212 Trans. Acad. Sci. of St. Louis. 



5 5 



in which ^ GAO = :j-^ tt. If in place of t^ tt we put the 



angle which a tangent at A to another part of the curve 

 makes with AO^ the new equation is still the equation of the 

 curve shown in Fig. 5. Each of the 2m parts of the complete 

 curve represented by equation (1) is a line of force. This is 

 shown in Fig. 5, which may be considered as representing 

 2m = 16 lines of force. 



In order to investigate the curve for asj'^mptotes, put for 

 the 



1*' position of parallelism «' = « 



3'^ " " " w = (o + 27r 



n*^ " " " (U' = ft) + (?l 1) TT, 



in which n is an integer. For the n*'' position of parallelism, 

 equation (1) becomes 



VI m' 



; (J. H ; (n — l)7r....(18), 



° m — m ■ tn — in 



in which w^ is the special value of oa when the rotating lines 

 are parallel. If in equation (18) n = 1 



m 

 cor, = 6 = — —7 «, 



"0 



m — m 



which is the same as equation (6). Equation (18) shows that 

 the angle between two consecutive positions of parallelism is 



In this fraction the numerator m'lr is the angle 



m — m' 



swept through by AP before a position of parallelism coin- 

 cides with a previous position of parallelism, and the denomi- 



TT 



nator m — m' is equal to the number of asymptotes. — — — , 



is the angle between two adjacent asymptotes, such for exam- 

 ple, as II and IV, Fig. 5. 



