}. . . (28). 



192 Sir W. Thomson on the Rectilineal Motion of 



We proceed to find the complete solution of the problem 

 before us, consisting of expressions for u, v, w, p satisfying 

 (22) . . . (25) for all values of x, y, z, t ; and the following 

 initial and boundary conditions : — 



when t = : u, v, w to be arbitrary functions 1 ,nn\ . 



of x, y, z, subject only to (1) / \ '» 



w=0, v = 0, w = Q, for y = and all values of x, z, f\ 

 u = 0, v=0, w = 0, for y = b „ „ J 



33. First let us find a particular solution u, v, w, p, which 

 shall satisfy the initial conditions (26), irrespectively of the 

 boundary conditions (27), except as follows : — 



v = 0, when t = and y = 



v = 0, when £ = and y = b 



Next, find another particular solution, a, V, XV, p, satisfying 

 the following initial and boundary equations : — 



U = 0, 10 = 0, W = 0, when* = . . . (29); 



U + u=0, » + v=0, «j + w = 0, when*/ = ) 



and when y = b J 



The required complete solution will then be 



u = U + u, v = V + v, w = W + w . . . (31). 



34. To find u, v, w, remark that, if fi were zero, the com- 

 plete integral of (21) would be 



a = arb. func. (x—fyt) ; 



and take therefore as a trial for a type-solution with //. not 

 zero, 



(J . = .r£ e i[mx+(n—mpt)y+qz} (32)' 



where T is a function of t, and i denotes V — 1- Sub- 

 stituting accordingly in (21), we find 



|i = -/.[m 2 +(n~m^) 2 + ^]T. . . (33); 



whence, by integration, 



T=Ce L 3 J 



By the second of (21), and (32), we find 



i.[mx+(n-mpt)y+qz] 



V= ~ T m' + (n-ml3ty + f ' ' ' (36); 



(34). 



