632 



PROCEEDINGS OF THE .AMERICAN ACADEMY. 



and thus each of the C's is a solution of equations (6). Therefore, 

 each of the C's is inexpressible linearly in terms of Bi, B2 . . . B^'- 

 Let 



(10) Bj = 6,1^1 + bj,e2 + . . . + bjmCm U = 1, 2, . . . m'). 



We may take the b's to be rational functions with respect to the 

 domain R (1) of the constituents of Ai (or of A2) which are integral 

 quadratic functions, rational with respect to R (1), of the constants 

 of multiplication of the number system {ei, eo. . .e^)- If this number 

 system belongs to the domain R', that is, if its constants of multiplica- 

 tion lie in the domain R', the b's may be so chosen as to lie in this do- 

 main. We may take the B's as m' new units of the number system. 

 Thus let 



(11) 



e'm r^'^i = Bi 



(i- 1,2, ... mO, 



and let e'l, e'l . . . e'^-m' be any m-m' numbers of {ey, e^ . . . Cm) which 

 constitute with the .B's a set of 7?i linearly independent numbers. By 

 what has just been said the coefficients of the transformation 



(12) e'i = Tiiei + T(2e-2+ ... + Tf^<'m (i = 1,2, ... m) 



of the number system can be taken rational in any domain to which 

 the number system belongs. 



If the number system is transformed by the preceding substitution 

 (12), and if we put 



(13) 



A'i = 



Si e'i e'j 



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

 then, since 



OT m 



Sie'ic'j = L L TihTjkSiChek (r, ; = 1, 2, . . . m) 



h = l k=l 



we have 



(14) A'l = rAi, 



where T is the determinant of the substitution. Similarly, if 



(15) A'o = 



we have 

 (16) 



Therefore, the equations Ai = 0, Ao = are invariant to any trans- 

 formation of the units of the svstem. 



