September 22, 1923] 



NATURE 



449 



itself, for the determination of the 40 unknown functions 

 r^" . The following brief statement will serve to show 

 how I have endeavoured to fill in this gap.^ 



If the German capital ,S^ be a scalar density that 

 depends only on the functions V^^, then Hamilton's 

 principle 



S{/§jTi=o .... (5) 



supplies us with 40 differential equations for the 

 functions F, when we stipulate that during the variation 

 the functions V are to be treated as magnitudes in- 

 dependent of each other. Further we assume that § 

 depends only on the magnitudes 7^^ and ^^^, and 

 thus write 



where we have 



'=9''''«rM^+f'"'^s 



0^ 





(6) 



(7) 



At this point it should be noticed that in the theory- 

 developed here, the small German letters respectively 

 represent the contra variant density {o}^") of the metrical 

 tensor, and the contravariant tensor density iY") 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), 

 and a metric is introduced which rests exclusively on 

 the affine relation. 



By performing the variation we obtain after some 

 amount of calculation 



where 



cfx„ 



= IM 



(9) 



Equation (8) shows that ojr extension of the theory, 

 which appears to be so general, leads to a structure of 

 the affine relation that does not deviate more strongly 

 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 



yM.= -'£'^+r;, r/,+i(?^I^+^i^;«)-r;,r<J (10) 



cx„ 



dXu ?x.. 



• Herr Droste of Leyden hit upon the same idea independently of the 

 present writer. 



,pa -pa 



(") 



In these equations the T^^ on the right-hand side are 

 to be expressed by means of (8) in terms of the 

 i]!^" and f*^^. Moreover, if ^ is known, then on the 

 basis of (7) 7^„ and (f)^^, i.e. the left-hand sides of (10) 

 and (11), can also be expressed in terms of g'^" and 

 f**". This latter calculation can be simplified by means 

 of the following artifice. Equation (6) is equivalent to 

 the statement that 



^§* = yM.Sr" + '/'M^sf'''' 



(6a) 



is also a complete differential, so that if ^* is an 

 unknown function of the g*^" and f*^", the following 

 relations will hold : 



of'*"! 



4>(iv = 



iia) 



We now have only to assume §*. The simplest 

 possibility is obviously 



)*=-('/..?'" 



(12) 



In this connexion it is. interesting that this function 

 does not consist of several summation terms which are 

 logically independent of each other, as was the case 

 with the theories hitherto proposed. 



In this way we arrive at the field equations 



^M^=-<teJ<../<^^ -/../:) +7/m/.] . (13) 



whereby R^^ is the Riemann tensor of curvature, k 

 and y are constants, /^^ is the electromagnetic potential, 

 which is connected with the field strength by the 

 relation 





(14) 



and with the electrical current density by the relation 



iM=-ygMO-/^ .... (15) 



In order that these equations may be in accord with 

 experience, the constant 7 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. 



Further Determinations of the Constitution of the Elements by the Method 



of Accelerated Anode Rays.^ 



Hy Dr. F. W. 



jyy further use of the method of accelerated anode 



■L>' rays, results have been obtained with a number 



of elements since the publication of the isotopes of 



copper (Xature, Aug. 4, p. 162). DeUiis of the 



« A paper re.-id on September i8 before Section A of the BritUh Association 

 Meeting at Liverpool. 



NO. 



2812, VOL. 112] 



Aston, F.R.S. 



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 



