428 



Transactions of the Society. 



formity with the law of its propagation upon its centre C. Here the 

 radius r becomes evanescent, and we have for the axial distance 



c = sin 6 r = 0. 



But the angular magnitudes remain unchanged, and when the 

 spherical wave-front is re-formed after passing the centre, its finite 

 magnitudes are still determined by the formula 



Vl =sin0 1 (-r). 



Therefore the new spherical wave-front is an inverted image of the 

 original wave-front. 



We have next to suppose that in passing the aperture A 2 , this 

 expanding spherical wave-front is flattened by any process which pre- 

 serves the resemblance. Then it will follow that the axial distances of the 

 resulting plane wave-front must be given by the formula 



Vl = sin0 1 (-r 7j ). 



2 



Fig. 100. 



Now it is to be observed here that there is no question of focal 

 planes. The image at every point in the system is perfectly correct and 

 perfectly defined. The correctness depends upon the preservation of the 

 sine relation which makes the axial distance equal to sin 6 r and the 

 definition results from the circumstance that the divergence angles u 



and u are each = 0. Hence the diameter of the diffusion disc at any 



77 _ •> 



point = 2 sin u r = ; that is to say, we have perfect definition as well 

 as true resemblance at every surface throughout the system. 



It is also worthy of remark that nothing turns in this demonstration 

 upon the particular magnitudes— linear or angular — employed. Thus, 

 figs. 100 and 101 give two modifications of fig. 99, in which the diagram 

 is varied in! an extreme degree, and every magnitude is changed except the 

 apertures and fields. But the relative proportion of the focal lengths r f 



