1887.] Sir W. Thomson on Stability of Fluid Motion . 367 
the manner of Fourier, we solve the problem of determining the 
motion produced throughout the fluid, by giving to every point of 
each of its approximately plane boundaries an infinitesimal displace- 
ment of which each of the three components is an arbitrary function 
of x, z, t. Lastly, by taking these functions each = 0 from t — - oo 
to t — 0, and each equal to minus the value of u, v, w for every point 
of each boundary, we find the U, ty, U) of § 33. The solution of 
our problem of ; § 32 is then completed by equations (31). To do 
all this is a mere routine after an imaginary type solution is pro- 
vided as follows : — 
38. To satisfy (21) assume 
v _ i {(ot-\-mx+qz^ 
= ^M+^+2^{ H€ ^(^ 3 +2 2 ) + K€-^ (m2+?2 ) + L/(y) + M F(y)} . (49), 
where H, K, L, M are arbitrary constants and /, F any two par- 
ticular solutions of 
+ </*)£ 
. (50). 
This equation, if we put 
mplfji = y, and ra 2 -}- cf + tw/'/x = A .... (51), 
becomes 
y^ = (\ + iy y)£ (52); 
which, integrated in ascending powers of (A + tyy), gives two par- 
ticular solutions, which we may conveniently take for our / and F, 
as follows : — 
/(s/)=i- — 
F (y) = A + tyy 
(A + iy y)\ y~ 4 (A + tyy) 6 y - g (A + ty//) 9 , ^ 
T2 — 6.5. 3.2 9. 8. 6. 5. 3. 2^ ' 
y- 2 (A + ty yY y- 4 (A + ty yf y “ 6 (A + tyy ) 10 
O 7. 6. 4. 3 10.9.7.6.4.3 
+ &c. 
(53). 
39. These series are essentially convergent for all values of y. Hence 
in (49) we have a solution continuous from y = 0 to y = b ; and by 
its four arbitrary constants we can give any prescribed values to V, 
and — for y = 0 and y = b. This done, find p determinately by 
dy 
(24) ; and then integrate (25) for w in an essentially convergent 
series of ascending powers of A + iyy, which is easily worked out, 
VOL. xiv. 31/12/87 2 A 
