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



5 5 



in which ^ GAO = rr. tt. If in place of ^i "^ 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 

 2771 = 16 lines of 'force. 



In order to investigate the curve for asymptotes, put for 

 the 



P* position of parallelism co' = co 



3'* " " " o)' = CO + 27r 



'* ft)' = ft) + (H 1) TT, 



(( (( 



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

 equation (1) becomes 



m m 



-a + ;(n — 1) 7r....(18), 



m — m m — m 



in which Wq is the special value of « when the rotating lines 

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





" 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 wj'tt is the angle 



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 as3''mptotes. — -^ — , 



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

 ple, as II and 7F, Fig. 5. 



