Rectilinear Vortices in Ring Formation. 632 
further, there is always an instability associated with the 
mode k=0. 
8. It remains to be seen whether we can obtain a greater 
degree of stability by a suitable choice of the ratio of «! to x, 
that is, with vortices of different strengths in the two rings. 
To avoid complicating the discussion we shall assume n even. 
Then the previous discussion suggests that we make the 
central mode stable, that is, we fix «’ by the condition that 
P,=P,' at k=3n. From (43) this gives 
2p” ram Qprt2 38 
a Epenya ~ +p") 4P Pop Oo 9 (47) 
The ratio of «' to «and the value of p are now determined 
by equations (37) and (47). 
Without examining the expressions in general a numerical 
example will show the nature of the results. 
Taking n=10, the appropriate roots of (37) and (47) are, 
approximately, 
p—0;1840 61 yamc)/ic—O728 enema neen (43) 
The following table shows the values of QQ’ and RR’ and 
of X2 for all the modes, calculated from (43) and (45) ; the 
values for k=6,7, 8, 9 are omitted, as they are the same as 
toe (oath By By al, 
k. QQ’. RR’. n3, 
—SS 
0) Sancabondeso 0 —411 0 85 
II sopseaanecce —61 —350 —318 —142 
Da daues esas -177 —72 — 307 —95 
ecueteees sas —292 —48 —545 —143 
AS eNer at eteecat — 369 —5 —475 —316 
Beconaq0co0as —396 0 —418 —418 
We see that the motion is stable in all the possible modes 
with the exception of k=0. Reverting to (40), we find that 
Kd/4ra2 =27rA/6°3T approximately where T is the period of 
rotation of the rings in the steady state ; thus the periods 
of the small oscillations in the stable modes range from about 
two-thirds to one-quarter of the period of rotation. 
It is easily verified that A2=0 in the mode £=0 corre- 
sponds to a neutral displacement of the system, consisting 
of a rotation and dilatation of the rings without alteration of 
the ratio of their radii. On the other hand, the root 
A2=85 in this mode gives rise to definite instability. 
