The m matrices E1, E2, ... Em which we thus obtain have the same multiplication table as the units of the system (e1, e2, ... 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', ... e'm) reciprocal to (e1, e2, ... em). We now have We may also proceed as follows. Let n = m + 1, and let (i, u, v = 1, 2, ... m), The m matrices E1, E2, ... Em thus obtained are linearly independent, as are also the m matrices E', E'2, ... E'm; and the former have the same multiplication table as the units of the system (e1, e2, ... em), while the latter have the same multiplication table as the units of the reciprocal system. We now have m m i=1 u=1 The fundamental properties of the scalar functions given in theorem I are more readily proved for the special case in which the number system is a quadrate than in the general case. What precedes in this section indicates how the properties of these functions may be made to depend upon the properties of the single scalar function of a quadrate. CLARK UNIVERSITY, WORCESTER, MASS. Proceedings of the American Academy of Arts and Sciences. VOL. XLVIII. No. 18.-APRIL, 1913. PRELIMINARY STUDY OF THE SALINITY OF SEAWATER IN THE BERMUDAS. BY KENNETH L. MARK. |
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