TABER. — SCALAR FUNCTIONS OF HYPERCOMPLEX NUMBERS. 67 



the term "quadrate" has been suggested by Clifford.* A quadrate with 

 n = nr units may be termed as a quadrate of order m. The units of a 

 matrix of order m constitute a quadrate of order in. If we denote by 

 £ y the units of the reciprocal system, we have 



**>•= c # (hj= h 2, • • • m). 



A quadrate is thus equivalent to its reciprocal. For the units of a 

 quadrate we have 



S? a = — = Ss it (i = 1, 2, . . . n), 



in 



S?,j = = Se tj (i,j = 1, 2, . . . n; i $j) ; f 



m m 



and, thus, for any number 2- 2 a »>" £ '}' ^ sucn a s y stem we have 



i ' i ' 



m m m i in 



sa = 2, 2,-% ^ = 2, a » ^«« = - 2/*"' 

 i i i w i 



•S-4 = 2, 2> a » ^ = 2< a « ^ f " ~ ~ 2, a »- 

 ii i m i 



Therefore, 



Theorem IT. Jf t ti ,for i,j = 1, 2, . . . m, are <Ae n = m 2 units of a 

 quadrate of order m, that is, if 



S ij £ jk — £ :k> £ ij £ hk 



(i,j, k, h = 1, 2, . . . »i; /« ^7), 



then, if 



m m 



A = 22,^-% 

 1 1 



is any number rf the quadrate, 



* Peirce, loc. cit., p 217. 



t For, for 1 ^5 i <^ m, e^ is iflempotent ; there are just m independent numbers 

 idemfaciend to e ;j , namely, t. (j = 1, 2, . . . m) ; and just m independent numbers 

 e.. (j = 1, 2, . . . m) idemfacient to e ;j .. Moreover, for 1 ^ t '^ m, 1 5* ,/ ts; >«, 

 i tJ, e . is nilpotent. 



