212 UNDULATORY THEORY OF LIGHT. 



PP' will there be about commencing the return movement, in the direction 

 denoted by P", while QQ' will be in the height of its activity. 



The actual position of" the molecule in the orbit described on QQ' as a dia- 

 meter will in fact be at M ; but QQ' and P" may be taken as representing the 

 directions of motion at the instant supposed. Let ACBD be a plate of the 

 crystal cut across the axis, and let the analyzer (not represented) have the plane 

 of its free molecular movement parallel to AB. Draw CD at right angles to 

 AB, dividing the crystal into four quadrants. 



As all the molecules in the wave front are actuated by similar movements, it 

 will be sufficient to consider one of them in each quadrant. Let their several 

 component forces be represented by the arrows marked p q, each pair having 

 the same relations to each other as PP' QQ'. It will be possible, in every 

 quadrant, to draw a radius parallel to p or q. Let these radii be drawn. Now, 

 the radii being principal sections of the crystal, that component in each case 

 whose direction of movement coincides with the radius, will (if at all inclined to 

 the axis) be an extraordinary ray, and will be retarded behind the other compo- 

 nent. An inspection of the figure will show that this will happen for p in the 

 first and third quadrants, and for q in the second and fourth. Let the inclination 

 of the rays to the axis be such as to cause a retardation of the extraordinary ray 

 of one quarter of an undulation behind the ordinary. Then, by comparing the 

 position? of the arrows which represent the relations of the components after 

 emergence, it will be seen that the effect has been to bring the planes of maxi- 

 mum molecular movement into coincidence in the second and foui'th quadrant, 

 and to increase the distance between them to half an undulation in the first and 

 third. 



But these changes are just what is required to obliterate the nodal dislo- 

 cations in both cases, so that the waves will emerge plane. The resultant 

 molecular movements in the second and fourth quadrants will be obviously 

 vertical and parallel to AB. At the inclination or distance from the centre, 

 therefore, which produces this amount of retardation, will be seen in these 

 quadrants the first bright ring. Had the incident light been plane-polarized, 

 however, the first ring would not have appeared until after a retardation of a 

 half instead of a quarter undulation had takeu place ; it would have conse- 

 quently required a greater inclination of the incident ray, and its apparent 

 distance from the centre would have been increased. In the first and third 

 quadrants, the resultant molecular movement may be inferred by referring the 

 two components p and q to a common plane. If q be referred to the plane of 

 p, for example, then, as the distance between them in the figure is half an un- 

 dulation, the arrow q must be reversed, and the resultant movement will be 

 horizontal. The analyzer will suppress this movement ; or, in other words, at 

 this distance in the first and third quadrants will appear a dark ring. In these 

 quadrants there will not appear a bright ring until the retardation is increased 

 half an undulation more ; that is to say, to'the total amount of three quarters 

 of an undulation. Plane polarized light would have required a total retardation 

 of only a half undulation to exhibit this ring. In the first and third quadrants, 

 therefore, the briffht riug-s are removed outward, and in the second and fourth 

 they are removed inward, from the places they are seen to occupy in plane- 

 polarized light, for a distance corresponding to a difference of a quarter of an 

 undulation. 



From what has been said, it will be easy to undei'Stand why two crystalline 

 lamina?, of equal thickness, and cut from a crystal parallel to its axis, or equally 

 inclined to the axis, when crossed upon each other, neutralize each other's 

 effects. For the original polarized ray is, by the first lamina, divided into two 

 which we have represented by Ro, He- I^i the supposed relative position of the 

 two lamina? the ray Yx^. passes without double refraction in the principal plane 

 of the second crystal, and the ray Il„ in the conjugate plane. After emergence, 



