588 PHYSICS: E. B. WILSON 
RADIATION LESS ORBITS 
By Edwin Bidwell Wilson 
Department of Physics, Massachusetts Institute of Technology 
Read before the Academy, November 10, 1919 
The reaction on an electric charge due to non-uniform motion has 
been calculated (by Lorentz, for example) and found to be 
F = ^^ (1) 
this term F is merely the first of a series which converges with extreme 
rapidity provided the dimensions of the charge are small compared with 
the distance light would travel in the time required for a sensible 
charge in the velocity.^ The activity of the force F determines the 
rate R of radiation of mechanical energy as 
--i?^F.v=-^v.^ (2) 
What plane orbits are such as to make the radiation as determined 
by (2) identically zero? The answer to this question is contained in 
the general integral of the quadratic differential equation of the third 
order 
dH dx dH . dy d^y ^ ... 
V. — = + — — ^ = 0. (3) 
df" dt dt^ dt dt^ 
The integration may be obtained easily by considering v as the radius 
vector in the hydrograph. Then d^v/df is the acceleration of the moving 
point in the hydrograph and its radial and tangential components are 
^-.f^-^Y and l^U^A 
df \dt/ V dt\ dt/ 
of <p be the incHnation of V to a fixed direction. The condition of 
perpendicularity (3) may thus be written 
d^v I d(pV ^ d(p 
— — V [ ~] = 0, or — 
dt' I AdtJ dt 
Then 
x = f[vcosf Vv'/v dt] dt, (4) 
y=flvsinf VV> dt] dt 
