Al^RIL 27, 1905] 



NA TURE 



607 



in the moving matter. Ttiis means a moving force 

 -(U,/c,)vi;,. But if there is compression, c, probably 

 always varies intrinsically as well. 



It will be found that the omission of the auxiliary h 

 has the result of complicating instead of simplifying the 

 force formula;. Similarly the omission of e complicates 

 them. Now the use of e is founded upon the idea that 

 the electric polarisation is produced by a separation of 

 ions under the action of E, for E, is the moving force on 

 a moving unit electric charge. Analogously h, is the 

 moving force on a moving unit magnetic charge or 

 magneton. If there arc really no such things, the inter- 

 pretation must be made equivalent in other terms. But 

 the categorical imperative is not "easily to he overcome. 



The application to plane waves I described in a recent 

 letter (Nature, March g) will be found to harmonise with 

 the above in the special case. 



But a correction is needed. In the estimation of the 

 moving force on " glass " receiving radiation, the assump- 

 tion was made that the electric and magnetic energies in 

 the transmitted wave were equal. .So the result is strictly 

 limited by that condition. The conditions E = ™B and 

 U=T are not coextensive in general, though satisfied 

 together in Lorentz's case. When U not =T, we have 

 instead of (8), p. 439, 



/> I V - /) ji; — />ji«/ = ^(T, — U ,) , 



and the rate of loss of electromagnetic energy is 



2/jH,H,«-|-(ii;-«)(T,-U,). 



Now this is zero when e = o, or the polarisation is pro- 

 portional to the electric force. The question is raised how 

 to discriminate, according to the data stated above, between 

 cases of loss of energy and no loss. To answer this ques- 

 tion, let o and h in the above be unstated in form ; else 

 the same. Then, instead of (^), thi' activity equation 

 will be 



- vw=U-i-t+{JE-(aV30 + -} + ('oq + 'iu)-(eJi fhG,), (5) 



wlure W is as in (4), whilst f„ and f, are the forces 

 derived from the stresses specified (not the same as F„ and 

 F,), and J|, Q, are the electric and magnetic polarisation 

 currents, thus, J,=b||Vvh,, &c. It follows that it is 

 upon e and h that the loss of energy depends in plane 

 waves, when u .ind q are constant. For the stresses 

 reduce to longitudinal pressures, so that by line integration 

 along a tube of energy flux we get 



2(.J, + /!Gi) = 2(U + T). 



(6) 



Thus, when a pulse enters moving glass from stationary 

 ether, the rate of loss of energy is 2(-eJ,). If e is zero, 

 so is the loss, as in the special case above. There is also 

 agreement with the calculated loss in the other case. 

 That the moving force on the glass should be controlled 

 by e is remarkable, for it is merely the small difference 

 between the electric force on a fixed and a moving unit 

 charge. The theory is not final, of course. If the electro- 

 magnetics of the ether and matter could be made very 

 simple, it would be a fine thing; .but it does not seem 

 probable. Oi.ivi;r Heaviside. 



April 5. 



The Dynamical Theory of Gases. 



In a letter to Nature (April 13) Lord Rayleigh makes a 

 criticism on my suggested explanation of the well known 

 difficulty connected with the specific heats of a gas. He 

 considers a gas bounded by a perfectly reflecting enclosure, 

 and says " the only effect of the appeal to the aether is to 

 bring in an infinitude of new modes of vibration, each of 

 which, according to the law (of equipartition), should have 

 its full share of the total energy." 



The apparent difiiculty was before my mind when 

 writing my book. Indeed, as Lord Rayleigh remarks, 

 something of the kind had already been indicated by Max- 

 well. (I think the passage to which Lord Rayleigh refers 

 will be found in the " Coll. Works," ii., p. 433 : — " Boltz- 



NO. 1852, VOL. 71] 



mann has suggested that we are to look for the explanation 

 in the mutual action between the molecules and the 

 .'Ethereal medium which surrounds them. I am afraid, 

 however, that if we call in the help of this medium, we 

 shall only increase the calculated specific heat, which is 

 already too great.") It seemed to me, however, that the 

 difficulty was fully met by the numerical results arrived at 

 in chapter ix. of my book. 



.Suppose, to make the point at issue as definite as 

 possible, we take a sample of air from the atmosphere, 

 say at 15° C. Almost all the energy of this gas will be 

 assignable to five degrees of freedom — so far ;is we know, 

 three of translation and two of rotation. Let us surround 

 this gas by an imaginary perfectly reflecting boundary. 

 The total energy of matter and a;ther inside this enclosure 

 will remain unaltered through all time, but this total 

 energy may be divided conveniently into two parts : — 



(i) The energy of the five degrees of freedom, say A. 



(2) The energy of the remaining degrees of freedom of 

 the matter plus the energy of the ;ether, say B. 



As Lord Rayleigh insists, the system is now a con- 

 servative system, so that according to the law of equi- 

 partition, the total energy A-)-B is, in the final state of 

 the i;as, divided in the ratio 



A : B = 5 



(>) 



whereas observation seems to suggest that the ratio ought 

 to retain its initial value 



A:B = s:o (2) 



This I fully .ulmit, but a further point, which I tried to 

 bring out in the chapter already mentioned, is that the 

 transition from the ratio (2) to the ratio (i) is very slow 

 — if my calculations are accurate, millions of years would 

 hardly suffice for any perceptible change — so that, although 

 (i) may be the true final ratio, it is quite impossible to 

 obtain experimental evidence of it. 



If the sample of gas were initially at a much higher 

 temperature than we have supposed, the transition would 

 undoubtedly be much more rapid ; but even here we could 

 not hope for experimental verification. For the assumed 

 boundary, impervious to all forms of energy and itself 

 possessing none, cannot be realised in practice, and as 

 soon as the energy of the enclosed ;ether becomes appreci- 

 able, the imperfections of our apparatus would become of 

 paramount importance in deterinining the sequence of 

 events. J- H. Jeans. 



Growth of a Wave-group when the Group-velocity is 

 Negative. 



The following may be of interest in connection with the 

 recent discussion on the flow of energy in such cases. 



Let the energy of an element of a linearly arranged 

 mechanical system be 



\{il-yli/xiil)'- +y\c/x/2. 



Such a system can be approximately realised by taking 

 a bicycle chain, loading it so that the radius of gyration 

 of each link has the same large value, and suspending it 

 by equal threads attached to each link so that the chain 

 is horizontal and the axes of the links vertical. By the 

 principle of least action we immediately find the equation 

 of motion to be d*y/dx'dt' = y. A simple harmonic wave 

 is given by y = sir\ (pt-x/p). The group velocity is -/>^ 

 and is negative. Let such a system, extending from x = o 

 to »: = <», be at rest in its position of equilibrium at time 

 t = o, and then let the point x = o be moved so that its posi- 

 tion at any subsequent time is given by y=i-cos(. 



By application of the usual method vii Fourier's in- 

 tegral, the motion of the system is found to be given by 

 either of the equivalent formulae 



/=2( -l)"(//j-)"+'Jj„+j2V(/x), 

 or 



y=l- cosU f a:) - I -I- 2( - I )"(a-//)"J.^,.2 ^Ux). 



where the J's are Bessel's functions and the summations 

 extend from rt = o to n = v>. There are some doubtful 



