216 Trans. Acad. Sci. of St. Louis. 



In order to investigate the curve for asymptotes put for the 

 1*' position of parallelism w = w, 



2°'* '* " " ft,' = ft, + TT, 



n*^ " '« ♦* w' = ft) + (?l 1) TT. 



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



m m' 

 ft)n = — i > « — — i , {n — 1 )7r. . .(21). 



To investigate the curve for tangents at A put for the 

 1«' tangent a = 0, 



2°* " ft)' = TT, 



th <( 



ft) 



' = ( "l — 1 ) T^. 



For the u^ tangent equation (2) becomes 



m 



wt-a— — {n, — l) TT (22), 



m 



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

 1** tangent &) = 0, 



2°d '« ft) = TT, 



Dj* " ft) = (^2 — l)7r. 



For the n^^ tangent equation (2) becomes 



^' = ^«-;;7^^'^-^)^ (^^)- 



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



the number of branches passing through A is m, 

 " «' " '< " " A is ?w', 



" •' " asymptotes is m + m\ 



" angle at A between adjacent branches is — , 



m 



t ( a a j^ <* it (( (< _ 



(( (( (( 



m 



TT 



O " '< asymptotes ** 



