242 



CENTRAL FORCES; 

 , ,x 



ydx xdy 



dac* 



y 



Also r = SP = 



f j 2 t J 2\ ^ 7 



Then the radius of curvature K = - ^ --- -\J- (X being - 



dx* x c?X & <:/# 



in terms of a?, and having no differential in it when the sub- 

 stitution for dy is made). Therefore, the expression for the 



centripetal force becomes - -. , in which, 



when y and dy are put in terms of x, as both numerator and 

 denominator will be multiplied by dx 3 , there will be no 

 differential, and the force may be found in terms of the 

 radical that is, of r, though often complicated with x also. 

 It is generally advisable, having the equation of the curve, 

 to find p, r, and E, first by some of the above formulas, and 

 then substitute those values, or dp and dr t in either of the 



* -n dp r 



expressions for F, ; or ^-^5. 

 3 8 



2p 3 dr 



To take an example in the parabola, where S being the 

 focus, and S = a, y 2 = 4 a x, and T M = 2 x, and p = Y S 



= V (a + x) a; r = SP = a + x, and E = = 2 (a + x) 



la + x 



V~^~ ' 



we have therefore F as 



Fur. 8. 



. E 



S M N 



a + x 



a (a + x~) )|- = 2 (a -f a?) /a + x _ a + a: 



/a + ^ _ a + ^ 

 v a 2 a (a + a?) 8 



