DETERMINATION OF THE PATH OF KAYS. 



the same side of the principal planes, which always happens when 

 p < /. The ratio of p to p* gives, in the same way as with the 

 single lens, the linear magnifying power, so that its value, as 

 found above, remains correct for all cases. - 



.. II. 



ANALYTICAL DETERMINATION OF THE PATH OF 

 THE RAYS IN REFRACTING SYSTEMS. 



WE have now to determine the points E and E*, and F and F* 

 (which are called the optical cardinal points) for a given refracting 

 system, in which the curvatures and distances of the refracting 

 surfaces, as well as the refractive indices of the media, are known. 

 This problem can, however, be reduced to a much simpler one in 

 our special case, where we have to consider only lenses and 

 combinations of lenses. It is sufficient to extend the theoretical 

 consideration to two refracting surfaces, and to work out the 

 calculation according to the formulae already obtained for the 

 separate lenses, or for the pairs of surfaces, of which the system 

 is composed. Each two systems of cardinal points may then be 

 combined in the same way as the effects of two refractions were 

 combined for the determination of their cardinal points. The 

 same, of course, holds good for the resulting systems, and the 

 process of combination can be continued, until, at length, the 

 total effect of the refracting surfaces is reduced to one system of 

 principal and focal points. 



Let* the points ^V and N' (Fig. 5) be the so-called vertices of 

 a lens, that is, the points in which its limiting surfaces are inter- 

 sected by the optic axis X ; r and / the respective radii of 

 curvature, J/, M f the corresponding centres, and 71, ri, ri* the 

 indices of refraction of the media in succession from left to right : 

 in front of, within, and behind the lens. If the first and the last 

 medium is air, as is the case with most optical instruments, then 

 n and n* may be taken as equal to 1. We will, however, leave 

 these quantities provisionally undetermined for the sake of 

 symmetry in the form of the equations. 



If S P is the line along which the incident ray is transmitted, its 

 position in the plane of the paper may be determined by means of 



