82 Proceedings of the Royal Irish Academy. 



Eealizing by adding solutions of this type for ± % and + m with proper 

 values of G, one obtains types of complete real solution 



, , iExii.{ - vt{l^ + m^ + n-" - lm(3t + I'm-ji)] cos ,, , ,r, 



V = lie \-^ — f+(,.-//JO-..- sin ^^-^ ^ ^''' - '^'^y ' ^'^^] 



_ E.2M-vt(P^.n^^n^J _ M^tU^(5Hy^^^ cos _ ^ _^ ^ j 



(14) 



where k is an arbitrary constant. This gives, when t = 0, 



V = -y^ = --, „ : sin77i7/ (Ix + W2;), (15) 



^ l" + m^ + n^ -^ cos ^ ^ ^ ^ 



which fulfils (6) if sin nib = 0, and allows us, by proper summation, for the 

 different admissible values of on, and summation or integration with reference 

 to / and n, with properly determined values of k, after the manner of Fourier, 

 to give any arbitrarily assigned initial value to v for every value of x, y, z 

 from X = - <x> to a? = + oo , y = to y = 'b, and z = - cc to + oo , The same 

 summation and integration applied to (13) gives v for all values of x,y,z,t. 



It remains to find the value of i) which must satisfy (2), (7), (8). To 

 do this Lord Kelvin first finds a real (simple harmonic) periodic solution 

 of (2), fulfilling the conditions 



V = G cos (1)1 + D sin wt 



dv ^, ^, . \ when y = 0, (16) 



-^— = G cos (lit -^ D sm (Dt 

 dy 



V = a cos (Dt + '^ sin (i)t 



dv ^, [ when y = h, (17) 



— - == (i cos (i)t + ^ sm (,)t 



dy 



where G, D, G', D', S, 2), S', 3D' are eight assigned arbitrary functions of x, z. 



Then, by taking d(i)f{(i)) of each of these after the manner of Fourier, 



Jo 

 one solves the problem of determining the motion produced throughout 



the fluid, by giving to every point of its 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 = - co to t = 0, 



and each equal to minus the value of v or dY/dy, as the case may be, for every 



point of each boundary, when t > 0, we find to of equations (2), (3), (7), (8). 



* As far as v is concerned we bave only to deal with arbitrary boundary -values of v and of dv/di/, 

 the latter being obtained from those of u, w by the equation of continuity. 



