206 



Trans. Acad. Sci. of St. Louis. 



Fig. 4. 



force. The angle 6 is measured 

 from the same line, A AX, and 

 in the same direction as a. 



The asymptote passes through 

 the centre of gravity of the masses 

 m and — in' . 



In order to show this draw a 

 tangent PT to a line of force 

 iP at P. (Fig. 4.) 



Also draw the two radii vectores 

 AP — r and AP = r', and the 

 two polar subtangents AT and 

 AT'. 



Then, from the figure, 



and 



A T = AP tan TPA = r tan << = 



<;> = r 



r dco 

 dr 



AT = AP tan TPA = »■' tan <l>' = i 



, r' dw' 

 ~dV" 



Whence 



AT r 2 dr'ih 



AT'~ r'^drdto 

 But from equation ( 1 ) 



As r approaches r' 



dco _ m' 

 dco' m 



AT 



r 2 dr' 



AT' r' 2 dr m 



AT , m' 



approaches — 



AT PH m 



