148 Mr Swan*s Formulce for constructing 



2. To find the radius of curvature on the supposition that A B 

 t* the osculating circle to a parabola at the point B. 



It has already been shewn that the true form of the arc 

 A B is a parabola whose focus is F, and whose axis is a line 

 at right angles to F B. Then F B is an ordinate through 

 the focus, and the equation to the parabola will be 



g^=4:mxj where FB = 2m; 

 so that m is determined by assigning the diameter of the 

 mirror from the flame to the outer edges of the zones. 



From the equation to the parabola 



dv 2m -, d^ y 4: m^ 



-— — ; and ~-% = ^ 



dx y dx^ y^ 



Then, if r is the radius of curvature, 

 d^ if 4:m^ 



dx^ y 



•2 ix-vrn)^ 



and if a and b are the co-ordinates of the centre of curva- 

 ture, 



/7 2 , , m 



2x^ 



From which a = 



m ^ 



also X— b =— ~ (y — a) 

 dx ^^ 



Hence 6 = 3 a; 4- 2 m 



But at the point B, ar=m therefore substituting, we dbtaiti 



6 = 5w; a= —2 m; r =t 4t m >^ 2 ; 



and if the origin be now transferred to F, so as to have ike 



same axes as formerly, 



6=~4w; a = 2 w; r = 47W -v/2 . . (2) 



