208 Trans. Acad. Sci. of St. Louis. 



For any point of the critical line 



r _ sin CO siniir — w') 

 r sin CO ~ sin ( tt — co)* 



from this and equation (9) 



sin — -, (tt — co) I 

 r' sin (tt — co) 



As CO approaches tt 



r , on 



— approaches 



r' m' 



Hence in the limit 



r AO' _ m 

 ? ^ ZX>' ^ m' 



or 



AO'Xm' = A'O'X m (10). 



If m = 2m', numerically, as in Fig. 2, equation (9) becomes 



TT — co' = 2 {tt — co) (11). 



This is the equation of a circle having its centre at A' and 

 a radius A' A = 2a. 



If m = m', numerically, equation (2) becomes 



io + co' = a (12). 



This is the equation of an equilateral hyperbola referred to 

 the poles A and A'. A line of force is only an arc of this 

 curve. 



If CO = co' = d, then from equation (2), 



m 

 d = -r — 70. (13), 



which determines the direction of the asymptote. 



