466 Transactions of the Society. 



is truly periodic, and may therefore be expanded by Fourier's 

 theorem in periodic terms : 



(34) = A + i B + (Aj + i BJ cos (2 it nfv) 



+ (Cx + * D x ) sin (2 7T %/■»)+ 



+ (A_ 4- »B,) cos (2r ttu}v)+ (C r + iD) sin (2rww/v)+ . . . (35) 



As before, if s = 2 r irjv, 



. co e — imu g [ Q u cos su 



iv(A r + iB r ) = J* 



% 



ow 



so that B r ss 0, while 



(** °° cos mu sin w cos su , /0 ~ x 



i v . A = «w. • (36) 



J —CO U 



In like manner C r = 0, while 



^ r + °° sin »*w sin w sin sw , /ot7N 



- £ v.D r = J ^ a«. • (37) 



In the case of the zero suffix 



^ f + °° cos mu sin u , ,., ON 



B = 0, « A = I ^ aw. . . (38) 



When the products of sines and cosines which occur in (36) 

 &c, are transformed in a well known manner, the integration may 

 be effected by (15). Thus 



cosmw sinwcossw = i {sin(l -f m -f s)u + sin(l - m - s) u 



+ sin (1 + m - s) u -f sin(l - m + s) u} ; 

 so that 



£ y. A r = i 7T {[1 + wi + s] + [1 - m - s] + [1 + m - s] 



+ [l-m + ,s]} . (39) 



where each symbol such as [1J+ m + s] is to be replaced by ± 1, 

 the sign being that of (1 + m + s). In like manner 



- i^.D r = lir{[l + m - s] + [1 - m + s] - [1 + m + a] 



-[1-m-s]} . (40) 



The rth terms of (35) are accordingly 



~ I e isu ([1 + m'-f s] + [1 - m> - *]) 

 2 v [ " 



+ <?- l ' s " ([1 -r m - ■»] + [1 - m + sj)} 



