of having identical spectra in all particulars between 1000 Å and 235 Å, while showing altogether distinct characteristic spectra between 1700 Å and 6000 Å, a behavior which could only mean that they themselves had no lines in the region, between 235 Å and 1000 Å, all of the observed lines being those of the common impurity oxygen—an interpretation supported by the fact that the well known oxygen lines of the visible region were also found in both spectra. It was with the aid of this discovery and the careful comparison of the spectra of otherwise pure metals, or of metals containing known impurities that the origins of practically all of the hundreds, even thousands, of new lines have been established with but very little uncertainty. 6 b. The spectrum due to the 4 L-ring electrons of the carbon atom begins upon our plates at 360.5 Å and extends with much complexity and strength up to 1335.0 Å where its strongest line, which, in harmony with the foregoing convention will be called its La line is found. Above 1335.0 Å, the carbon lines are widely scattered and relatively weak. Since the K line of carbon is accurately computed to be at 44.4 Å, the ratio of the K to the L frequency in carbon is about 30. It is interesting that practically the whole group of lines which Lyman obtained from his condensed discharge in helium' are due to carbon, oxygen, nitrogen and hydrogen. Thus those whose wave-lengths he gives as 599.0, 643.7, 702.9, 718.2, 796.8 and 834.1 are all due to oxygen; those whose wave-lengths he gives as 904.6, 977.2, 1010.6, 1037.0, 1175.9 and 1247.9 are due to carbon; those whose wave-lengths he gives as 916.7, 991.1 and 1085.5 are due to nitrogen; while those whose wave-lengths he gives as 972.7, 1026.0, 1216.0, are due to hydrogen. να C. The spectrum due to the 5 L-ring electrons of the nitrogen atom (atomic number 7) was obtained by using Al electrodes with ammonium nitrate in their cores and observing the new lines which were not due to aluminium, oxygen or hydrogen. Before this experiment was tried, it was predicted that the La line of nitrogen would have to lie between the La line of oxygen and that of carbon. It was found that the spectrum due to the nitrogen atom was very simple, beginning on our plates on the short wave-length side at 685.6 Å and reaching a maximum in the line of wave-length 1085.3 Å, precisely as predicted. This 1085.3 line is taken, in accordance with the foregoing convention, as the La line of the nitrogen atom. The only other strong lines which we have obtained which are due to this atom have the following wave-lengths: 685.6, 916.2, 991.1. The pair of nitrogen lines found by Lyman at 1492.8 and 1494.8 appear upon our plates, but they are very faint in comparison with the foregoing lines, the nitrogen spectrum thus showing a behavior quite like that due to the atoms of oxygen and carbon. Since the Ka line of nitrogen is at 31.2 Å, the ration of the K, to the La frequency is 34.8. α α d. In the case of fluorine a strong line has been found through the use of sodium fluoride at 657.2 Å. This is in about the position predicted by the foregoing mode of approach for its La line. Another strong fluorine line appears at 607.2 Å. These are the only lines thus far definitely α α identifiable as coming from the 7 L- ing electrons of the fluorine atom. The longer of these wave-lengths is taken as the La line of fluorine. Fluorine probably has, however, other lines of shorter wave-lengths but of such intensity that we have not yet been able to obtain them. α α α e. The spectrum, due to the 3 L-ring electrons of boron (atomic number 5) is especially interesting because of its simplicity. It contains less than 10 strong lines all told. It begins on the short wave-length side at 676.8 Å and has only the following strong lines: 676.8, 760.0, 1624.4, the two doublets at 2164.2, 2166.2 and 2496.9, 2497.8 and the single spark-line near the visible at 3451.5. According to the foregoing convention, its La line should be the doublet at 2497 Å. Since the Ka line of boron is at 67.2 Å, the ratio of the K to the La frequency in boron is about 37. f. The spectrum due to the 2 L-ring electrons of beryllium begins on the short wave-length side, according to all the data available up to date, at 2175 Å and reaches its maximum, its La doublet, at 3130.6 and 3131.2. The entire absence upon our plates of any lines whatever due to beryllium between 230 Å and 2100 Å is a conspicuous illustration of the wide gaps in spectra obtained with ordinary gratings in or near the visible region. g. Similiarly the present experiments with lithium (atomic number 3) reveal no lines whatever between the shortest wave-lengths measurable on the plates used in the case of lithium and the familiar series due to its single L-ring electron whose La line is at 6708 Å and whose convergence wave-length is at 2299 Å. α 5. The graph of the La lines of the elements from lithium up is shown in the accompanying figure, along with the corresponding graph for the Klines from helium up taken from the work of others. The only point on the La graph which has not been directly observed is that corresponding to neon which has been inserted from the resonance potential of that gas. It is to be observed that the method here employed gives, as I think, the characteristic spectra of the atoms of each element, not of the molecules. If it were possible to work with the atomic gas of each element, the L lines given herewith would be the resonance potentials of these atoms. a The progression thus revealed in these optical spectra is exceedingly interesting and simple, and very like that exhibited by X-ray spectra. The reason it has not been observed before is clearly because hitherto only the upper ends of these optical spectra have been observable, so that the unfolding of simple relationships between spectra and atomic number had to await the development of an ultra-violet technique. α A glance at the figure shows that we now have the complete outline of all of the types of radiations which are emitted by atoms of small atomic number. It remains only to fill in the details of their K. and La spectra. One interesting fact which appears from a glance at the whole series of spectra of elements from hydrogen to neon is that atoms of odd ordinal all appear to have simple, few-lined spectra, while those of even ordinal number have much more complex spectra. 