Lord Rayleigh's Investigations in Opti 



405 



in which the upper limit of the integral is the cube root of the 

 extreme aberration expressed in quarter-periods. For example, 

 the upper limit is unity when the phase at one extremity is a 

 quarter-period in advance, and that at the other extremity a 

 quarter-period in the rear, of the phase at the centre. 



The influence of aberration may be considered in two ways. 

 We may suppose the aperture (a) constant, and inquire into 

 the effect of an increasing aberration (c) ; or we may take a 

 given value of c (i. e. a given wave-surface), and examine the 

 effect of a varying aperture. To the latter comparison Airy 's 

 results are more immediately applicable. The following Table, 

 easily derived from that given by him, exhibits the values of 

 cos ^7r(o> 3 — mco)dco, between the limits specified in the head- 

 ings of columns 2, 3, 4, 5. The results are applicable at once 

 to the comparison of the amplitude-curves corresponding to 

 various apertures, since the relation of m to f in (3) is inde- 

 pendent of a. To obtain intensities, it would be necessary to 

 square the numbers given in the Table. 



5 



Value of m. 



From to 

 1-00. 



From to 

 1-26. 



From to 

 1-44. 



From to 



00. 



-4-0 



+ 0929 



-•0692 



+•0588 



+•0030 



-3-6 



•0783 



-•0467 



+ •0197 



•0062 



-32 



+ 0343 



+ •0142 



-•0309 



•0124 



-2-8 



-•0203 



•0849 



-•0461 



•0239 



-2-4 



•0563 



•1320 



+ •0018 



•0444 



-2-0 



-•0430 



•1399 



•1009 



•0791 



-1-6 



+ •0411 



1263 



•2095 



•1346 



-1-2 



•1997 



•1377 



•2906 



•2184 



-0-8 



•4140 



•2266 



•3462 



•3362 



-0-4 



•6449 



•4185 



•4211 



•4886 



0-0 



•8422 



•6873 



•5672 



•6653 



+ 0-4 



•9570 



•9538 



•7898 



•8404 



+0-8 



•9559 



1-1120 



1 0157 



•9701 



+ 1-2 



•8307 



1-0748 



1-1141 



•9979 



•fl-6 



•6024 



•8170 



•9681 



•8705 



+2-0 



•3161 



+ •3952 



+•5569 



•5649 



+2-4 



+ 0303 



-•0679 



-0060 



+ •1172 



+2-8 



-•1988 



•4290 



•5110 



-•3624 



4-3-2 



•3309 



•5826 



•7545 



•7087 



+3-6 



•3521 



•5028 



•6485 



•7652 



+4-0 



-•2761 



-•2525 



-•2725 



-•4745 



The second column relates to the case where the aperture is 

 such that the aberration between the extremities and the centre 

 is one quarter of a period, or (which is the same thing) where 

 the wave-surface at the extremities deviates by a quarter wave- 

 length from the tangent plane at the central line of inflection. It 

 will be seen that the position of maximum illumination deviates 

 sensibly from the centre {m = 0, £=0). This is no more than 

 might have been expected, since the plane which most nearly 

 coincides with the actual wave-surface is inclined to the cen- 



Phil. Mag. S. 5. Vol. 8. No. 50. Nov. 1879. 2E 



