The mathematical translation of this principle of physics yields us the following equation, which must remain invariably zero in value for all Galilean frames: When we pass from one Galilean frame to another, , , , may vary in value; but the variations must be so connected that the sum total of the above mathematical expression remains invariably zero. Such is the mathematical condition which expresses the principle of the invariant velocity of light. [Pg 194] Now it is our object to determine exactly in what measure these different magnitudes , , , must vary in value when we pass from one Galilean frame to another, moving with a constant velocity with respect to the first, if the previously written mathematical condition of invariance is to remain satisfied. From a purely mathematical standpoint problems of this type form part of a branch of mathematics known as the theory of invariants. Such problems had been studied many years before, and it was known that the relations between the variables , , , , or, what comes to the same thing, the transformations to which it was necessary to subject these variables (in order to satisfy the condition of invariance set forth above), were given by a wide group of transformations known as conformal transformations.[62] But when, in addition, the relativity of velocity is taken into consideration it is seen that conformal transformations are far too general. We must restrict them; and when the required restrictions are imposed we find that the rules of transformation according to which the space and time co-ordinates of one Galilean observer are connected with those of another depend in a very simple way on the relative velocity existing between the two systems. These rules of transformation are given by the Einstein-Lorentz transformations.[63] Now these transformations are, as we have said, more restricted than the conformal transformations; and this lesser generality of the Einstein-Lorentz transformations has, as a consequence, the further restriction of the conditions of invariance of the mathematical expression mentioned previously. Not only will this expression have a zero value for all Galilean frames when it has a zero value for one particular Galilean frame, but in addition, if it does not happen to have a zero value in one frame but has some definite non-vanishing numerical value, it will still maintain this same definite non-vanishing value in all other Galilean frames. In other words, Einstein’s premises are represented mathematically by the invariance of the total value of for all Galilean frames, regardless of whether this value happens to be zero or non-vanishing. The deep significance of this condition of invariance was first noted by Minkowski, and it led, as we shall explain in the next chapter, to the discovery of four-dimensional space-time. [Pg 195] CHAPTER XVIII THE DISCOVERY OF SPACE-TIME THE discovery of this invariant