Central Forces. 47 



at the origin of the motion, then by substituting these values in (2) 

 — is found, and it is evident by (7), (8), (9), given in Vol. XVIL 



1 . . .^ 1 . 



page 73, that if — , is positive, the curve is an ellipse; if —-,=Q, it 



1 . . . . 



is a parabola ; but if — , is negative, it is an hyperbola ; or since p' 



is a given positive quantity, the curve is an ellipse, if a is positive ; 



a parabola, if-=0, or a= infinity; and an hyperbola, if a is nega- 

 tive. I have thus far supposed A to be positive, or the central force 

 to be centripetal ; but should A be negative, or the force centrifugal, 

 the signs of the terms involving p' in (1) are to be changed, and it 



1 12 



becomes —^r-- — rT= — ',~ — > (3); which is the equation of an hy- 

 r^ sin. ^4/ ap' rp' ^ '^ ^ •' 



perbola, which shows that the particle is moving in one of the hy- 

 perbolas, and is acted upon by a centrifugal force situated in the focus 

 of the opposite hyperbola. (1) is easily changed to (2ar-r^) 



1 — COS. 24^ 

 sm.^'^=ap' (4); substitute for sin. ^4^ its equal ^ ' then 



multiply both sides of the equation by 4, and there results (2a - r)X 

 (2r — 2rcos.24')=2ax2p'', or 2a :2a — r: :2r — 2rcos.24^ :2p' (5); 

 (5) agrees with Newton's proportion, (Prin. book I. sec. iii. prop. 

 17,) for in his figure, 2a=SP4-PH, 2a-r=PH, -2rcos.2-^= 

 2SP. sin.PSK=2PK, 2r=2SP, 2y=L. 



But Newton's 17th proposition admits of another very simple con- 

 struction. For suppose L or 2p' to be found, (see his figure,) then 

 cut off on the line SP, from P towards S, a distance =p^; through 

 the point of section erect a perpendicular to SP, also draw a perpen- 

 dicular to the tangent through the point of contact P, and these per- 

 pendiculars will intersect at a point in the axis ; hence a straight line 

 drawn through S, and the intersection of the perpendiculars gives 

 the position of the axis, and PH will intersect the line thus drawn 

 at the other focus H of the conic section, except in the parabola, 

 when PH will be parallel to the axis ; hence every thing else sought 

 in the problem is readily found. The proof of this construction is 

 easy, (see fig. 2 to plate 4, prop. 11, Prin.) admitting what has been 

 proved in prop. 11, we have PE=AC, suppose PF cuts the axis 

 in r, then PFxPa — CB" (Viiice's Con. Sec. ellipse, prop. 15,) 



