451 



and eliminating Xi, yi in like manner, we get 



K = K{\-K)xi + ^{\--B)yi; (7) 



from which expressions it appears that, every thing else re- 

 maining, the focus and directrix may be changed without 

 changing the surface described. For in order that the sur- 

 face may remain unchanged, it is only necessary that K should 

 remain constant, since a and b are supposed constant. This 

 condition being fulfilled, the focus may be any point F whose 

 coordinates xi, y^ satisfy the equation (6), and A (the foot of 

 the directrix) may be any point whose coordinates x.j, yi 

 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/oca/ curve, 

 or theybea^ line; and the locus of A is another elhpse or 

 hyperbola, which may be called the dirlgent 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 z, describes a cylinder which may be called 

 the dirigent cylinder. 



Since, by the relations (4), the corresponding coordinates 

 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 



^2 - ^1 = j^ x„ y.i ~yx- j— ^2/1, (8) 



* In the Proceedings of the Academy, vol. i. p. 90, it was stated inadver- 

 tently that " if we confine ourselvea to the central surfaces, the locus of the 

 foci will be an ellipse." 



