JoLY — The Associative Algebra applicable to Huperspace. 113 



Pq =/^, and (Tq = -/'fl, the systems referred to the new origin hecome 



r =/^ + F^le = Fi, and cr = -/'O + FiOe = - i^'O, 



where i^and F' are still conjugate functions. In fact, 



/S'p (/'fi - Fifie) = .S'Q/p - >S'epO = &Q. {fp + F:,p€), 



because V^pe = - Kep. 



52. I shall now show that e can be chosen so that^for any units 



%VAFi, = 0. 



In the first place, it is necessary to show that % Viiifii is an in- 

 variant, or that it is independent of the particular system of units 

 chosen. Consider the quotient of determinants of order m involving 

 m arbitrary vectors (X), 



Q- 



Ki K, Kj K 



Kj Ki K} ■^n 



^1} ^2, ^3) 



^1) ^Zi K, 



K) K) ^3y • • ■ K 



here the first row of the dividend consists of the results of operating 



by / on each of the vectors in a certain order, and the m - 1 rows 



which remain are alike, and formed by the vectors in the same order ; 



the divisor consists of to equal rows the same as the equal rows in the 



dividend. In determinants of this kind, it is lawful to add the columns 



when multiplied by suitable scalars.^ Consequently, if %ti\i = X', any 



column in the dividend may be replaced by f\', A', X', . . . X', provided 



the corresponding column in the divisor is replaced by A', A', A', . . . A'. 



In fact, everything turns on the distributive property of the function/, 



expressed by the equation %tifX-^ = f%tiXi. Thus the quotient Q is 



independent of the vectors A, which may consequently be replaced by 



any other set of independent vectors. 



Eeplacing X-^Xo, . . . A^ by i^i^ . . . v, and remembering the rules of 



expansion, it is easy to see that the invariant 



\ n=; 1 

 Q = {f{h) . i^iz '"im -/(4) . «>3 . . . 4 + &c.} 



= ±-%f{i,).i^. 



rntit^ 



1 Compare the Paper already cited "On Quaternion Invariants'^ fProc. Eoy. 

 Irish Acad., 1896). 



E.I.A.PEOC, SEE. IIJ., vol. V. I 



