( 483 ) 



A^ to this curve, are the i)()iii(8 of intersection of k'' will» tiie tirsi 

 polai' curve p^"~^ of A^ witli respect to /l". The locus of all tiiese 

 curves p,"~' is a surface, wliich for convenience' sake we siiall call 

 "first polar surface of /*i with respect to <fi and 7\" ; the intersection 

 of this surface and <I> is the curve of contact to be found. 



It is easy to see that the first polar surface of /•, with respect to 

 and )\ is of order n and contains the line r.^ as single line. A 

 plane through )\ namely contains the first polar curve />i"~^ of the 

 point of intersection A^ of that plane with ^i ; if now the |)lane 

 rotates round )\, then the points of intersection of />,"""' and r.^ will 

 travel in general along the line r.^, from which ensues that )\ itself 

 lies on the polar surface to be found; so the question is only how 

 many different polar curves />i"~' pass through an arbitrary point 

 of i\. We choose as this point one of the points of intersection S^ 

 of 7\ and 'P. If the first polar curve /)/'— ^ of a certain point A^ of 

 i\ is to pass through S^, then one of the tangents drawn in the plane 

 A^)\ to the curve /;" lying in that plane must have its point of 

 contact in S^, and it must therefore touch the surface in S^- Now 

 the tangential plane in S.^ to intersects the line )\ only in one 

 point; so only one curve />i"^^ passes through 6'.^, and so also through 

 an arbitrary other point of r.^. 



Each plane A^)\ contains thus of the surface to be found a curve 

 />,"—' and the single line r./, the surface is thus of order n. We 

 shall indicate it by the symbol //,". It intersects in a curve of 

 order if, and this is the required curve of contact c"^ o J S2 and 

 'P. Also r.^ possesses of course a first polar surface, TJ^", but now 

 with respect to and r, ; it intersects *P according to the same 

 curve c''^ It is clear that c"* contains the n points of intersection 

 ;S'^ of ?\ and as well as the n points of intersection S^ of r^ and 

 0; the torsal lines thi'ough these points (ouch here c"", because they 

 touch as well as 11^ and 77^. In a point >S'j namely the torsal line 

 touches a curve k'\ thus 0, and a curve />i"~^ thus 77,, and therefore 

 also the section c"" of these two surfaces. 



We control these results analytically. Let r, coincide with the 

 edge A^A^ U\ = x.^ == Ü), and r., with the edge A^A^ {x^ ='^'4 ^=0) of 

 the fundamental tetrahedron, and let be a homogeneous polynomium 

 of order n in .v^,...x^, and let = be the ecpiation of the 

 surface 0. 



F'or a plane through r,^=:A^A.^ the two homogeneous coordinates 

 §1 and are zero, so the equation rutis: 



