298 Trans. Acad. JSci. of St. Louis. 



or if we consider the equation tto- (x^ — x^^) = mtrav eo put 

 for the 



1** asymptote trav o) ^ 

 2°'^ " trav CO = 2 



n*j^ " trav ft) = (?i^— 1)2 



then from the above equation 



m 



xj = 2{n,-l) l^ + x,^ (48). 



in which cc„ is the distance from to the u^^ asymptote. For 

 n^ = I equation (47) becomes the same as equation (40). 



For ^a = equation (48) becomes x^^ = — + x^^. The 



dashed curve (Fig. 7) is a portion of the complete curve of 

 which the limiting line is a part. For a == equation (47) 



becomes cc'„ = 2^^ — . and for x.=^0 equation (48) becomes 



771 



x'a = 2 (7i„ — 1) — , in which x'^ is the distance from O to the 

 7r<T 



n^ asymptote of the dashed curve. 



A comparison of case (6) with case (a) will show a complete 



analogy of the corresponding equations. 



Issued January 19, 1897. 



