406 Sir J. J. Thomson on the Scattering of 



Hence 



2(a 2 + b 2 + c 2 -ab-ac-bc) _ 5_ 

 4(a 2 + b 2 + c 2 ) + ab + ac + be "" 98* 



So that by equation (2), p. 397, the minimum intensity 

 would in this case be about 5 per cent, of the maximum. 



Let us take next the case of* an electron describing a 

 circular orbit under the attraction of two equal positive 

 charges, the plane o£ the orbit bisecting at right angles 

 the line joining the charges ; we have found the values 

 of <x, b, e as functions of 6. 



We see from the expression for c that there cannot be a 

 state of stable steady motion unless 1 — 3 cos 2 6 is positive ; 

 so that must be between 90° and 54° 42'. We have 

 already considered the case when = 90°. The values of a, 

 the ratio of the minimum to the maximum intensity, for 

 some intermediate values of 6 are given in the following- 

 table : — 



6. a in per cent. 



80 22 



70 16 



60 -46 



58 2 



55 50 



We have supposed hitherto that only one electron was 

 describing a circular orbit, and with a law of force of the 

 type we have assumed it is easy to show that if we have 

 more than one electron travelling round the same circle steady 

 motion is unstable. For take the case of two electrons 

 travelling round a circle of radius a : the steady motion, 

 if possible, will be one where the two electrons are at 

 opposite ends of a diameter. If r l5 r 2 , l9 9 2 are the co- 

 ordinates of the two electrons when disturbed, we may put 



r 1 = a + l v 1 , r 2 = a-f^ 2 , 0i = nt+y l9 2 = nt + ir+y 2 , 



where x l9 x 2 , y±, y 2 are small quantities. 

 The equations of motion are 



*?»_, /^V- -J^l 4.P 



dt* *\dt) [ ri 2+>> ? 



r.dty 1 dt) " 



where P and (?) are the radial and tangential forces exerted 



