DOC. 123 THEORY OF AFFINE FIELD 201 [5] SEPTEMBER 22, 1923] NATURE 449 itself, for the determination of the 40 unknown functions The following brief statement will serve to show how I have endeavoured to fill in this gap3 If the German capital § be a scalar density thatIn depends only on the functions )), then Hamilton’s principle *{ƒ§*}= o . . . . (5) supplies us with 40 differential equations for the functions , when we stipulate that during the variation the functions are to be treated as magnitudes in- dependent of each other. Further we assume that §the depends only on the magnitudes yF„ and and thus write 8§=G‘“%"+f'“'s4V" • • • (6) where we have 36 £mr\i 3§ --H_r= • (7) At this point it should be noticed that in the theory developed here, the small German letters respectivelypossibility represent the contravariant density (gt»") of the metrical tensor, and the contravariant tensor density (ff“) of the electromagnetic field. Thus in a well-known manner is given the transition from tensor densities (expressed by German letters) to contravariant and covariant tensors (expressed by the corresponding italic letters),logically and a metric is introduced which rests exclusively on the affine relation. By performing the variation we obtain after some amount of calculation where ~ è&yî“ + + 48%, (8) • (9) Equation (8) shows that our extension of the theory, which appears to be so general, leads to a structure of the affine relation that does not deviate more stronglyIn from that of the geometry of Riemann than is required by the actual structure of the physical field. We now obtain the field equations in the following manner. From (3) and (4) we first derive the relations 3 Herr Droste of Leyden hit upon the same idea independently present writer. . (11) these equations the *, on the right-hand side are to be expressed by means of (8) in terms of the ge" and fe". Moreover, if § is known, then on the basis of (7) yh„ and ýF„, i.e. the left-hand sides of (10) and (11), can also be expressed in terms of ge" and fe". This latter calculation can be simplified by means of the following artifice. Equation (6) is equivalent to statement that 8¡f?« = 7ev8ge"-Hre"Sfe" . . . (6a) is also a complete differential, so that if is an unknown function of the ge" and fe", the following relations will hold : (7a) = 3fe" We now have only to assume jp*. is obviously The simplest • (12) In this connexion it is interesting that this function does not consist of several summation terms which are independent of each other, as was the case with the theories hitherto proposed. In this way we arrive at the field equations A"-f^fiï + y/Ar] (13) whereby i?M„ is the Riemann tensor of curvature. K and y are constants,/M is the electromagnetic potential, which is connected with the field strength by the relation (14) f JthJh. dx, 3XM and with the electrical current density by the relation V*= -yWh .... (15) order that these equations may be in accord with experience, the constant Y must be practically in- definitely small, for otherwise no fields would be possible without noticeable electrical densities. The theory supplies us, in a natural manner, with the hitherto known laws of the gravitational field and of the electromagnetic field, as well as with a connexion as regards their nature of the two kinds of field but it brings us no enlightenment on the structure of electrons.theof Further Determinations of the Constitution of the Elements by the Method of Accelerated Anode Rays.1 By Dr. F. W.ASTON, F.R.S. BY further use of the method of accelerated anode rays, results have been obtained with a number of elements since the publication of the isotopes of copper (NATURE, Aug. 4, p. 162). Details of the 1 A paper read on September 18 before Section A of the British Association Meeting at Liverpool. NO. 2812, VOL. 112] method will be published later. Most of the following results were obtained by the use of fluorine compounds of the elements investigated. The mass-spectrum of strontium shows one line only, at 88. This was obtained in considerable intensity. If any other constituents exist they must be present