Roever — Geometrical Constructions of Lines of Force. 225 



for the n"*, 



&)' = o) -f- (« — 1) TJ", or trav (o = Irav a> + (n — 1) s, 



in which s is the stroke and n is au intesrer. Then, for the 

 n"" position of parallelism 



m 77i' 

 trav «„ = ; versin a H ; (n — 1)5 ( '^9 ) > 



which is analogous to equation (18). 



To investigate the curve for tangents at -4, put for the 



P' tangent w = 0, or trav w' = 



2°'^ " (w' = TT, or trav w' = s 



n* '< &>' = (»! — l)"^* or trav (o' = {ii^ — I) s. 



Then, for the i\^^ tangent at A 



m' 

 irav (at = versin a-\ {u-^ — 1 ) .s ( 40 ) , 



which is analogous to equation (19). 



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



l^' tangent &» = 0, or trav co = 



2°^ " (^ = TT, or trav co = s 



n^^ " ft) = (?«2 — 1) Ti", or trav w = {% — 1) s. 



Then for the n/'' tangent at A 



m m 

 trav CO,' = — — r versin a + — r (n„ — \) 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^ is m, 

 " " " loops is m' , 

 " " " asymptotes is ??? — m' . 



