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



The asymptote passes through the centre of gravity of the 

 masses 77i 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 

 ^Tand^'7". 



Then from the figure 



AT= AP (an TPA = r tan 4> = —. — , 



^ dr ' 



and 



r'2 dco' 

 A'T' = AP tan TPA = r' tan <p' = 



dr' 



Whence 



AT r'^dr'da 



AT' r"^drd<o' 



Differentiating equation (24), 



sin (o dm in' 

 sin co' dco' m 



But 



szn (o _ r 

 sin 0)' r 



AT _ r^ dr' d(o _ r^ dr' in^ 



'** AT^ ~ r"' dr day' ~ V^r ^ 



In the limit, when r = r', 



AT AG m' 



AT' ~ A'O m 

 or 



AOXm = A'OX m' (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. 



