Transactions of the American Mathematical Society Volume 7 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1906 Excerpt: ... of I) and hence is not zero: this requires that the classes (' and C shall contain the same number of variables. Thus pl must be a root of A(/j ) = 0 of the same multiplicity as the root p. A first necessary condition for an invariant t under S' is: (11) The characteristic equation of S must be a reciprocal equation. A second condition for the existence of CrC is (jordan, 13): (12) Classes Cx. C mvst be of like type as to number and length of series. 12. When these conditions on S are satisfied, invariants (7.C of nonvanishing discriminant exist (jordan, 12, case I alone occurs for bilinear functions), the general one being . Sr-"8, /a8ri where f is a definite bilinear function and the -'s are any polynomials in p, p, p _, for which the discriminant of C C 1 is not zero and the following; " reality" conditions hold: Since t is to be equal to a function of the initial variables with coefficients in F, we must have 1 = Y. +, where E-c, + c;c; +... + C;-ic;--) C'rC, ', being derived from C/7, by interchanging p and pn, p and pt. j, respectively, /, pt, _t, being the remaining roots of the same irreducible factor of A(p). Thus a"P must equal a polynomial in p. Let these conditions be satisfied. Then (jordan, 14-16) by a linear transformation leaving the canonical form of S unaltered (not necessarily the same transformation on the //'s as on the r's), we can reduce $ to a unique canonical form. An obvious correction (not altering the argument) is to be made on p. 237, 1. 11. The expression in brackets should be.v; (; + c, Li l_i-- +;)+-' ' (" + ci_t /;, _, + + /; )+-. 13. Conditions (11) and (12) depend upon quantities irrational in general with respect...