1878.] on certain Definite Integrals. 131 



Now let us define four quantities, pi, p 2 , jii, thus 



^1=P2^2>— Pi Mj y «2 = />2 + />lV / 3— 



then we shall have 

 C (a x 3 + a 3 dx 



( 75 Jj^Z^ y^3| = n sm 1 a + sm- 1 ( / > 2V / 3-^i)a}. 



d6cosr6 7T r y iZI^2~_ 1 1 r 2tt 



' = 37l^l ^ J 3(2a) 2 0>i~ 



(76.) 



o 1 + a? cos 3 0~3 yiZv 1 S J 3(2a)V 2 + /?2 2 ) 



/ r-1 r-2 r _ 3 3 , \1 

 —p^rtf > 2 — r.— _/* 1 r V+ • •) j- 



( 77 -) Jol + ^cos^Sv/T^ 6 a 



2tt 



(78.) log e (1 + ci 3 cos 3 0) = 7t log e l±-^ll 



J° 8 



(79.) 



+ 7rlog e (l+ y(a 9 +2\/l + a»+a*+2) + v^L + ^+a 4 ). 

 cos r<9 log e (l + a 3 cos 8 #) j ^T^j: j * + 



2 / r r—1 r _ 2 _ . r— 1 r— 2 r— 3 r , \ 



(80.) From (79.) we can immediately deduce 



[V m 2 + m cos 7 „, ~ . ' 



- — «0 log. (1 + a 3 cos 3 0) . 



Jo l + 2mcos0 + m 3 Be v 7 



(81.) If we denote integral (78.) by jp, and (80.) by q, we have at 



once 



dO log e (1 + « 3 cos 3 <9) jc» — 2q 



l + 2mcos0 + m 2 1—m 2 ' 



K 2 



r 



Jo 



