118 
DR. G. E. GOLDSBEOUGH ON THE INFLUENCE OF 
§ 4. Discussion of the Solutions of the Equations for the Case of Equal Particles. 
(i.) The complementary function. 
Equations (17) and (18) which determine the value of the exponent c, may be 
. re-written here, 
(e li0 —w 2 ) (0,, o —?T) = 4/cV+{|(e 5i0 -n 2 )e li2)l 
— KnQ 2 ,2n + KnQ 4 '2n — \ («1,0 _ ^) ©5,2/i} C0S 2 t 
2c((q, u 0 r ,, o -W 4 ) = {^( 05 ,o-W 8 ) 0i,2#-^ 3 0a,2» 
+ KU-e h 2n - f)i (a lt . 0 - n 2 ) e 3i 2 n\ sin 2 r J 
In these a 1<0 is determined by the relation 
(«i,o-W 2 ) (e 5 ,n-W 2 ) = 4d)f. 
It is noticeable that the coefficient of sin 2r in (26) is n times that of cos 2 t in (25). 
Owing to the smallness of the quantities 0 (excepting 0 1O and 0 5O ), it is clear that 
the coefficients of cos 2 t and sin 2r are both very small quantities. Now real values 
of c are only given by real values of r, and conversely. Hence in order that (25) 
may give real values of r it is necessary that the expression 
(0i,o-^ 2 ) ( 05 ,o-^ 2 ) -4 K 2 n 2 . .(27) 
should be less than, or at most equal to, the coefficient of cos 2t. That is, the real 
values of c will be in the vicinity of these values of k that make (27) vanish. The 
actual limits of the zone in which real values of c are found will be given by 
(0i,o — n 2 ) ( 00,0 — rf2 ) = 4/obr± ||-(0 5 , o —w 2 )0i, 2 « + /cr60o 2 „±^n0 4i2 „- 
i(«i,o — 005 . 2 ,,} • ( 28 ) 
There are four groups of signs possible in this expression, and there will result four 
values of k. The outermost and innermost of these will define the zone in which 
some real value of c appears, and this zone will be the zone of instability. Owing, 
however, to the extreme smallness of the coefficient of sin 2 t in (26), it is clear that c 
will be extremely small, in general; that is, the modulus of instability will be small 
and departure from the zone will be slow. In one case, however, c may be quite 
large. That is, when the coefficient of c, a liO 0 5)O —w 4 , is exceedingly small. 
Each of the quantities 0 is a function of or of k. Further 0, 0 and 0, 0 involve 
v a 
both the mass of the particles and the number of them. Both of these are entirely 
unknown. All that can be said is that Maxwell’s criterion, # that is, 
< 
P 
3’ 
* Tisserand, ‘ Mec. Celeste,’ vol. ii., p. 184. 
