Roever — Geometrical Constructions of Lines of Force. 223 



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



and when d = ir, 



versin Aq ~ -'■ — ^'^^ % 



m — in ^ ^ m — m' , qi \ 



= CI — cosir) = 2 . . . . ( oi;. 



m m 



Putting this value of a in equation (24) gives 



m ( 1 — cos 00 ) — m' (1 — cosco') = 2 ( m — m' ) 

 or 



7)i' COS co' — m cos CO = m — m' ( 32 ) . 



This is the equation of the limiting or critical line. 



If m = m', numerically, equation (25) becomes 



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

 or 



cos CO + COS co' = cos a + 1 ( 33 ) . 



If O) = &)' = ^ 



071 



versin d = I — cos d = j r ( 1 — cos a ) . . ( 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 



771(1 — cos co) + m' (l — cos co' ) = 2?;^ 



or 



m 



cos CO + ni' cos co' = — ( wi — 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'. 



