MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
171 
Hence we have 
■’r r.i. 
27r 2ir 
CJ 
1 f "■ 
{l+iz)^{\+iz] 
M+iz)(l+iz').2^ 
f2i0 
Hence 
77 77 
COS 9 cos (J cos (0 + (p) 
cos i &co^~^<p.d&dp, 
cosf I cos 6 cos ^ sin {d-\-<p) +2+?*— (tan ^+tan <p) 
4 -V^TT 
c 
fx cos 20 
^^gacosd ^os a sin ^.s cos 
fx, sin 2 fl 471-2 
=2 • 
(A.) 
But we may effect these reductions systematically by means of the following- pro- 
position due to M. Smaasen: — 
If ao+fli 4?-|-a2 x^-l-ag ^®-i-&c.=<pi(x), 
and Z>o+6i ^-4-^2 •^^+^3 ■3^^+&c.=<p2(‘2^), 
then «o ^ 0+^1 ^1 4:’+«2 ^2 
— 2wJ] '0)('P2 (0+92(2 ’®))}. 
M. Smaasen has also proved in the same paper, that if the sums of the three series 
Oo+ai ,r+«2 •3;"+ -[-&c. 
j:- 1-62++^3 .a:®+&c. 
Co + Ci x-\- C2 ■X’^+Cg x^-f&c. 
are known, we may determine the sum of the series 
^0 Co + «i &1 Cl X-{-a2 62 C2 4?^ + &C. 
by means of a double integral, but we shall not want this in what follows. 
'y> /y)2 
w , ^ Jl, 
Now 
^■^1.20 1.2.3.4"^ 1.2.3.4.5.6"^^^' — 
1 -b&c. = ; 
£ _f_ £“ 
consequently 
*+T 
\XX 
A. 12.22 i. + 12. 22. 24 
2 22 
-&C. 
1^ C- 
“2^ Jo 
d6 
j As'® e ■''^■£*9 ^ j— — i9 g V yj 
-I- ^ 3re~^^ -1- f= As 
■Aci® I .-A.-*® I .— d Xt-i9 
_2g'^i>;cosY j^Qg ^ g-j^ 0 _j_2g 
— cos '^r 
2 COS 
JT sin 2 ^ ; 
Now 
