TABER. — SCALAR FUNCTIONS OF HYPKR COMPLEX NUMBERS. G31 



By theorem I e\ery solution of equations (8) is nilpotent. Let B' 

 be any solution of this system of equations. Precisely as above, 

 we may show that B' is nilpotent, and that both B'A and AB' 

 are also solutions of these equations for any numl)er .1 of the system 

 (ei, €■:, . . . Cm); and, therefore, both B'A and A B' are nilpotent. Since, 

 in particular, for 1 ^ i ^ m, B'ci is nilpotent, it follows from theorem I 

 that B' is a solution of the system of equations (G). 



Let now the nullity * of the determinant Ai be vi' , where 0<m'<m. 

 There is then a set of just vi' linearly independent numliers, 

 Bi, B^ ... Bn' of the system (ri, fo • • • ("„) satisfying equations (6) ; 

 therefore, just m' linearly independent numbers satisfying equations 

 (8), whence it follows that the nullity of A2 is m'. For l^j^ m', 

 the product of Bj in either order with any number A of the system is 

 a solution of equations (6) and, therefore, both BjA and ABj are 

 expressible linearly in terms of Bu Bo • • • Bm'', otherwise, there is a 

 set of more than in' linearly independent solutions of equations (6) 

 which is contrary to supposition. Moreover, since 



81 (pi Bi + P2 Bo + ... + Pm' Bm') Ci 



= piSiBiCi + piSiBoCi + . . . + Pm'SiBm'Ci = 



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



every number linear in the 7i's is a solution of equations (6), and is, 

 therefore, nilpotent. Whence it follows that Bu Bo. . Bm' constitute 

 an invariant nilpotent sub system of (o, Co. . .c„). 



Further, the sub system (Bi, B^ . . . Bm') contains every invariant 

 nilpotent sub system of {ex, e^ . . . Cj^, and is therefore the maximum 

 invariant nilpotent sub system of the latter. For, let (Ci, C2 . . . Cp) 

 be any invariant nilpotent sub system of (t'l, c-i . . . Cm)- Since every 

 number of this sub system is nilpotent, in particular, 



S,Cj = (j= [,2,...p). 

 Moreover, since 



CjCi = gjiiCi + fiTjiiC'o 4- • • • + OjipCp 



(i = 1,2, . .. in; j = 1, 2, . . . j)), 

 we have 



SiCjCi = (jjiiSiCi + (ijiiSiCo + . . . + OjipSiCp = 

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



4 The nullity of a matrix or determinant of order m is m' if every {la' — l)th 

 minor (minor of order ni — m' + 1) is zero but not every »i'th minor (minor 

 of order m — m^). Nullity of order m' is equivalent to rank (Rang) m — m' . 



