192 Lord Rayleigh on the Theory of Optical Images, 



we get 



A r =7r(y I= _,+y„ +8 )=2^ =s . . . . (62) 



From (62) we see that the occurrence of the term in A r , 

 i. e. the appearance of the spectrum of the rth order, is asso- 

 ciated with the value of a particular ordinate of the object- 

 glass. If the ordinate be zero, i. e. if the abscissa exceed 

 numerically the half- width of the object-glass, the term in 

 question vanishes. The first appearance of it corresponds to 



ia = 2r7r/p l = r\f/% 1 , 



in which a is the entire width of the object-glass and fj_ the 

 linear period in the image. By (17 a), 



X/_ Xf sin 13 _ J«X 

 £, e sin a e sin a, ' 



so that the condition is, as before, 



e sin a = rX. 



When A r has appeared, its value is proportional to the ordinate 

 at x=.s. Thus in the case of a circular aperture (a=2R) we 

 have 



y z=t = Ti\/{l-r 2 \ 2 /e 2 sin 2 *} (63) 



The above investigation relates to a row of luminous points 

 emitting light of the same intensity and phase, and it is 

 limited to those points of the image for which 7} (and q) 

 vanish. If the object be a grating radiating under similar 

 conditions, we have to retain cos qy in (12) and to make 

 an integration with respect to q. Taking this first, and 

 introducing a factor e ±k ? } we have 



+°° 2k 

 J ±kq cos qy dq = p-—t (64) 



This is now to be integrated with respect to y between the 

 limits —y and -\-y. If this range be finite, we have 



Limit* =0 f JV+p = 2lT > ' ' ' ' (65) 

 independent of the length of the particular ordinate. Thus 



Ci = I Cdq = 2ir\ cos pxda:, .... (66) 



the integration with respect to x extending over the range for 

 which y is finite, that is, over the width of the object-glass. 



1 



