[3 
ith 
its 
mg 
3] 
ON CERTAIN DEFINITE INTEGRALS. 
In general if u be any function of £, a, b... 
d 2 cl 2 
. d 2 u „du ^ 1 . d 2 u ^ a 
4 + 2 S -irT-r„ + 4 -1 
. _ d% 2 di% %+ii 2 d%da % + h 2 y d 2 u 
da 2 db 2 / ^ a 2 da 2 ’ 
(Z+hJ 
from which the values of the second side for q = 1, q = 2, «fee. may be successively 
calculated. 
/ d\ p f d\Q ( d \ r 
The performance of the operation (-5-) I ] ..., upon the integral V, leads in 
\da) \dbj \dc 
like manner to a very great number of integrals, all of them expressible algebraically, 
for a single differentiation renders the integration with respect to </> possible. But 
this is a subject which need not be further considered at present. 
We shall consider, lastly, the definite integral 
(a -x)f(ri + ri + — ) dxd y 
U= ... (n times)- 
Ì1 2 h 2 
{(«a-x) 2 + (b — y) 2 ...} in 
limits, &c. as before. This is readily deduced from the less general one 
(a — x) dxdy ... 
... (n times) 
where 
and 
Hence 
or 
proper limits, 
C. 
{(a-x) 2 + (b-y) 2 ...} in 
For representing this quantity by F(h, h / ...), it may be seen that 
U=\ f (m 2 ) — F(mh, mh,...) dm ; 
but in the value of F (h, h / ...), changing h, h,... into mh, mh,... also writing m 2 <p 
instead of </>, and m 2 ^ for £, we have 
, hh / ...ir in 
d (mh, mh / ...) = — a 
r(*n) Jo (r + ^+4>)V (<*>') 
«F «(*' + ** + *) (f + V + tf,)... 
f+A i+ f+V + "' 
L F ^ mh --) =Ê £i F{mh ’ mh --)' 
hh ... 7r‘ n 
— L // _ 
T(in) “dm 
, observing that is equivalent to ^, and effecting the integration between the