Roever — Geometrical Constructions of Lines of Force. 223 

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

 and when 6 = tt, 



versin a — 1 — cos a 



= (1— costt) = 2 (31). 



Putting this value of a in equation (24) gives 



m ( 1 — cos co) — m' ( 1 — cos w) = 2 ( m — m' ) 

 or 



w' cos w' — m cos co = m — rn' ( 32 ). 



This is the equation of the limiting or critical line. 



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



1 — cos at + ( 1 — cos a>') = 1 — cos a 

 or 



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



If tO = <Q = 



m 



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



m -j- 7n i \ / 



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 7r, and when a = it, equation 

 (25) becomes 



m ( 1 — cos co ) + m' (1 — cos co' ) = 2m 

 or 



m cos co + 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'. 



