66 Central Forces. 



Art, VII. — On Central Forces; by Prof. Theodore Strong. 



(Continued from Vol. XX, p. 294.) 



I WILL now reverse the question which I have considered in Vol. 



XVII, p. 329. Put A= const, and F=Ar: but the general form 



c'^ j l\ 

 of F (given at the place cited) is F=— -^£? — , hence by compar- 



dr 



ing these values of F, I have —c^^d—=2Ardr, and by integration 



g/2 c &- 



c — — =A?--, or ■a~T~j=^^? (l)j (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 c is easily found in terms 

 of A, c', and the initial values of r and p. 



c / c^ d^ c /c" d^ 



Put a^=arT+V2Tl--^,«p'=^-VTT7-X' •'•«' + 



c d"^ a^v' 



^P'~A' ^^P'~~\"^ hence (1) becomes a~-{-ap' — — ^=r2, (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 of a have a contrary sign from what they 

 have when the force is attractive ; hence (2) becomes in this case 



a'^p' 

 a'^ — ap'-\- — ^ = r-, (3), which is the equation of an hyperbola, the 



origin of r being at the centre, a=the semiaxis, ^'=the semipara- 

 meter. It is evident that the equation a'^p' = -t-i (4), has place, 



whether the curve is an ellipse or hyperbola; in the ellipse, F=Ar 



d ' r 

 becomes, by substituting the value of A from (4), F=-y~/; and in 



g/ 2 ^, 



the hyperbola, F= — Ar becomes F= — -r— .• 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. 10. 



