The Inverse Method of Central Forces. 107 
4 1toc+ S? 
ee ee 
'SS— Se ——— 
(mw — 1*1t+c+S—2—C¢ *n—1)" — 
pee Ne aie a fe 
(rpc Sern — 3) Oar 
where it is evident that if we make »—1+1-+¢+ 
See ni an — 18°, then. the denomina- 
tors will be equal, and therefore F*+ G* 33 s°¢ 
1. + c+ S*, which is a constant ratio; and. there 
fore when g = 3, or when the additional force va- 
ries as the cube of the distance reciprocally, the an- 
gles, which lines drawn from the bodies to the centre 
of force, make with the line passing through the 
apse, are in a constant ratio to each other, whatever 
be their common distance (the condition mentioned 
being observed). 
Cor. 2. Becausen—1+1-+0+S*?—con—1= 
ae SoC 
n—1 * s*, therefore sSlae Let R = rad. of curva- 
ture and v = velocity at the apse in the orbit describ- 
ed by one force; then the centripetal force at the dis- 
2 2 
tance n=l = 0» therefore a ‘R: ane ts: = 
a ne 
Ste R+re RR’, R+7-¢ 
ai ete hence F? ¢ G* $$ —+} Roa 8 
it+c yrrite Y UP 
Git 5 a 
{ee tt ees 
rT ' 
C= 
op) 
a. x R= additional force at the apse. But 
Y 2 
because = 3,9? 3 7? 33 
