1879.] certain Definite Integrals. 363 



cos 20 . cos 30 . cos 46> . Q . . o 0, / . 6\ tt—0 . , 



+ + + • . . = cos + 2 sin~-loo' e I 2 sm- sm&. 



1.22.33.4 2 ° V 2/ 2 



We shall call this expression 9, and shall write more generally 



(r) _GOs r0 J _ cos (r-fl)< 



mn (m+l)(n + l) 



then it is manifest that Qi% Q^l, .... may be found immediately in 

 terms of 0. 

 Again, since 



# 3 # 4 or 5 . a; . a? 2 , 1— a;? , ... N 



we may find a precisely similar expression for Q[*\ and consequently 

 for ei*i, 0f 5 ... 

 We thus obtain 



<92.) \Q cos r0.d0= 



1 ' Jo 2(r + l)r 



/no \ I"" ^ 7/1 a 2 COS 26? — a 3 COS tt ( . /1 A , ... . , 



(93.) ede.—— j— r-= 5 {*+(l-*)log e (1-*)}. 



Jo 1 — 2a COS + a 2 2 



(94) T 0d0e« cos * cos (x sin 6? + 26?) = 1 ! ^-1 + * j . 



JO 2 I a 2 a 2 J 



(95.) J^f^jloge (l + 2^COS0 + t 73 2 )+2 t 6'COS£? + ^ 2 COs20} 



= - {( l_^2 logg ( l_ ; ,)_ ( , 2 + <(0} . 



<96.) r dee§>e* cos * cos (* sin + 30) = —{ 2) -f (* + 2 ) } . 



J 2a 3 



(9 7.) \ d0ei%e« cos a cos (« sin 6? + 46?) = Jl- ( 6 «(* 2 - 3a + 3) + — - 3 1 . 



Jo 2a 4 I 2 J 



(98.) £<w.e(i>e<l>=^. (99.) (^e 1 <?e^=^. 



More generally we may find 

 (100.) ^d0Q% n Ql*l, 



where viz, r, s, are in order of magnitude. 



The instance I have mentioned at the commencement of this paper, 

 is not the only instance of my obligation to the kindness of Professor 

 Stokes. He prevented me on another occasion from being led to a 

 false conclusion, by taking for granted without due examination an 

 integral which had appeared in print. 



2 c 2 



