152 Trans. Acad. Sci. of St. Louis. 



If we now make x^, y^ variables, equation (34) becomes 

 a^y + y^rix = (d^ + b"^) xy. (35) 



This equation is satisfied by the co-ordinates ?, t; and also by 

 the co-ordinates x^, y^ ; and, therefore, represents a curve 

 which passes through the given point P and through the in- 

 tersections with the hyperbola of the normals through P. 



Equation (35) represents an equilateral hyperbola, which 

 will be designated in what follows as the auxiliary hyperbola. 

 The equations of its asymptotes are 



and 



a'- -\- 0^ 



^ = aib'''- ^^'^ 



Equation (36) represents a straight line parallel to the axis 

 of Y and at a distance 9 , ,0 ^ from it. 



a^ . ., . , . «^ p. 



As ■ 2_L 7)2 ^s necessarily positive, the expression 2 a. 12 ^ 



always has the same sign as ^; therefore, the asymptote rep- 

 resented by equation (36) lies on the same side of the axis 



of Y as the point P. And as 2x12 i^ ^^ss than unity, the 

 asymptote lies between the axis of I^and the point P. 



The distance from the point P to this asymptote is 



(38) 



Equation (37) represents a straight line parallel to the axis 



&2 



of -Z'and at a distance ., , ,, v from it. 



As 2 _L A2 '^ necessarily positive, the expression 2_|_ 7.2 '^ 

 always has the same sign as r); therefore, the asymptote rep- 



