On the Surfaces of the Second Order. 283 



When the two directive planes coincide, and become one 

 directive plane, the umbilicar property is reduced to this, that 

 the distances of any point in the surface from the point F and 

 from the directive plane are in a constant ratio to each other 5 

 and therefore the surface becomes one of revolution round an 

 axis passing through F at right angles to that plane; the point 

 F being a focus of the meridional section, or the vertex if the 

 surface be a cone. When the directive planes are supposed to 

 be parallel, but separated by a finite interval, we get the same 

 class of surfaces of revolution, with the addition of the surface 

 produced by the revolution of an ellipse round its minor axis ; 

 the point F being still on the axis of revolution, but not having 

 any fixed relation to the surface. 



10. If in the equation (30) we supposed the right-hand 

 member to have an additional term containing the product of 

 the quantities x - x z and y - y^ with a constant coefficient, all 

 the foregoing conclusions regarding the geometrical meaning 

 of that equation would remain unchanged, because the addi- 

 tional term could always be taken away by assigning proper 

 directions to the axes of x and y. If, after the removal of this 

 term, the coefficients of the squares of the aforesaid quantities 

 were both positive, the locus of F would be a modular focal of 

 the surface expressed by the equation ; but if one coefficient 

 were positive and the other negative, the locus of F would be 

 an umbilicar focal. The equation in its more general form is 

 evidently that which we should obtain for the locus of a point 

 S, such that the square of its distance SF from a given point F 

 should be a given homogeneous function of the second degree 

 of its distances from two given planes ; the plane of xy being 

 drawn through F perpendicular to the intersection of these 

 planes, and x 2 , y* being the co-ordinates of any point on this 

 intersection, while x ly yi are the co-ordinates of F. The point 

 F might be any point on one of the focals of the surface de- 

 scribed by S; the intersection of the two planes (supposing 

 them always parallel to fixed planes) being the corresponding 

 directrix. 



