1920-21.] Stable Flow of a Fluid in a Uniform Channel. 145 
Equations (24) and (26) indicate that if the initial displacement is 
dr\ , dii 1 
dt and w 
yj yi 
purely in rj and rj', then -U and -A. are both zero. Similarly (23) and (25) 
show that if the initial disturbance is purely in £ and then ^ and ( ~~ 
are both zero. In each of these cases the system is neutral to the dis- 
turbance in question. This corresponds, of course, to the two zero values 
for X. 
The other expressions for X 2 being real, it follows that the arrangement 
is stable or unstable according as 
A 128U/ > , 2 7ra^- f , 
4 ( a — a) tan 2 — ^0, 
K7T 2 a 
%.e. as 
XJ (a -- a) ^ 7 r 7 ra 
— v ’ ^ — cot 2 
<32 
(29) 
If 2 h is the distance apart of the two vortices, h = a — a, and the criterion 
(29) takes the form 
UA •> 71 ■ , 9 7rh /OAN 
V<32 tan 2^ (30) 
according as there is stability or instability. 
§ 10. The inequalities (29) and (30) provide the criterion sought for, 
and furnish the analogue of the experimentally known critical value for 
TJa/v already referred to. At first sight there appears a serious dis- 
similarity between the two critical conditions; whereas Reynolds has 
found that for Ua/v less than a definite number stability existed, the 
analysis of the present paper shows that for U ol/k (or what is in effect the 
same, TJh/ic) greater than a definite quantity a stable state of affairs would 
exist. The inconsistency is, however, only apparent. For a legitimate 
comparison of the two conditions, equations (1) or (2) expressing k in terms 
of v and TJajp are required; k/v may be a comparatively complicated 
function of JJa/p, and the inequalities (29) and (30) would require to be 
transformed accordingly. If, for example, it were found that under certain 
circumstances k oc p(Ulfp 2 ), the inequalities in question would immediately 
revert to the Reynolds form. How to establish the appropriate relation 
of the form (1) or (2) is, however, a question for a future paper. For the 
moment it suffices to state that the criterion (29) or (30) determines the 
maximum intensity permissible for the two eddies, that they may stably 
maintain their arrangement for a given forward velocity of fluid. It should 
be noted that where TJh/ic is already above the critical, and therefore stability 
already exists, the gradual decay of the vortex strengths due to the 
VOL. xli. 10 
