Kansas University Science Bulletin. 



Vol. I, No. 12. DECEMBER, 1902. |^M: 



ON MONOIDS. 



BY JOHN N. VAN DER VRIES. 

 With Plate XV. 



The object of this paper is, first, to consider the general monoid, 

 and, second, to classify the quartic monoids in detail. 



A monoid M a of order a is a surface of order a that has an 

 {a — 1) -tuple point. If the point (0, 0, 0, 1) is taken as this 

 (a — 1) -tuple point, the equation of the monoid can be written 



U, + Ua -lS = 0, (1) 



where u a and u a -i are homogeneous functions of x, y and z of degrees 

 a and a — 1, respectively. The point (0, 0, 0, 1) is called the vertex, 

 and the cones u a and u a -i the superior and the inferior cones, re- 

 spectively, of the monoid. There are a(a — 1) lines on M a passing 

 through its vertex, viz., the lines common to the superior and inferior 

 cones. These lines are commonly called "the lines of the monoid."* 

 It is evident that if a line is a &-tuple line on M a , it is a line of multi- 

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

 cone. Four kinds of lines are thus seen to be possible on the monoid, 

 according to the relative multiplicities of the lines on the two cones of 

 M a . We can represent them by the following equations of monoids 

 containing them : 



I. (w k zP + 1 + W k+ iZP+ ) + s(v k +1ZP- 1 



+V k+2 zP- 2 + )=0, 



where w k is not identically equal to ; v k + i, . . . . v k + g _ i may be zero ; 



II. (w k zP 41 + w k + izP + ....)+s(v k zP 



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



where w k is not identically equal to 0, and v k does not contain w k as a 

 factor ; 



*This nomenclature is due to Cayley, who uses this surface in his consideration of twisted 



i, 672. 

 305) 



curvGS 



Comptes Rendu?, t. L1V ( 1862 ), pp. 55, 396, 672 



