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PHYSICS: E. B. WILSON 
— — 2a (1 + aA) t — — laH = A — , 
dt^ dt dt^' 
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and has the roots a{l — i) and \/A in addition to —a{\ — i); two of 
the roots therefore give increasing motions. The root 1 /A is the large 
extraneous root that comes in when the force (1) is used;^ the root 
a(l — i) gives a non-radiating orbit. Any orbit of the whole set 
is also non-radiating. 
The orbits (4) have been described as non-radiating because the 
force (1), being normal to the path, does no work. It is only in this 
sense that they are free from radiation: the equation of energy holds, 
the sum of the kinetic and potential energies is constant. (As the 
motions are supposed to be small, quasi-stationary, relativity effects 
have been ignored.) From the point of view of electro-magnetic theory 
the rate of radiation varies as the square of the acceleration and can 
never vanish in accelerated motion; and there is a coordinate radiation 
of momentum. The relation between the two rates of radiation is 
shown by the equation 
The total radiation is the same according to both points of view when- 
ever estimated between two instants for which y.dv/dt has the same 
values; but the instantaneous rates cannot be indentically equal except 
for orbits in which Y,dv/dt is always the same constant independent of 
the time. I have discussed elsewhere the effect of replacing the law 
(1) by its alternative in the case of the rectilinear oscillation,^ — which 
then ceases to be an oscillation. 
From the point of view of relativity the rate of radiation of energy 
(and momentum) has been treated by several authors and has been 
shown to vary with the square of the generalized curvature of the path 
when regarded as a space- time locus in four dimensions.^ For any 
real motion, where the velocity is less than that of Hght, the radiation 
must be positive and radiationless orbits, other than straight lines 
uniformly described, are impossible. If, however, velocities greater 
than light were possible and if the formula for the rate of radiation still 
held, the orbits would be those for which the curvature vector was a 
minimum line. Although this state of affairs may have no present 
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