292 Central Forces. 



e^ — 1 e 

 = 1. By comparins!; (2) and (4)1 have -7"^ = — -1, 



•^ r o \ / V / 1+eCOS.V COS. 9 



hence tan. |= s/^^ X tan. ^ (6) ; (see Mec. Cel. p. 187, Vol. I.) 



I will now suppose that the parameters of the conic sections are 

 indefinitely diminished, so that they may be considered as differing 

 insensibly from right lines. Let a— half the transverse axis of the 

 section, (if it is an ellipse, or hyperbola,) which is supposed to be in- 

 variable, when the parameter is diminished; r= the distance of the 

 particle from the centre of force at any time i, V= the velocity of 

 the particle, V'= the velocity of a particle of matter describing a 



A 



circle around the centre of force at the distance r, F=— = the cen- 



tral force, (A= const.) Then by (10) and (11) given at pp. 331, 

 332, Vol. XVII. V=V' \/^l:zI (7), when the section is an el- 



/2a-{-r _ 



lipse ; V= V \^ (8) when it is an hyperbola ; and V=VV2 



V'2 A /A 



(9), when it is a parabola. Now — =F=-j' or V'=\/ — , let 



V''= the velocity of a particle describing a circle about the centre 

 of force at the distance a, in the ellipse or hyperbola, and at any as- 



y// 2 A 

 sumed distance, p, in the parabola; then =~' or A = aV^, 



when the section is an ellipse, or hyperbola,' and A=pV^^^, 

 when it is a parabola ; hence by substitution V'=V'''\/ - in the el- 

 lipse, or hyperbola, and Y'=V''\/E in the parabola; by substi- 



r 



tuting these values of V'' in (7), (8), (9), they become V= 

 y,,^2a-^r ^jQ^^ v=V^'\/?^' (11), V=V^'\/^ (12,) 



respectively. Now when the parameters of the sections are indefi- 

 nitely diminished, it is evident that the focus in which the centre of 

 force is situated may be considered as coinciding with the nearer 

 vertex of the ellipse, or hyperbola, and with the vertex of the par- 

 abola, also the ellipse may be considered as coinciding with its trans- 

 verse axis, and the hyperbola with its transverse produced, also the 



