306 KANSAS UNIVERSITY SCIENCE BULLETIN. 



III. (\V k + i ZP + W k + 2 Z p _ * + ) + S(v k ZP 



+ v k + iz p ~ 1 + )==0, 



where v k is not identically equal to 0, and w k — i does not contain 

 v k as a factor ; 



IV. (wk + 2Z p_1 + Wk + 3zP- 2 + . . ..) + s(v k zP 



+ v k + 1 zP- 1 + ) = 0, 



where v k is not identically equal to 0; w k + 2, w k + 3, . ...w k+g _i 

 may be zero. 



In these equations, Vk, Vk— 1, Wk, etc., represent functions of x and y 

 of degrees k, k-{-\, k, etc., respectively; also, k -\- p -\- l = a. The 

 line xy is a #-tuple line on each of these monoids, and differs in the 

 four cases only in its relative multiplicities on the two cones. We shall 

 hereafter distinguish between the different £-tuple lines and designate 

 them as lines of kinds I, II, III, and IV, respectively.* No &-tuple 

 line of kind I cr II has a point on it of higher multiplicity, whereas 

 every line of kind III or IV has on it a point of multiplicity k + 1 

 (e. (/., the point (0, 0, 1, 0) on the monoids III and IV above). Every 

 line of kind I, III or IV is torsal on every sheet of M a that contains 

 it, whereas a line of kind II is scrolar on at least one sheet of the 

 monoid that contains it.f In the case of a line of kind II the scrolar 

 sheets are inseparably connected and the tangent plane to them re- 

 volves through 180° as we pass along the line. The lines of kinds I, 

 II, III and IV count as k(k-\- g), k' 1 , k(k-{-\), and k(k-\- g) lines, 

 respectively, of the monoid. Lines of kind III differ from lines of 

 kind IV only in the fact that the tangent cone at a (A"+ 1) -tuple 

 point on a line of kind IV breaks up entirely into planes. 



It is evident that all multiple lines on M a pass through the vertex. 

 The only lines that M a can have not passing through the vertex are sim- 

 ple lines. For, a plane passing through the vertex and such a line of 

 multiplicity greater than one would meet the monoid, in addition to 

 this line, in a curve of order less than a — 1 having an {a — l)-tuple 

 point at the vertex ; this is impossible. A plane through the vertex 

 and a simple line not passing through the vertex meets the monoid, 

 in addition to this line, in a curve of orders — 1 having an {a — 1) -tuple 

 point at the vertex ; that is, in a — 1 lines passing through the vertex. 

 Such simple lines on the monoid that do not pass through the vertex 

 are called transversals by Cayley.J Thus, for every transversal on M a 

 it is necessary to have a — 1 of the a(a — 1) lines of M a in one plane. 



*Fora fuller explanation of these lines, see a paper published in the proceedings of the 

 American Academy of Arts and Sciences, 1902 and 1903. 



t A line is torsal or scrolar on a sheet of the surface containing it according as the tangent 

 plane to that sheet of the surface is or is not the same at every point of the line. 

 See Cayley, Collected Papers, vol. VI, p. 334. 



IComptes Rendus, t. LVIII ( 1864), pp. 994-1000. 



