228 Trans. Acad. Sci. of St. Louis. 



Equations (42), (43), and (44) show, in a manner similar to 

 that shown in case (a), that 



the number of branches passing through A is w, 

 *' *' •' branches passing through A' is m', 

 *♦ ** <* asymptotes is m + m\ 



the difference in irav between the angles made with 



AAX hy two adjacent tangents at A is — , 



the difference in trav between the angles made with^'^X 



by two adjacent tangents at ^1' is — , 



m' 



the difference in trav between the angles made with^'^X 



o 



by two adjacent asymptotes is 



m + m' 



In Fig. 9, where m = 4 and m' = 1, 



the number of tangents at A \a m = 4, 

 " '♦ "■ " " A' is m' = 1, 

 •* " *' asymptotes is m + ^' = •'>• 



Is$ued June 30, 1896. 



