12 Transactions of the Society. 



lens, or downwards towards the objective and out through the front 

 lens, in either case we shall have to deal with the same limiting 

 aperture, and both the actual antipoint in the real image and the 

 theoretical antipoint produced in the object by Reverted beam 

 will depend simply on the divergence angles in the front and back 

 of the instrument respectively. 



We thus see that a simple numerical relation can be established 

 between conjugate antipoiuts, but for the fuR significance oi this 

 proposition we must consider it in the light of another, which also 

 we owe to Helmholtz and find in this paper, and which connects 

 conjugate images by the same law. For Helmholtz shows that i 

 any Optical system-no matter how simple or how complicated 

 -yields a correct (i.e. a flat and aplanatic) image ot a plane object, 

 the law of magnification in that system will be— 



YII The conjugate images will be proportioned to one another 

 inversely in the ratio of the sines of the divergence angles ol the 



Fir,. 7. 



beams by which they are severally formed. Diagrammatically, if 

 in fig. 7 e be an object (or an image) and 77 its conjugate image 



sin ue 



€ " sin «t) 



(9) 



It is evident that a similar rule with regard to antipoints can 

 be deduced from equation (8). For , putting p, and p v tor the 

 diameters of the conjugate antipoints, we have 



<p 1 £_ 



P<:-= - 1 "- 2 2~Sin ^ alld ** = ^ 2lK U V 



Pv 



sin Ue 



= m by equation (9) 



sm u v 

 From this proposition several important conclusions can at 



•once be drawn. 



In the first place we perceive that— _ 



VIII The state of resolutidn of a correct image cannot be 

 either improved or impaired by mere change of scale, whether 

 effected by eye-piece magnification or otherwise. It you magnity 

 the irna^e you magnify the antipoint in the same proportion, and 



