( 359 ) 



d.v : dy : dz z= p : q : — 1 , 



dz dz 



where »= r— , g z= r—. 



OW 01/ 



So the differential equation of the surface becomes : 



— pqz {B - C) f yp {A - B) -{- ,vq {C — A) — 

 or 



X \ y R _ 



The complete integral with two parameters C and C^ becomes : 



It represents a surface of order four. 



It is evident out of the equation that for R =—, the surface re- 

 mains the same; only the A- and the I^-axes have been interchanged. 

 (This is geometrically immediately made clear). So we have but to 

 examine the sui'face for, let us say, i? ]> 1. 



^ 3. It must be possible to find the equation of tlie cone of the 

 complex in a definite point out of the equation of the surface because 

 that cone is the locus of the normals to the gd^ number of surfaces, 

 passing through the point under consideration. If «,/?, y represent the 

 cosines of direction of a ray of the complex in the point .I'l, yi, -Sj then 

 a ^ 



Substituting this in the differential equation and eliminating a and 

 /? by means of the equations of the ray of the complex, namely 



'^■—•'•1 y—i/i ^—^1 



« /? y 



we find for the cone of the complex : 



(R— 1) z, (a-— -t'l) (,v— /a) ~ Ry, i'V—.v,) {z- z,) -\- ,v, (y—y,) (z—z^) z= 0. 

 The planes of tlie coordinates forming the singular surface of the 

 complex, the cone of the complex must degenerate for each point 

 of one of these planes. For the point P{.i\, 7/^^=0, z^) the cone 

 breaks up into y = and into .i\ z -f- [R — 1) z^ x = R z^ x^, i. e. a 

 plane passing through P and parallel to the y-axis. This plane is at 



right angles to OP, if this line has for equation ^ = ± .i' I y/ — . 



(Comp. § 4). 



