the Resoli'iny Power of Objectives. 169 



achromatic ; while mathematicians refused to believe them, 

 and maintained that achromatism could be of no advantage, 

 seeing that the sole purpose of a condenser was to give wide 

 pencils of strong light. 



There is another advantage from sharp focussing by the 

 condenser, which may be regarded as the complement of that 

 above indicated. Jf the focussing were perfectly sharp, the 

 waves from one point of the object could not interfere with 

 waves from another point. Such interference gives rise to 

 spurious diffraction patterns, liable to be mistaken for structures 

 existing in the object. The two components of a double star 

 exhibit no mutual interference in a telescope ; and different 

 points of a microscopic object cannot produce mutual inter- 

 ference if they send light which has come from completely 

 distinct sources. Lord Rayleigh (Phil. Mag.xlii. 189(>) was, 

 I believe, the first to indicate this advantage. Abbe, in his 

 paper on microscopic perception, makes no allusion either to 

 the focussing of the source on the object, or to the finite size 

 of the spot which (with its surrounding rings) is the diff- 

 raction image of a single isolated point. 



The following explanation of the advantage of oblique 

 illumination is. 1 believe, new. 



Perfect sharpness of focussing by a condenser is un- 

 attainable ; and two points of the object which are not 

 further apart than twice the limiting distance of separability 

 will inevitably have a portion of the source in common, as 

 regards their illumination. Let /3 denote the obliquity of the 

 illumination, the two object-points in question being supposed 

 to be in a plane which contains the illuminating rays and the 

 axis of the objective. The difference of optical path for rays 

 coming from the same point of the source to the two object- 

 points is .s sin /3, s denoting the distance between the two 

 points. The best condition for separation is, that this dif- 

 ference of path shall be half a wave-length in the medium in 

 which the object is immersed (say JXi), for this gives the 

 most complete extinction in the overlapping portion of the 

 two diffraction spots which are the images of the two points. 

 Putting then 



s sin /3 =4X3, (5) 



and assigning to 5 the value A^/sin a u which being double of 

 the accepted minimum value may be taken as representing 

 .an ordinal-}- test, we deduce 



sin/3 = ^sina 1 (6) 



If we put s equal to the accepted minimum itself, we obtain 

 sin |S= sin**! (7) 



These conclusions agree with the received view among 



