TABER. — SCAI-AK ir\(TIt)\s oi ini'KK COMI'LKX NUMBERS. 659 



in which case, for any number 



m m n 



»=1 »=1 u = l 



of the system {ei, e.o, ... Cm), toe have 



^ m n n 



ii>lA = Z Z Z aiGiuJun 

 ''^ t=l u=l »=! 



1 



m n n 



S2A = X Z Z (liOiuJ tut. 



II 



i=l u=l »=! 



Finally, let X = J] •'^«^'» "^^^ ^^^ 



Vi = 



V2 = 



S-ieiBj 



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



SiCiCj I, 



(ij, = 1,2, ... m) \ 



be, respectively, the resultants of the systems of equations 



(a) SiXei = xiSiCiet + X2Sie2ei + . . . + XmSiemet = 



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



(/S) S^Xci = XiSieoCi + xoSoe-iCi + . . . + XmSiemet = 



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



Then, if the number system {ci, c-2, ■ . . Cm) is transformed by the substi- 

 tution 



e'% = Til ei + r,-2 ^2 + • . . + Tim em 

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

 and if 



V'l = Sis'ie'j 



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



, V'2 = Sie'ie'j 



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



ice have 



Vi'= PVi, V'2= PV2, 



jr^erf r w /^^ determinant of the substitidion. If Vi 9^ 0, in which case 

 V2 5^ 0, a/ic/ conversely, the system {ci, ci, ... e„) contains a modulus, 

 hit no invariant nilpotent sub system; and in this case, if for any two 

 numbers 



^ = Z ««'^»» 

 »=i 



B= Z hiCi 

 »=i 



