1887.] Sir W. Thomson on Stability of Fluid Motion. 365 
i[mx+ (n - mpt)y+qz ] 
v= -T-i 7 
m 2 + (n — m/3t ) 2 + q 2 
whence, by (22), 
i[m.r+(n - mpi)y+qz\ 
v = - 2/3miT r - e - . 
[m 2 + (w - m/3t) 2 + g' 2 ] 2 
Using this in (25), and putting 
w = }y e L[mx+(n-mpt)y+qz] 
we find 
dW 
dt 
- fx[m 2 + (/? - m/3t) 2 + q 2 ~\ W 
2/3mqT 
\m 2 + (n- mj3t) 2 + g 2 ] 
which, integrated, gives W. 
Having thus found v and w , we find u by (i), as follows 
(35) ; 
(36) . 
(37) , 
(38) , 
W = 
(n - 7nfit)v + qiv 
m 
(39). 
35. Realising by adding type-solutions for ± i and ± n, with 
proper values of C, we arrive at a complete real type-solution with, 
for v, the following — in which K denotes an arbitrary constant : — 
■ |ui[m2+rc2 + g2 _ nmpt+^rrfipW CQg 
m 2 + (n + 7n(3t ) 2 + q 2 s i n 
\mx + (n + 7n/3t)y + qz\ 
- y.t[m' 2 +n2+q‘ 2 +nmpt+im2pW] 
. \mx + (n + mpt)y + qz] J- (40). 
m 2 + (n + 7n(3t) 2 + q 2 sm 
This gives, when t — 0, 
v — 
+ K 
m 2 + n 2 + q 2 
which fulfils (28) if we make 
n = iiry/b (42); 
sin ny ^ 0 s( mx + F) ■ • • ( 41 )> 
and allows us, by proper summation for all values of i from 1 to oo , 
and summation or integration with reference to ra and q, with 
properly determined values of K, after the manner of Fourier, to 
give any arbitrarily assigned value to v t=0 for every value of a?, y, z, 
from x = — co 
„ y = ° 
to X = + 00 , 
„ y=b, 
„ z = + co . 
. . ( 43 ). 
