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. XVII. 



1 1 



page 72, that if—, is positive, the curve is an ellipse; if - / = 0, it 



is a parabola; but if — > is negative, it is an hyperbola; or since/ 

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



1 



an 



live. I have thus far supposed A to be positive, orthe 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 1 ^ 



becomes . • 2 , = — ;-IZ)(3); which is the equation of an hy- 



T Sill. Np ilJJ ijf 



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 2 ) 



1 — cos. 24, 

 sin. 2 -\,=ap' (4); substitute for sin. 2 4> its equal ^ "' " ien 



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

 (2r — 2rcos.2+)=2ax2p', or2a :2a- r: :2r-2r cos. 24, :2p' (5); 

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

 17,) for in his figure, 2a=SP+PH, 2a-r=PH, ~2rcos.2i 

 2SP. sin.PSK=2PK, 2r=2SP, 2p'=L. 



But Newton's 1 7th 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 —jf\ 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 gin 

 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 i 

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



proved in prop. II, we have PE=A(\ suppose PF cut* the ni 



then PFxPi=<-'B- (Vince's Con. Sec. ellipse, prop. 15,) 



in r 



