1 2i'(2t , + 2 —n) 
JN ow it is immediately seen that — = —^ ; 
from which we may deduce 
A a 1 _ 2i (2i + 2) ... (2i + 2q — 2) (2i + 2 — n)... (2i + 2g — n) 
1 p?~ p/ +q 
or, restoring the value of p 1} and forming the expression for the general term of (yfr), this is 
k-P** 1 f 
A? 
P2i(2i + 2 n)A^ 1 ( a * + p + m " + i a , + m p + mm ')< 
4- &c. 
p representing the quantity a? + b 2 + &c. 
Hence, selecting the terms of the s th order in l, m, &c. the expression for the 
part of which is of the s th order in l, m &c. may be written under the form 
(a? + b 2 ... + la 2 + m& 2 4- &c.) { 
1 
a s(-l) S p^& 
1 . 2 ... s 
multiplied by 
( 
»’(» + !) ...(¿4-s-l) AP-^- s 
2i (2i 4- 2 - n) (i + 1) ... (% + s) A ÿ_1 -j 
,i+*+l 
2i ^ + 2) (2 ' + 2 “ n > ( 2i + 4 - n) (i + 2) ... (» + 8 4 1) A 
— &c. 
[¿a 2 4- w6 2 ... = U suppose] 
which for conciseness we shall represent by 
(- 1 ) 8 a v +i t 
1.2 
p i+s 
-f 
i pj£r±) yiA p-*JL 
+ 1.2 7s P i+S+2 
— &c. 
= suppose.