194 SirW. Thomson on the Rectilineal Motion of 



^-component of velocity, for every value of x, y, z, add to the 

 solution (w, v, w) , which we have now found, a particular 

 solution (w', i/, w) fulfilling the following conditions : — 



v = for all values of t, x, y, z ; 



w f =zw — w for t = 0, and all values of x 



,V, Z J 



(44), 



and to be found from (25) and (1), by remarking that t>' = 

 makes, by (22), p' = 0, and therefore (23) and (25) become 



/iVV 



du' ~ du 



dw' ~ dw' _, , 



(45), 



(46). 



Solving (46); just as we solved (21), by (32), (33), (34); and 

 then realizing and summing to satisfy the arbitrary initial 

 condition, as we did for v in (40), (41), (42), we achieve the 

 determination of w' ; and by (1) we determine the corre- 

 sponding u', ipso facto satisfying (45). Lastly, putting 

 together our two solutions, we find 



TL — u + u' y v = v, w=w + w' . . . (47) 



as a solution of (26) without (27), in answer to the first 

 requisition of § 33. It remains to find U, V , H>, 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 



m = A cos cot-{- B sin cot \ "\ 



v = C cos cot + D sin cot 



w? = E cos cot + ¥ sin cot 



u = 51 cos cot + $8 sin cot 



v = 6 cos 0)^ + 2) sin cot 



w=(£ cos cot + § sin cot J J 



where A, B, C, D, E, F, % #, <S, ©, g, % are twelve 



arbitrary functions of (<#, z). Then, by taking I dcof(co) 



of each of these after 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 displacement 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 £ = 0, and 



when ?/ = 



when v = b 



>■ 



(48), 



