148 Trans. Acad. Sci. of St. Louis. 



The equation of the tangent to the hyperbola is 



aW\ {a*-V)y' 2 c25 ) 



y ~ b 2 V x' 2 &V 



in which x , y' are the co-ordinates of the point of contact on 

 the hyperbola. 



If equations (24) and (25) are made to represent the same 

 line, which signifies that the tangents to the two curves coin 

 cide,and if, at the same time, x', y' in (24) and (25) represent 

 the same point, which means that the curves are tangent to 

 each other at that point, we have as equations of condition 



b-x' tf-y" 1 



a^y' b 2 r)x' 2 



(26) 



2 



(27) 



y' b 2 v 



from which for the co-ordinates of the point of contact, 



y - = , JX^- (29) 



a 2 — b'l 

 If these values are substituted in equation (15), we have 



This, if £> v are regarded as variables, is the equation of the 

 evolute of the ellipse, which is therefore the locus of the 

 point P when the hyperbola and ellipse are tangent to each 

 other. 



From the property of an evolute already cited (pp. 141-2) 

 and by reference to the figures, it is apparent that for any 

 point on the evolute (Fig. 6) one branch of the hyperbola is 

 tangent to the ellipse while the other branch cuts it in two 

 points and there are three normals, PN~ y , PiV 2 , PJV 3 ; for 