1 Physic. Rev., 10, 1917 (205). 2 Astroph. J. 52, 1920 (47). A detailed paper giving the full spectra of the light atoms with photographs will soon be published in the Astrophysical Journal in collaboration with Mr. I. S. Bowen. 4 Zeit. Physik, 1, 1920 (439). 5 Ibid., 2, 1920 (470). 6 This convention is more logical than that used in a former paper (cit.) in naming line. the La 7 Astroph. J., 43, 1916 (102). THE AVERAGE OF AN ANALYTIC FUNCTIONAL AND THE BROWNIAN MOVEMENT BY NORBERT WIENER Department of MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY The simplest example of an average is the arithmetical mean. The arithmetical mean of a number of quantities is their sum, divided by their number. If a result is due to a number of causes whose contribution to the result is simply additive, then the result will remain unchanged if for each of these causes is substituted their mean. Now, the causes contributing to an effect may be infinite in number and in this case the ordinary definition of the mean breaks down. In this case some sort of measure may be used to replace number, integration to replace summation, and the notion of mean reappears in a generalized form. For instance, the distance form one end of a rod to its center of gravity is the mean of its length with reference to its mass, and may be written in the form fldm÷fdm where I stands for length and m for mass. It is to be noted that l is a function of m, and that the mean we are defining is the mean of a function. Furthermore, the quantity, here the mass, in terms of which the mean is taken, is a necessary part of its definition. We must assume, that is, a normal distribution of some quantity to begin with, in this case of mass. The mean just discussed is not confined to functions of one variable; it admits of an obvious generalization to functions of several variables. Now, there is a very important generalization of the notion of a function of several variables: the function of a line. For example, the attraction of a charged wire on a unit charge in a given position depends on its shape. The length and area of a curve between two given ordinates depend on its shape. As a curve is essentially a function, these functions of lines may be regarded as functions of functions, and as such are known as functionals. Since a function is determined when its value is known for all arguments, a functional depends on an infinity of numerical determi nations, and may hence be regarded as in some wise a function of infinitely many variables. To determine the average value of a functional, then seems a reasonable problem, provided that we have some convention as to what constitutes a normal distribution of the functions that form its arguments. Two essentially different discussions have been given of this matter: one, by Gâteaux, being a direct generalization of the ordinary mean in n-space;1 the other, by the author of this paper, involving considerations from the theory of probabilities. The author assumes that the functions ƒ(t) that form the arguments of his functionals have as their arguments the time, and that in any interval of the small length as many receive increments of value as decrements of equal size. He also assumes that the likelihood that a particle receive a given increment or decrement is independent of its entire antecedent history. When a particle is acted on by the Brownian movement, it is in a motion due to the impacts of the molecules of the fluid in which it is suspended. While the retardation a particle receives when moving in a fluid is of course due to the action of the individual particles of the fluid, it seems natural to treat the Brownian movement, in a first approximation, as an effect due to two distinguishable causes: (1) a series of impacts received by a particle, dependent only on the time during which the particle is exposed to collisions; (2) a damping effect, dependent on the velocity of the particle. If we consider one component of the total impulse received by a particle under heading (1), we see that it may be considered as a function of the time, and that it will have the sort of distribution which will make our theory of the average of a functional applicable. It will result directly from the previous paper of the author that if f(t) is the total impulse received by a particle in a given direction when the unit of time is so chosen that the probability that f(t) lie between a and b is A + Saf (t) G (t)dt + SaSaƒ (s) f (t) H (s,t) ds dt__[H(s,t) = H(t,s)] will be A+ SoS &t H (s,t) ds dt. (1) We now proceed to a more precise and detailed treatment of the question. 2. Einstein has given as the formula for the mean square displacement in a given direction of a spherical particle of radius r in a medium of viscosity ʼn over a time t, under the action of the Brownian movement, the formula |