The Hclmholtz Theory of the Microscope. By J. W. Gordon. 437 



Now this, it will be seen, falls very far short of being a proof of the 

 sine law, as established by Helmholtz. And the reason is quite clear. 

 Hockin has endeavoured to establish the relation between a plane object 

 and its spherical image, and between them the relation actually does not 

 hold. It is true, as we now know, for the axial distance of rj u but for 

 the distance rj . . . ■q l measured on the sphere, if that distance has any 

 finite magnitude, it is not true. 



But Hockin, although he failed of proving the sine law, did not fail 

 by much, and a very slight modification of his proof makes it perfectly 

 valid. In fig. 108 the essential parts of Hockin's diagram are reproduced 

 with the addition of a second aperture and lens to give a flat form to 

 the image rj . . . r) x , and then it will be seen that it can be used for 

 the proof of the sine law. 



Fig. 108. 



Here the proof concerning the equality of all the optical paths remains 

 unaltered, for the second lens has only altered the form of the trans- 

 mitted wave-fronts, they still remain the identical wave-fronts and 

 therefore equidistant at every point from their foci c x and c 2 . It follows, 

 therefore, that rj 1 . . . -q 2 is in this arrangement, as in the original, equal 

 to e x . . . « 2 , and the two wave-fronts -q . . . ^ and •*/ . . . rj 2 are, by 

 hypothesis, perpendicular to the rays along which they travel. Here, 

 then, we have a rectilinear, right-angled triangle, having its perpendicular 

 Vi' • 'V2 equal to e 1 . . . e 2 > and its angle ^i^^ equal to the divergence 

 angle u . Therefore 



sin u v (77 



Vl ) = sin u (e 



Os 



and the proof under these conditions" applies to conjugate images. 

 c . . . «! and t\ . . . t] x of any magnitude. But the conditions are that 

 there shall be an optical centre C l5 at which the positioning angle d can 

 be measured, and on either side of this centre a refracting surface 

 distant from it by its principal focal length to render the direct and the 

 reverted wave-fronts flat before they reach the image-plane. Hockin's 

 construction does not satisfy this latter condition, and hence the failure 

 of his proof. 



