to be published] will be improved further by obtaining bounds for the components of the characteristic vectors. Let A = (a K \) be an arbitrary matrix, co a characteristic root, 36 =» (xi, xi, • • • , x n ) a characteristic vector belonging to co, and x r the absolute greatest component. Then co lies in the circle C r with center at a rr and radius P r = ]Cx^n | a r \ | • Let R be a closed subregion of C r containing w and d K be the minimum distance of a KK from R. Set /« -l if d K~0 , and

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... rwise ^ = min (1, -PK/^K) for K = 1, 2, • • • , n; K7*r. It is shown that \x K \ ^U^Xr], and this result is improved by iteration. It follows that co lies in the circle \z-a rr \ ^ X^n |flrX| A-x contained in C r . Moreover, if the inequalities for the x K contradict the rth. of the corresponding linear equations, then co does not lie in R. If A is non-negative, co the greatest positive root, and x m the smallest component, then lower bounds for x K /x m are obtained. Using these results it is often easy to compute co as exactly as wanted. (Received October 13, 1954.) 73. C. C. Buck: The algebraic aspect of integration in space. By "integration in space" is meant the w-dimensional analogue of the notions of integration along a curve, integration over a surface, etc. The term "algebraic" indicates that the discussion is restricted to topics that can be studied without the use of a limit process. An algebraic definition for space integrals has been improvised from a 57 58 AMERICAN MATHEMATICAL SOCIETY [January set of postulates for the notion of integral which were stated by Lebesgue in 1904.