Dynamical Motions of Charged Particles. 549 



We next solve for the motion of the centroid. As X 

 and Y are of the order C~ 2 , it will be sufficient to use large 

 order values for %, etc. Then changing the independent 

 variable to 6 and making use of the known value of w 2 > 

 (17) gives 



dX 1 M-m i dj _ Eei /d± | dr\ ) 



do 20 2 (M+m)4 l } ae \de + r de) j > 



and this is directly integrable in the form 



If « is the half major axis a = q/(q 2 — s 2 ) and W = E<?/2« 

 and we have 



1 M-m Ik; , _ /rtfN 



If the motion of the centroid is to be valid for many revolu- 

 tions, we must replace cos 6 by cos\#. Both are the same 

 to the degree of approximation considered, but cos X0 will 

 enable the centroid to keep pace with the motion of the apse. 



From (23) we see that both particles and their centroid 

 always remain collinear with the origin, which is the in- 

 variable point of the system. Observe that this invariable 

 point cannot be calculated by taking the centre of mass of 

 the particles as though each had its mass increased separately 

 by the effect of velocity. Such a process would give X 

 proportional to w*% or [r — 2a) cos 6. There is in fact no 

 simple definition for the invariable point. 



Expressed in polar coordinates the centroid describes the 

 curve 



_ Ee M-m e + co sfl 



n ~ 2C 2 (M + m) 2e 'l + eeos~6 , * ■' *<■> 



where e — \/(l — s 2 /q 2 ) is the eccentricity of the relative orbit 

 of the particles. If M>m, R is negative at perigee, that is 

 to say, the centroid is towards M. At apogee it is an equal 

 distance towards m. If the time average of II be taken it is 

 found to be 



E. M-m . y 

 _2C 2 (M+m) 22<r ' ^ > 



that is, on the average it is towards m. As the velocity of 

 the lighter particle is the higher, so its mass is the more 

 increased by the motion, and so (25) is directly contrary to 



