216 Trans. Acad. Sci. of St. Louts. 



In order to investigate the curve for asymptotes put for the 

 1 st position of parallelism &>' = <o, 



2 nd " " " ft)' = ft) + 7T, 



n* << " " &)' = co + (?i — 1) w. 



For the n th position of parallelism equation (2) becomes 



m m' 



o} n = — j , a — — i , (n — 1)7T...(J1). 



m + m m + m v v 



To investigate the curve for tangents at A put for the 



1 st tangent &/ = 0, 



2 nd " ft)' = 7T, 



n. th " &)' 



(«!—!) 7T. 



For the n x th tangent equation (2) becomes 



COt = a—— (», — 1) tt (22). 



wi 



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



1 st tangent &> = 0, 



2 nd " ft) = TT, 



n 2 th " &) = (w 2 — 1) tt. 



For the n 9 th tangent equation (2) becomes 



- m m / IN /9*\ 



Equations (21), (22), and (23) show that 



the number of branches passing through A is m, 



t< «< " »< " " ^4' is ??i', 



asymptotes is m + m', 

 " angle at A between adjacent branches is 



7T 

 — » 

 111 



IT 



J^ C ( 4 4 < « " } 



m 



7T 



O " " asymptotes " 



m + m' 



