1884.] homogeneous polarized light seen in a uniaxal crystal. 75 



so that r may still be treated as a radius vector the effect of 

 putting r 2 for — being merely to alter the size of the image 

 in a certain ratio. 



The object of the present paper is first to obtain the mathe- 

 matical properties of these curves and secondly by means of actual 

 calculation to trace with accuracy the curves and thus obtain their 

 exact form. 



Some of the properties have been given in a paper by 

 Mr R. T. Glazebrook, Proc. Comb. Phil. Soc, Vol. iv. Part VI. 



General Properties of the Curve. 



k 2 = sin 2 29 sin 2 r 2 



or k = + sin 20 sin r 2 . 



The equation k = ± sin 29 sin r 2 is in Polar Co-ordinates. When 

 in the investigation which follows Rectangular Co-ordinates are 

 mentioned it is to be understood that the origin is the same as for 

 Polar Co-ordinates, that the Prime Vector is taken as the axis of x 

 and a line perpendicular to it through the origin as the axis of y. 



Proposition I. The curve is symmetrical in the four quadrants. 

 This is obvious from the form of the equation. 



Proposition II. The curve {for a given value of k) consists of an 

 infinite number of closed curves. 



When k = the parts of the closed curves combine so that the 

 system of curves may be regarded as a number of (double) quadrants 

 of circles and (double) straight lines Ox, Oy. See Glazebrook loc. cit. 



These curves we shall term ovals, and for the sake of distinction 

 we shall give them the names first, second,. . .nth. ovals of intensity k. 

 Thus the first oval of intensity k is given by 



r 2 = p 2 and r 2 = 7r — p 2 . 

 The second by 



r 2 = 7r + p 2 and r 2 = 2ir — p 2 , 



and the nth. oval will be given by 



r 2 = n— 1 7r + /3 2 and r 2 — mr — p 2 (a), 



