404 Lord Rayleigh's Investigations in Optics. 



intensity-curves, and especially to ascertain at what point a 

 sensible deterioration of definition ensues. The only work 

 bearing upon the present subject with which I am acquainted 

 is Sir G-. Airy^s investigation " of the intensity of light in the 

 neighbourhood of a caustic"* ; but the problem considered by 

 him relates to an unlimited beam. 



Considering in the first place the case of a beam of rectan- 

 gular section, let us suppose that the aberration, or error of 

 phase, is the same in all vertical lines, so that the actual wave- 

 surface is cylindrical. With origin at the centre and axis of 

 x horizontal, the aberration may be expressed in the form 



C0 8 +/tf* + (1) 



No terms appear in x or x 2 : the first would be equivalent to a 

 general turning of the beam ; and the second would imply im- 

 perfect focusing of the central parts. In many cases the cir- 

 cumstances are symmetrical with respect to the centre ; and 

 then the first term which occurs is that containing x*. But 

 in general, since the whole error of linear retardation which 

 we shall contemplate is exceedingly small in comparison with 

 other linear magnitudes concerned in the problem, the term 

 in x d is by far the more important, and those that follow may 

 be neglected. 



As in the case of no aberration (treated in § 2), the distribu- 

 tion of brightness in the image of a point is similar along every 

 vertical line in the focal plane ; and therefore the image of a 

 vertical line follows the same law of brightness as applies in 

 the case of a point to positions situated along the axis of f. 

 The phase of the resultant at any point f is by symmetry the 

 same as that of the secondary wave issuing from the centre 

 (#= 0) ; and thus the amplitude of the resultant is proportional to 



\ +ja cos27r(^,+cx^dx (2) 



In Sir G. Airy's problem the upper limit of the integral (2) 

 is infinite. Fortunately for my purpose the method of calcu- 

 lation employed by him is that of quadratures, and the inter- 

 mediate results are recorded (p. 402) in sufficient detail. In 

 order to bring (2) into conformity with Airy's notation, we 

 must take 



o <? i i 2ttxP it /on 



2<7rcr = i7rft> d , -r^~ = — m^co ; ... (3) 



Kj z 



we thus obtain 



(4c)~*i 2 c cosi7r(a> 3 — mco)dco, . . . (4) 



* Cambridge Phil. Trans, vol. vi. 1838. 



