Eoever — Geometrical Constructions of Lines of Force. 225 



for the n th , 



to' = co -\- ( n — 1 ) 7r, or trav to' = trav to + ( n — 1 ) s, 



in which s is the stroke and n is an integer. Then, for the 

 n th position of parallelism 



trav to n = ; versin a. -\ ; (n — 1 ) s (39 ), 



o m — m ' m — m v ' v 



which is analogous to equation (18). 



To investigate the curve for tangents at A, put for the 



1 st tangent to' = 0, or trav to' = 



2 nd " to' = 7r, or trav to = s 



n/ h " to' — (n 1 — l)7r, or trav to' = {n i — 1) s. 



Then, for the n^ tangent at A 



m' 

 trav to t = versin a-\ ( n a — 1 ) s ( 40 ) , 



which is analogous to equation (19). 



To investigate the curve for tangents at A', put for the 



1 st tangent to = 0, or trav to = 



2 nd " to = 7r, or trav to = s 



n 2 th " to = (n 2 — 1) 7r, or trav to = (n 2 — 1) s. 



Then for the n 2 th tangent at A' 



m m 

 trav to,' = — — t versin a + —r(n 9 — l)s (41), 



which is analogous to equation (20). 



Equations (39), (40), and (41) show, in a manner similar 

 to that shown in case (a), that 



the number of tangents at A is m, 

 " " " loops is m', 

 " " " asymptotes is m — - m' . 





