120 
DR. G. R. GOLDSBROUGH OX THE INFLUENCE OF 
Whence, by (ll), 
a 
) 0 e 5 , 0 -^ 4 = -0 5 , o {22« ri 
s^r, S T 
This right -hand member is of the same order of value as the factor of sin 2t in (26). 
In this case, then, c may assume a high value. But it is noticeable that only at the 
limit of Maxwell’s relation is great instability to be found. 
When expression (27) has a value far from zero, either by virtue of the value of 
k/oi or the value of it is clear from (25) that 2 t is imaginary and hence c is 
imaginary, the solution being stable. 
It might be inferred from this that if values of rL 5 were chosen such that 
Maxwell’s relation were not fulfilled the effect of the satellite would be to stabilise 
what would otherwise be an unstable system. As pointed out already, however, the 
original equations and their solutions, as given here, simply give the motion of the 
particles in the vicinity of certain circles. In some cases the motion may be such 
that the particles depart rapidly from this zero circle; this we have termed 
instability. In other cases the solutions may indicate that the particles never move 
far from the zero circle ; and this type of motion we have termed stable. But it is 
clear that if a small arbitrary displacement were given to each of the particles in the 
latter case, nothing in this paper precludes the possibility of their departure finally 
from the zero circle. That is, they may be again unstable. What we have found 
here is a series of orbits for the particles when subject to the attractions of Saturn, a 
satellite, and one another. Those in which the particles have large inequalities result 
in collisions with the neighbouring rings of particles and hence a complete departure 
from their former positions. Those which have no large inequalities and hence avoid 
collisions with neighbouring rings of particles may yet prove unstable when an 
arbitrary disturbance is further imposed upon them. 
(ii) The particular integral. 
In the expression (24) there appears a denominator of the form 
(e li0 -on 2 ) (0 5il) -m 2 ) -4Af.(30) 
Here on takes all positive integral values including zero. When the conditions are 
such, therefore, that expression (30) is approximately zero, the term in the particular 
integral will become very great and departure from the orbit will result. This 
expression is the same as (29), which, it has been pointed out, gives the positions of 
the unstable solutions of the complementary function. It may therefore be said that 
all the unstable positions are in the vicinity of the zero values of (30), and the 
following remarks apply equally to both parts'of the solutions. 
Referring to the form (29) it is seen that there are two variables, k/oi and rL^.. For 
a given value of rL s there are in general two values of k/ n , and for a given value of k/oi 
there are two values of iL s . In the figure (p. 125), the relation between k/oi and kL s 
is shown graphically, only those values of vL s which satisfy Maxwell’s criterion being 
