134 Central Forces. 
To find the effect of T, I take the differential of (1), considering 
d(rdv ay 
ie but it is 
d(rd 
evident (supposing that T tends to increase rdv,) that ed = 
-. e’'de’=Tr*dv, and by integration ce’? =h? +2fTr?dv, (2), fbe- 
ing the sign of integration, and A? the arbitrary constant which equals 
e’* when there is no disturbing force ; also by substituting c’2 in (1) 
1 
Put r=7> then (2) and (3) 
¢e’ and rdv as alone variable; hence e/de/=r*dv X 
2 
there results dt= ke (3). 
Vk? +2/'T redo 
Tdv dv (5) 
us ; Tdv’ 7 
u ee hs a ie 
denote the resultant of the force towards the first centre, and of the 
resolved part of the disturbing force, which acts in the direction of r; 
Let F 
become ¢/? =h?42f 
2? (4), dt= 
ce’? 2 
then I have by Vol. XVI, p. 286, 5-4 a) =F, (6), which by 
2dr 
d ed 
2 E r? yp? ce’? r2 
substituting va for dt? becomes yd ( I 7a )=F, (7). 
: 
By putting r==7? and making dv constant, (7) is easily changed to 
cd/édu F 
dty — dy? ~ 4? 
doz Fut ad a =0, or since c’dc’ =Trido=— 5) it becomes 
d : 
Pa T>.—Fu 
doa +U+ TF’ (8)> by substituting the value of ¢?. 
us (ee : 
The equations (5) and (8) are sufficient 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 r to the fixed plane; put ’=rcos.é, P=F= the 
resultant in the direction of r’ in the fixed plane. 
nen 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 
