TABER. — SCALAR FUNCTIONS OF HYPER COMPLEX NUMBERS. G51 



we derive 



(59) m"'Ai = I mSiieiCj) 



I (i, ./ - 1 , 2, . . . m) 



'^h'^kyijk'Ykhh 



{%,} = 1,2, ... m) 



(ii = 1, 2, ... m) 



7iU» ■ • ■ Tilm, • . . Timl. • • • Timm 

 [i = 1,2, ... 7/0 



(60) vl'"^2 = mSiejCi 



(i,j = 1,2, ... m) 



^h'^k'Yjikyhkh 



(/,.;•= 1,2, ... m) 



Till) • • ■ Turn, • • • ImiU ■ ■ ■ Jmim 



(i = 1,2, ...m) 



JjU, ■ ■ ■ yjml, ■ ■ ■ Jjun, ■ ■ ■ limm 



ij =1,2,... m) 



2/,S/fc7A;fc7foVi 



(i,j = 1,2, ... m) 



Jljl, ■ ■ ■ Jmjl, • • • lijm, ■ • ■ Jmjm 



{j= 1,2, ...m) 



A number system containing no invariant sub system is termed by 

 Cartan a sunplr si/strin {sy,'itcme simi)h'), and he shows that such a 

 system is what is here termed a quadrate. A non-simple system 

 containing no invariant nilpotent sub system Garten terms semi- 

 simple. ^'^ Such a system is constituted by nilfactorial quadrates of 

 which the invariant sub systems are any p{l'^ p < v) of these quad- 

 rates. B\- what is shown above it appears that Ai 5^ or A2 5^ 

 is the conchtion necessary and sufficient that a number system shall 

 be either simple or semi-simple. We have, therefore, the following 

 theorem : 



Theorem IV. Let e\, 6'2, . ■ . Cm he the units of any hyper complex 

 number system, (uid let 



Ai = SiCiej 



I {i,j = 1,2, ... /•) 



SiBiej 



(i,j= 1,2, ... r) 



Then Ai = A2. If Ai 9^ 0, the number system contains a modulus and 

 is either simple or semi-simple, that is, is constituted by v ^ 1 mutually 

 nilfactorial quadrates; and, conversely, in this case, Ai = A2 7^ 0. 



17 Comptes Rendus, 124, 1218 (1897). 



