394
Prof. de Sitter, On the bearing of the Principle
LXXI. 5,
The values of and the higher differential coefficients must be taken for the time , defined by (8). We consider as a small quantity of the first order, and we wish our formulæ to be exact to the second order inclusive.
We find easily
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where
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and are of the first order, is of the second order.
In the equations of motion there appear the invariants—
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To express these in simultaneous relative coordinates we have—
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(10)
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6. The equations of motion can be given in three forms:—
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(11)
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(12)
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(13)
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and similarly for the other coordinates.
is the force according to the ordinary definition, or "Newtonian" force; is called the "Minkowskian" force.[1] The mass is a constant.
Differentiating the formula , we derive a fourth equation analogous to (11) or (12), viz.:
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(11′)
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- ↑ See Lorentz, Göttingen Lectures, October 1910 (Physikalische Zeitschrift, vol. xi. p. 1234). The names quoted above occur on page 1237.