Roever — Geometrical Constructions of Lines of Force. 207 



Hence in the limit 



AT _ A0_ _ m ' 

 AY' ~ A'O ~ m' 



or 



A0y<m = A0y.m' (7). 



O is the point in which the asymptote cuts A' A produced. 

 Equation (7) shows that the moments taken about are 

 equal ; hence is the centre of gravity. 



The angle d, equation (6), cannot be greater than tt, and 

 when it has this value {d = ir) 



m 



TT = 



m — m 

 from which 



m — m 



«„ = TT (8), 



in which a^ is the special value of a that makes d = tt. 

 For this value of « equation (1) becomes 

 ma) — m'(o' = (??j — tn' ) tt 



or 



m 



'^ — «' = ;;/ ('^ — <") (^)- 



m 



This is the equation of the limiting or critical line which 

 separates the lines of force that go to infinity from those that 

 go to A. The dashed line Fig. 2 is the critical line for that 

 system. 



A positive particle at 0\ the point in which AA! produced 

 is cut by the critical line, is in unstable equilibrium, being 

 attracted as much by A as it is repelled by A. The point 

 0' and the centre of gravity O are symmetrically situated 

 with respect to the points A and A.* 



Electricity and Magnetism," Mascart and Joubert, Vol. I, 1883. 



