284 On the Surfaces of the Second Order. 



These considerations may be further generalized, if we 

 remark that the equation of any given surface of the second 

 order may be put under the form 



'(x-% 2 ) (z-z 2 ) +N'(x-x 2 ) (y-y a ), (38) 



where L, M, N, L', M', N' are constants, and x l} y^z^ are con- 

 ceived to be the co-ordinates of a certain point F, and # 2 , y 2) ^2 

 the co-ordinates of another point A. The constants L', M', JV' 

 may, if we please, be made to vanish by changing the directions 

 of the axes of co-ordinates ; and when this is done, the new co- 

 ordinate 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 expressed 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 

 lines. The surface on which A lies is therefore also of the 

 second order, and the right line AF is a normal at F to the 

 surface which is the locus of this point. Moreover, if through 

 the point A three or more planes be drawn parallel to fixed 

 planes, and perpendiculars be dropped upon them from any 

 point S whose co-ordinates are #, y, z, the right-hand member 

 of the equation (38) may be conceived to represent a given 

 homogeneous function of the second degree of these perpendi- 

 culars ; and the given surface may therefore be regarded as the 

 locus of a point S, such that the square of the distance SF is 

 always equal to that function. 



11. In the enumeration of the surfaces capable of being 

 generated by the modular method, we miss the five following 



