Diffraction of an Anmdar Aperture. 3 



We shall, however, at present confine our attention to the 



case where <f> ( ^Ir + ;-) may be represented by / g . 



The disturbance of ether produced by the small area 85 xrSr 

 at the point in question, is now represented by 



. 2-nvt ., 27r . rlr 



s,n -^ X Sfl X cos -^A^ + r- . ^^7== 



9.i:vt -,. . 27r /_ rlr 



— cos — - — X 8 9 X sm — V u^ + r- 



the integral of which, with respect to r, is 



\ . 2T:vt .^ . 2 



sin 



X 8 9 X sin ^ -/A- + ?•« 



2w X X 



+ — .cos — - — X 8 9 X cos -— V ^2 + r% 



27r X X 



which, taken between the limits r, and /•„, is 



— . sm -^^ . (^sm — VA2 + r,;-^- sm— \/ h^ + vA .1 



, X 2'Trvt / 2 77 /-3 



27r 

 — cos— - 



^h^ + rf\l^. 



We have now to find the values of r^ and r,, in terms of d. 

 For this purpose we have only to remark that the equation of 

 the smaller circle, referred to the foot of the normal as origin, 

 and expressed by means of rectangular coordinates {x being 

 measured through the centre of the circle), is 



(.r — cf + y"- = a\ 

 or (r^ cos ^ — cf -\- [r, sin 9)^ = a% 

 or r^ —2c cos 9 . r^ = «2 _ ^9^ 



the positive solution of which is 



r^ = c . cos 9 + 'V^a^ — c^ sin^ 9 ; 

 or supposing c so small that the first power only is worth re- 

 taining, 



ri = a -\- c cos 9. 

 Therefore 



2 TT , 2 TT , 2 7r.ac.cos9 



-r ^*' + -.'= V ^"' + «' + x^i»T^ 



to the first power of c. 



B2 



