Harry Bateman

“ The theory of Lorentz's transformation has been developed very considerably by Einstein, Planck, and Minkowski. The transformation has become a powerful instrument of research, inasmuch as it provides a means of transition from the mathematical expressions of physical quantities connected with a system at rest to the corresponding quantities for a similar system in uniform motion. ”

“ The transformation of Lorentz only enables us to pass from a system at rest to one in uniform motion, but it has been postulated that there is a more general transformation which can be applied to systems moving in a more general manner. ”

“ The transformations which leave the electrodynamical equations for ponderable bodies unaltered in form must depend upon the form of the constitutive relations connecting the magnetic induction and electric force with the magnetic force and electric displacement when the bodies are in motion. Minkowski, Einstein, and Laub have proceeded in the opposite way and constructed a set of constitutive relations for bodies in uniform motion, by assuming that the transformations are Lorentzian transformations and transforming the known constitutive relations for bodies at rest. ”

“ The mathematical analysis on which the principle is based was introduced by Lorentz in 1892. It consists of a certain space time transformation of the coordinates in which the fundamental equations of the electrodynamical field are unaltered in form.
The principal features of the transformation are the introduction of a local time, and the hypothesis that a body moving through the aether with a constant velocity v suffers a contraction in the direction of motion: c being the velocity of light. ”

“ When multiplied by this yields and it is clear from this that the multiplication of the integral forms by Grassmann's rule is justified.
Taking the case of a transformation from four variables (x, y, z, t) to (x', y', z', t') . If
and
we obtain, on multiplication,
while, if either form be multiplied by itself, we obtain
These equations indicate the invariance of the property that at a surface of discontinuity, or in a spherical wave, the electric force is equal in magnitude to the magnetic force and perpendicular to it. ”

“ Further, when x = ut, we have
hence every point of the plane is transformed into itself. [18]
The formulae of transformation of the electromagnetic vectors are found from the first set of equations for a transformation with negative Jacobian.
The general infinitesimal spherical wave transformation is
where the coefficients are all constants, and ε is a quantity whose square may be neglected. Since there are fifteen arbitrary constants the group is a fifteen parameter group. ”

“ The double integral expresses the total charge as a surface integral, but here again the particles over which the integration extends are not all on the surface at the same time, but at different times.
In the same way equation (II) may be regarded as a general way of expressing the fact that there is no free magnetism.
If Φ is the scalar electromagnetic potential and the components of the electromagnetic vector potential, we have
These equations are equivalent to the single integral equation provided a suitable interpretation is given to the integrals. ”

“ Let the coordinates of points on S be expressed in terms of two parameters .
A term such as may now be understood to mean [9]
and a term such as to mean
The equation (II) may now be written
Transforming the left-hand side by means of Green's theorem, we get
Now
hence the above integral may be written
and this is equivalent to
if equations (I) are satisfied and the axes form a right-handed system. In the same way, it can be shown that the integral on the left-hand side of (II) is zero, if equations (I) are satisfied. ”

“ When the Jacobian is negative and , the axes are righthanded in the original system and left-handed in the system specified by the dashed letters. To obtain the correct formulae of transformation we must change the sign of H' in equations (2) and (8) and put . There is also a doubt about the sign in the equation
the sign depending upon whether positive electricity is transformed into negative electricity or positive electricity. ”

“ Multiplying conjugate pairs together, attributing proper signs to each product, and adding, we find from the properties of the minors of a determinant that the coefficient of a term such as , is zero, unless the quantities s,t,p,q,r,... are all different. When these quantities are different the coefficient is simply the determinant with a proper sign. ”

“ With the help of this hypothesis a satisfactory explanation of the results of the Michelson-Morley experiment can be given on the supposition of a stationary aether. The results of the experiment seem at first sight to indicate that the aether is carried along with the earth, but on such a supposition a satisfactory explanation of aberration would still be wanting. ”

“ These equations may be replaced by the two integral equations [6]
provided the integrals receive suitable interpretations. The interpretation that first suggests itself is obtained by regarding (x, y, z, t) as the coordinates of a point in a space of four dimensions. Let any closed two-dimensional manifold in this space be assigned by equating x, y, z, t to one-valued differentiate functions of two parameters , and let be the boundary of a three-dimensional manifold in which the coordinates are like functions of three parameters , of which on , and on . ”

“ Proc. London Math. Soc., Ser. 2, Vol. 7, p. 70 (1909) . ↑ Proc. London Math. Soc., Ser, 2, Vol. 8, p. 77. ↑ Geometrically the transformation is equivalent to a reflexion in a moving plane mirror. ↑ Jahrbuch der Radioaktivität, Band iv. (1907) , p. 457. ↑ Proc. London Math. Soc., Ser. 2, Vol. 7, p. 70. ↑ The theory of integral invariants has been developed by Poincaré, Mécanique céleste, t. 3 ”