Lord Rayleigh's Investigations in 0}rtics. 407 



quantity of light in the beam is independent of the aberration; 

 and this area is equal to that of a rectangle whose height is 

 the maximum ordinate A of the first curve, and width the 

 distance B between and the first position of zero intensity. 

 It appears that aberration begins to be distinctly mischievous 

 when it amounts to about a quarter-period, i. e. when the 

 wave-surface deviates at each end by a quarter wave-length 

 from the true plane. The most marked effect is the increased 

 importance of the lateral band on one side, and the approxi- 

 mate obliteration of the lateral band on the other side. 



When the aberration is symmetrical about the centre of 

 the beam, the term in a? vanishes, and the whole effect is of 

 higher order. In general the term in x* will preponderate ; 

 and thus the problem for a symmetrical beam resolves itself 

 into the investigation of aberration varying as x A . In one 

 respect the problem is simpler than the preceding, on account 

 of the symmetry of the intensity-curves ; but in another it is 

 more complicated, since the phase of the resultant does not 

 correspond with that of the central element. The intensity is 

 represented by 



[ ( +a °cos 2tt (^ +fi*j dxj 2 



+ [p%in2^(g+> 4 )^] 2 , .... (5) 



and requires for its calculation two integrations. These could 

 be effected by quadratures ; but the results would perhaps 

 scarcely repay the labour, especially as the practical question 

 differs somewhat from that here proposed. The intensity- 

 curve derived from (5) represents the actual state of things on 

 the supposition that the focusing adopted is that proper to a 

 very small aperture; whereas in practice the aberration would 

 be in some degree compensated for by a change of focus, as 

 it is obvious that the real wave-surface, being curved only in 

 one direction, could be more accurately identified with a sphere 

 than with a plane. 



Some idea of the effect of aberration may be obtained from 

 a calculation of the intensity at the central point (f=0), 

 where it reaches a maximum ; and this can be effected with- 

 out quadratures by the aid of a series. In this case we have 

 instead of (5), 



4 I" ( '"cos (2irfo:*)dx\ + 4 [" f '"sin (2irfaf)da!] . . (6) 



Now by integration by parts it can be proved that 



\ X e ihxi dx = e ih3:i lx-^ih %- + 



{Uhf x 9 

 5 9 



(Uhf x u 

 5.9 13 + ' 



1 

 "7' 



2E2 



