PHYSICS: W. DUANE 
Proc. N. A. S- 
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/ w (#) instead of a broken straight line is any broken line with 
corners at (0, 0) and at (x {K) J(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 Gateaux (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 t u <h< . . . < t„. 
4 Cf. V. Volterra, Fonctions des Lignes. 
5 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 
Communicated July 23, 1921 
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 com- 
putations the electrons were supposed to revolve in orbits which lay in 
planes passing through the nucleus of the atom. 
Later I presented 2 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- 
