( -t84) 



if this plane is to pass through a point (.t-',, r\) of A^A^, then we 

 tind 



^,.r', + ^X. = 0, 

 so that finally the equation of this plane runs: 



.rVr, — .c'yi'^ = 0. 

 If we now take of the point {x\, w\) the first polar surface 



0.1', d.r, 



with respect to <P, then the section of this surface with the plane 

 a;\a;^ — x'yV^zizzO is the polar curve pj"~^ ; the locus of these, hence 

 the surface T/j", we find by elimination of ,r', and .r\ out of both 

 equaiions; so the equation runs: 



"■ = ■'• a:^ + ■'• a:^ = ''' 



really a surface of order )i containing the line r.^ {.v^ = .v^ := 0) as a 



single line. 



The equation of 'I* can be written in tlie form 



4 Ö0 



,=1 oa-i 

 SO the coordinates of the points of intersection with r^ {.x^ =: .r, ^ 0) 

 satisfy 



i.e. the equation of TT^. 



In the case of the right sphere conoid one of the two polar surfaces 

 is a parabolic cylinder, the other a cylinder of revolution. Let us 

 call the director line )\, the line at infinity of the director plane r,^ , 

 then each plane through a point A^ of /', and through r.^^ intersects 

 the sphere according to a circle, so that the first polar curve of .Ij 

 becomes a line normal to the plane through )\ and the centre of 

 the sphere; this line as well as r^^^ form the complete intersection of 

 the considered plane with 77,. If however we consider in particular 

 the plane at infinity we have to take the polar line of the point of r^ 

 at infinity with respect to the absolute circle, which coincides 

 with )\j^ ; so /7i is indeed a parabolic cylinder whose generatrices 

 are normal to the plane through /\ and the centre of the sphere. 

 In the planes through )\ on the other hand we have to take the 

 vertical diameters of the circles of intersection with the sphere lying 

 in that plane, from which ensues immediately that 77, becomes a 

 quadratic cylinder with vertical generatrices. The points of intersection 

 of r.^^ with the sphere are isotropic points; the circle lying in the 



