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



when we now take p 2 to be the least positive value of r 2 which 

 satisfies 



sin r = + 



~ sin 20 ' 



Proposition III. Form of the curve near the origin. 



The form of the curve near the origin is a rectangular 

 hyperbola. 



The general equation of the curve is 



h = ± sin 20 sin r 2 . 



For points near the origin r is very small so that for sin r 2 we 

 may put r 2 neglecting quantities of the sixth order. Thus the 

 equation of the curve near the origin is 



k = ± sin 29 . r 2 



or transforming to Cartesian co-ordinates 



h - 



2 = ± X V- 



The equation of the rectangular hyperbola referred to the axes 

 of x and y as asymptotes. 



Proposition IV. Form of the curve at points remote from the 



origin. 



The ovals become arcs of circles except towards their extremi- 

 ties which are rounded off (provided the ovals are not too near the 

 point curves). 



The equation to the curve is 

 sin r — + 



sin 20' 



As in Prop. II. let p 2 be the least positive value of r 2 which 

 satisfies this equation. 



Then by equations a the general value of r 2 is 



r 2 = nir ± p 2 



7j- 



where p 2 is less than -^ . 



Thus 



/— . P 



r = Jmr + p 2 = Jmr±~r=±&c (D). 



~ r 2sjmr K J 



