10 
ON THE PROPERTIES OF A CERTAIN 
[2 
7] being determined by the equation 
a 2 6 2 _.. 
tf + a 2 rf + /S 2 " " * ’ 
•or, as it may otherwise be written, 
1 = a 2 + b 2 4-... — 
a-a- 
6 2 /3 2 
&c. 
rf + at 2 rf + /3 2 
rt, it will be recollected, denotes the number of the quantities a, b, &c. 
Now suppose 
V= ff... cf) (a —cc, b — y, ...) dx dy ... 
{the integral sign being repeated n times) where the limits of the 
by the equation 
x 2 y 2 
h 2+ hf 
+ &c. 
= i; 
integral 
are given 
and that it is permitted, throughout the integral to expand the function <f>(a — x,...) 
in ascending powers of x, y, &c. (the condition for which is apparently that of $ 
not becoming infinite for any values of x, y, &c., included within the limits of the 
integration): then observing that any integral of the form jj ... x p y q ... dxdy & c.... where 
any one of the exponents p, q, &c. ... is odd, when taken between the required limits 
contains equal positive and negative elements and therefore vanishes, the general term 
of V assumes the form 
1.2... 2r. 1.2 ... 2s ... 
m 
...</> (a, b ...)//...x^y 28 ... dxdy ... 
Also, by a formula quoted in the eleventh No. of the Mathematical Journal, the value 
of the definite integral //... x™'y 2S ... dx dy ... is 
M h 2S+1 r (r + £) r ($+ j) ... 
n ' ••T(r + «+...+in + l)’ 
{observing that the value there given referring to positive values only of the variables, 
must be multiplied by 2 n ): or, as it may be written 
h** i V* +1 • • 
1 1.3... (2r — 1) . 1.3 ... (2s — 1)... 
17 ■ 2 r+s - in (in + 1)... (in + r + s ...) r (in) ‘ 
hence the general term of V takes the form 
kh / ... 7r* n 1 1 1 
r(in) |n(î n + !) ••• (in + r + s ...) ' 2 2r+2S - 1.2.3 ... r. 1.2 ... s... 
x fi 
da- 
l2 dl 
h ‘ db 2 
(f>(a, b, ...); 
and putting r + s + &c. =p, and taking the sum of the terms that answer to the same 
value of p, it is immediately seen that this sum is 
hh,... 7T* n 1 
r(in) •2^:i.2...p.in(in + l) ...(in+p) 
d 2 , h2 d?_ 
da* + h ' db*" 
<f>(a, b...).