170 Theory of the Resolving Power of Objectives. 



microscopists, that the obliquity o£ illumination should be 

 rather less than the obliquity of the extreme rays o£ the 

 incident pencil. 



Xote onHochin's proof of the Sine Condition. 

 Various proofs have been given of the sine condition 

 expressed by equation (3), which must be fulfilled in every 

 case in which a sharp image, in a plane perpendicular to the 

 axis of the instrument, is formed of a small flat object whose 

 plane is perpendicular to the axis. By far the simplest is 

 that given in an article " On the Estimation of Aperture in 

 the Microscope," published after the author's death in the 

 Journal of the Royal Microscopical Society (1881. ser. 2, 

 lv. p. 337), where he is described as the late Mr. Charles 

 Hockin, junr., an electrician and mathematician of repute. 

 Appreciative notes by Abbe are inserted in the article. 

 Strange to say, the proof does not seem to have been 

 reproduced in any English publication, though it is to be 

 found, modified for the worse, in German optical treatise-. 

 In Muller-Pouillet it is erroneously described^ and the 

 author's name is given as John Hockin. These circumstances, 

 in conjunction with the great importance of the theorem 

 itself, are my reasons for reproducing it. I have corrected a 

 clerical error of — for + in the two principal equations. 

 Let PP' in the figure represent the axis of an optical 



system which gives the linear image P'Q' of the small object- 

 line PQ, both the lines PQ and P'Q' being perpendicular to 

 the axis. The incident pencils may be of large angle : and 

 the image is supposed to be aplanatic, that is to say, all rays 

 sent by P pass through P', and all rays sent from Q pass 

 through Q'. Let PS be any one of the rays sent from P, 

 and QS a ray from Q intersecting it at S. 



Since PQ is small, the angle PSQ is small, and the plane 

 pencil bounded by PS, QS will give an emergent pencil 

 bounded by P'S', Q'S', the optical path from S to S' having 

 the same value for all the rays of the pencil ; denote this 

 value by (SS'). 



Them if (a be the index of the first and yl that of the last 

 medium, the optical path from P to P' is 

 /x.PS + (SS')~^.P'S', 



