VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 493 



factors in common. It may count for more lines in the same way as the 

 line xy in the preceding case. 



III. The third case is where the line x 7/ is a ^-tuple line on the 

 inferior cone and a (^ -f- l)-tuple line on the superior cone. The equa- 

 tion of the monoid will then be of the form 



(vk zP + Vk+i zP-'^ + ) s + {wk+i zP + Wk^2 zP-^ + ) = 0. (2) . 



We assume w^. ^ 0, otherwise the line will belong to the preceding case. 

 We also assume w^jf-i v^ (and .'. w/,--j_i ^ 0), otherwise we shall have 



{Vk zP + y^.+i zP-"^ + ) s + {vkazP + m^^.+o zP-^ + ) = 0, 



where a is a linear function of x and y. Substituting s — a for s, this 

 reduces to our next case. The line xy is a X--tuple line on the monoid 

 whose equation is (2) above, and has a point (0, 0, 1, 0) of multiplicity 

 k -\- \ on it. The tangent planes along xy are given by r^. ; the line is 

 thus torsal on all the sheets that pass through it. The tangent cone at 

 the {k + l)-tuple point is given by the terms Vk s + wi-j^i. The line xy 

 is also a ^-tuple line on this tangent cone. This cone can never break 

 up into ^ + 1 planes, as this would require wa-(-i O Vk. Nor can it 

 break up into factors of which more than one are proper cones, as it 

 could not then contain the line xy as a Z;-tuple line. If it breaks up at 

 all, it must break up into factors of which all but one are planes, and 

 this remaining factor a proper cone of a certain order, say o-, having the 

 line xy as a (o- — l)-tuple line. The plane or planes into which the 

 tangent cone at the {k + l)-tuple point may break up belong to the k 

 tangent planes through the >t-tuple line. The line xy counts in general 

 for k {k 4- 1) lines of il/^, but may count for more as in the preceding 

 two cases. 



IV. The next case is where the line is a /---tuple line on the inferior 

 cone, and a (k -f 2)-tuple line on the superior cone. The equation of 

 the monoid can then be put into the form 



{Vk zP + Vk^x ZP-^ + ) S + (Wk+2 =P-^ + WA-+3 ZP-^ + ) = 0. 



We assume vk ^ 0, otherwise the line will belong to the preceding case. 

 The tangent planes along the line xy are given by the term t'^ ; the line 

 is thus torsal on all sheets that contain it. The point (0, 0, 1, 0) is a 

 (k + l)-tuple point on j'l/^, the tangent cone at it being given by Vk s. 

 The tangent cone thus breaks up into the k tangent planes along the 

 A'-tuple line, and the plane s. The line xy counts here for k {k -f 2) 



