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



when the axis of x is the tangent to all the ovals and the origin 

 is at the usual place. 



This follows from the value of p found in Prop. XVI. 



Proposition XVIII. If a straight line be drawn through the 

 origin to meet ovals of intensity k, the area of the triangle 

 formed by the vectors to points of section and corresponding 

 subnormals is constant. 



The equation to the curve is 



sin r — + 



k 



Therefore 



r = + sm" 



" sin W 

 h 



sin 20 



+ M-7T, 



n being an integer. 



Therefore by differentiation 

 rdr 



dd 



= + function of 0, k. 



Kg. 3. 



Let OP (fig. 3) be the line through origin meeting the oval of 

 intensity k in P, and PS the normal, OS the polar subnormal. 



Then, if OP = r, Pox = 6, 



rdd 



cot OPS = 



dr ' 



rdr 

 Id 



= r 2 tan OPS 

 ^r.OS, 



