492 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The line is still a £-tuple line on il/„ , torsal on every sheet through it, 

 but counts as k (k + g) lines of M^ . The polynomials vt+<? and w k may 

 have any number * (where k < k) of their linear factors equal, i. e. k 

 sheets of M^ may touch « sheets of both m w — l and u u along x y. The 

 line in this case counts as k (k + g) + k lines of M^ . There is no diffi- 

 culty in seeing the effect of any combination of tangeot planes on the 

 number of lines on M^ . The quantity w k may break up io a great 



variety of ways, e.g. x a y s ; xy being torsal on all k sheets of 



M^ , the number of lines adjacent to xy alone being affected by the way 

 in which v k+g and Wk break up. 



II. The next case is where the line xy is a i-tuple line on both cones. 

 The equation of the monoid can then be put into the form : — 



(v k zP + v k+ i zP-* + ) s + (w k zP+> + w k+i zP\ ) = 0. (1) 



We assume w k 4= ; otherwise the line will not be a &-tuple line on both 

 cones, but will belong to a later case. We also assume v k w k * (and 

 ,\ Vk ^0); otherwise we shall have 



0* zP + i'jfc + i zP~ ] + ) s + (a v k zP+* + w k+i zP + ) = 0. 



Substituting s — a z for s, this reduces to 



(v k zP + Vjfc +1 2P-» + ) s + [(w k+ i - a v k+ i) zP + ] = 0, 



which is a later case. The line xy is a £-tuple line on the monoid whose 

 equation is (1), and has no point on it of multiplicity greater than k. 

 The tangent planes along this line are given by the terms v k s + w k z. 

 As we have assumed v k (J) w k , s and z cannot factor out, and the line xy 

 must be scrolar of the first kind on one or more sheets of M^. If v k and 

 w k have no common factor, that is if the line x y is torsal on no sheet of 

 Mn it is scrolar on all k sheets. These k sheets, however, are insepar- 

 ably connected, that is the tangent planes to all of them together revolve 

 through 180°. If v k and w k have a factors in common (where a < k — 1), 

 the line is torsal on a sheets of M^ and scrolar of the first kind on the 

 remaining k — a sheets together. The inferior and superior cones then 

 touch the monoid along this line on a sheets. The line xy counts in 

 general for k' 2 lines on M^ but for F -f a lines if v k and w k have a 



* The sign O should be read : contains as a factor ; the sign O should be read : 

 does not contain as a factor. 



