362 
Proceedings of Royal Society of Edinburgh, [july 15 , 
This and (8) are two equations for tire determination of vainly. 
Eliminating between them, we find 
a single equation which, with proper initial and boundary condi- 
tions, determines the one unknown, v. When v is thus found, (8), 
(7), (9) determine p, u, and w. 
30. An interesting and practically important case is presented by 
supposing one or both of the bounding planes to be kept oscillating 
in its own plane ; that is, E and § of (6) to be periodic functions of 
t. For example, take 
F = a cos wt , § = 0 . 
The corresponding periodic solution of (4) is 
(13). 
(b - y) +/£. 
to 
2;u 
v — a- 
b ./ - b . / 
E v 2j a - e v 
to 
2^ 
COS 
• • (14) 
In connection with this case there is no particular interest in sup- 
posing a current to be maintained by gravity ; and we shall there- 
fore take c — 0, which reduces (7), (8), (9), (11), (12) to 
du 
du 
dv 
dp 
dt 
+ 
v dx + 
dy V 
= fX\/ 2 U- 
dx 
dv 
+ 
dv 
0 
dp 
dt 
V dx 
= p v v - 
dy 
dw 
dw 
dy 
dt 
+ 
V dx 
II 
1= 
<1 
bs 
§ 
dz 
~dv dv 0 
d\] 2 v d 2 v dv d\? 2 v 
dt + dig 2 dx V dx 
fA\7*V 
• (15), 
■ (16), 
. (17), 
• (18), 
. (19); 
in all of which v is the function of (y, t ) expressed by (14). 
These equations (15) . . . (19) are of course satisfied byw = 0, 
v = 0, w - 0, p = 0. The question of stability is, Does every possible 
solution of them come to this in time ? It seems to me probable 
that it does ; but I cannot, at present at all events, enter on the 
