Roever — Geometrical Constructions of Lines of Force. 225 



for the u"', 



co' = 0) -f ( n — 1 ) TT, or trav <o' — trav « + ( n — 1 ) s, 



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

 n"* position of parallelism 



m , m' 

 trav Wrt = versin a H ; (n — 1)5 ( 39 ) , 



which is analogous to equation (18). 



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



I*' tangent a>' = 0, or trav w' = 



2°* '♦ o,' = TT, or trav aj = s 



r\^ " a)'=(w, — l)7r, ov trav (o = {n^ — 1) .«. 



Then, for the n^ tangent at A 



m' 

 trav (Ot = versin a-\ {n^ — 1 ) a- (40), 



which is analogous to equation (19). 



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



l"* tangent a> = 0, or trav &> = 



2"* *• ft, = TT, or trav © = s 



n^ *' (o = {n^ — 1) TT, oxtrav(o = {n^ — l)s. 



Then for the n^"* tangent at A 



m m 

 trav G)/ = — — r versin a + — r (w, — 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 w, 

 ** " '* loops is m\ 

 '* <' " asymptotes is m — ni'. 



