400 PROCEEDINGS OF THE AMERICAN ACADEMY 



stitutc an independent quadruple algebra of the same form with qua- 

 ternions. Thus if i, j, k, are the primitive quaternion vids and w the 

 new vid. Let 



tCt - ■■- Cit A/« #Vrt ■■■ - cc,2 fO* . 



Then since 



a^ = Kj. a\ = «2. 



• 2 • 2 /, 2 • 2 * 2 /, 2 



*i — 7i — "-1 — — '^i* ^2" — J2~ — "2 — — ''^l* 



J I "'1 = i'i= —^i Jl- J.2 k.^ = ?2 = —k, j^. 



kit,i^=-Ji = 1 1 k^. k.j, «2 =J2 = *2 ^V 



«j «2 = = ^^2 "l* 



in which J^ denotes any combination of the vids of the first algebra, and 

 iV^ any combination of those of the second algebra. It may perhaps be 

 claimed that these algebras are not independent, because the sum of the 

 vids «^ and a^ is absolute unity. This, however, should be regarded 

 as a fact of interpretation which is not apparent in the defining equa- 

 tions of the algebras. 



