44 Mr. W. Baily on Starch and Unannealed 



of M(0), and let s + % be the value of a- where the same radius 

 intersects a branch of K(±a); then by (8) we have 



| (sin 2/3 sin 2s + cos 2/3 cos 2s sin 2<fi) cos 2f 



+ (sin 2p cos 2s— cos 2p sin 2s sin 2<£) sin 2f) J- 2 



= cos 2 2p cos 2 2£ tan 2 2a (14) 



But by (13) the coefficient of sin 2£=0, and f is determined 

 by cos 2f only, from which it appears that f has pairs of 

 values of equal magnitude and opposite sign. Hence, if we 

 have drawn a part of K(±a) on one side of a branch of M(0), 

 we can draw the corresponding part on the other side of M(0). 



A similar property of M(±/3) with respect to K(0) may be 

 proved in the same manner. 



Where K(±a) intersects M(0), £ vanishes and the diameter 

 becomes a tangent. Its position is determined by putting 

 0=0 in (10). This gives 



cos 2 2(j> cos 2 2p= cos 2 2a (15) 



The position of the diameter tangents to M(±/3) is determined 

 by putting a = in the same equation. We get 



cos 2 2(f> cos 2 2/3= cos 2 2(3 (16) 



Hence the same diameters are tangents to K(±7), M(±y) 

 at the points where they intersect M(0) and K(0) respectively. 



If we put cr±45° for a in K(0) we obtain M(0), and con- 

 versely. See equations (12) and (13) (17) 



• If we put o-±90° for a in K(±a) or M(±/3), the equation 

 is unaltered; and by this means from one branch of one of 

 these curves we can obtain all other branches of the curve. (18) 



We will now consider the forms of the loci, dealing first 

 with figs. 3 and 4, in which the incident light is elliptically 

 polarized. 



To obtain the isomorphals, we have the equation (7) for 

 M(/3) and (13) for M(0). 



Putting (3 = /3, we obtain the following equation to M(p), viz. 



sin o-(sin 2/) sin cr + cos 2/3cos o-sin 2<£) = ; . (19) 



and putting /3=— p, we obtain the following equation to 

 M(— p), viz. s 



cos a (sin 2p cos a— cos 2/3 sin a sin 2<p) = 0. . (20) 



M(p) therefore consists of circles for which sin<r = 0, and 

 ovals intersecting those circles on the diameters SS / and TT^ 

 and M(— p) consists of circles for which cos<r=0, and ovals 

 intersecting those circles also on SS' and TT 7 . The outer 

 circles in the figures are circles of M(p) ; and the middle circle 

 is one of the circles of M(— p). 



