18 Central Forces. 



AC X p', the length p' being cut off from P towards H on the line 

 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, yPz are similar, hence the 

 proportion PE : PF : : Pz J Py or AC : PF : : Pz J p' .• . AC Xp' 

 PFxP^r, but AC Xp'=PF xPx, hence Px=Pz, 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 v= 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 Hzy gives 



Hy p'—r 

 cos ' v ~^ c ~~^~' > Dut since Pa? bisects the angle SPH, (Euc. 



SH 

 6. 3.) SP+PH : SH : : r : Hx; put spq_-p|j=e, then Hx=er, 



(5) ; (see Mec. CeL Vol. I. p. 191.) By substituting 



. * . COS. V 



p' — r 



er 



O* . . O /2 



tya C /a /2 1\ 



in (1) for p' its equal -x-i it becomes -?—. — rr=V 2 =A X ( ) 



^ r~ sm. ^%p V r a* 



V= the velocity, let V / = the velocity of a particle describing a circle 



A V' 2 



at the distance r from the centre of force, then F— — = — J hence 



r 2 r 



A=.V'» ..V==V"x(2— J, or ^=2-~ (6). 



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



ya 



the curve described by the particle is an ellipse, if 2 — v.— is posi- 



tire; 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. I. 



I. p. J 90.) Again, by substituting 



squab V-r- sin. 2 ^ and rV' 3 it be- 



Mec 



,n p'= x 



V* 



comes p'=^-rsin. 2 4- (7). At the aphelion or perihelion, sin. 4,= I 



V* 



and p f =y,^r (8) ; hence 2p J>»; ;V« : V'« ; this proportion agrees 



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

 at the aphelion, rind the aphelion distance r are known.) 



