332 Mr. W. H. L. Russell on [Jan. 27, 



Hence we have 



2 3 l1r 



— " d0{ 6- 4 ^/(cos 0e®) + e 4 ^/(cos 6e-to) } 



==— I d0{ Aq cos 40 + cos cos 30 -f A 2 cos 2 cos 20 4- A 3 cos 3 cos } 

 2A 2 -3A 3 =1 4 (a 3 . -^+A 3 . ^)+/"|-2A 3 -3A 3 . 

 Hence we shall have : — 



[ 2 ^{ e -^/(cos^) + e ^/Ccos^)} =-/"(i) • • (133), 



This formula was obtained by differentiating f(x) twice, but similar 

 formulae may be obtained by differentiating any number of times. 



By analogous processes we may obtain likewise the following in- 

 tegrals : — 



i ie i w 7t r 1 1 1 1 •) 



j 2 d0{e-±J(coS*0eT)+e^f(cOV0e--i)} = Q j ^- ? =__ / /— J 



. . . (134). 



This integral requires the evaluation of 



J* 



cos" cos 00(20 



when /5 is greater than n, and consequently the usual formula does 

 not apply. We may, however, proceed thus ; since 



fd0 cos" cos (30=%fd0 cos» +1 cos (0- Y)0-\fd0 cos" +1 cos (0-3)0 

 -/d0cos«0cos (0-2)0+/ c?0cos" +2 cos (0-2)0 .... (135), 



we are able by successive reductions to reduce the required integral to 

 known forms. 



[*d6 l0g e COS COS 2 0{e^ (^)/ 6 2;(f+0 + ^(M^-Ji J 



Jo 



=>(1) where ^{x)=ffff(x)dx^ .... (136), 

 4 



|^0 sin | («¥/«"- e-?/ 6 -*0 = *V0(1), where 0(®) =fff f(x)dx\ 

 and O is the quantity denned in the fifth paper of this series . (137). 



