Central Forces. 293 



parabola coincides with its axis ; hence supposing the particle to re- 

 cede from the centre of force, it may be considered as moving in the 

 axis of the ellipse at the distance (r) from the centre of force, or at 

 the distance (r) from the centre of force in the axis produced, in the 

 case of the hyperbola j and at the distance (r) in the parabola ; 



dr 



hence Y=-r:^ by substituting this value of V in (10), (11), (12), 



rdr rdr 



and by reducins; they become —. — ^ =.Y"dt (13), . ^ —- .■=. 



rdr 

 Y"dt (14), . — = V"dt (15). It may be well to observe that V, 

 V 2pr 



Y', Y", &;c. are not supposed to have the same values in the three 



cases treated of; but supposing them to be adapted to any one case, 



their values are supposed to be altered when they are applied to the 



other cases, so as to suit those cases also. Put cot. ? = ■*■ 



(16), cot.cp'=\/?^+^(17), andcot.(p''=\/?? (18), then 

 r r 



r- cosec.2 (pdf(p aY" 

 (13), (14), (15) are easily changed to ^ ' = "2"'^ 



r^cosec-ffi't?®' dY" r^ cosec.^ cp'^d'/' pY" 

 (19), ~-^^-^=-^xdt (20), ^^-^=^ Xdt 



Sr^ cosec.^cpc??! 

 (21); or by integration (19), (20), (21) become ^ = 



aV Sr^ cosec^ffl'^a'' aY'' Sr^ cosec.^ a)''d(p" 

 -^Xt (22), ^~-^-^=—xt (23), 2 = 



"2" X t (24) ; S being the sign of integration, and it may be observed 



that no correction is necessary supposing t to commence when r=0. 

 (22), 23), (24) indicate Newton's constructions, (Prin. Vol. I. Sec. 

 . ' ' - , p 



7, prop. 32.) (22) gives his case 1. for (see his fig. 1,) -^ — cp= his 



angle CBD, r=CB, 2a=AB,rcosec.(p=-.BD, and — — ^— ^ = 



S?'2 cosec."9^.p 

 the difiierential of the area BD, .'. ^ -=■ the area BD, 



the integrals commencing when r or CB = 0, hence in his fig. L 



VoL= XX.~=No. 2, 38 



