170 



NATURE, 



[April 7, 192 1 



in no way alter the apsidal progress determined by 



Einstein from (i). The "measuring rod," however, 



does not now alter in length for different orientations, 



.^j., which is a somewhat comforting result. 



1*^, , The gravitational potentials in (2) are not additive, 



ff^and Prof. Eddington ("Report on Relativity," p. 59) 



'^proposes to get over the difficulty by neglecting squares 



of mjri in (2), which would then give 



ds*= -|i+?^)fsri2 + ri28^« + 



r/sm'' 



+ 



O-f) 



if ■ (3) 



so that the contributions of potential would be 

 additive. 



Unfortunately, neglecting squares of m leads to a 

 change in the apsidal progress, and it appears that 

 treating (3) as exact gives 4/3 times the apsidal pro- 

 gress calculated from (i). We cannot, therefore, 

 neglect squares of tn at an early stage without vio- 

 lating the observations which (i) or (2) was called in 

 to explain. The adjustment is, in fact, so delicate 

 in (i) that we may not approximate at all until the 

 end of the calculation. 



What, then, are the exact equations for two finite 

 bodies m^ and m,, both mobile? Here we are not per- 

 mitted to superpose any velocitv which would reduce 

 one of the bodies to rest. (jEORge W. Walker. 



Portsmouth, March 30. 



I AM indebted to the Editor's courtesy for the oppor- 

 tunity to comment on the foregoing letter. In decid- 

 ing whether an approximation is allowable, regard 

 must be had to the problem to which it is to be 

 applied. It is true that equation (3) neglects a term 

 of importance in the motion of the apse, and is there- 

 fore not valid for the problem of the perihelion of 

 Mercury ; but there may be other problems for which 

 the approximation can "be justified. One of these is 

 the calculation of the Cm^ for continuous matter on 

 p. 59 of my "Report." My proof starts with the 

 approximate calculation of the line-element in a sphere 

 which is ultimately made infinitely small ; I think that 

 the justification of the neglect of m% given in § 36 

 is correct, though the argument is intricate, and I 

 would welcome detailed criticism. But, for example, 

 my formulae are not sufficiently accurate to give the 

 rotation of the apse-line of a particle moving freelv 

 through a diffuse spherical nebula. 



Dr.^ Walker goes on to ask : What are the exact 

 equations for two particles moving freely? He who 

 can supply the answer will have solved one of the 

 bifff^est mathematical problems of the theory. The 

 problem of two bodies in Einstein's theory is an out- 

 standing challenge, like the problem of three bodies in 

 Newton's theorv. The solution will give ds'' through- 

 out all time, and therefore incidentallv the tracks of 

 the particles which are the singularities of the solu- 

 tion. I am not satisfied that it has vet been proved 

 that the tracks are periodic — that there is no dissipa- 

 tion of energy by the gravitational waves set up. 



A. S. Eddington. 



Observatory, Cambridge, April 2. 



Atomic Structure. 



My letter in Nature of November 25 last has served 

 a useful purpose by evoking the verv interesting 

 account of his new line of work which Prof. Bohr has 

 given in the issue of March 24. But since he did not 

 deal, and scarcelv professed to deal, with my sug- 

 gestion, perhaps I mav trv to make clearer what that 

 suggestion 5s. 



NO. 2684, VOL. 107] 



The success of Prof. Bohr's theory, and of Sommer- 

 feld's developments of it, is sufficient evidence of the 

 truth of its general assumptions, and therefore of the 

 reality of the ideas on which it is based. " Fixed 

 electron " theories have nothing to set against (e.g.) 

 the weighing of the helium atom by means of its 

 spectrum or the detailed prediction of the structure 

 of the L-rays ; moreover, those theories, as Prof. 

 Bohr points out, are empirical and based on no 

 general principle. But the superiority of the " orbital 

 electrons " theory does not alter the fact that there 

 are things which it is very difficult to reconcile with 

 the view that the stationary states of an atom con- 

 sist of electronic orbits of which the dimensions are 

 comparable with i A., and of which the periods are 

 comparable with 10- " sec. 



The suggestion that I made is that, by means of a 

 generalised principle of correspondence, the distinction 

 between moving and fixed electrons might be abolished 

 and the conceptions that have proved so fruitful in 

 explaining spectra made available immediately for 

 explaining also such things (if there are such things) 

 as are only explicable by fixed electrons. Thus the 

 distinction would be abolished if " time " had no 

 meaning inside the atom. For the difference between 

 electrons following an orbit and electrons fixed at 

 points on that orbit can only be expressed in terms 

 of temporal conceptions ; if all such conceptions are 

 totally invalid in dealing with problems of atomic 

 structure the distinction vanishes. 



Expressed in the very crude form demanded by 

 brevity, such a suggestion will doubtless be deemed 

 unacceptable, or even unintelligible. Here I would 

 only mention two considerations, one special and the 

 other general, that have led to it. First, very difficult 

 questions can be asked (and have been asked by Stark) 

 concerning what happens in the interval during which 

 an atom passes from one stationary state to another, 

 and during which it emits or absorbs homogeneous 

 radiation. We might deny that such questions have 

 anv meaning, because there is no such thing as an 

 interval during which the transition takes place. It 

 is not merely that the interval is infinitesimally small 

 or zero; it is that the conception of a time interval 

 is not permissible when we are considering the process 

 which we observe as change of radiant energy and 

 explain as change of atomic structure. Secondlv, the 

 conception of continuity is very closelv associated with 

 that of time. The assumption of the older physics, 

 that all fundamental theories (usually mistermed 

 "laws ") were to be expressed by means of differential 

 equations, involved in all but a few instances (which 

 can be explained awav) that the variable with respect 

 to which the integration of the equations was to be 

 made, in order to compare the theory with experiment, 

 was the "time." Now it is the characteristic and 

 essential feature of Prof. Bohr's theory that the emis- 

 sion and absorption of homogeneous radiation, which 

 is the outward expression of change of atomic state, 

 is not to be described by a differential equation. Con- 

 sistencv seems to compel us to conclude that it is 

 also not to be described ultimately in terms of con- 

 ceptions in which "time" plays any part. 



Norman R. Campbell., 



British Plants Available as a Source of Industrial 

 Alcohol. 



The production of cheap alcohol for industrial pur- 

 poses is a subject much under- discussion at the 

 present time, and in considering the question of 

 available materials from which it could be obtained 

 the following notes mav be of interest. 

 • Apart from the mangel and sugar-beet, it is im- 



