96 Lord Kayleigh on Pin-hole Photography. 



lated by means of (12), (13) the value of pi 2 , that is of 



8 



7T* 



(U'.+iv), 



corresponding to z=0, 1, 2, 3, 4. 

 The results are as follows : — 





y=i7r. 





z. 



V2.s. 



^M 2 . 







1 

 2 

 3 

 4 



•000 

 1-414 



2-828 

 4-243 

 5-657 



•4748 

 •3679 

 •1590 

 •0272 

 •0041 



The various curves, or rather the halves of them, are plotted 

 in Plate IV., and exhibit to the eye the distribution of light in 

 the images corresponding to the different apertures. It is at 

 once evident that y = ^ir is too small, and thaty = 37ris too 

 great. The only question that can arise is between y = ir and 

 y=2ir. The latter has decidedly the higher resolving power, 

 but the advantage is to some extent paid for in the greater 

 diffusion of light outside the image proper. In estimating 

 this we must remember that the amount of light is represented, 

 not by the areas of the various parts of the diagrams, but by 

 the volumes of the solids formed by the revolution of the 

 curves round the axis of I 2 . In virtue of the method of con- 

 struction the total volume is the same in all cases. The best 

 aperture will thus depend in some degree upon the subject to 

 be represented; but there is every reason to think that in 

 general y = 2ir will prove more advantageous than y — ^r. It 

 will be convenient to recall that 



yJTT= 



2r 2 a + b 

 X ab 



or, if we write a = co , b=f, 



y/7r = 2ry\f. (25) 



The curve # = 7r thus corresponds to (1) ; and we conclude 

 that the aperture may properly be somewhat enlarged so as 

 to make 



„2_ 



=x/. 



(26) 



In the general case when a is finite, yjir represents four times 



