366 Proceedings of Royal Society of Edinburgh, [july 15 , 
The same summation and integration applied to (40) gives v for 
all values of t , x, y, z; and then by (38), (37), (39) we find corre- 
sponding determinant values of w and u. 
36. To give now an arbitrary initial value, w 0 , to the 2 -component 
of velocity, for every value of x , y, z, add to the solution ( u , v, w ), 
which we have now found, a particular solution (u\ v', w') fulfilling 
the following conditions : — 
v = 0 for all values of t , x, y, z ; 
id — W 0 - iv 0 for t = 0, and all values of x, y, z 
and to he found from (25) and (1), by remarking that v =0 makes, 
by (22), y/ = 0, and therefore (23) and (25) become 
du' 
dt 
did 
dt 
du' o , 
+/3 ^ =mvV 
• • ( 45 ), 
• - (46), 
Solving (46); just as we solved (21) by (32), (33), (34); and then 
realising and summing to satisfy the arbitrary initial condition, as 
we did for v in (40), (41), (42), we achieve the determination of 
id ; and by (1) we determine the corresponding u\ ipso facto satisfy- 
ing (45). Lastly, putting together our two solutions, we find 
u = u + u\ v = v, w = w + id , . . . (47), 
as a solution of (26) without (27), in answer to the first requisition 
of § 33. It remains to find u, t), It), in answer to the second 
requisition of § 33. 
37. This we shall do by first finding a real (simple harmonic) 
periodic solution of (21), (22), (23), (25), fulfilling the condition 
u = A cos wt + B sin wt 
v — C cos wt -j- 1) sin wt 
w = E cos wt + F sin wt 
u — Q l cos + sin wt 
v = (S, cos wt + 2) sin wt 
w = (£ cos wt + § sin wt 
when y = b 
■ (48), 
where A, B, C, D, E, F, 33, (£, 2), (S, § are twelve arbitrary 
functions of ( x , z). 
Then by taking 
of each of these after 
