362 On certain Definite Integrals. [June 21, 



(34.), (35.), and (50.) arise from the integrals 



r w cos 2ra6.ddd f " cos2rad.dd 

 J o sina0 (l + 2 ) an J Q cosa0(l + d 2 ) 2 ' 



(53.), (54.), and (55.) are obtained by using series analogous to 



e cosO cossin0 + /ie 2cos0 cos(2sin0)+/i 2 e 3cos0 cos(3sin0)+ . . ., 



and (59.), (60.), and (61.) are found by the series 



a? sin 0(1— cc 2 ) 



sin0-2^ 2 sin20+3^ 3 sin30 — . . .= 



(l+2x COS0 + O7 2 ) 2 



The series employed are considered as convergent, according to the 

 rule given by Cauchy. On account of the great importance of the sub- 

 ject of convergent series, I have thought that mathematicians would be 

 interested to see how the results of Cauchy are confirmed by a method 

 given by me in the ' Proceedings of the Eoyal Society' for 1872, vol. xxi. 

 page 20, and which was afterwards discovered independently by Sir ~W. 

 Thomson. Let u Q +u x x + w 2 .r 2 +..'.+ u n x n + ... be any series, then 



I call ^Htll when n increases without limit the ultimate ratio of the 

 u n ^ 



series, and denote it by p. Then if cc is less than -, the series will be 

 convergent. 



Now consider the function which we have employed in these 



6*4-1 



investigations. 



1 1 ■ ■ . . 1 



Since h 7 =1? the series comprising can contain no 



e x +l e _x +l e x -f-l 



even powers, and therefore we may assume 

 1 



- = u + x 4- 4- u 5 x 5 4- . . . 



Expanding e*4-l in terms of cc, and multiplying the series together, 

 we obtain 



u 2n -i u 2n -3 

 2u 2n+1 + T - J + 1 2 3 4 4- 0, 



or 1 -f 14- z — 7, + • • i o q a + * * — u * 



U 2 n+1 I- 2 U2n-1 V>2n+l l.J.d.4 



Let n increase without limit, and then 



U2n+l «*2n-l _ _ 



U2n-1 U2n-3 



