210 UNDULATORY THEORY OF LIGHT. 



fore, the ligbt liomogeueous, the apparent phones of pohivization of the emergent 

 rays will be in azimuths 2a and 2a+9()°. 



Ninth, if A=A, 21, ^l, &c., sin''^-- =0 ; and the equations become simplified 



A 



to the forms 



A2==V2cos/3 ; and A'^^V^sin^^?, 

 which are a reproduction of the law of Mains. The interposition of the lamina 

 produces, therefore, no apparent change in the plane of original polarization. 



Tenth, if A=4"A, f A, -f^ &c. — that is to say, an odd number of quarter undula- 

 tions — sin^-- becomes sin'45°=J. If, then, a=45°, the equations become — 



A2=V2[cos2,3+i(sin2/3— cos2/5)]=iV^cos^3+sin2/3)==W^ 

 A'2=V^[sin--/3+l (cosV— sin'/3)]=-^-V2(sin2i3-f cosV3 =i V^. 



This result, being independent of the value of /3, indicates an apparent depo- 

 larization of the light. But in fact, it is the case of circular polarization, which 

 we have already considered. It will be seen that it is necessary to the produc- 

 tion of the effect that a should be 45°, in order that the two normally polarized 

 rays may be equal to each other. In any other azimuth of the lamina, the 

 polarization will be elliptical. 



If a lamina of crystal cut at right angles to the axis be employed, then in 

 the direction of the central incident light the two rays are of equal velocity, 



and the factor - =0. No chromatic efiects will therefore be perceived in the 



centre. But the rays which come to the eye converging from points not central, 

 will differ in velocity, the difference increasing with the obliquity. As every 

 plane which passes through the axis is a principal plane, there will be an infinite 

 number of principal planes intersecting each other in the line which forms the 

 path of the central ray, the projections of which upon the surface of the lamina 

 will form so many radii diverging from a centre. And as all planes which are 

 parallel to the axis, however placed, are principal planes also, it is obvious 

 that the planes normal to these radiating principal planes will form cylindrical 

 principal sections having a common axis. The plane ef polarization of the 

 incident light can only coincide with one of the radiating principal planes. For 

 that plane, the valtie of « in our formuhTc will be 0°. For the principal plane 

 at right angles to that, the value of a will be 90°. But we have seen that when 

 a=0° or 90°, the value of the chromatic term is 0. Hence there will be two 

 planes in which no color will appear for any position of the analyzer — that is, 

 for any value of /3. But the brightness of the light seen in those planes will 

 undergo variations of intensity, as /3 varies, according to the law of Malus. 



For every plane except the two which have just been mentioned, the chromatic 

 term will have a value — very slight in the neighborhood of those planes, and 

 maximum at 45°. Very near to the centre, converging rays will have but a 

 slight obliqtiity to the axis ; and as a difference in length of path of oue-qtiarter 

 of an undulation or less fails to produce color in white light, there will be a 

 central area which will be alternately white and black as the analyzer turns. 

 From this area will proceed at right angles the arms of a cross, alternately 

 dark and bright, which, from the faintness of the color in the neighborhood of 

 azimuths 0° and 90°, will have a very sensible breadth. 



At that degree of convergt^ncy which makes the amount of I'etardation for 

 the most refrangible rays =.]/, will appear the first decided chromatic effect. 

 And as, in a plate of unifnrm thickness, this convergency must be the same on 

 every side, the color will take the form of a ring. This ring will be bright if 

 the analyzer is crossed upon the polarizer; in the opposite position, dark. In 

 order to observe the phenomena to the greatest advantage, it is best to employ 

 homogeneous light. Then at greater couvergencies, corresponding to retarda- 



