222 Trans. Acad. Set. 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 A'P = r', and the two polar subtangents 

 ATandA'T'. 



Then from the figure 



AT= AP tan TPA = r tan 4> = ^-^ , 



and 



r'^ dm 

 AT' = A'P tan TPA' = r' tan 4>' = 



dr' 



Whence 



AT _ r^ dr' dco 

 AT ~ r"^ dr d(i>'' 



Differentiating equation (24), 



But 



AT' ~ r'^ dr dco' ~ r'^ dr m 



In the limit, when r = ?•', 



AT _ AO _ m' 

 AT' ~ A0~ ^ 

 or 



AOXm=AOXm' (30). 



is the point where the asymptote cuts A A produced. 

 Equation (30) shows that the moments taken about are 

 equal. Hence O is the centre of gravity. 



