527 



§ 6. Ill order to sliow that tlie result found in ^ 5, is in accordance 

 witli the result of Schubkrt, mentioned in § 1, we have to know 

 the rank of the congruence r (mm', 7» in') tiiat two complexes C, 

 and 6', of the orders m and m' have in common. It might suffice 

 to refer to Schubkrt, Kalk'dl der Abzahlenden Geometrie, where there 

 is found on p. 330 a derivation of this number. We shall however 

 show that the order of F may also be found by the aid of the 

 representation used in this paper. 



The surface ii consisting of the straight lines of r which cut the 

 axis a of C, is of the order 2mm' and has a as an 7)i??i'-fold straight 

 line. It is the intersection of the two congiaiences 2^ [m, vi) and 

 S^ {in' , in') consisting of the straight lines out of 6\ and C', that 

 cnt a. 



S^ and 2^ are represented resp. on two surfaces .S, and S, in 

 R,. As C,, hence also ^,, contains 7/1 geneiat rices of an arbitrary 

 plane pencil of C\ all points of />, and r, are ?H-fold points of 5, 

 and all straight lines cutting /;, and r, have m more points in 

 common with S^. .S', has accordingly the order 2m. and p^ and i\ 

 are /;?-fold straight lines ot .S'j. In the same way S, has the order 

 2m' and p^ and v^ me in'-i'o\d straight lines of this surface. The 

 intersection of S^ and ,S', consists of the straight lines y>, and i\, 

 each counted mm' times, and the curve y on which ii is represented. 

 This curve has the order 2inm' and has mm' points in common 

 with each of the straight lines p^ and v^. We first determine the 

 number of apparent double points of y. 



The cone A projecting y out of an arbitrary point L of /?,, is 

 of the order 2mm' and has in common with »S, besides y a curve 

 Q of the order Am'm' — 2mm' = 2mm' {2m — ^). The curve (,» has 

 {m — l)-fold |)oinl8 in the 2mm' points where y cuts the lines p^ or 

 v^, l)ecause the entire intersection of A and 5, must have there 

 ?H-fold points. Further A cuts each of the lines p^ and r, in mm^ 

 more points, that are «i-fold points for p. As all these points are 

 Hi'-fold for .S',, y has 4/h?/i'' (2m — 1) — 2'inm"' {m — J) — 2?«'»i'' =: 

 =z2mm"^{2m — 1) points of intersection with 5, outside /;, and i;,. 

 These belong to y and lie partly in the points where a generatrix 

 of A touches the surfaces 5, on y, hence in the points of intersection 

 with y outside p^ and v^ of the first polar surface of L relative to .S,. 

 As this polar surface is of the order 2m — 1 and has (m — I)-fold 

 straight lines in/;, and ?»,, it cuts y outside y:», and i;, in 2mm'(2??? — 1) — 

 2min'{m — \) =2in'm' points. The remaining 2nnn''{27n — 1) — 2?»'/»'^ 

 = 2inm' {2inm' — m — m') points where q and y cut each other 

 outside /J, and v,, are points that the bisecants of y through L have 



