Boever — Geometrical Constructions of Lines of Force. 223 

 The angle 6, equation (29), can not exceed tt ; 

 and when 6 = tt^ 



versin a^j = 1 — cos a^ 



= (l_co«7r) = 2 (31). 



Putting this value of a in equation ( 24 ) gives 



m ( 1 — cos (a) — m' (1 — cosco') = 2 (m — m') 

 or 



m' cos a> — m cos m = m — m' ( 32 ) . 



This is the equation of the limiting or critical line. 



If w = m\ numerically, equation (25) becomes 



1 — cos 0) + ( 1 — cos ft)' ) = 1 — cos a 

 or 



cos o> + cos Q)' = cos a+ 1 (33). 



If G) = fit)' = <? 



versin 6 = 1 — cosd = ; r ( 1 — cosa).. (34). 



This gives the direction of the asymptote. 



In this case also may be shown, as before, that the 

 asymptote passes through the centre of gravity of the masses 

 m and m'. 



The angle a can not exceed tt, and when a = tt, equation 

 (25) becomes 



or 



m(l — cos ft) ) + wi' ( 1 — cos a) = 2m 



m cos Q) -\- m' cos co' = — (m — m') (35). 



This is the equation of the limiting or critical line ; it cuts 

 A' A (Fig. 9) in O', a point at which the repulsion due to m 

 equals that due to m'. 



