.-'-" ivith Special Reference to the Microscope. 187 



The series (32), as it stands, is not periodic with respect to 

 u in period v, but evidently it can differ from such a periodic 

 series only by the factor e imu . 



The series 

 <?-*"'" sin u e ~ im(u+v) sin (u -f v) 



u u-\-v 



+ e + — rV — + 34 



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

 theorem in periodic terms : 



(34) = A + t'B + ( A x + iB x ) cos (2iru/v) 



+ (Ci + e'Dj) sin (2ttu/v) + 



+ (A r + iB r )cos(2nn*/t') + (C r + iT) r )sm{2<nru/v) + . . . (35) 



As before, if s = 2rir/v } 



~ e~ imu sin u cos su 



i^(A r + iB r )=j _^ du; 



so that B r =0, while 



i a P +c0 cos mw bid u cos su 1 " x 



i V ; A r= | - — <**• ' ' ( 36 ) 



In like manner r = 0, while 



-*M- 



sin mu sin w sm su 



u 

 In the case of the zero suffix 



-r, rt A P +Q0 cos ??2w sin m 7 

 B = 0, t?A = 1 du. . 



J -go M 



<£«. . . (37) 



(38) 



When the products of sines and cosines which occur in 

 (36) &c. are transformed in a well known manner, the inte- 

 gration may be effected by (15). Thus 



cos mu sin u cos su = \ {sin(l + m + s) u + sin(l — m — s)u 

 -t-sin(l + m — s)w + sin(l — m + s)u} ; 

 so that 



±v. A r = \ir{ [1 + m + s] + [1 — m — s] + [1 + m — s] 



+ [l-m-fs]} . (39) 



where each symbol such as [l + ?n-fs] is to be replaced by 

 + 1, the sign being that of ( L + wi + s) . In like manner 

 — 1?,. D r = i7r{[l + m-s] + [l-m-f s]-[l-H>* + s] 



-[l-wi-a]} . (40) 



