The Helmholtz Theory of the Microscope. By J. W. Gordon. 429 



and r v being preserved, the argument applies equally well to these so 



greatly altered figures. This will be evident without discussion. 



In all cases the axial distance of the ray c L . . . ^ whether on one 

 side or other of the optical centre, is sin 6>, r. If, therefore, we write 

 e for the diameter of the object and rj for the diameter of its image, we 

 shall have 



6 = sin0r € 77 = sin0r7j • . . . (10) 





1. 



(10a) 



This construction therefore can be employed to determine the 

 magnifying power of any optical system, if we know the positions of 

 the focal planes and optical centre, for we can always in theory place 

 upon that optical centre a system such as that shown in fig. 101, and 

 then, as we have seen, we shall get the dimensions of conjugate images 

 in the two focal planes by the equation (10). 



Fig. 101. 



Now the optical centre can be at once determined from observations 

 made upon the divergence angles u e and u v . Fig. 102 will make this 

 clear. 



Let it be supposed that we have two images formed at e and 77 

 respectively, and that the beam which focusses in them has the divergence 

 angle u € in the one plane and Ur) in the other. The positions of these 

 focal planes may then be taken to be given by direct observation. 



Next let the rays from e and rj be prolonged until they intersect one 

 another in the point A. Then it is clear that the rays c . . . A and 

 A ... r] must be proportional in length to the distances of the points 

 € and rj respectively from the optical centre, for both these are edge rays 

 of their respective divergence angles, both therefore must by definition 

 touch the edge of the aperture, and there is no other position for a 

 common aperture which will satisfy this condition. If, then, we draw 

 «« . . . «ij parallel to the optical axis, and at such a height that the 

 circular arcs cie . . . C, a v . . . C, drawn about the centres e and rj 



Aug. 19th, 1903 2 f 



