* Journal of the London Math. Soc. 10 (1935), 184Google Scholar. A brief account of my results was published previously in the Comptes Rendus de l'Académie des Sciences, Paris, 200 (1935), 107.Google Scholar

[Added in proof, 17th January 1936. Prof. Trefftz obtains in a recent paper (Zeits. für angewandte Math. 15 (1935), 339Google Scholar) nearly the same inequalities for the minimum in Problem 1. The method is slightly different and does not give the connection between the Problem 1 and the membrane problem, established in my papers. Prof. Trefftz states that he became aware of my results only while reading the proofs of his paper.]

* We assume the existence of the minimum. It is greater than the minimum in the “membrane problem”:

as can be seen from § 4 of my previous paper.

The direct methods of the calculus of variations would most probably prove the existence of the minimum in a way often used in other problems. It is even likely that the existence of continuous derivatives up to the second order would be sufficient. See for instance Friedrichs, K., Math. Annalen, 98 (1928), 205.CrossRefGoogle Scholar

* Bryan, , Proc. Lond. Math. Soc. (1), 22 (1891), 54.Google Scholar

† The corresponding result is true in the case of one dimension. The first characteristic value of the boundary problem of the buckling of a clamped bar

is 4 (the first characteristic function being 1 + cos 2x), i.e. is equal to the second characteristic value of the problem of the vibrating string

in the same interval.

‡ The upper limit for λ was obtained by the Ritz method. It is a slight improvement of a previous result of G. I. Taylor, who obtained the inequality λ < 5·33.… The object of my previous paper was to obtain a lower limit for λ.

§ The evenness of one of the minimizing functions would have been sufficient.

* It is well known that the evenness of the minimizing functions does not necessarily follow frbm the symmetry of the conditions in the problem. Consider, for instance, in the interval (− π/2, π/2) the one-dimensional problem

The minimizing function is the odd function f (x) = √(2/π) sin 2x. The evenness and uniqueness of the minimizing functions in our problem would be obvious, if, as in the case of the membrane problem, we knew that the minimizing functions do not vanish in the interior of S.

† In order to satisfy the condition (2) we have to multiply both terms by constants not equal to 0.

* Courant, R. and Hilbert, D., Methoden der mathematischen Physik, 1 (2nd ed. 1931), 353.Google Scholar

† The condition (14) is satisfied for a suitable choice of the constant C ₀.

‡ Zentralblatt für Bauverwaltung (1909), p. 93Google Scholar. See also Nadai, A., Elastische Platten (Berlin, 1925), p. 239.Google Scholar

§ Without loss of generality we can assume that the rectangle in this problem is S ⁺.