486 EEPOKT— 1887. 



On the Vortex Theory of the Luminiferous ^Ether. (On the Propa- 

 gation of Laminar Motion through a turhnlently moving 

 Inviscid Liquid.) By Sir William Thomson, LL.D., F.R.S. 



[A communication ordered by the General Committee to be printed in extenso 

 among the Reports.] 



1. In endeavouring to investigate turbulent motion of water between two 

 fixed planes, for a promised communication to Section A of the British 

 Association at its Meeting in Manchester, I have found something seem- 

 ingly towards a solution (many times tried for within the last twenty 

 years) of the problem to construct, by giving vortex motion to an incom- 

 pressible inviscid fluid, a medium which shall transmit waves of laminar 

 motion as the luminiferous aether transmits waves of light. 



2. Let the fluid be unbounded on all sides, and let «, v, w be the 

 velocity- components, and^ the pressure at (.t, y, s, t). We have 



^ + ^;i + ^=o .... (1), 



dx dij dz 



du I du , du , dn , dp\ /c>\ 



^= — I M-— 4- V — + w— ■{---] . . . (2), 



dt \ d.c dj dz dxj ^ ^" 



d^ / dv dv dv dp\ ^o^ 



dt \ dx dij dz dyj 



dw I dio , du) , dw , dp\ .,^ 



iir-Vd^^''dy-^''d^^£) • • • (^'^^ 



From (2), (3), (4) we find, taking (1) into account, 



_ 2 dii^ dv'^ div- p A^w dw dw du dn dv\ .^^ 



dx"^ dy"^ rfs^ \dz dy dx dz dydzj ' 



3. The velocity-components u, v, w may have any values whatever 

 through all space, subject only to (1). Hence, on Fourier's principles, 

 we have, as a perfectly comprehensive expression for the motion at any 

 instant, 



«=2SSSSS a[l;X"^ sin (^mx + e) cos (ny+f) cos (q^ + g) . (C), 



■y=22:SSSE;8[';,'^,^^cos (mx + e) sin (ny+f) cos (q~ + g) . (7), 



tt-=SSSS2S y['^/,;,f)Cos (mx + e) cos (ny+f) sin (qz + g) . (8) ; 



where a^J, J ^j, ^(^m^,]^^-), 7(„',,„',5) are any three velocities satisfying the 

 equation 



and 2SS2SS summation (or integration) for different values of m, n, q, 

 e, f, g. The summations for e, /, g may, without loss of generality, be 

 each confined to two values : 6=0, and e=T,7r;/=0, and /=^7r ; g=0, 

 and gr=^7r. We shall admit large values, and infinite values of m~', n~', 

 q~^, under certain conditions [§ 4 (10), (11), (12), and § 15 below], but 

 otherwise we shall suppose the greatest value of each of them to be of 

 some moderate, or exceedingly small, linear magnitude. This is an 

 essential of the averagings to which we now proceed. 



