Formation by Crystalline Media. 



251 



readily demonstrated by means of a microscope ; for example, 

 in the case of calcspar it is necessary to move the microscope 

 along its axis a distance of about one-seventh of the depth 

 of the object-point in the crystal in order to change the focus 

 from the one image to the other.) 



The formation of this second image arises from the fact 

 that although the waves have equal velocities along the axis 

 they have not the same curvatures, and the position of the 

 inuige is determined not only by the change of velocity on 

 refraction but also by the change of curvature. Since the 

 wave has travelled a distance^ along the axis, the radius of 

 curvature of the wave-front is c 2 pja 2 as it reaches the 

 refracting surface, by (3). Now, when refraction takes 

 place at a plane refracting surface, the ratio of the curvatures 

 of the incident and refracted waves is equal to the ratio of 

 the velocities of light in the two media. Hence the radius 

 of curvature of the wave as it enters the air is given by 



s a . c 2 p 



c 2 ^Jo7 2 ~Y ,1 ' e ' S ~aY ; 



and since the image in air is situated at the centre of 

 curvature S of the w T aves coming from it, the distance s must 

 be the required distance from the refracting surface to the 

 image. In the diagram, P'M is the incident wave, B/M its 

 circle of curvature, and S'M the refracted wave. 



These results may be used to measure the principal extra- 

 ordinary index of the crystal by the microscope method. 

 Thus, s/p which is equal to c 2 /aV is also equal to rc 3 /rc 2 2 , (5), 

 by (1). The image is therefore at the same depth as it 

 would be in a substance of index n 2 2 /n l , and the value of n 2 

 is given by 



" 2 =5 or f? by(4 )- 



The values of p, q, and s will then be found by focussing 

 the microscope in turn on P, Q, and S, and finally on M. 



No one appears to have called attention to the fact that 

 the two images cunnot be distinguished from one another by 

 the usual polarization test if the eye or microscope is placed 

 symmetrically over the axis of the crystal through the 

 source. For, in the wave-front of the extraordinary pencil 

 the vibration directions radiate out equally in all directions- 

 from the axis, so that an analyser will transmit the same 

 amount of light in all azimuths and the source will appear to 

 be of the same brightness for all positions of the analyser. 

 On the other hand, in the wave-front of the ordinary pencil 



