498 PROCEEDINGS OF THE. AMERICAN ACADEMY. 



the tangent cone in the line xy h times and in one other line, which is a 

 tangent to the curve. If the tangent cone breaks up into g planes and a 

 cone of order k — g + 1 (where 2 = k — g + 1) that has the line xy 

 as a (k — </)-tuple edge, the k tangents to the curve will lie on this cone 

 of order k — g -f 1. In this case it is only necessary to have the line 

 xy as a (Jc — ^)-tuple line of kind III on M^ . We need not consider the 

 case where one or more sheets of K m touch M^ and therefore the tangent 

 cone at the multiple point along the line xy, for this line is then tangent 

 to the curve. Nor can any sheet whose tangent plane is one of the com- 

 ponents of the tangent cone at the multiple point cause a branch of the 

 curve by having contact with a sheet of the cone. 



If Mn contains a line xy of kind IV, every sheet of k m that passes 

 through this line cuts out of M^ a branch of the curve that passes through 

 the multiple point on M^_ . The tangent lines to the curve in this case 

 all lie in one plane, viz. the plane that is tangent to M^ at the multiple 

 point but does not contain the line xy. Thus, if the line x y is a K-tuple 

 edge on A' m , the curve of intersection will have a K-tuple point, the tan- 

 gents at which all lie in one plane. As the only sheet of M^ that affects 

 the curve is the sheet that does not pass through the multiple line, it is 

 not necessary to have the line from the multiple point to the common 

 vertex of K m and M^ as a line on M^ . The single sheet of M^ simply 

 cuts the k branches of the curve out of the k sheets of the cone, its tan- 

 gent planes intersecting the k tangent planes to K m in the k tangent 

 lines to the curve at that point. 



4. We can therefore always obtain a curve of order m having a K-tuple 

 point, the tangents at which lie on a cone of order k + 1 that has the 

 point as vertex and has an arbitrary line through this point as a £-tuple 

 line, as the partial intersection of a particular cone and a particular 

 monoid. The cone is of order m, has its vertex on the &-tuple edge of 

 the cone of order k + 1, and has this edge as a &-tuple edge. The 

 monoid is of order fx, and has the multiple point of the curve as a (k + 1)- 

 tuple point and the line from this point to the common vertex of the two 

 surfaces as a £-tuple line. The line from the common vertex to the 

 multiple point counts thus as k< lines common to K m and M^ , and as 

 h (k + 1) lines of M^. The cone and the monoid have m (/x + 1) — k k 

 additional lines in common, whereas there are only ^ (fi — 1) — k (k + 1) 

 additional lines on M^ . The cone K m must therefore have at least 



m Qt - 1) - k k - (i (p — 1) + k (k + 1) 



i. e. (m — fj) (ji — 1) — Tc (k — h — 1) = a 



