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m/m₀ found for the different observed values of β in the second column. The third column shows those values of ft which would correspond with the same values of m/m₀ according to equation (15).

It will be seen that the observed values of β follow to a remarkable degree the same trend as those which are calculated by equation (15), but are in every case six to eight per cent higher.^[1] I believe that these differences lie within the limits of experimental error of Kaufmann's measurements. It is true that he claims a higher degree of accuracy, but, notwithstanding the extreme care and delicacy with which the observations were made, it seems almost incredible that measurements of this character, which consisted in the determination of the minute displacement of a somewhat hazy spot on a photographic plate, could have been determined with the precision claimed. Moreover, Planck^[2] and Stark^[3] have pointed out certain corrections which probably should have been made by Kaufmann and which would produce a material change in his results.^[4]

That a charged particle must possess mass in virtue of its charge, and that this mass must vary with the velocity of the particle, was shown to be a consequence of the electromagnetic theory by J. J. Thomson and by Heaviside, and numerous attempts have been made to find the exact expression for the change of mass with the velocity. But before this can be done some assumption is necessary as to the shape of the particle and the distribution of its charge. The three theories of the simple negative particle or electron which are now most discussed are due to Abraham, Bucherer, and Lorentz.^[5] The first assumes that the electron is and remains a rigid sphere, the second assumes an electron which is spherical when at rest but which in motion contracts in the direction of its translation and expands laterally so as to keep a constant volume. The third assumes an electron similar to the second, which contracts in the direction of translation but which does not change its other dimensions. On the basis of these theories and from known electromagnetic principles, three equations have been obtained for the value

1. The constancy of the difference between the observed and calculated values of β is striking, and would alone indicate some constant error in Kaufmann's results.
2. Verhandlung Deutsch. Phys. Ges. ix. p. 301 (1907).
3. Ibid. x. p. 14 (1908).
4. In reply to Planck see Kaufmann, ibid. ix. p. 667 (1907).
5. For a discussion of these theories see Abraham, Theorie der Elektricität, vol. ii. Leipzig, 1905; and Bucherer, Mathematische Einführung in die Elektrontheorie, Leipzig, 1904.