Boever — Geometrical Constructions of Lines of Force. 209 



The asymptote passes through the centre of gravity of the 

 masses m and m'. . 



This is proved in a manner similar to that in the last case 

 (Fig. 4). In this case 



dw m' 



dm' ~~ m 



. ^ o — do) , ^ dio' 



AT=r'' —r-,A'T' = r'^-rT 

 dr dr 



AT —r^dr'dco r^ dr' m' 



A'T' ~ r'2 dr dco ~ r"' dr m 

 In the limit, when r = r'y 



AT _ AG _m' 

 A'T'~ A'O ~ mT 

 or 



AOXm = A'OXm' (14). 



The point O in this case is between A and A'. (Fig. 3.) 

 The angle a, equation (2), cannot exceed tt, and when it 

 has this value (a = tt) equation ( 2) becomes 



mo) -\- in'w = mir 

 or 



m 



CO 



' = W (^-«) (1^)- 



m 



This is the equation of the limiting or critical line, which 

 separates the region occupied by the lines of force that start 

 at A (Fig. 3) and go to infinity, from that occupied by the 

 lines of force that start at A' and go to infinity. 



The point O', in which A'A is cut by the critical line 

 (dashed in Fig. 3), and the centre of gravity O, are symmet- 

 rically situated with respect to the points A' and A. 



For any point on the critical line 



r _ sin co' _ sin co' 

 r' sin (o sin{7r — a>) 



