628 PROCEEDINGS OF THE AMERICAN ACADEMY. 



invariant nilpotent sub system of (ci, c-i,... c^, and a metlaod of 

 determining the maximum invariant nilpotent sub system, if any 

 exist.^ These results are embodied in theorem II. 



Theorem I. Let 7^, for i, j, Jc = 1, 2, . . .m, be the constants of 

 multiplication of ariy given hyjjer com 2>Iex number system (ci, e^, . . . e^). 

 Let 



A = aici + a2C2 + . . . + amC„ 



be any number of the system; and let 



■.mm 



SiA = - J^ X «J7t;j, 

 ^ t=l i=l 



■, m m. 



1=1 y=i 



Then both Si A and So A are invariant to any linear transformation of 

 the system: that is, if 



e'i = T,ifi + TaCo + . . . + TimCm a = 1, 2, . . . m), 



the determinant of transformation not being zero, and if 



m 



^'i^'i = Z T'y'fc^'fc (ii = 1. 2, ... m), 



k=l 



and 



A = Old + a-(^2 + • • • + fim<'m = «'it''i -!- a'oe'o + . . . + a'^e'^n, 

 then 



■^ m m 



SiA = - Y. Z (I'lYnj. 



-.mm. 



S2A = — Y. Z «Wjy- 

 m ■ 1 1 

 1=1 j=i 



2 A sub system Bi, B2, . ■ ■ Bp of any hyper complex number system 

 (ei, 62, ... Cm) is said to be invariant if the product in either order of each 

 number of (ci, e-i, . . Cm) and each mmiber of {B\, By, ■ . ■ Bp) belongs to the 

 sub system, for which the necessary and sufficient conditions are 



eiBj = g'lijBi + g'njBi + . . . + g'jnjBp, 



BjCi = g'lijB). + g">ijBi + . . . + g"pijBp 

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



An invariant sub system {Bi, B^, ... Bf^ is an invariant nilpotent sub system 

 if its units by themselves constitute a nilpotent system; and in that case 

 is a maximum invariant nilpotent sub .system if it contains every invariant 

 nilpotent sub system of (ei, co, ... em)- 



