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



In order to investigate the curve for asymptotes put for the 

 1*' position of parallelism w' = co, 



2°'* " *' " O)' = CO + TT, 



n**' " " " 0)' = o) -\- (n — 1) TT. 



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



771 m' 

 OJn = — j , « — — ■. , (u — l)7r...(21). 



To investigate the curve for tangents at A put for the 



For the n^*^ tangent equation (2) becomes 



«^ 



iO 



, = «— — ('^, — 1) TT (22). 



m 



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

 1*' tangent w = 0, 



2-^^ " CO = -TT, 



n^**^ " « = (w,— 1)77. 



For the n^"" tangent equation (2) becomes 



m ill 

 co't = — ;« — — T (?i„ — 1 ) TT (23). 



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

 the number of branches passing through A is in, 



.( «< 4t 4« 44 44 ^' is WJ'^ 



*' *' " asymptotes is m + m', 



rrr 



" angle at A between adjacent branches is — , 



m 



TT 



4 4 4 4 4 4 J^' <4 (( (( H 



in 



" *' " " " asymptotes " "^ 



??i + m 



