660 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



of this system, we have 



or 



SiAe-i = SiBei (i =1,2,... m), 

 'SoAci = SoBei {I = 1,2, ... m), 



then A = B. If the nullity of Vi is vi' (0 < m' = m), in which case 

 Vi = 0, the nullity of V2 is m' , and conversely; and the system 

 {ei, Co, ... Cjn) then contains a maximum invariant nilpotent sub system 

 constituted by any m' linearly independent numbers of {ci, e-i, ... e,^ 

 satisfying equations (a), or equations (/?), every solution of equations (a) 

 bei7ig a solution of equations (/?), and conversely. 



§ 4. 



Let {ei, ei, ... e^^) be any given number system; let e,;i„ for 

 u, V = 1, 2, ... n, constitute a quadrate of which (f], Co, ... r,„) is a 

 sub system; and let 



(81) 



Ci= L L 9uv^'^^r> 



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



u=l v=l 



The units of the system (^i, eo, ... Cm) are then represented, respec- 

 tively, or may be identified, respectively, with the m linearly inde- 

 pendent matrices defined by equations (63). 



The number system (e'l, e'2, ■ ■ ■ e'^) reciprocal to (fi, e-i, ... e,^ 

 is then also a sub system of the quadrate: that is, 



(82) 



e'i = Y^ Y. Vuv^'huv 



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



u—l v=l 



for a proper choice of the 77's. For the m numbers c'l, e'o, . . . e'„, of 

 the quadrate defined by equations (82) may be identified, respectively, 

 with the VI matrices E'l, E'o, . . ■ E'^, where E'i, for 1 ^ / = m, is 

 defined by the equations 



(83) E'i (ti, ^o, . . . ^„) = ( rjn^'\ rj,^'), . . . rj,n^') l^„ ^„ . . . t,,) ; 



Vni^'K Vn^^'K ■ ■ ■ Vnn^'^ 



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



