Conversely, if = a;e; 0 and 4X 0 for every number X of A Σae; AX = (e1, e2, i=1 .em), equations (89) are satisfied for at least one system of values a1, a2, am not all zero, and we cannot assign to the 's, nor to the n's, the values given by equations (88). In this case, we have ... It is to be noted that equations (89) are the conditions, necessary and sufficient, that the reciprocal system shall contain a number in which case E1, E2, ... Em are not linearly independent, nor are E'1, E'2, E'm linearly independent.24 In this case, there is some B' = Σ b;e'; # 0 i=1 m Exe; of this system; and there is also a number i=1 of the reciprocal system such that B'X' = 0 for every number X' of the reciprocal system. Conversely, if there is for every number X of this system (or if, for B' for any number X' of the reciprocal system, we have B' X' = 0) equations (91) are satisfied for some system of values b1, b2, .bm not all zero, and we cannot assign to the 's, nor to the n's, the values given by equations (90). When equations (91) are satisfied, to satisfy equations (89) nor equations (91). We may distinguish three cases. First, the given number system em) may contain both a number A m Σae; 0 and (e1, e2, = number I = Σe of the system, in which case the system does i=1 = not contain a modulus and A1 A2 0. In this case it is not possible to assign to the 's the values given by either equations (88) or (90), nor to assign to the 7's the values given by either of these equations. Nevertheless, it may be possible in this case to put n = m, provided m> 2, but not otherwise. Thus let m = 3, and let On the other hand, let m = 2 and let (c1, e2) contain a number A # 0 such that In this case, we may, without loss of generality, put A =e, when we have where k 0 and the determinant of the matrix is not zero; and, therefore, since e1e2 = 0, = where, without loss of generality, we may put a 1, 80, giving = This system, however, contains no number B0 for which Second, the number system (e1, e2, ... em) may contain either a number A0 such that A¤¡ B0 such that e; B = 0 for i = 0 for i = 1, 2, ... m, or a number m, but not both. In this case, we may put n = m and assign to the 0's and 7's either the values given by equations (90) or equations (88) respectively. 0 for i = Third, the system (e1, e2, ... em) may contain neither a number 1, 2, ... m nor a number B 0 1, 2, ... m, for which a sufficient, but not. A0 such that Ae; such that e, B = 0 for i = necessary condition, is the existence of a modulus, and, a fortiori, that A10. In this case, we may put n = m and assign to the 's the values given by equations (88), and to the n's the values given by equations (90). We then have On the other hand, if we assign to the e's the values given by (90) and to the n's the values given by (88), which is now possible, we shall have When either the representation of the number system (e e2, (m) and its reciprocal system given by equations (88) or by equations (90) fails, and indeed in any case, we may proceed as follows. Let n = m+1, and let Our (i) (96 a) (96 b) 0m+1,m+1) = 0 Om+1, (i, u Oi,m+1 (1) = 1, 2, ... m; u#i); The m matrices E1, E2, 1, 2, ... m; u ‡ i). Em which we thus obtain have the same multiplication table as the units of the system (e1, 2, ... em) and are, moreover, linearly independent. For, if Further, the m matrices determined by the above values of the n's are also linearly independent and have the same multiplication table as the system (e'1, e'2, ... e'm) reciprocal to (e1, e2, ... em). We now have (98) A = Σ αi li = Σα; Σ Σ yivu eur + eim+1), u=1 v=1 m = A' are= and, therefore, i=1 Σai (Σ Σ Yviu €uv + €i,m+1); We may also proceed as follows. Let n = m + 1, and let |