JoLY — The Associative Algebra applicable to Syperspace. 85 



For seven points, in addition to the four triangles in the first case 

 for six points, (167), (257), and (347) are obtained. In addition to 

 the triangles (123) and (456) in the second case, the triangles (716), 

 (734), and (752) may be constructed. In the first of these cases for 

 seven points, every point of the seven is joined to every other point, 

 so that three new points 8, 9, and 10 must be added before a new 

 tidungle can be constructed.^ 



15. It is possible to derive from a given homogeneous function q 

 ■of order in a series of self -conjugate functions analogous to that of 

 Arts. 9 and 13, which I shall only mention here. 



The function referred to is $p = ViC[ V„^iqp. The next function of 

 the series is i^2^2 = ^2$' T^t,i-2qP2, where 2h is a homogeneous quadratic in 

 the units. The general function is 



FnPn= VN^Vm-Nq-pN^ 



It may also be noticed that the functions /of Art. 13, defined by 

 relations of the type I^ = V,n_iqii, lead to the following equations : — 



Fj^Ji = ViqV„^iqii = ^ii = c^ii, 



Thus $ and F,„_i have the same roots. This holds good also for Fjf 

 and F,^^. 



Further, the series of linear functions defined by the equation 



^NPm-N = V^qPm-N 



may be noticed. These convert a homogeneous function of order m - N 

 into one of order iV, and 



shows that ^m^N^^ is the self -con jugate function F,n_N. 



1 After this Paper was read, I saw that if a function can he written in the form 

 q - iili + izh + . . . + imim, 

 in which none of the /involve any of the m units, iiiz . . .im, these m units belong 

 to tlie canonical system, provided Slilz = &c. = 0. 



In particular, the cubic 



q = «i [aisii + bi^ie) + xizibi^ii — ai^ie) 



is in the canonical form, although it cannot be tjrpified by triangles having no side 

 common. 



