471 



obtain for the locus of a point S, sucl) that the square of 

 its distance SF from a given point F should be a given ho- 

 mogeneous function of the second degree of its distances 

 from two given planes ; the plane o{ xy being drawn through 

 F perpendicular to the intersection of these planes, and 

 X2, 2/2 being the coordinates of any point en this inter- 

 section, while a;, , ?/, are the coordinates 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 correspond- 

 ing directrix. 



These considerations may be farther generalised, if we 

 remark that the equation of any given surface of the second 

 order may be put under the form 



(^ - ^O^'+CZ/ - Z/if +(^— «i)''=l(^- ■>fa)' + m(?/ - ?/2)'4-n(3!— a;.,)'' 

 4-L%-i/2)(sr-5r,)+M'(.r-^-2)(^--22) + NV-:r,)(y-?/,), (38) 



where l, m, n, l^ m', n' are constants, and a"i, y, , z^ are con- 

 ceived to be tlie coordinates of a certain point F, and 

 ^2 5 y^i ^2 the coordinates of another point A. The con- 

 stants l', m', n' may, if we please, be made to vanish by 

 changing the directions of the axes of coordinates ; and when 

 this is done, the new coordinate planes will be parallel to 

 the principal planes of the surface. Then, by proceeding as 

 before, it may be shown that, without changing the surface, 

 we are at liberty, under certain conditions, to make the 

 points F and A move in space. The conditions are ex- 

 pressed geometrically by saying that the two surfaces, upon 

 which these points must be always found, are reciprocal 

 polars with respect to the given surface, the points F and A 

 being, in this polar relation, corresponding points ; and that 

 the surface which is the locus of F is a surface of the second 

 order, confocal with the given one, it being understood that 

 confocal surfaces are those which have the same focal hues. 

 The surface on which A lies is therefore also of the second 



VOL. II. 2 s 



