270 ON CERTAIN FORMULAE FOR DIFFERENTIATION WITH APPLICATIONS [41 
which however is only a particular case of 
dx (1 — a?) 1 i (1 — 2 yx + y 2 )‘ 
d/3 
/3* (1 - 2@-x+- 2 
7 7 
= r 2 r 0 + I) Q _ ^x_2i 
I> + 1) p K P) 
.(10), 
/3 
which supposes 7 and - each less than unity. This formula was obtained in the case of 
(i + an integer, from a theorem, Leg. Gal. Int., tom. 11. p. 258, but there is no doubt 
that it is generally true. 
From (9), by writing x — cos 6, we have 
sin 2i 6dd rir(f+!) 
•(H). 
(1 — 27 cos 6 + y 2 ) 1 r (i + 1) 
which may also be demonstrated by the common equation in the theory of elliptic 
functions sin (c/> — 6) = 7 sin cf), as was pointed out to me by Mr [Sir W.] Thomson. It 
may be compared with the following formula of Jacobi’s, Grelle, tom. xv. [1836] p. 7, 
sin 21-1 6 d6 _ 1 f n cos (i — \)ddd ^ 
.(13), 
0 (1 — 27 cos 0 + y 2 ) 1 r (i + J 0 \/(l — 27 cos 6 + y 2 ) 
Consider the multiple integral 
w= i dxdy... 
J fix-a) 2 
{(iC — a) 2 + ... u 2 Y 
the number of variables being (2i +1) (not necessarily odd), and the equation of the 
limits being 
x 2 + y 2 . 
then, as will presently be shown, W may be expanded in the form 
(-)M* 
W = 7T i+ * & 
A 2- A r (A + 1) r (i + A + 
where A = a 2 + b 2 + and A extends from 0 to 00 . Suppose next 
dx d y ... 
£i-i(£ + w 2)-^£ (14), 
V 
.[ f 
J {(x-a) 2 ... 
u 2 Y {x 2 + ... v 2 ) u 
.(15): 
the number of variables as before, and the limits for each variable being — go , ao. We 
have immediately 
1 dW 
V 
-J 
J 0 
(£ + v 2 ) i+1 d% 
d£; 
W as before, i.e. 
V = 7T i+i S K 
(-)*M A 
d 
d£ 
2 2A r (A + 1) T (i + A + i) \duj J 0 (| + u 2 ) 1 (f + v 2 ) i+1 ’