66 Central Forces. 
Art. Vil.—On Central Forces; by Prof. Toropore Srrone. 
(Continued from Vol. XX, p. 294.) 
[ witx now reverse the question which I have considered in Vol. 
XVII, p. 329. Put A=const. and F=Ar: but the general form 
1 
ide 
dr 
of F (given at the place cited) is F= 5d | ); hence by compar- 
¥ 
1 
ing these values of F, I have —¢2d-3=2Ardr, and by integration 
e 
e ee 
e——,=Ar’, or Aap” (1); (1) is the equation of the curve 
described by the particle, the origin of r being at the centre of force, 
and p =the perpendicular from the centre of force to the tangent at 
the extremity of r; the arbitrary constant ¢ is easily found in terms 
of A, c’, and the initial values of r and p. ~ 
c ot c oe 
¢ ate ¢c/2 : at! 
ap’ =? a*p’= 7-3 hence (1) becomes a?--ap/——-=r?, (2); 
which is evidently the equation of an ellipse, the origin of r being at 
the centre, a=half the greater axis, and p’= the semiparameter- 
If the force is repulsive, the sign of A must be changed; which re- 
quires that the odd powers ofa have a contrary sign from what they 
have when the force is attractive ; hence (2) becomes in this casé 
a*p! 
a? —ap’+ mS =r?, (3), which is the equation of an hyperbola, the 
- origin of 7 being at the centre, a=the semiaxis, p’= the semipart 
ya c'? 
meter. 2 Fit ea 
r. It is evident that the equation a’p’= Xx’ (4), has place, 
whether the curve is an ellipse or hyperbola; in the ellipse, F=Ar 
Ged & . 
ap’? and in 
becomes, by substituting the value of A from (4), F= 
the hyperbola, F=—Ar becomes Fas If the centre of 
force is removed to an infinite distance, the ellipse denoted by (2) 
becomes a parabola, and F'=const., the direction of its action being 
in lines parallel to the axis of the parabola. (Prin. B. I. prop. 1% 
