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 analysis. We begin with a "substantial point," which represents the location of a definite particle, together with the instant at which the particle occupied this location, so that four scalar quantities are required to specify a substantial point. We then proceed to define various four-dimensional vectors, the "absolute velocity," "absolute acceleration," and "absolute force," and formulate the law of motion in the form

This law may be expressed analytically in terms of the ordinary coordinates (x, y, z) of the particle m, in the form

$$m\frac{d^{2}x}{dt^{2}}\left(1-\frac{v^{2}}{c^{2}}\right)^{-\frac{1}{2}}=X$$, and two similar equations,

where v denotes the velocity of the particle and (X, Y, Z) the force acting on it, in the ordinary sense of the term. These equations differ from those given by Newton’s laws owing to the presence of the factor $$\left(1-v^{2}/c^{2}\right)^{-\frac{1}{2}}$$, and it thus appears that Newton’s laws must henceforth be regarded as only approximately true. E. T. W.