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



For any point of the critical line 



r sin co sin ( it — to') 

 r sin to ~~ sin (nr — to ) ' 



from this and equation (9) 



sin ( — ; (it — to) 



r \ m K ' ) 



r ~ sin ( 7r — to) 

 As to approaches it 



r , m 



— approaches 



r' m 



Hence in the limit 



r A O' m 

 r ~ A'O' ~ m' 



or 



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



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



TT — tO = 2(7T — 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 



to + 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 to = co' = 6, then from equation (2), 



= x — '« ( 13 )' 



m + m ' 



which determines the direction of the asymptote. 



