November 6, 1902] 



NA TURE 



But if Au be distributed uniformly over the time 2A/z>, the 

 shell will be doubled in depth, and H will rise at uniform rate 

 from o to the same full value in the middle of the shell and 

 then fall similarly to zero in the second half. Now if a second 

 equal Au acts in the same way, beginning as soon as the first Au has 

 made H reach full strength, H will continue of that full strength. 

 And so on with a third Au. Finally, if 2A = vAt, and An/ At is 

 steady, and allowing for the variable depth of the shell according 

 to (11) below, we come to 



a/ MQ! /A«y _ _J (S) 



6« 



("-£)' 



to represent the waste in time At. 

 waste 



W 



_p.Qr(du 



6-nz\dt 



('-£)" 



Or, if W is the rate of 



(9) 



This holds when the acceleration and the velocity are parallel. 

 By the manner of construction, it is necessary that dujdt should 

 not vary sensibly in the time taken by light to traverse the 

 diameter 2A. 



By a fuller analysis, allowing for change of direction of motion, 

 I find that the waste of energy per second from a charge Q with 

 velocity u and acceleration a is 



„„ , I - - sin-9, 

 fit^t-a- v- 



(10) 



W 



5) 



(12) 



E, = 



Q 



4ttR 3 <: 



cos 8 



B.= -"Q 



ttR I / _ U 



R I cos (J) 1 



cos 8 



(1 - \ cos «) 



(13) 



(14) 



This is referred to origin at the virtual position of the charge, 

 not the actual. The actual is best for the steady state, the 

 virtual to show the waves emitted. The factor ( I - 11/v cos 8) ~ 1 

 expresses the Doppler effect. Divide by uc to obtain the 

 scalar potential *. Then 



H = curlA, E=-^A-V* 



in Maxwell's manner. The trouble here is the differentiations, 

 which require great care, since u, R and 8 all vary in a 

 rather complicated way as Q moves. The relations (12) exhibit 

 the field clearly. 



For an infinitely small sphere of Q, the energies in the 

 shell at distance R corresponding to the displacement udt of Q 

 are 



T = T 1 +T„ + 2T,„, 



U = U 1 + U„ + 2U,,, 



where 1 relates to the EjH, part and 2 to the other part, whilst 

 12 refers to the mutual energy. They are connected thus : 



when 8] is the angle between the velocity and acceleration 

 (absolute). The dimension A does not appear. W is the same 

 for any size, subject to the restriction mentioned. The smaller 

 A the better, of course. It is exactly true with A = 0, only then 

 the motion would be impossible. 



This calculation of the waste may be confirmed by following 

 up my investigation of the electric and magnetic field by the 

 method I gave in 1S89 ("Elec. Pa.," vol. ii. p. 504). 



The waste is greatest when the velocity and acceleration are 

 parallel, and least when perpendicular. There is another reserv- 

 ation, viz. 11 must be less than v. If not, special treatment is 

 required, after the manner I have already published. 



The meaning of waste is this. When Q moves through the 

 distance udt, it casts off a spherical shell of depth 

 vdt 



(II) 



I cos 8 



v 



and the energy of this shell when it has gone out to an infinite 

 distance is Wdt. 



When at a finite distance, E and H in this elementary shell 

 are given by 



E = Ej -4- Eo, H — H] + H.i, 



Hi = VuD„ H. 2 = VvD„," 



Here the part Ej, Hj belongs to the steady travelling state of 

 steady u, whilst the other part E 2 ,H 2 is electromagnetic, and 

 represents the waste. The angle between the acceleration a and 

 R is if, 1 . The waste part has E 2 ,H„ tangential, that is, 

 perpendicular to R. H T is also tangential to the sphere, but 

 Ej is radially directed from the point which Q would reach at 

 the moment in question (belonging to the sphere R) if it were 

 not accelerated at all. This means the steady travelling state 

 (see " El. Pa.," vol. ii. p. 511, equation 29). There is another 

 way of treating the question, viz. by the vector and scalar 

 potentials. The vector potential of the impressed current Qu is 

 not Qu/47rR, but {Joe. cit.) 



Qu 

 A= ( I5 ) 



47rR( I - - cos 8 J 



NO. 1723, VOL. 67] 



U 2 = T 2 , U 12 = T 12 , U 1= T 1 + g|, 

 liQPa udt cos 8 X 



T _ Q/W/ u"-!v- T ^ _ /*Q 2 a 

 1 I2irR-i k-' ' '•»■"■" 



I2ttR 

 U 



T, 



jiQW/| 



sin-8, 



(16) 



(17) 



(18) 



where k 2 = I - a 2 /z> 2 . 



The corresponding " momenta," or force-impulses, say 



M 1 = 2VD 1 Bj, M 2 = 2VD„B 2 , M I2 = 2VD,B 2 , M 21 = VDnBj, are 

 given by 



2T, 2T.,u 2T,oU 



— , M„ = — rp, Moj = ™- 



M, 



(19) 



These are all parallel to u. But M 12 is not, though it is in the 

 plane of u and a. Its components parallel to u and to a are 



2T lg , 



and 



2 _Jl! 



a 



I - — sin-8. 



cos 8, 



(20) 



With the previous restriction, these are independent of the 

 size of the sphere of Q. But to obtain exact formulae without 

 this restriction, either a very difficult integration must be effected 

 over the surface of the sphere of Q, every element of which will 

 usually have (effectively) a different velocity and acceleration, 

 on account of the Doppler effect, or we may derive the resulting 

 formula; by a differentiating operator. Thus, for example, 

 exhibiting it fc>r * only, let * be the formula when A = o, 

 then the real * is, by a previous investigation, 



shin </A 



outside the sphere, and 



(21) 



(22) 



inside the sphere, where q is the differentiator d\d(vt^, and * U a 

 is the common value of both *'s at R = A. But this ^ is not 

 the same as the previous t ; it is the corresponding value ; the 

 place where the differentiations are performed is at the end of 

 R. The differentiations are troublesome. Thirdly, we may 

 calculate the time integral of * , and then apply Taylor's 

 theorem. Nearly all the trouble in the electronic theory is 

 connected with the necessity of making A finite to have finite 

 energy (though this does not apply to the waste) and finite 

 moving forces, with the consequent resulting two superposed 

 waves, one outward from the surface of Q, the other inward, 

 and then outward again. The results for impulses work out 

 easily enough, but not for continuous accelerations. 



Details of the above will be published in vol. iii. of " Electro- 

 magnetic Theory" (and perhaps elsewhere), which is, as the 

 advertisement says, "in preparation." 



Returning to the waste formula, an electron revolving in a 

 circle of radius r has 6 l = iir, and u-jr = a. So we want an 

 applied force along u varying as 11 s to maintain the motion, 

 since the waste varies as « 4 . This revolving electron has some- 

 times been supposed to be a circular current. But it is really 

 a vibrator. The free path followed under decay of energy 

 without fresh supply would perhaps be difficult to follow 



