﻿based on Free Electrons. 1085 



Table IV. — Some values of N c /w for a multiple core. 



u. cnf'2. ~Nc/n. 



10 3 862 -614 



20 0935 -503 



30 16-834 -439 



60 40-286 '329 



90 66-254 -264 



Table V. — Solutions for two axial ions. 



n. y. x N. »/N. 



2 4-588 1-803 -986 2027' 



3 3-378 1-360 1-448 2072 



4 2-786 1-210 1-865 2145 



5 2-356 1-138 2-331 2-145 



6 2-033 1-099 2-877 2-130 



7 1757 1-075 3-158 2-217 



8 1-493 1-060 3-595 2-225 



8* -754 1082 3-720 2-150 



7* -639 1-125 3-417 2-049 



6* -556 1-208 3-165 L896 



5* -460 1-552 3-233 1547 



The figures starred belong to the closer position of axial ions. 



Part II. — Natural Oscillations and Stability. 



§ 16. One main object in dealing with the natural oscil- 

 lations of a system is to ascertain whether it is intrinsically 

 stable or unstable. In the present case it is also important 

 to ascertain the periods, with a view to comparison with 

 spectroscopic results — electrical and optical. 



For the two-ring scheme the number of variables in the 

 oscillation problem is 6>i, of which the 2n referring to axial 

 oscillation stand separate from the rest. Variables attaching 

 to ions and electrons have very different coefficients of inertia, 

 but enter on like terms into the forces derived from potential 

 energy. As a consequence, the periods fall into two classes 

 (Pj, P 2 say) specially related to ions and electrons respec- 

 tively; and if 27r //?!&>, 27r/p 2 co are the periods, then m x , p\ 

 and m 2 p2 have values on the same scale dependent on n. 

 This allows us to halve the number of the equations required 

 to determine the separate types. Notwithstanding this 

 reduction, the problem is very laborious, and the amount of 

 work required for the case n = 2 suggested the inquiry 

 whether it would be possible to base a solution on asymptotic 

 forms, and so general rather than individual, though 

 restricted in application to larger values of n. 



