( 488 ) 



contact is simply tlie locus of the points of contact of the generatrices 

 of Si witii 0, and the satellite curve is the locus of the points of 

 intersection of the same generatrices with 0. However, if we start 

 from i\, we find as satellite surface ^^ a surface of order 2;z(?2 — 2) 

 with an \n{n — 1) — 2j-fold line i\ , from which ensues that the 

 satellite curve of C'" has also {??, {n — 1) — 2}-fold points in the n 

 points of intersection S^ of i\ and 0. This result is also easy to 

 control with the aid of ^ ^ ; this ^ ^ namely does not contain the 

 line }\ , but it does the points .S',, and it has in these points a 

 contact with of higher order, and inversely 2^ does not contain 

 the line i\ , but it does the points S^, and it has likewise in these 

 points a contact of higher order with «f>. 



Let us imagine a point Si and the section k'' of the plane Sii\ 

 with <!*. The point ;S, lies on k" ; so through *Si pass, besides the 

 tangent in S^ itself, n{n-~^) — 2 tangents more, from which ensues 

 that the satellite curve .s-, of >Si has in this point wüth t' an {?i(n — 1) — 2|- 

 pointed contact. If we allow the plane under consideration to revolve 

 a little about t\ in one sense as well as in the other, then *Si passes 

 into a point A^ ; the tangent in S^ itself passes in one case into two 

 different real ones, in the other into two conjugate complex ones; on 

 the reality, however, of the other tangents the slight difference in 

 position of the plane will have no influence, and so we see by 

 direct observation that through S^ pass 7i [ji— 1) — 2 branches of the 

 satellite curve of c"'. So the points S^ must lie also on ^Jj : the 

 remaining points of r, however lie in general not on it, because the 

 satellite curve .s\ of an arbitrary point A^ does in general not pass 

 through Ai itself; so the points S^ must thus be either singular points 

 of I^i, or iTj and «/^ must have in those points a contact of higher 

 order. If >Si were a singular point, thus a multiple point with a tangen- 

 tial cone of order n {n — 1) — 2, then each plane through this point 

 would have to cut I^j according to a curve with an |»(?z — 1) — 2j-fold 

 point in <S', ; we saw, however just now that the plane ^SjT, ,cuts 

 the surface ^^ according to a curve, which has in Si an ordinarj^ 

 point, but with k'^ an \n [u — 1) — 2|-pointed contact; so S^ is also 

 an ordinary poijit of I^j, but an \n (ji — 1) — 2|-fold point for the 

 intersection with *F*. 



We control the preceding results in the following way. The complete 

 intersection of i2 and <^ is a curve of order 2)f{n — 1) : it consists 

 of the curve of contact c"' , counted double, and of the satellite curve; 

 and 2n^ -\- In'' {n — 2) really furnishes 2/i* {n — 1). 



§ 6. The surface 52 contains in general a certain number of 



