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



Proposition XVI. The .curvature of the ovals at points where 

 tangents from the origin touch them varies directly as the cube 

 of the distance of the points from the origin. 



Let OQ (Fig. 2) be the tangent to the nth. oval, OP a vector 

 consecutive to OA. 



Let POQ = d9. 



We have sin 20 = k for the line OQ, 



OQ 2 = r 2 = 2n - 1 % 



Fig. 2. 



Now at the outset we may observe that it follows from the* 

 geometry (i.e. since OQ is a tangent) that if dr the increment 

 of OQ be of the first order of infinitesimals dd will be of the 

 second order of infinitesimals. Hence we may expect the relation' 

 between dd, dr to assume the form 



d~r\ 2 = Add. 



Consequently in the expansion that follows we keep the terms 

 dr up to the second order, the terms dd to the first order of 

 infinitesimals only. 



The general equation of the curve is 



smr 



20 



To find the relation between dr and dd of point Q we sub- 

 stitute for r 2 (r + dr) 2 , for 6 d + dd. 



Thus we obtain 



sin (r + dr) 2 = ± 



k 

 sin 2 (d + dd) ' 



