30 Transactions of the Society. 



The successive values of (10) obtained by giving successive values 

 to n in that expression are not capable of being simply added together, 

 for here the undulation is resolved in a direction perpendicular to the 

 D M . . vi axis ; which varies in position with the change of n. It is 

 desirable, therefore, to obtain a resolution of these impulses in certain 

 specified directions, and we may, for this purpose, select rectangular 

 axes coinciding with the line through the normal point and the point yj l 

 and with the edge ray through 8 S respectively. So resolved, A if/ will 

 obviously yield two resultants, as follows, if we write x and y for these 

 two axes in the order named. 



Aif/ X = sin ttAif; A if/,, = cos a A if/, 

 that is to say, in place of (10) we obtain two equations as follows— 



xf/ x = 2 (A V* ) = s . M x 2 ( -J?-- • ^ ^ . sm v — ^J— . 



, _, , . . N ,, _, /nc 4- s 1 • CN — n)ir sin n 6\ I 



or, more compendiously, 



A^=^A</,; A^=_A</r. 



If the conclusions now reached are sound, it would seem to follow 

 that much misunderstanding exists as to the distribution of light in the 

 antipoint. It is, therefore, proper to say that these results are not put 

 forward as being more than an approximation to the actual facts. To a 

 certain extent the mode of computation now proposed is open to the 

 same objection as that advanced against Sir Geo. Airy's method, namely, 

 that it yields discrepant results in the region in which beams going to 

 different points in the focal plane interpenetrate one another. But it 

 will hereafter appear that a correction can be applied upon the present 

 plan which gets rid of that difficulty, and yields a strictly coherent 

 result. Furthermore, the ultimate test is experimental, and to me it 

 appears that observation strikingly confirms the results to which these 

 calculations point, and is as strikingly at variance with Sir Geo. Airy's 

 curve. How far these impressions may be due to imperfect observations, 

 I cannot presume to say. Some of the results which have emboldened 

 me to submit the present hypothesis to public criticism are described in 

 the foregoing paper, and some of these will be exhibited at the meeting 

 of the Society by means of the apparatus with which I have observed 

 them. Assuming, for present purposes, the approximate accuracy of the 

 above expressions for if/ x and xj/,,, the following seem to be legitimate 

 inferences and matters of interest. 



(«) The value of if/ is ^ = at the focal point. This establishes a 

 broad distinction between the direct and the diffracted light — ^the light 

 that comes to focus within the cone and the light that strays outside it. 



(&) The periodic factor- — — will cause certain regions in the 

 radiant wedge to be wholly ineffective, and will limit the effective part 



