1884.] homogeneous polarized light seen in a uniaxal crystal. 81 



But OR^ = mr-p\ 



Therefore OR n ^ + OR,; 2 = 2mr 



= a constant independent of the in- 

 tensity k. 



Proposition XIII. If a vector be drawn through the origin 

 meeting any kth intensity ovals in R k , R,/, 



OR k ' 2 - OR k 2 = constant. 



This can be proved in a manner similar to Prop. XII. 



Proposition XIV. If a circle be described from the origin O as 

 centre meeting the nth ovals in points A 1} A 2 . . . and A t , OA 2 . . . 

 be produced to meet the nth ovals again in A/ A/ . . . A/A/ . . . 

 will lie on another circle. 



For by Prop. XII., 



OA 1 2 + OA 1 ' 2 = 2n-l7r. 

 But OA 2 = a constant = a 2 say ; 



.-. OA* = 2w -lir-a 2 . 

 Thus A[A^ ... lie on an nth oval, 



OA t ', OA 2 ' . . . &c. are all constant. 



Therefore A^, A 2 ' ... lie on a circle-centre the origin and of 

 radius J(2n — 1 it — a 2 ). 



Cor. I. If A/, OA 2 ' ... be produced to meet the next set of 

 w + lth ovals in J./', A I' ... A"A 2 " lie on a circle. 



Cor. II. If production be continued the radius of the next 

 circle is 



s/(2n + 1 7r - 7r - a 2 ), 



or V 2?wr — a 2 . 



Thus we see the squares of the radii of alternate circles increase 

 in Arithmetical Progression. 



Proposition XV. The moments of inertia of the ovals of 

 same intensity considered of uniform density form a series 

 of terms in Arithmetical Progression, the moments of inertia 

 being supposed taken about an axis through the origin perpen- 

 dicular to the plane of the curves. 



This follows by the aid of Prop. XIII. 

 VOL. V. PT. II. 6 



