VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 491 



of the position of the point of tangency on the line. A multiple line 

 may thus be torsal on all sheets, torsal on some and scrolar on the re- 

 maining, or scrolar on all sheets of the surface that contain it. A line 

 that is scrolar on any or all sheets containing it may, moreover, be 

 scrolar of different kinds. By scrolanty of the first kind we mean that 

 the tangent plane revolves through an angle of 180° as we pass along 

 the line. A generator, say the line x y, of the quadric x (linear function 

 of z and s) -f y (linear function of z and s) is a line of this kind. A 

 /{.'-tuple line on a surface may be scrolar of the first kind on each of a 

 number of sheets separately, or it may be scrolar of this kind on a num- 

 ber of sheets taken together. In the latter case the sheets are inseparably 

 connected, the tangent planes to them together revolving through 180°, 

 If a line is scrolar of the kih kind, the tangent plaue revolves through 

 k X 180°. There are then k points along this line that have the same 

 tangent plane, that is, every tangent plane is a &-tuple tangent plane. 

 A number of sheets in this case may also be inseparably connected, as 

 in the case of lines scrolar of the first kind. 



As the lines of M u are the lines common to the inferior and superior 

 cones of M^, it is evident that for a line to be a line on M a it must be a 

 line on both cones, and for it to be a £-tuple line on M^ it must be a line 

 of multiplicity k on one cone and of multiplicity not less than k on the 

 other cone. A number of cases must therefore be considered according 

 to the relative multiplicities of the line on the two cones of M^ . 



I. We shall first consider the case where the line xy is a &-tuple line 

 on the superior cone and a (k -\- l)-tuple line on the inferior cone. The 

 equation of M^ can then be put into the form : — 



(y k+ i zP- 1 + v k+2 zP- 2 + ...)«+ (w t zP+ l + w H i zV + ...) = 0, 



where p = /x — k— I; v a and w a being homogeneous functions of de- 

 gree a in x, y, and z. The tangent planes along the line xy are given 

 by Wk — 0; the line is therefore torsal on all k sheets passing through it. 

 There are no points on the line of higher multiplicity than k distinct 

 from the vertex. We assume w^ ^ 0. otherwise the line xy is a (k + 1)- 

 tuple line on both cones, and therefore also on M^ ; this is a case to be 

 considered later. The line xy therefore counts as k (k + 1) of the 

 fi (/a — 1) lines on M^ . This case includes the case where 



»*+l = «>fc+2 = = l 'k+g-l — 0, 



i. e., where the line is a (k + <7)-tuple line on the inferior cone, while 

 only a £-tuple line on the superior cone. We assume 2<y<|u — k — 1. 



