ON THE PROPERTIES OF A CERTAIN SYMBOLICAL 
EXPRESSION. 
[From the Cambridge Mathematical Journal, vol. ill. (1841), pp. 62—71.] 
The series 
Spl’tp (®' J + 6 ' 2 ••• n terras)^ (- 
l 
d 2 
+ 
m 
æ 
+ l'da 2 1 + m'db 2 "'J {(1 + l)a 2 + (l + m) b 2 ...j i 
2 2i?+1 1.2 ... p. i (i + 1) ... (i + p)J ' 
possesses some remarkable properties, which it is the object of the present paper to 
investigate. We shall prove that the symbolical expression (\{r) is independent of a, b, 
&c., and equivalent to the definite integral 
x 2i ~ l dx 
f 
J 0 
{(1 + lx 2 ) (1 + mot?) ... ’ 
a property which we shall afterwards apply to the investigation of the attractions of 
an ellipsoid upon an external point, and to some other analogous integrals. The 
demonstration of this, which is one of considerable complexity, may be effected as 
follows: 
Writing the symbol 
l d 2 
+ 
d 
1 + l ' da 2 1 + m ' db 2 “ 
under the form 
d 2 d 2 
da} 
+ 
d 2 \ 
db 2 "') 
d 2 
+ 
d 2 
1 + 1' da 2 1 +m' db 2 '" 
= A — 
d 2 
+ 
d 2 
1 +1' da 2 1 + 7ii ' db 2 
suppose, 
let the p ih power of this quantity be expanded in powers of A. The general term is 
(-1)* 
which is to be applied to 
p(p-l) ...(p- q + 1) 
1 . 2 ... q 
1 
{(1 + V) a 2 ... Y ' 
d 2 
1 + 1' da 2