THE OBJECTIVE. 33 



practicable to the second lens. This approximation of the lenses 

 which naturally finds its limit in the contact of the two lenses, 

 necessitates at the same time a diminution of the focal length, since 

 the expression found for the latter varies with t'. The approxi- 

 mation of the lenses gives, therefore, a double advantage : a 

 greater magnifying power, and, at the same time, a greater focal dis- 

 tance. This not only holds good for the first and second lenses, but 

 also for the second and third ; for, since the formulae just obtained 

 are applicable to any two systems of cardinal points, the combination 

 of the third lens with the two anterior ones must lead to the same 

 results with regard to the magnitude t' which here appears. This 

 follows also from the formula for the focal length, applied to (/), 

 which is the shorter the less the distance of the principal points 

 (here T and E"). The smallest value of (/) corresponds to the 

 greatest object-distance (if the allowable supposition is made that 

 (/) > t'), as a glance at the value of N F shows. 



It is evident that the approximation of the principal points, above 

 discussed, may be extended to every combination of refracting 

 surfaces, and, therefore, also to single crown and flint lenses. 

 Since with the latter the distance of the principal points from 

 the surface of contact varies with the thickness, it follows that 

 they must be made as thin as possible so as to increase the focal 

 distance ; by these means, at the same time, a higher magnifying 

 power is obtained. 



The focal distance and the object-distance, therefore, under similar 

 circumstances, vary inversely as the distances of the refracting 

 surfaces. 



By this it is not meant that the sum of the distances can be 

 taken as the absolute measure ; it is self-evident that the stronger 

 lenses, in this respect also, have a preponderating influence. 



The refractive indices of flint and crown were taken above 

 at 1*6 and 1-5. It is, d priori, clear, that an alteration of these 

 figures will not be without influence upon the position of the optical 

 cardinal points, and it is not difficult to understand the general 

 nature of the effect of the increase or decrease. If the relation 

 between the crown and flint remains the same, and still more if 

 the crown only is regarded as variable, an increase of the indices 

 of refraction will increase the optical action of the single lenses, 

 and involves shallower curvatures with equal effect. It acts, 

 therefore, in the same way as an approximation of the refracting 



D 



