and is a function of limited variation in x1, ..., X. What we wish to prove is that n If in this latter expression the total variation of , is less than a quantity independent of x, we can permute the and the lim, and get In this we suppose ƒ uniformly continuous. It is easy to show that on (x1,...,x,, x)du is of limited variation. Conse our assumptions quently we obtain A further transformation just like the preceding turns this into so that we have now a sufficient condition for the validity of our theorem. The extension to non-homogeneous terminating analytic functionals is obvious. The extension to non-terminating analytic functionals may be deduced with the help of (3) and a well-known theorem on the integration of uniformly convergent series, and reads as follows: let F. be an analytic functional of the form where the total variation of each is less than some quantity independent of x, and let each yn be uniformly continuous in x over the interval (a, b). Let u(x) be a function of limited total variation in x over the same interval. Let A{F,du} exist, and let As to (5), let us begin as above with a functional of the form where A is taken in the original sense as an n-fold integral. By definition xXn, I = lim A{ƒ(§,a) ... f(Eno) } *, a + 1 α, ...! where xko =0,x1,...,xku-1 is an increasing sequence of numbers, Ex lies between x and xk,K+1 and lim is taken as max(x,K+1-**K) approaches 0. Let V be the total variation of as its arguments range from 0 to 1, and let M stand for max(xk,K+1-XkK). Let Q stand for the least upper bound of the variation of A{ƒ(x1)... f(x,)} as the point (x1,...,x«) wanders over an interval (*+*+1). Then Now, let fm(x) be that function whose graph is the broken line with corners at (x(K), f(x(x)), where 0≤K≤μ, x(o)=0, x()=1. Then if lies between x(k) and x(x+1), fm(x) is of the form afm(x(K))+bfm(x(x))÷a+b. It follows that if (...) lies in the interval ((a+1). (8 + 1) ;* A {fm(§1a)...ƒm(Eno)} is of the form +ap *(a)...*(0) where each C is the value of some A{ƒ(a)...f(§,s)} such that (...) is a corner of the hyperparallelopiped ((a+1)(+1)). It readily results from considerations of continuity that A{ƒm (§1)...ƒm()} is of the form A {f(n,a)...ƒ(ʼnns)}, where (a, ..., nn) also lies in the interval ((a+1)...(+ \x (a)...x (0) +1)). is an increasing functional of p(x1,...,xn), we can draw the conclusion that if it exists, lies between the uppermost and lowermost values of This proves our theorem for homogeneous analytic functionals. In precise terms, then, our general theorem will read: let be an analytic functional. Let Vn stand for the total variation of Fn as its arguments range from o to 1: Let (x(o),..., x()) be a set of numbers in ascending order from 0 to 1, inclusive. Let M stand for max (x(x+1)−X(K)) and Qn for the upper bound of the variation of A{f(x1)... f(x)} as the point (x1,...,x) wanders over the interval (x+1). (0+1)). 1)*(0+1)). Let Let fu(x) ... X(a) ... X(9) be the function whose graph is the broken line with corners at (0, 0) and (x(K), f(x(K)). Then if (b) A{Fn{f}} exists for every μ and n according to the definition lim A {F{fm}}, where the first A is defined in the sense of the average of an analytic functional, and the second as a multiple integral. A precisely analogous theorem holds when f(x) instead of a broken straight line is any broken line with corners at (0,0) and at (x(K), f(x(K))), and consists of monotone arcs between these points. This last theorem makes our average of a functional the limit of the average of a function of a discrete set of variables, and justifies our use of the term average. 1 The problem of the mean of a functional has been attached by Gâteaux (Bull. Soc. Math. de France, 1919, pp. 47-70). The idea of using the analytic functional as a basis is there found. The actual definition, however, is essentially different, and does not lend itself readily to the treatment of the Brownian Movement, for which the present method is especially adapted. 2 Einstein, Leipzig, Annalen Physik, 17, 905. 3 We here take t1, <t< ... < In. 4 ♦ Cf. V. Volterra, Fonctions des Lignes. 'Cf. P. J. Daniell, Annals of Mathematics, Sept., 1919, p. 30. ON THE CALCULATION OF THE X-RAY ABSORPTION FREQUENCIES OF THE CHEMICAL ELEMENTS BY WILLIAM DUANE JEFFERSON PHYSICAL LABORATORY, HARVARD UNIVERSITY The K critical absorption frequency of a chemical element is the highest frequency of vibration known to be characteristic of that element. In our laboratory we have measured the K critical absorption frequencies of most of the chemical elements by the ionization method. This data, together with measurements made elsewhere by the photographic method, may be found in Table 2 of a report by the author on Data Relating to X-Ray Spectra, which has been published by the National Research Council. At a symposium on Ultra-Violet Light and X-Rays, held at the meeting of the American Association for the Advancement of Science at St. Louis in December 1919,1 I presented a set of computations of the K critical absorption frequencies based on the Rutherford-Bohr theory of atomic structure and the mechanism of radiation. The computed values equalled the observed values to within one or two per cent. In these computations the electrons were supposed to revolve in orbits which lay in planes passing through the nucleus of the atom. Later I presented2 to the National Academy of Sciences and to the American Physical Society computations of these K critical absorption frequencies, calculated on the assumption that the orbits did not all lie in planes through the nucleus. I assumed that the orbits were cir cular, that they occurred in pairs and that the two orbits in a pair lay opposite to each other in parallel planes equi-distant from the nucleus, as represented in the figure. This gives a volume distribution of electrons. The mutual repulsion of the electrons for each other keep the orbits apart. It is necessary, however, to suppose that the electrons in the two orbits of a pair revolve in opposite directions. Otherwise they would be pulled together to form a single orbit in a plane through the nucleus (at least for elements of high atomic numbers). The revolution in opposite directions has the advantage, among others, of reducing the magnetic field due to the electrons for points at a distance from the atom to a very small value. The theory contains three fundamental laws. The acceleration law, the angular momentum law and the frequency law. The acceleration law states that the centripetal acceleration of each electron revolving in its orbit equals the centripetal force acting on it, due to the attraction and repulsion of all the electrical charges in the atom acting according to Coulomb's inverse square law. The angular momentum law states that the angular momentum of each electron equals a whole number (called the quantum number), 7, multiplied by Planck's action constant, h, and divided by 2. According to the frequency law, the product of h into the frequency of vibration, v, of the radiation emitted during a shift of the electrons from one position of dynamic equilibrium to another equals the difference in the amounts of energy in the atom before and after the shift. The first two laws cannot be true at every instant of time. One or both of them must represent average values. In the modern development of the theory a definite integral of certain generalized coördinates is equated to a multiple of h. The theory does not determine the numbers of electrons in the various orbits. In making calculations, however, we must know how the electrons are distributed. Several authors have calculated X-ray frequencies by choosing distributions of electrons in the orbits that best fit the X-ray data themselves. I have taken a distribution suggested by the intervals between the inert gases in the sequence of chemical elements. It has long been supposed that these intervals correspond to groups of electrons in the atom that are completely filled up. From this point of view we get as the numbers of electrons in the various groups the following: the inner orbit contains two electrons. The next group consists of a pair of parallel orbits containing in all eight electrons, four in each orbit. The third group contains eight electrons, four in each of the two parallel orbits. The next group contains eighteen electrons in all, nine in each of the two parallel orbits. The fifth group also contains eighteen, nine in each orbit. The outside pair of parallel orbits contains thirty-two |