VAN DER VRIES. — MULTIPLE POINTS OP TWISTED CURVES. 497 



always be obtained as the partial intersection of K,n with a monoid of order 

 m — 1 that has a line of kind III or IV. We shall therefore not use a 

 line of kind I to obtain a curve with a multiple point. 



If Mfj, contains a line of kind II, every sheet of Km that passes through 

 this line intersects M^ in a branch of a curve that crosses xy &t the 

 point where the tangent plane to the sheet of A',„ coincides with the 

 tangent plane to the scrolar sheet o^ M^. If x j/ is a K-tuple line on Km, 

 there will be k branches of the curve crossing the line due to the scrolar 

 sheet of 3f^. No torsal sheet of J/^ can cause an additional branch of 

 the curve ; for this would necessitate the line to be of a multiplicity greater 

 than K on Km, which in turn would cause more than k branches of the 

 curve to be due to the scrolar sheet alone ; and so on ad infinitum. We 

 therefore need a line on M^ that has torsal sheets passing through it in 

 addition to the scrolar sheet. If two sheets of Km unite to form a cus- 

 pidal sheet along xy, there is either a branch of the curve touching xy 

 at the point where the tangent plane to the cuspidal sheet coincides with 

 the tangent plane to iT^, or a branch of the curve passing through the 

 vertex. Both of these cases may be avoided. A number of conditions 

 must be imposed on the coefficients of M to make the k points of cross- 

 ing of the line by the curve coincide. We shall therefore not use a line 

 of kind II to obtain a multiple point on the curve. We need not there- 

 fore in general consider the case where the line a:y is more than a double 

 line on Km. Lines of kind II are only necessary on 71/^ to produce the 

 apparent double points of the curve, and only ordinary lines of this kind 

 are then needed. The line to an apparent double point counts thus in 

 general for two lines common to Km and M^, while only for one line on 

 3f^. If we consider M^ as of order m — 1, it is evident that Km must 

 have at least m — 3 double edges in addition to the double edge caused 

 by this apparent double point. The curve must therefore have at least 

 m — 3 apparent double points in addition, i. e. at least m — 2 in all, as 

 on page 490. 



If M^ has a line xy of kind III on it, there will be a branch of the 

 curve through the multiple point for every sheet of Km that passes 

 through xi/. If xi/ is a K-tuple line on Km and a ^"-tuple line of kind 

 III on M^ , there will be k branches of the curve through the (k + 1)- 

 tuple point of M^ . The point will thus be a K-tuple point on the curve 

 having its k tangents lying in general on a cone of order k + \ that has the 

 line xy as a ^-tuple edge. For the tangents to the curve at this multiple 

 point are the lines of intersection of the tangent cone at this multiple 

 point by the k tangent planes to Km along xi/, each plane intersecting 

 VOL. xxxviii. — 32 



