Prof. A. Schuster's Electrical Notes. 5 



we may therefore write 



T=^{[[(a 2 + b* + c 2 )dxdydz. ... (5) 



This integral may be transformed either in the manner 

 indicated by Heaviside or in that of J. J, Thomson, and it is 

 found that, retaining only that part of the energy which 

 depends both on the outside magnetic force and the moving 

 sphere, the result is 



T=gw¥ v 



5. J. J. Thomson's value differs from this because he 

 wrongly applies equation (4) to the case in which the field is 

 due partly to permanent magnets, substituting for a, b } c their 

 values a -f 47rA, &c. 



Equation (4) thus becomes 



T 1 =i(ft[(a' 2 + 6 2 + c 2 )-47r(aA + 5B + cC)]^^^. . (6) 



In the subsequent investigation Thomson puts a, b, c equal to 

 pa, fi/3, fiy, where fi is apparently the magnetic permeability 

 of the medium in which the charge moves. It is easily seen 

 that if under these circumstances (6) is compared with (2) 

 and (5), a relation is obtained between T 1? T, and W, viz., 



I 1 =KI + |W)=^T=i ww F s . 



This is J. J. Thomson's value, which differs therefore from 

 that of Heaviside by the factor %fi. It follows that J. J. 

 Thomson's forces, which are calculated from this value of the 

 energy, only give half the correct results in a medium of 

 unit permeability ; but the appearance of fju in the expression 

 for the energy is alone sufficient to show that there is some- 

 thing wrong with it. 



6. It has already been pointed out that in his ' Recent 

 Researches in Electricity and Magnetism,' chapter i., J. J. 

 Thomson obtains a value for the magnetic force which is only 

 a third of that deduced by Heaviside and verified above. I con- 

 fess I find it difficult to follow the method of " moving tubes " 

 employed in that investigation. I speak with diffidence on the 

 subject, but the investigation on page 22 of the work quoted 

 seems to me to be obscure and incomplete. In the equations 

 for U, V', W the components /, g, h are taken to represent 

 the polarization of the moving sphere only. Should there not 

 be some additional polarization due to the outside magnetic 

 forces ? In the meantime all recognized methods give 

 Heaviside* s value for the force which acts on the sphere, viz. 



