JoLY — The Associative Algebra applicable to Hyperspace. 83 



Lastly, if an odd number of units is involved, the common perpen- 

 dicular to all these planes satisfies ij/p = 0. 



There is, of course, a certain indeterminateness about the axes 

 of <E>, The general quadratic function is split up into a sum of arett 

 vectors in determinate planes which have no line common (and which 

 are hyper-perpendicular) ; any pair of perpendicular vectors in one of 

 these planes may be taken as canonical units, but the products of 

 pairs are definite. 



As a corollary, if the sum of two area vectors is an area vector, 

 the planes of the two given areas have a common line. 



13. In the general case for homogeneous functions of order m, if ii 

 is an axis of the self -conjugate function $/) = ViqV,n_iqp, and if Ci is 

 the corresponding root, the series of functions (/), defined by equa- 

 tions of the type /j = Vm-iqii, obey the equations 



SI,'' = (-)'»ci, and Sill, = 0. 

 Por SI-' = S V„,_,qiy r,^,,qi, = (-)'"-l.S^\<D^\ = (-)'"Ci, 



and SI Jo = S K^^i^q K^J^q = {-)"'-^Si,^i^ = 0, 



since V„^^iqi, = (-)"'-i V„_ii,q. 



As an example, consider the general cubic in four of the units, 

 q = ^a,tJsitiu-^ 

 From this il = - ^ciiJ.it, and 1^ = - ^«2si«>'«> 



and >S/i72 = - 2«is««2s( = - «i34«23i, since (ve)^ = -l. 



The six equations aiaiffos^ = &c. = 0, which result if the units are 

 canonical, require all but one of four coefficients a^tu to vanish. Thus 

 the cubic reduces to «i23h44- 



Again, for the cubic in five of the units, ten equations of the type 



((iZi^hzi + <''l45<*^245 + ^153^253 = 



are found connecting the coefficients when the units are canonical. 

 For variety, instead of solving these ten equations, multiply the cubic 

 into the product is^^ . . . in of all the units not involved in it. Now, 



ayi-Siiii'i . ieii ... 4 = - «i23«i44(«'44)%«7 ... 4 = - (imiiio^^) 



in which w is the product ^l^2^3 . . . 4 of all the units. The result is 

 consequently a quadratic in five units multiplied into o). 



1 It is convenient to suppose 



«123 = «231 = «312 = — «321 = — fl213 = — «132. 

 G 2 



