164 
MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
The use of this integral will give an important extension of the method I have 
employed for expressing the integrals of differential equations by means of definite 
integrals. For in order to the success of that method, it is necessary, as is shown 
in my paper in the Cambridge Mathematical Journal before alluded to, that the 
magnitude of the factorials (if any) in the numerator of each term of the series to be 
summed, should be less than that of the corresponding factorials in the denominator ; 
whereas this integral enables us to sum series in which the reverse is the case. I 
shall now apply the series, whose sum we have just found, to the evaluation of definite 
integrals, using series VI. and VII. Hence 
“TT 3'7r TT 
Mdv t;(l — vf cos {2^v sin ^ cos 
2 
d6dv ^;(l — cos^^ g 2 ^«cos 20 sin & cos ^+2^) 
2 
STT r ^ \ O ■ ^ \ STT 
By a process similar to those used above, we find 
14 
2.3 
jW-V 
5.3.1 
2^ 
5.5. 3. 4. 1.2 
2 2 
2 
24 
&C. 
■ 4 ^ 4 (/>'' V — 2[mx 4 2 ) 5 '^ V + 2 / Ad : + 2 ) £ 
Hence 
.a cos (p 4 2/4 cos B cos {B 4 (p) 
cos d cos (a sin (p) cos cos 6 sin (^4'?)) 
“~^+^2(2//-a— 24//Aa+l) £'*'''^“ 4 ^^( 2 fAa 42 ^//Aa+l) s 
The following formulae are found in Crelle’s Journal; — 
I cos“ ‘^0 cot^0 cos a0d0- 
Jo 
i 
- I 
cos" ^0 cot^0 sin a0d0 
cos‘‘~^0 cot*^ s.‘'‘^d0 
a — r(a + b— 1) 
7C 
r(«)r(z.) 
2 cos— 
,_r(«+6— 1) 
TT 
r(fl)r(6) 
^ . bn 
2 sin- 
1 
1 
% «(1- 
- i? 
TaVb sin^TT 
whence we find 
r(fl + i— 1) rZ»sin^7r 
Ta TT ^ 
cos® ‘^0 cot*^ 
In this formula we suppose {h) to be less than unity. 
