» - 
= 7 8 
- 
292 Central Forces. 
€ 
ss : 3 = : : ’ at 
=}. ees (2) and (4) I have i ra soti Soe —1, 
henge tin. 5 gm V5 dee ix tan. 5 (6); (see Mec. Cel. p. 187, Vol. 1.) 
‘I will now al it 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 t, V= the velocity of 
the particle, V’= the velocity of a particle of matter desert a ae 
. Grele around the centre of force at the distance Vrs the cen- 
iS tral force, (A= const. ) = (10) and (11) given at pp. 331, 
a Bos Vol. XVII. v= se <*—* (7), when the section is an el 
A A 
ai when it is a aS st Now = ph, Ta? of Ve 4/7 let 
V”= the velocity of a particle describing a circle about the sas 
of force at the distance a, in the ellipse or hyperbola, and at any @& 
: 5) when it is an hyperbola ; - ihe vi | 
3 A eh yi, 
sume Eiience, p, in the parabola; then an ae’, or A=a 
: 
when the section is an ellipse, or hyperbola, and A=pV"" 
y when i it is a parabola ; hence by substitution V’=V” JS4i in thee 
“Tipse, or hyperbola, and V/ =v'n/? P in the nisl substi- 4 om 
a these values of V’ in (7), (8), (9), they. + V=" 
vin /24= (19), Varn / 20 (11); VV ioe i 
saapaitinety. Now when the Sieaies of the sections are 
nitely diminished, it is evident that the focus in which sits le 
force is. situated may be considered as coinciding w. Loa 64 
vertex of the ellipse, or hyperbola, and with the verte ‘of ee Ps 
abola, also the ellipse may be considered as coinciding with its a 
verse axis, and the hyperbola with its transverse produced, alee a | 
* 
