268 On the Intensity at the Focal Point of a Telescope. 

 is the coefficient of xp~ 1 in the product 



(-ir , ^{ 1 +^+o;f + ra-f. + -"}{ 1 -( 2 " + ^ 



+ (2n + i)(2n-i)-^-...}, 



and this product is equal to 



(-l)?- 1 .^(l-o ? )-*(l-^) 2w+ "=(-l) p -\7r t (l--^) 2B . 

 Hence 



g l° < 1Y ' 2 * + * -A 2 >- igg 



g=P -i ^ ' |jg — g — 1 | 2n + j-p + y + l I? | p-l [ 2n + l— j? 



and the general term of the integral is 



7r. a 



4«+2 



0/3) ra p= 





We thus obtain for the amplitude of the displacement at 

 the focal point an expression of the form 



A = 7r{C + C 1 (t/3) + C 2 (^) 2 + ...f, 

 where C„ is the sum of the expressions obtained from 

 a 4»+ 2 P =2»+i {^} 2 ( r \ 2p 



by giving r and « their proper values : and the intensity is 



I=^{C 2 +(C 1 2 -2C 2 C )^+(C 2 2 -2C 3 C 1 + 2C 4 C )/e 4 + --.K 



An interesting case is that in which the apertures consist 

 of x equal circles, of which one has its centre at the middle 

 point of the object-glass, while the centres of the remaining 

 x— 1 are equidistant from this point. In this case ; writing 



r=fiR, a = vTi, 



where R is the radius of the full aperture of the object-glass, 

 the value of the intensity at the focal point is 



I = 7T 2 . x V R 4 { D 2 + (D x 2 - 2D 2 D ) /3 2 R 8 



+ (D 2 2 -2D 3 D 1 + 2D 4 D )/3 4 R 16 + ... }, 



