818 Transactions of the Society. 



case the blue oblique ray would not meet the axis at 0* but in a 

 more distant point (not indicated in the diagram), owing to the 

 spherical over-correction of the more refractive rays. 



Consider now the action of the under-corrected anterior part of 

 the system and the effect arising from the supposed distance between 

 M and N. Owing to chromatic under-correction, the axial pencil 

 emanating from will have two different virtual foci E and B 

 for red and blue after its passage through M ; and owing to spherical 

 under-correction of the anterior system, the oblique pencils O P 

 after their passage will yield two other virtual foci R' and B' at 

 a greater distance from and from each other than E and B. 

 The intervals EE' and BB' will indicate the linear amount of 

 spherical aberration for the extreme colours, introduced by the front 

 system M. 



Owing to the positions of these four points the pencil of blue 

 rays emanating from M will be confined to a smaller divergence 

 than the red, and the sectional diameter of this pencil will be 

 diminished, in comparison with the other, more and more from M 

 to N. If now both pencils reach the concave surface of the cor- 

 recting flint lens in the posterior system N, every blue ray will 

 meet this surface at a smaller distance from the axis than the corre- 

 sponding red ray, and therefore with a smaller angle of incidence ; 

 the difference increasing as the distance from M to N is increased. 

 Owing to this difference the negative spherical aberration arising 

 from the concave surface will be of less amount in the blue pencil 

 than in the red, the action of this surface on the blue rays being 

 somewhat similar to the action of a surface of less curvature ; and 

 thus the increase of negative aberration for the blue which at the 

 same time arises from the greater difference of the refractive indices 

 between flint and crown, may be exactly balanced. The whole 

 pencil of blue rays will then be gathered to the same point 0* 

 where the red rays are collected. 



As will be understood, this demonstration does not pretend 

 to prove that the method here considered will attain the object in 

 view within practicable limits in respect to the necessary amount of 

 under- and over-correction, and the necessary distance between both 

 parts of the system. This could only be done after an exhaustive 

 mathematical analysis of the problem. The foregoing discussion is 

 intended only to afford a clear idea of the method in general and to 

 estabhsh the principal direction in which the computations must be 

 made in devising an objective according to this plan. This direction 

 is readily expressed by the rule : combine a strongly under-corrected 

 anterior system with an over-corrected posterior at such a distance 

 that the chromatic difference of spherical aberration may be over- 

 come — all other conditions of optical performance being taken in the 

 ordinary way. 



