134 Central Forces. 
To find the effect of T, I take the differential of (1), considering 
dv) 
d(r sue 
ce’ and rdv as alone variable; hence e/dce’=r?dv X = » but it is 
, d(rd 
evident (supposing that T tends to increase rdv,) that ade) 
*. de’ =Tr*dv, and by integration ce’? =h? +2f'Tr*dy, (2), f be-: 
ing the sign of integration, and A? the arbitrary constant which equals 
c’? when there is no disturbing force; also by substituting c’? in (1) 
2 (3). a = then (2) and (3) 
vas ee 
denote the resultant of the force towards the first ee and of the 
resolved part of the disturbing force, which acts in the direction of 7; 
dr 
then I have by Vol. XVI, p. 286, <j 5-4 7) =F, (6), which by 
2dr 
is ¢c/2 12 2 
substituting —7a_ us for dt? becomes — 5—a(S55 =F, (7). 
” Odr 
= 
there results aa ae ae 
become c/?=h? ofan (4), dt= 
(5). Let F 
1 
By putting r=» and making dv constant, (7) is easily changed to 
cde'du F 
d?u dv? ~ u? a RNY eter Ia 
qr tut Te =0, or since ¢ de =Tr?du=Ta 2 it becomes 
au 
Ee by substituti lue. of e!2 
ape + Tdo\’ (8), by substituting the value of c’?. 
corer 
The equations (5) and (8) are sudicient to find the place of the par- 
ticle at any given time. 
Again, if the disturbing force is not in the plane of the curve de- 
scribed by the particle around the first centre, then imagine a fixed 
plane drawn at pleasure through the first centre, and let 6= the vari- 
able inclination of 7 to the fixed plane; put 7’=rcos. 4, P=F= the 
resultant in the direction of r’ in the fixed plane. 
Then change r into 7’, F into P, and the preceding results will be 
true in this case ; as is evident by supposing the moving particle and 
the forces which act on it to be orthographically projected upon the 
