48 Central Forces. 



= AC X//, the lengthy' being cut off from P towards H on the Una 

 PH (produced if necessary,) let y denote the point of section, let the 

 perpendicular through y to PH intersect PF in Z', then since PF bi- 

 sects the angle EPH, the triangles EPF, yVz are similar, hence the 

 proportion PE : PF: :Pz : Vy or AC : PF: :P^ : /?' .'. AC Xp'= 

 PF xP^:, but AC xy=PF xPa?, hence Vx=Vz, or the points x 

 and z coincide, and the perpendiculars PF, yz intersect at a point in 

 the axis; a similar demonstration is applicable to the parabola and 

 hyperbola. Again, by using the same figure, and supposing the cen- 

 tre of force to be at H, P being the place of the particle at any time, 

 and PH being denoted by r; let«7=the angle PHA= the angle 

 made by r and the perihelion distance HA; then v is easily found by 

 the above construction. For the right-angled triangle Hxy gives 



Hw p' — r 



cos. « = f5--=-TT — ' but since Pa; bisects the angle SPH, (Euc. 

 Ha? rix ° ^ 



6. 3.) SP+PH : SH : : r : Ha;; put op , pTj = e, then Ha;=e/-, 



p' ~r 

 .'.COS. v=—r- (5); (see Mec. Cel. Vol.I. p. 191.) By substitutmg 



C'- . c^^ /2 n 



in (1) for »' its equal — t-j it becomes -r^- — 7T=V^=Ax l / 



Y= the velocity, let V^= the velocity of a particle describing a circle 



A Y'\ 

 at the distance r from the centre offeree, then F=-t= — ^ hence 



(A r y^ , , 



A=rV'- .'.V-=V'2xf2--J5 or ~=2-y,-^ {&). 



It is evident by (6), and by what has been previously shown, that 

 the curve described by the particle is an ellipse, if 2—^77- is posi- 

 tive; a parabola, if it =0; but an hyperbola, if it is negative. These 

 results are manifestly the converse of cor. 7, prop. 16, sec. iii. b. 1. 

 Prin. (See also Mec. Cel. Vol. I. p. 190.) Again, by substituting 



in 'p'=~\~ for C'2 and A their equals Y-r- sin. "4^ and rY'^ it be- 

 comes j9'=Y7-rsin. ^4^ (7). At the aphelion or perihelion, sin, -1 = 1 



and p'=y7-r (8) ; hence 2y : 2r: : V- \Y'- ; this proporiion agrees 



with Newton's cor. 2, prop. 17, (supposing that the velocities V, Y' 

 at the aphelion, and the aphelion distance /• are known.) 



