cos rO _ (163) 



; (164), 



1881.] certain Definite Integrals. 335 



['do ™* (161), [de ■ cosr * _ . (162), 



Jo (l-2*sm0 + <* 3 )'* V J Jo (l-2*sm0 + * 2 )" K J 



may be reduced to integrals 132, 153, 154, and other known forms, 

 and that consequently (resolving into partial fractions) 



l; 

 f: 

 J; 



Jo"" ' (1 — 2« sin 6> + * 2 )™ (1 — 2/3 sin 6> + /3 2 ) w . . . (1 — 2X sin fl-hA 2 )' v 

 may be ascertained. We may also find 



^log e ^2cos0^=g (167). 



I" S( log %T( log6 ^ + 3%3 * M + 2w) r (n) * (168) ' 



If we expand the denominator of the integral 



fa__52££?__ where . . . . (169), 



Jo 1 — 2a COS + a? n 



and integrate the terms in succession, we shall have to determine the 

 integral series of the form 



(1- 



-2a cos0 + « 2 ) wl (l- 



-2/3 cos0 + ^) n .. 

 sin rO 



• a- 



-2\ COS0 + A. 2 )'' 



'a- 



-2a cos6> + a 2 ) w (l- 



-2/3 cos6> + /3 2 )". 

 cosrtf 



..a 



— 2Xcos0+V 2 ) r 



(i- 



i 



■2asin6> + a 2 ) m (l- 



-2/3 sin<9 + /3 2 ) K . . 

 sinr0 



•(!■ 



^XsinS + X 2 )' 



which may always be found, when the values of (m) and (n) are as- 

 signed, from the expanded form of log e (l + x) by the method of 

 summation of the equidistant terms of series. Similar reasoning will 

 apply to 



fa . (170). fa oospe 



Jo l-2«cos0 + * 3 v ' Jo l-2*sintf+a 3 V ' 



J. 



i0, f n f° 5 (172). 



o l-2asin0 + « 2 v ; 



This method of summing the equidistant terms of series may be 

 applied to the determination of the values of other integrals, as for 

 instance 



