TJie Helmholtz Theory of the Microscope. By J. W. Gordon. 435 



Or substituting for these position angles the divergence angles equal to 

 them we have for (16) 



ft = sin u n r v . 



Furthermore, if we write ft for the image in the sine field produced 

 in the r v plane, we shall have 



ft _ft 



5 



ri r v 



and multiplying both sides of the last equation by this constant, we 

 obtain 



P±- = sin iir, ft. 



?i 



Here the first member of the equation is a constant, and as the 

 equation holds cfor all values of u n the second member must be a 

 constant also. Therefore 



sin it v ft = a constant, .... (17) 



which again is Helmholtz' law in a slightly altered form. 



From this result it is a simple matter to infer the corresponding rule 

 concerning images upon the tangent scale. For we know that ft the 

 image on the tangent scale, formed in the same plane of the same object, 

 is related to the sine image ft in the same way as the tangent and sine 

 scales. Therefore by (15) 



ft = cos Ur, ft. 



Substituting this expression in (17), we obtain 



sin u v cos iir, ft = a constant 



= \ sin 2 Ur, ft. . • . sin 2 u v ft = constant. 



The angle 2 u (= the angular aperture) therefore takes the place of 

 the divergence angle when we pass from the centred to the uncentred 

 system, and it is of interest to note that upon the analogy of Helmholtz' 

 law for the centred system we have now obtained for the uncentred 

 system the rule 



n e sin 2 m € ft = w, sin 2 w, ft. . (18) 



It forms no part of my present purpose to pursue the discussion of 

 these matters beyond the point now reached, but I have in conclusion 

 to consider, and in the light of these results to offer a few observation^ 

 upon, Hockin's proof of the sine law. 



Hockin's proof may be shortly stated thus (fig. 107) : * 

 The line e . . . rj is the optical axis ; A x . . . A 2 is an image-forming 

 aperture ; c x . . . e 2 is the principal focal plane ; e ... e x and c . . . c 2 



* I borrow this statement of Hockin's proof with slight modification from Prof. 

 Sylvanus Thompson's translation of Lummer's work on 'Photographic Optics.' but 

 since this paper was written have become aware that it does not quite accurately 

 represent the original. This is fully explained in the additional note at the end of 

 this Note I. 



