222 Trans. Acad. Sci. of St. Louis. 



The asymptote passes through the centre of gravity of the 

 masses m and — m' . 



In order to show this draw a tangent PT to a line of force 

 AP at a point P. (Fig. 4.) Also draw the two radii vec- 

 tores AP — r and AP = r', and the two polar subtangents 

 ^Tand^L'7". 



Then from the figure 



r 2 dco 

 AT = AP tan TPA = r tan 4, = -j— , 



and 



r' 2 dco' 



AT' = A' Plan TPA' = r' tan 4' = -5-7- " 



ar 



Whence 



AT r 2 dr'doo 



AT ~ r 2 dr d<»' 



Differentiating equation (24), 



sin co dco m' 



i' 3 dr' m' 



or 



AO Xm = AOX m' (30). 



is the poiut where the asymptote cuts A A produced. 

 Equation (30) shows that the moments taken about are 

 equal. Hence O is the centre of gravity. 



