﻿based on Free Electrons, 1097 



The second pair gives /r = or p 2 -f- 1 = 0. The first pair 

 gives p 4 + o> 2 /7 - 23/49 = 0, 



jt> 2 =(-3±4\/2)/7 = -3795 or —1*2367, 



the first root implying instability which attaches to radial 

 and tangential coordinates. Axial oscillation is here stable. 

 With core + 3e the quadratic. is altered to 



p 4 + 10p 2 /ll -35/121 = 0. 



It is not proposed to write out the work for three electrons 

 with cores + 2e, +3e, + 4e. The final equation is a quartic 

 o£ simple character. 



By way of exploration of the source of instability, these 

 problems were also solved with the single modification of 

 repulsion to attraction in the mutual action of electrons, 

 masses and intensities of force as before, with the result 

 that complete stability was found. 



§ 29. A search for further information in respect to R n (0) 

 may be pursued in two ways — either by examination of 

 individual cases as ?i = 4, 6, 8..., or by use of asymptotic 

 formulae. The tedious work which the former course would 

 entail seemed prohibitive, and in respect to the course 

 actually taken it must be understood that approximations are 

 based on the treatment of 1/n as a small quantity. The 

 treatment of axial displacement is simple, and yields 



TJ 2 /6 2 = 2{z-~ r 'y/2DrZ-Z(z-z s y/2I) s s . (35 a) 



r s 



as expression for the terms containing any one z, D s then 

 being distance between this element and any repulsive 

 element, or D s = 2asin sirjn ; and D r distance between 

 the same element and any attractive element, or approxi- 

 mately D r = 2a sin (2r + l)ir/2m 



The second order terms in potential energy which contain 

 displacements in the plane are more complicated, and here 

 approximation is needed. For two points whose distance in 

 steady motion is given by D 2 = a 2 + a' 2 — 2aa' cos^,-and for 

 which the increments of coordinates are (p, 0), (p , 0'), we 

 find the terms of second order in D -1 to be 



- O 2 + p" 2 - 2pp' cos ^)J2D Z + 3{ap + alp' 



— (ap' + a 1 p) cos ^Y J '2D 5 , 

 + (6-6')sin^[ap' + a'p-Zaa{ap + a'p' J> . (36) 



- (V + a' p) cos ^}/D 2 ]/D 3 , 

 -aa'(6>-6'') 2 [cosi/r-3«a'sin^/D 2 ]/2D 3 . 

 Phil. Mag. S. 6. Vol. 44. No. 264. Dec. 1922. 4 B 



