Roever — Geometrical Constructions of Lines of Force. 207 



Hence in the limit 



AT _ A0_ _ 711^ 

 AT ~ AO ~ m' 



or 



AOy.m = AOXm' (7). 



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

 Equation (7) shows that the moments taken about are 

 equal ; hence O is the centre of gravity. 



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

 when it has this value (d = tt) 



m 



TT = T a, 



m — m 



from which 



m — m' 



«„ == 



m 



•^ (8), 



in which a^ is the special value of a that makes Q — it. 



For this value of a equation (1) becomes 



mo) — m'w' = (??z — m' ) tt 

 or 



m 



IT 



— ^' = w' l'^ — ^) C^)' 



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 O', 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. 



