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#### §12. – ELECTRIC OSCILLATIONS. GENERAL CONSIDERATIONS.

WALTER RITZ

##### Translated from Recherches
critiques sur l'Ėlectrodynamique Générale,

*Annales de Chimie et de Physique,* Vol. 13, p. 145, 1908.

We have seen in the preceding sections, that for very rapid
oscillations, only the accelerations are

Annales 249

to be retained
besides the electrostatic term, the (effect of the**)** speeds being
negligible compared to them. It will no longer be permitted to refer to
developments which we have used up until now, and we must go back to formula
(VI), using *u*=0. It then becomes

(31) _{}

In Lorentz’s theory, only the electric force will play a role

(32) _{} [*]

(Oeuvres 404)** **and we can
also neglect the terms in _{}so that it (the** **expression) becomes

(33) _{}

and this is for
any value of R which would not be incomparably smalle**r** than 1 cm, and *a
fortiori* (even more so) for large values.

The second terms of the two formulas (31), (33) correspond to a force
perpendicular to the radiation vector,

Annales 250

which in optics
(where the term in 1/r^{2} disappears) plays the role of *Fresnel’s
vector*. It depends only on *x, y, z, t*, and not on *v*; the
notion of field may thus apply, that is to say that we may consider the
distribution of this force in space independently of the presence or absence of
an electron, which would no longer be the case if it depended on the speed of the
latter.

*General discussion*. –** **Formulas (VI) and (31) demand a
general remark. They contain, in effect, the factor_{}, which becomes infinite for sufficiently large values of *r*.
I am going to show that there nevertheless result from these expressions only
finite values.

Let us consider a charge *e' *vibrating
parallel to the x-axis and whose speed *v**'=** v**'(t')** is nil for **t'=**0 and
afterwards positive.* The wave
emitted at*
t'=**0 *propagates[**]
the length of the axis with the speed *c*, and reaches a fixed point *P*
situated on the positive axis at an instant *t>t'>**0. Then
the speed of propagation increases, and will be **c+v' **for the
wave emitted at *_{}**.** This wave will therefore overtake the previous one
at a certain distance *x*_{0}*, then will pass it***. **At point *x*_{0}* the two
waves emanating from **e'** at **t'=**0 and *_{}* will arrive
simultaneously. The speed **v' **remaining positive, the number of
waves arriving simultaneously at **x*_{0}* may still increase; but,
when **v'
**diminishes or changes sign, this case no longer presents
itself. Where*** **the position of *P** and the movement of **e'** are
given, the time of arrival **t** is a function **f(t')** clearly
determined from **t'**, given by the law of propagation [analytically by
formula (II*^{a}) (Oeuvres 405) *resolved in relation to t**]. On the contrary, the function t'=**φ(t)** may present, for a given value of t**, an finite number of distinct determinations *_{}*as we have just seen. Let us show **t'** *

Annales 251

*in ordinates,
**t** in *abscissas, and suppose the relative speeds are** **greatly
inferior to *c*; the curve *t'=**φ**(t)** will
oscillate from side to side of the straight line *_{}, where r_{0} is the average distance from *e'** to **P**.
Parallels to the t-axis will not cross it except at one point; parallels to the
t'-axis, at one of several points; among these latter, the tangents to the
curve correspond to the points where*_{} becomes infinite and where the two values _{}*become
equal. If *_{} remains finite at
this point (that is to say if there is no inflexion), the values of _{}*will tend towards great values and of opposite signs.*

These considerations are clearly general and apply even when *P* is
in movement, __as soon__ as the relative speeds become less than *c*.
We have seen [formula (6) obtained by differentiation of (II)] that

_{}

The denominator therefore __annuls __itself and changes sign precisely
at the instant when two different values _{}_{ }*are corresponding to a same
instant of reception, and tend to become equal. But it has been specified (*§*2) that
the action of **e' **on **e** is the **sum** of the
actions corresponding to *_{}*Let us multiply and divide F*_{x} (VI) by _{}**.** This expression will be written

_{}

*where Φ remains finite and *__continuous__ for
the instant where_{}* and*_{}* become infinite. The sum of the terms*

Annales 252

*relative to *_{}* and *_{}

*(34) *_{}

(Oeuvres 406) will remain finite^{2},
because _{} and _{}*for the critical instant.*

*------------------------------*

_{}

*is unlimited in *_{}, this reasoning is not applicable any more; but these cases
are exceptional, and we have to keep account here of the finite dimension of
the electron: the *impetus* because it remains finite in all these cases.

Translation editor notes:

[*] In expression (32) there needs to be a c^2 in the denominator of
the third term inside the first upper parenthesis.

*[**] Even though Ritz uses the word **propagation**, which
would seem to imply a wave action impressed on a deformable medium, his
mathematical reasoning is more in line with the idea of ballistic **projection **of his
fictitious particles. (In Lorentzian
ether there is only **one** speed of **propagation, i.e., **the speed
of light**.**) *

** **