Lord Kayleigh on Phi-hole Photography. 93 



In general by (10), (11), 



C 2 + S 2 =^{U 1 2 + W}=7rV 4 .M 2 , . . (21) 

 if with Lommel we set 



M2= 6 Ui ) 2+ fi u *) 2 (22) 



Also 



l2 =^' m ( 28 > 



In these formulas U, 2 , U 2 2 , and therefore by (22), (23) M 2 

 and I 2 are known functions of y and z. The connexion with 

 r and f is given by the relations 



2wf_a + b 2^ . 



1J ~ X ab> 2 ~ \b W 



In Lommel's memoir are given the values of M 2 for integral 

 values of z from to 12 when y has the values 7r, 27t, 3-7T, 

 &c. If we regard a, 6, X as given, each of these Tables affords 

 a knowledge of the distribution of illumination as a function 

 of f for a certain radius of aperture by means of the two 

 equations (24). In each case J is proportional to z\ but in 

 comparing one case with another we have to bear in mind 

 that the ratio of J to z varies. As our object is to compare 

 the distributions of illumination when the aperture varies, 

 we must treat £, and not z, as the abscissa in our diagrams. 

 Another question arises as to how the scale of the ordinate I 2 

 should be dealt with in the various cases. If we take (23) as 

 it stands we shall have curves corresponding to the same 

 actual intensity of the radiant point. For some purposes this 

 might be desirable ; but in the application to photography 

 the deficiency of illumination when the aperture is much 

 reduced would always be compensated by increased exposure. 

 It will be more practical to vary the scale of ordinates from 

 that prescribed in (23), so as to render the illumination cor- 

 responding to an extended source of light, such as the sky, 

 the same in all cases. We shall effect this by removing from 

 the right-hand member of (23) a factor proportional to the 

 area of aperture, proportional that is to r 2 , or y. Thus for 

 any value of y equal to sir, we shall require to plot as ordi- 

 nate, not M 2 simply, but sM 2 , and as abscissa, not z simply, 

 but z/ s/s. The following are at once deduced from Lommel's 

 tables III.-Y1. 



Phil. May. S. 5. Vol. 31. No. 189. Feb. 1891. I 



