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



For any point of the critical line 



r _ sin CO sin (tt — w') 

 r sin CO ~ sin ( tt — «> ) ' 



from this and equation (9) 



sin I — 7 (tt — co) ] 

 r' ~ sin ( TT — o)) 



As CO approaches ir 



r , m 



— approaches — . 

 r' m' 



Hence in the limit 



r AG' m 

 r' ~ A'O' ~ m' 

 or 



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



If 7n = 2m' J 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 



CO -\- 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. 



It CO = co' — 6, then from equation (2), 



m 

 d = J- J a (13), 



which determines the direction of the asymptote. 



