On the Surfaces of the Second Order. 265 



and eliminating #1, 1/1 in like manner, we get 



B}y?', (7) 



from which expressions it appears that, everything else remain- 

 ing, the focus and directrix may be changed without changing 

 the surface described. For in order that the surface may re- 

 main unchanged, it is only necessary that K should remain con- 

 stant, since A and B are supposed constant. This condition 

 being fulfilled, the focus may be any point F whose co-ordinates 

 #1, 7/1 satisfy the equation (6), and A (the foot of the directrix) 

 may be any point whose co-ordinates x 2) y 2 satisfy the equation 

 (7) ; it being understood, however, that when one of^these points 

 is chosen, the other is determined. The locus of F (supposing 

 K not to vanish) is therefore an ellipse or a hyperbola,* which 

 may be called the focal curve, or the focal line ; and the locus of 

 A is another ellipse or hyperbola, which may be called the din- 

 gent curve or line : the centre of each curve is the centre of the 

 surface, and its axes coincide with the axes of the surface which 

 lie in the plane of xy. Moreover, as the quantities 1 - A and 

 1-B are essentially positive, the two curves are always of the 

 same kind, that is, both ellipses, or both hyperbolas ; and when 

 they are hyperbolas, their real axes have the same direction. 

 The directrix, remaining always parallel to the axis of s, de- 

 scribes a cylinder which may be called the dirigent cylinder. 



Since, by the relations (4), the corresponding co-ordinates 

 of F and A have always the same sign, these points either lie 

 within the same right angle made by the axes of x and y, or lie 

 on the same axis, at the same side of the centre. And as these 

 relations give 



X*-KI= J3-J Xi > y* ~y^ = iirg yi * ^ 



it is easy to see that the right line AF is a normal to the focal 



* In the Proceedings of the Academy, VOL. i. p. 90, it was stated inadvertently 

 that " if we confine ourselves to the central surfaces, the locus of the foci will be 

 an ellipse." 



