DETERMINATION OF THE PATH OF RAYS. 15 



when f * JV* is positive, or P* lies in the last medium. In the 

 opposite cases object and image are virtual only, i.e., the rays do not 

 actually proceed from P nor actually converge at P*, P and P* 

 being merely the points of intersection of their prolongations. 



If we draw through the points P and P* planes at right 

 angles to the axis, it is evident that to every point in the one plane 

 there corresponds a co-ordinate image-point in the other ; for, if f 

 remains constant, f* undergoes no alteration, being dependent 

 in a given system upon alone. The distances of the respective 

 object- and image-points are to each other as the corresponding 

 ordinates ?? and ??* ; or, according to equation (13), as 



1- 1 



M - k (f - 



Every object of finite extent may be regarded as such a system 

 of points, and it would give rise, therefore, to a continuous image 

 whose linear dimensions are determined by the above-mentioned 

 relation, the object being taken as unity. If the quantity 



-QJ T~7t arm ^ e represented by m, then this is the co- 

 efficient of linear amplification. Its sign distinguishes whether 

 the image is erect or inverted; if it is negative, and with it there- 



17* 



fore : , it indicates that the object-point and the image-point 



n 



lie on opposite sides of the axis. 



The points P and P* may, of course, assume all possible positions 

 within the limits established in (13) as to their inter-dependence, 

 since for every value of f the corresponding f* can be calculated. 

 Of these positions three in particular deserve special consideration, 

 since they exhibit a more simple ratio between the incident and 

 the emergent rays. 



We will first bring the two points into such a position that 

 they are at equal distances from the axis. Therefore 77 = 77* ; or, 



m = j^ jr ,- yo\ = 1 > 



n (1 - I) 

 whence we obtain f A" = - -r ; 



consequently f = -V T (14) 



