( 485 ) 



plane tlu'ougli such a point and r, passes itself lliroiigh lliat point 

 and touches here the absolute circle, so thai the |)olar line of liiat 

 point become-; a tangent lo the absolute circle ; so the cylinder touches 

 the absolute circle twice and is therefore a cylinder of rotation. The 

 sphere and. these two cylinders intersect each other according to a 

 twisted cni-ve of order 4 and the 1-^ species, containing among others 

 the isotropic points of intersection of ?",^ with the sphere ; on the 

 plane through }\ and the centre of the sphere it projects itself as a 

 pai-abola, on a horizontal plane as a circle. 



§ 4. We a^ain imagine a point A, of r,, then a plane A^}\_, and 

 the section with 77^ lying in this plane and consisting of the curve 

 p/' ^ and the lijie r,. We take this system as a curve of order n 

 and we determine the first polar curve ^i"~^ for the pole A^, which 

 is of order n — 1, and contains the Ji — 1 points of intersection of 

 />/'—' and r.^, but moreover the points of contact of the (?2 — ^){n — 2) 

 tangents which can be drawn out of ^j to p^"'^' . We now look 

 for the locus of the curves q^""'' and show that this is again a 

 surface of order ?i, having )\_ as a single line. The first polar surface 

 of the point (a-' 3, a:' J with respect to 77^=0 has for ecpiation 



Ö/7, dn, 



hence (see § 3) : 



dfp d-<Ii d'<P dtp d'fP 



■''^ li, + •' «-^^ 0^, + ^-' ^''^ ^ ^' ^'^^'^ d^ö^ -^-"^'07,^' ^''^ 0^ = '' 



a surface of order n — 1 and which, cut by the plane .i"'^^,— .r'j.i^ = 0, 

 furnishes the curve '//'"^ The locus of this curve, found by elimination 

 of x'f and .v\ out of the last two equations, is therefore the surface 



o.7?3 ox^- o.v^ox^ ox^ ox/ 



it is indeed of order n and contains ]\^ {.i\ = ,%\ ^= 0) as a single line, 

 just as /7j . The section with IJ^ is therefore a curve of order /i% of 

 which r, forms a part ; it is however easy to show that i\ must be 

 counted twice, so that there remains a residual section of order ?i' — 2. 

 The section of /7j and K^ lies namely evidently also on the surface : 



ox/ Ox^Ox^ ox/ 



which 1ms evidently the line A^A^ as a double line. For the section 

 of li^ and /Ci*, or K^ and K^*, the director ?•, counts double ; thus 

 it must also count double for the section of IJ^ and K^, with which 



33 

 Proceedings Royal Acad. Amsterdam. Vol. XIV. 



