156 



Trans. Acad. Sci. of St. Louis. 



point JP when the auxiliary and given hyperbolae are tangent 

 to each other. From the property of the evolute already 

 cited (pp. 141-2) and by reference to the figures, it is apparent 

 that for any point on the evolute (Fig. 10) one branch of the 

 auxiliary hyperbola is tangent to one branch of the given 

 hyperbola and cuts the other branch of the given hyperbola in 



-N2F"-"-=-- 



Figure 10. 



a single point, while the other branch of the auxiliary hyper- 

 bola cuts one branch of the given hyperbola in a single point 

 and there are three normals, PiVj, PN^^ PN^\ for any point 

 in the plane on the convex side of either branch of the evolute 

 (Fig. 11), one branch of the auxiliary hyperbola cuts one 

 branch of the given hyperbola in two points and cuts the other 

 branch of the given hyperbola in a single point, while the 

 other branch of the auxiliary hyperbola cuts the one branch 

 of the given hyperbola in a single point and there are four 

 normals, PiVj, PN,, PN^, PNi ; and for any point in the 

 plane on the concave side of both branches of the evolute 

 (Fig. 12), one branch of the auxiliary hyperbola cuts one 



