492 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The line is still a ^-tuple line on 7^/„ , torsa) on every sheet through it, 

 but counts as k {k -(- g) lines of M^^ . The polynomials v^-^g and Wk may 

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

 sheets of M^, may touch k sheets of both M„_i and ?^„ along x y. The 

 line in this case counts as k {k -\- g) + k lines of Al^ . There is no diffi- 

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

 number of lines on M^. The quantity Wk may break up in a great 



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



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

 in which Vk-\-g and Wk break up. 



II. The next case is where the line xy '\» a ^-tuple line on both cones. 

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



{Vk ZP + i'fc+l 2^-^ + )s+{Wk 2P+' + M'/t+l 2^ +.....)= 0. (1) 



We assume ^^7^ 4. ; otherwise the line will not be a ^-tuple line on both 

 cones, but will belong to a later case. We also assume Vk C|) m'ji * (aud 

 ,'. Vk ^ 0) ; otherwise we shall have 



{vk zP 4- n-+i 2^-' + ) s ^- {avk zP+'^ + Wk+i zP i- ) = 0. 



Substituting s — a 2 for s, this reduces to 



{vk zP + Vk+\ zP-^ + ) s + [(m'/c+i - o ^^+1) 2^ + 1 = 0, 



which is a later case. The line ry\s 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 Vk s + WkZ. 

 As we have assumed Vk (|) M'a-, 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 vk and 

 Wk have no common factor, that is if the line xy is torsal on no sheet of 

 J4, 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 Vk and Wk have a factors in common (where a < i — 1), 

 the line is torsal on a sheets of M^j, 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 P lines on M^, but for F + a lines if Vk and Wk have a 



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

 does not contain as a factor. 



