JoLY — The Associative Algebra applicable to Hijperspace. 81 



If ii is an axis of this fimction <l>, Viq'zs-^ must vanish, and ^ii = i^v5^^ 



But F'l^CTi = -^lii^^i + F](^'t^i = - iirsi', 



and therefore $OTi = Viq Fi^ra = - V^qii . -zs^ - CTi . zs^^ 



so that t^i is also an axis of $. 



Pursuing this argument, it appears that the homogeneous quadratic 

 in iVof the units is reducible to the form 



in which ??^ is the lesser half of JV (viz. m = ^JV, if iV is even ; 

 m = i-(iV- 1), if iVis odd). For each binary product uses up two of 

 the iVnew units, and no unit can occur in more than one binary 

 product. 



Thus the inference in Art. 8 concerning the form of the quantity 

 O is verified, and it is proved that it is possible to reduce a homo- 

 geneous quadratic of iV^ units involving ^iV(iV- 1) arbitrary constants 

 to a form involving explicitly but ^H, or ^-(iV- 1) constants, and 

 when iVis odd, but iV- 1 unit vectors. 



In particular, when three units are involved, a^zhh + (inhh + «i2h4 

 may be reduced to a product of two units multiplied by a constant. 



11. In the particular case of a quadratic, the new linear vector 

 function defined by ij/p = V^qp may be profitably considered. 



Now Scrij/p = Scrqp = S V^crq . p - Sij/'cr . p, 



so ip'a = + Vicrq = - Viqcr = - ij/cr ; 



and this function i//- is the negative of its own conjugate. 

 Suppose ij/ satisfies the symbolic equation 



/(^) = ^» - Miij/''^ + m^il/''-^ - &c. = 0, 

 its conjugate satisfies the equation of similar form 



/(,/,') = ^'" - mii/.'"-i + m^""-'- - &c. = 0.1 

 But ij/' = -i{/, so i{/" + miif/''-'^ + mo\{/"-^ + &c. = ; 

 and this is consistent with the former equation for i// only, if 



Ml = mz = &c. = 0. 

 The symbolic equation is therefore 



i/^" + ??^2^/'"-2 ^ m-if'-^ + &c. = 0. 



' A general property for all linear vector functions. For, if 



this requii-es fW) a" = 0, or /(>^') = 0, as o- is arbitrary. 



E.T.A. PKOC, SEK, III., VOL. V. G 



