80 Mr Spurge, On the curves of constant intensity of [Jan. 28, 



Thus for example the areas between tangents and n p and n + 2 th 

 ovals are the same whatever be the value of n. 



Proposition IX. Each oval of the nth order is bisected by the 

 circle which is the locus of the points of contact of tangents 

 drawn to them from the origin. (See Prop. VI.) 



The proof of this involves no difficulty. 



Proposition X. Let tangents be drawn from the origin to the 

 ovals. The area contained by the tangents and the parts of any 

 two consecutive ovals they intercept will be bisected by the 

 quadrantal arc of intensity k = which passes between the two 

 consecutive ovals. 



This also admits of a very simple proof. 



Proposition XI. Generally if two consecutive lines be drawn 

 through the origin to meet two ovals each of the nth order and 

 of \th and rath intensities the area of the element formed by 

 the ovals and the parts of the consecutive lines intercepted by 

 them is independent of n. 



Proposition XII. If a vector be drawn in any direction through 

 the origin meeting any nth ovals in points R n , R n ', 



OR n 2 + OR n ' 2 = constant. 

 Let the line OR n R n ' meet an nth. oval of intensity k. 

 By Prop. II., equations a, 



OR n * = n^lTr + p\ 



OR„ 2 = nir - p 2 , 



where p 2 (only) depends on k. 



Therefore OR 2 + OR n ' 2 = 2n-lw. 



Since the right hand does not contain k the Prop, holds for 

 all ovals of the nth order. 



Also the constant is independent of 6 the direction of the 

 line, and the proposition holds for all directions of the line. 



Cor. If ORJR n be produced to meet the w + lth oval of in- 

 tensity k in R n+1 , we have 



OR n+ 2 = mr+p 2 . 



