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PHYSICS: L. PAGE 
a single oscillator, i.e., a single vibrating electron, and yet we can obtain a 
result that will be a perfectly general test of Larmor's expression. For the 
latter gives the retarding force as a function of the rate of total radiation and 
the velocity of the radiating body, and of these quantities alone. Hence if 
the ether exerts a reaction on a group of moving oscillators, it will exert a 
similar reaction on a single oscillator; and conversely, if there is no reaction 
on a single vibrating electron due to its drift velocity, there can be none on a 
group of such vibrators. 
A rigorous solution of the problem for this relatively simple case shows 
the existence of no retarding force. Larmor's result is found to be invalid 
because of a tacit assumption underlying his reasoning which was introduced 
substantially in the following manner. In order not to complicate matters 
by the introduction of terms in the inverse second power, the radiation reaction 
is calculated by applying the electrodynamic equations to the surface "of a 
moving sphere with the electron as center, whose radius is large compared to 
that of the electron (though small compared to a millimeter) . In this way 
only those terms in the expressions for the electric and magnetic fields which 
involve the inverse first power of the distance from the electron need be 
retained. But the result obtained really gives the force on the electron and 
the ether inside the moving sphere, not that on the electron alone. Now, 
if the motion of the electron were undamped, the field inside this moving 
sphere would remain unchanged, and consequently the force found by Larmor 
would be that actually exerted on the electron. But as the electron is radiat- 
ing, its motion must be damped unless energy is supplied from some outside 
source, and in that case it must be shown that no impulse accompanies the 
transfer of energy to the electron — a matter of considerable difficulty to 
treat rigorously. It is far simpler to consider the case of an oscillator left 
to itself and allowed to radiate at the expense of the energy of its vibration. 
For this case it is found that the force exerted on the electronic vibrator by 
the ether inside the moving sphere mentioned above is exactly equal and 
opposite to that due to the ether outside. Moreover, from the point of 
exchange of momentum, the law of conservation of momentum demands that 
Momentum lost by electron = Momentum gained by ether outside 
sphere — Momentum lost by ether inside sphere. 
The terms on the right hand side (the second of which is overlooked by Lar- 
mor) annul one another. Therefore a single moving oscillator is not retarded 
by its radiation field, and as already noted we can generalize this result and 
conclude that a moving body of any size and complexity suffers no retarda- 
tion as a result of its emission of radiant energy. 
In the analytical reasoning leading to this conclusion it is found necessary 
to develop the complete dynamical equation of the Lorentz electron through 
terms of the fifth order, for the most general type of motion. Previous deri- 
vations of this equation have been confined to some special case, such as 
