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
|
|
where
|
|
and
are of the first order,
is of the second order.
In the equations of motion there appear the invariants—
|
|
To express these in simultaneous relative coordinates we have—
|
(10)
|
|
|
6. The equations of motion can be given in three forms:—
|
(11)
|
|
(12)
|
|
(13)
|
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.:
|
(11′)
|
- ↑ See Lorentz, Göttingen Lectures, October 1910 (Physikalische Zeitschrift, vol. xi. p. 1234). The names quoted above occur on page 1237.