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TRANSACTIONS OF SECTION A. 633 
his investigation would apply, owing to the rapid diffusion of the motion, and he 
illustrates his paper by reference to a liquid m which separate vortex rings are 
arranged in a regular cubical order which would, as he says, be almost certainly 
subject to the diffusion of motion which would vitiate his investigation. A few 
‘years afterwards, however, Lord Kelvin published in the ‘Proc. of the R. I. 
Academy,’ 1889, vol. i. p. 340, a paper in which he described an arrangement of 
long, thin, empty, vortex filaments, which he considered would be stable, and not 
subject to the diffusion and mixing which would vitiate the application of his wave 
theorem to the first turbulent medium he suggested. I have this year, in the 
‘R. D. 8. Proceedings’ (p. 51), published a paper calling attention to the way in | 
which laminar waves might be propagated by this latter medium, and have sug- 
gested a way in which electrons might exist therein. The paperis only suggestive, 
and cannot claim to prove much. I now desire to call attention to the expressions 
that Lord Kelvin has given for the structure changes that take place when the 
waves he describes are being propagated through the medium, and to show how 
to calculate a quantity which must be proportional to the energy per c.c. of this 
wave-motion. I must refer to the paper itself for an explanation of the notation, 
as it would make this note very long to give it here. 
The equations from which Lord Kelvin deduces the possibility of the propa- 
gation of laminar motion through a turbulent liquid are two: 
(1) WD — axan — 
d aaa es 
2) 2Aav = —_ R= Fj 
In this, R? is the mean square of the velocity of the original turbulency of the 
liquid. 
The comparison of this with Maxwell’s equations is obvious, and f(y, t) may 
be either magnetic or electric action, and xzav(uv) will then be either electric or 
5) 
magnetic, It shortens matters to call 5 R=V*, f(y, =P, and rzav(ur)=y; 
so that the equations are 
aeney 
dt dy’ 
dy 2 dP 
heen ee 
and o a 
If now we take the quantity 
P? + a = 25 
and integrate it throughout space, and then determine its variations with time, 
we find a lee ne dP ady\ 107 70 
az ||) 22aayde= ||P + 7IV AG end 
= {fj Bee ny oy vada 
Integrating the second term under the integral by parts, and omitting the super- 
ficial terms which may be at infinity, or wherever energy enters the space under 
_ consideration, we get 
© || [2zacdyde= {\{P es + a andy =0. 
Hence we see that =, which is of the right dimensions, must be proportional 
