Harry Bateman

“ Going back to the case in which a quadratic form is sufficient to determine the optical properties of a medium, we may remark that if Einstein's idea of the gravitational equations is accepted, it is still by no means certain that his quadratic form from which the gravitational equations are derived is the same as the quadratic form which determines the optical properties of the medium. ”

“ The description of any series of phenomena is thus qualitatively the same in the two series of coordinates.
It is interesting to compare the result just obtained with some ideas with regard to the way in which experience is interpreted by the mind.
If we suppose that some physical process taking place in the brain is the physical adjunct of a sensation in the mind, such as visual perception, we may regard the mind's interpretation of the sensation as a transformation of the actual physical process as it would be described by external observers if observed directly by them. ”

“ Confining ourselves to the case in which C is constant, we may use the results of § 2 to obtain new solutions of this differential equation.
Let
be the formulæ giving a transformation which enables us to pass from one solution of the above equation to another; then
when expressed in terms of x, y, z, t, is a second solution of the equation, and if the equation
be solved for t, the surfaces t = const, will form a system of parallel wave surfaces and t considered as a function of (x, y, z) will be the characteristic function for them. ”

“ 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. ”

“ It is clear that if two observers are situated at different points of space a comparison of observations can only be made by means of something which travels from one to the other, and for the sake of simplicity it is convenient to choose something which can be supposed to travel in a straight line with constant velocity. ”

“ For this reason we shall call a transformation which leaves the differential equation invariant, a spherical wave transformation. Our analysis shows that a transformation which leaves the integral equations of the theory of electrons invariant is necessarily a spherical wave transformation. We must next inquire whether all spherical wave transformations are relevant for our purpose. If we call t' the local time, we must exclude transformations which make the local time run backwards as t increases. Hence, in order that a transformation may be relevant, the condition
must be satisfied. ”

“ A change in the sign of corresponds to an inversion, a change in the sign of coupled with a change in the sign of corresponds to the other transformation we have mentioned. It is evident that each of these transformations is of period 2. In general, a reflexion in a linear manifold in the a space corresponds to an inversion with regard to the corresponding circle, sphere, or hypersphere, in the space of four dimensions. A displacement of period n in the a space may be obtained by taking successive reflexions in two plane five-folds which cut at an angle π/n. ”

“ The general theory of space time transformations is discussed with the help of two theorems on the transformation of integral forms. It is shown that when a transformation of variables is performed, and the forms equivalent to two given integral forms are known, then the integral forms may be multiplied together by the rules of Grassmann's calculus of extension, and the resulting integral form is equal to the one obtained by multiplying the equivalent integral forms in the same way. ”

“ Let us suppose that a large number of observers find that their observations of one another's experiences give a consistent universal time as far as they are concerned; they can then regard themselves as being at constant distances from one another, the distance between two observers , being defined as , where is the time light takes to travel from to , and C is a constant called the velocity of light. ”

“ Since the transformation enables us to derive the surfaces which are parallel to one surface from the surfaces which are parallel to the inverse surface, it is natural to expect that the above relation between the direction cosines will make the normals to the two surfaces correspond. ”

“ 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. ”

“ On the above view Einstein's idea of an influence of gravitation on light is simply an hypothesis, but a very interesting and reasonable one. It may be remarked, however, that in the theory of surfaces there are two fundamental quadratic forms, and we may perhaps expect something similar in general relativity. ”

“ Systems of non-Newtonian mechanics and kinematics of a rigid electron have been based upon the theory of relativity for the case of uniform motion, but they can hardly be regarded as satisfactory, and difficulties arise as soon as a uniform motion of rotation about an axis is considered. [14]
If mechanics is to be based on the science of electromagnetism, we must make a complete study of the transformations connected with the fundamental equations of electromagnetism. ”

“ As an example of standardization, the Ford motor-car is not in it with the electron; and, according to the above view, we must regard the design of the electron as one of very great value in relation to the surrounding medium. ”

“ Since the angles between lines remain unaltered by the transformation, it follows that if (l, m, n) , (l', m', n') , (p, q, r) are the direction cosines of an incident ray, the refracted ray and the normal at a point on a surface f, the corresponding quantities (L, M, N) , (L', M', N') , (P, Q, R) are the direction cosines of an incident ray, refracted ray and the normal at the point of the inverse surface F. This is also easy to verify from the analytical formulæ.
The method of inversion can thus be applied to problems in geometrical optics. ”

“ These observers may then form a standard system for the measurement of time and distances at other points of space. The measurements of four standard observers should suffice to determine the position and time of any occurrence, if space is of three dimensions. If there are more than four observers in the standard system the relations between all the different observations of an event will depend upon the nature of space, and will take a comparatively simple form if the space is Euclidean. ”

“ 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. ”

“ We should say, for instance, that there is a correspondence between the disturbance running along a telephone-wire and the sound-waves which produce it, because we can pass from one to the other by mathematical equations of a definite type, or rather by solving the equations and the boundary conditions. A correspondence is, moreover, regarded as an entity which may have real existence and be capable of growth and variation. ”

“ It is by no means obvious that a universal method of comparing observations can be found which will lead to consistent results, for this presupposes the existence of a universal time, an entity which has sometimes been regarded as the psychological time of an infinite mind governing the whole of the universe. ”

“ 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. ”

“ The phenomena here considered were those occurring in the brain and body; and although the correspondence by which the universe is reconstructed, so to speak, may be totally different [5] from the type contemplated here, yet it was thought that some of the general conclusions might still be valid if a transformation of co-ordinates was adopted as a working model of the correspondence. ”

“ The advantage of using the age of a particle in forming the equations of motion is that there is a gain in simplicity. The analytical methods based upon the use of the age of a particle may be compared with Lagrange's method of dealing with problems in hydrodynamics, while the methods based on the use of a standard time may be compared with the Eulerian method. ”

“ During recent years mathematicians and philosophers have been much occupied in analysing the fundamental conceptions on which the different sciences are based, with the result that many things which were formerly regarded as quite simple and axiomatic can no longer be regarded as such. The tendency has, of course, been to make definitions as precise as possible, and to make descriptions of phenomena approximate to reality as we know it, and not to a preconceived idea of what the description ought to be. ”

“ 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. ”

“ In the case of time, for instance, it is found that we have to examine the connection between time as it is known to us by the mind's experience, i.e., psychological time, and time as it is measured by the course of physical phenomena, i.e., physical time. ”

“ Many difficulties arise, however, in a careful examination of the fundamental concepts of any science, and this is soon found to be the case when we commence to examine the ideas of space and time which are fundamental in all physical and metaphysical enquiries. ”

“ It is possible that when the mind is concentrated on a subject the type of transformation is practically constant, and so our interpretations of sensations become clearer as they become more of a quantitative nature, but it is dangerous to speculate and so I shall leave the subject at this point. ”

“ If we could represent an event by a point on a line, the idea of simultaneity would be quite simple, for two events could be regarded as simultaneous when their representative points were coincident. In reality such a representation is not valid, there is no sensation of such a simple nature that it can be represented by a point on a line. If we adopt a representation by means of an interval on a line, we obtain what is probably a truer representation of an event as regards its duration ”

“ At present the theory is being widely used as a working hypothesis, and is of great theoretical importance, as it enables us to pass from the known analytical specification of a phenomenon for a medium at rest to the corresponding case of a medium in motion. ”

“ An observer situated at a point A observes at time t an event which has taken place at another point B. If τ is the time which light takes to travel from B to A, the universal time to be associated with the event at B according to A's measurements is . ”