Hilbert’s double series theorem and principal latent roots of the resulting matrix

The inequality i "=1 j + n -S ir ap2, was proved by Hilbert and published by Weyl.1 Various proofs were given by Hardy, Littlewood & Plya.2 In this inequality tv is the best possible constant; that is, the maximum value of Y, JL-^-^-/ Z. av ror arbitrary {ap\ is 7r. It is no m " m + n -1/ p longer the best possible sum when the summation is finite; from 1 to N, say. In this case Frazer 3 has shown that (N + 1) sin [_ir/(N + 1)] is better. But this result is not the best possible, and Copsey, Frazer, & Sawyer have published investigations4 based on empirical values of the constant X for N = 1(1)5,10,20, computed by the Royal Aircraft Establishment. Further computations for N = 2(1)20 are being made by the National Physical Laboratory.