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Similarly equation (11) of the former paper becomes here

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From (5), (6), and from (8) in the former paper we obtain

o¿¿{u(u2+v2+w2)+u(352+n2+52)} +6p1⁄23u.

d

+ (3u2+v2+w2 +352+n2 + 52)

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a

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(p2)+2uvдл

a

(pE3) + 2 uv__ (pn2) + 2 uw
ду

+(3u2+v2+w2+352+n2 + 52)pX+2uvpY+2uwpZ,

whence, comparing with (4),

drx a

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дго

อะ

(p52)

(7)

a

P at +

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If the disturbance is not a very violent one, the first, the fourth, and the fifth member on the left-hand side may be omitted. Hence

Srz

P St

= ? ; (p§3(E2 +n2+5?) ) −(3E2+n2+52)

(p¥3), (9)

an equation which (under somewhat particular assumptions) was given by Maxwell. Let us write

D2 = §2 (§2+n2+52) — §2. (352 + n2 +52) ;

equation (9) becomes

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Now this equation would lead at once to results contrary to experience, as shown by Maxwell, unless D=0; accordingly the first term on the right-hand side may be dropped. And if 2, 77, and can be replaced each with sufficient approximation by (+n2+52), we shall have

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an approximate equation which is of secondary importance only in the subsequent calculation.

3. Let us now proceed to prove our principal equation.

Putting in (1) Q=(u+)2 and again Q=u+§ and multiplying by 2u we obtain (without neglecting any term)

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the symbols L, M, N being defined as follows :—

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We easily find, to a first approximation, L=0, and

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If we make 3p-p++) and introduce Lord Rayleigh's

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Returning to (29), multiplying by da dy dz and integrating throughout the volume occupied by the medium,

Jelly A dx dy dz=−

A dx dy dz = − SSA (lu+mv+nw) dS

ди де
дл

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+ § §}} (¥a2 + n2 + 52) F dx dy dz — § [ÎÏÑŒE 2 + n2 + 52 ) p { " + Z v

— § SSS p (E2 + n2 + 52) (1rz+mry+nr2)dS

a

a

+ 8 [ { pr= 3 + E + n2 + 5 ) + pr, dy (E+72+5)

ду

д
+pr. (P+7+52) } dz dy dz. (32)

Here the direction-cosines of the normal to the element dS of
the surface are denoted by l, m, n.

4. In order to interpret equation (32) let us adopt a some-
what generalized definition of "Kinetic Energy." Suppose
C (a vector) to represent any current or flux, and let q be its
velocity. We define then the kinetic energy per unit volume
to be the scalar product

S(Cq)=(C'g'+C"q"+C""q");

(33)

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C', C", and C being the components of C, and q', q", and q"
the components of q. Thus, if C means an ordinary flux of
matter, of density p, then C-pq, and the "kinetic energy
of matter, i. e. the kinetic energy in the case of a matter-flux,
is found as usually given. But we are now enabled to form
an idea of "kinetic energy" in other cases as well. Thus in
the case we are dealing with, the motion of a molecule
through space may be said to be equivalent to a "molecular
current" of the quantity Q carried about by the molecule;
and then Q, Q, and Q will be the values of the compo-
nents of that current. From (33) we conclude that the
kinetic energy of such a molecular current of Q is

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and the total kinetic energy of such currents per unit volume is

2

{N((§2+n2 +2)Q),

(35) N being the number of molecules per unit volume. Calling now M the mass of a molecule, let us consider the flux of the quantity Q=M(§2+n2 + 52): we see that (35) becomes equal to A as defined by (28); A therefore represents the total kinetic energy of molecular currents of (ordinary) molecular energy in unit volume, i. e. the kinetic energy of the motion of heat-energy in unit volume, and SA da dy dz represents the same quantity for the total fluid. The idea of "molecular currents" is likely to conserve a definite meaning even when the idea of "molecules" will be found to be superseded.

5. The energy of motion of the heat-energy is susceptible of several kinds of variation, from various sources, to which the consecutive terms of the right-hand side of (32) refer. The first term represents the loss by convection across the surface; the second the gain due to viscosity; the third expresses the reversible effect of the mean pressure doing work. The fourth term relates to the communication of heat through the surface, since pr., pry, and pr, are the values of the total component fluxes of energy. In order to find the meaning of the fifth term, let us substitute for the differential coefficients values from (12) and two other equations which can be written down from symmetry; then that term will be

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1 б

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8

§; ((pr,)") + 12. § ((pr.)")]dx dy dz, (36)

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pn2

عم

and represents therefore the source of variation due to interaction between molecules. This we shall call the "interior" source of variation, whereas the foregoing will be described as "exterior sources in fact the "interior" source remains active even in a fluid at rest when contained in a surface impermeable to heat. We now see that the direction of the interior variation depends on the nature of the mutual action between molecules. Since the quantities (pr), (pry)2, and (pr) are positive, the energy of the motion of heat-energy will be always decreasing if molecular interaction is such as to tend to diminish the absolute values of prz, pry, and prz; in the opposite case that energy will be always increasing. That it is the first case only that is realized in all fluids in Nature, as attested by the phenomenon of conduction of heat, cannot be deduced from Kinematical Theory. We have ascertained, as it were, the path of change of the energy of

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