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Between each grain and its neighbours there is a true surface, the tension of which probably depends in some measure upon

the relationship of their axes to each other. Now we know that a by no means unimportant portion of the energy of a solid body depends upon its surface area ; and I would suggest also that it depends in some measure upon the surfaces separating the crystal units, of which it is composed, from each other, even when they are of similar composition.

In a block of glacier-ice, therefore, or in a block of granite, the total energy depends, not only upon its temperature and volume, but also upon the energy of the surface-tensions of

, its component grains, whether they be similar or dissimilar in composition.

Now it is an important theorem of Dynamics that, for the stable equilibrium of a system, the potential energy of the whole must be a minimum. For instance, when immiscible liquids are intimately mixed, work is done in increasing their surface-energy, and the energy thus rendered potential serves to again separate them. Glacier-ice is in much the same condition. Its potential energy is always tending to a minimum, a condition which can only be reached by the disappearance of the interfaces between the crystal-grains. Being viscous, it is unable to do more than delay the disappearance of the interfaces, and the grains would continue to grow larger and larger with the lapse of time were it not for the fact that they are broken in the production of the ribboned or veined structure.

For the same reason masses of all liquids possessing structure, whatever their viscosities may be, eventually become single crystalline particles.

Yours truly,

R. M. DEELEY.

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XLV. On the Kinetic Interpretation of the Dissipation

Function. By Dr. LADISLAS NATANSON, Professor of

Natural Philosophy, University of Cracow *. 1. In

N the following paper we consider a fluid medium from

Maxwell's point of view (see the paper “On the Dynamical Theory of Gases,” “Scientific Papers,' ii. 26); we suppose it to be composed of a great number of moving molecules. Let g, n, & be the components of the individual (or “ molecular”) velocity of a molecule; and let u, v, w be

* Translated from “ Rozprawy(Transactions) of the Cracow Academy of Sciences, Math. and Phys. Section, vol. xxix. Communicated by the Author.

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the components of the mean (or “molar”) velocity within an element dx dy dz of volume ; ndx dy dz may represent the number of molecules within that element. The mass of a molecule being M, p=Mn being the density of the medium, the kinetic energy of a molecule is I M{(u+)2 + (v+n)2 + (6 + 5)2};

(1) and the total kinetic energy of a portion of the medium consists of the two following parts (1) the kinetic energy of the visible motion, K=}SP(+1° +242 +wo) dx dy dz ;

(2) it is this energy that, in Hydrodynamics, is taken into account ; (2) the molecular or heat-energy, E=WPC & + m2 + **) dx dy dz.

(3) In these expressions the integrations are supposed to be performed throughout the volume occupied by the medium ; and gand similar symbols represent the mean values of g &c. for the molecules within an element. We have &=0; n=0; [=0.

(4) At the point (x, y, z) we have the normal component pressures,

Px=p?; Pyy=pm2; p=p52; (5) and the tangential,

Pyx=Pay=pms; Pet=Pxz=PÉE ; Pay=Pg==pn. (6) Let Q be any property of the molecule which can be expressed as a function of (u + 5), (v+n), and (w+8). Then writing d/dt for the actual or total variation of Q, and 8/8t for

any alteration of Q that can be due to the mutual interference of molecules,

dQ а
+ (

да
80

дQ

; ди

(7)

du X, Y, Z denote here the components, at (x, y, z), of acceleration due to external forces. From this fundamental equation (7) the equation of continuity, as well as the equations of

а
a

a
Pat
+ (52) + ( m8) + (258)=pX, .. (8)

ду

дг immediately follow. Let us now put

Q=(u + 5)2 +(0+n)2 + (w+$)?;

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p at

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dx

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+pX

tord

+pZ Jw

motion, du

(9)

t

T

ду

р

ду

We obtain

d P at

(u2 + v2 + wo +52 + m2 +52)
a

(2ιρξ? + 2υρξη + 2υρέζ +ρξ +ρξη +ρξξ')
2.c
a

(2υρηξ+ 2ιρη2 + 2ωρηζ+ρηξ? +ρη +ρης?)
(2ηρίξ + 2υρζη + 2υρζ +ρζξ+ρζη? +ρζί)

2up 8 }

8 =2p(uX+vY + WZ) + (^2 + y2 + w? + $? + m2 + 5%). (10)

St By (8) this equation is reduced to a

du де dw ip (&? + m2 +5°) +PEP

де

ρζ

8
де
duo

dw di ди dol +ρης

+p

+ + 'da ду

da

дут д.е.

a + .. (P88 +PE=+ PEK") + (ρηξ? +ρη + ρηζ2)

a
+ 2 (PSE* +p 502 +p5")

8
Pštw+m
(u? + v2 + x2 +52 + m2 + 32). .

(11) All these equations are quite independent of any view we may entertain as to the intimate nature of molecules and as to the law of force between them ; accordingly they may be said to belong to the Kinematical Part of the Theory of constitution of fluids.

2. If we suppose that the molecules of the fluid do not contain any internal energy (or that they coutain such energy in absolutely constant amount) the energy of a molecule will be only that which depends on the velocity of its centre of inertia. Assuming this equation (11) may be transformed. Pat for brevity, au av

де =b;

(12)

8
де dw де, ди du, do
+
=A;
=B;

=

C; . (13 az dy 2.C 2:

dy DU and let us add to the left-hand side

de
? +
?+

(11) dt where 0=a+b+c. Multiplying by dx dy dz and integrating

- የ

=a;

=0;

ду

+

+

.

=

throughout the volume occupied by the medium, we have then
JE
+38p13* + m2 + 3) (lu+mv + nw)ds

+
at
+ $[p&a+pn26+pf*c+pingA+PEB+p&m C]dx dydz=0; (15)
here the direction-cosines of the normal to the element ds of
the surface are denoted by l, m, n. If the medium cannot
flow across the surface, the second term on the left-hand side
may be omitted. Assuming this and calculating dK/at from
equations (8) in a similar way we find

a OK

-MS (p&a+p+b+p 5*c+pNGA+PE B+p&C]dx dy da
at

= SS P(uX+vY+wZ) dx dy dx ; (16)
honce
aK JE

+ MSP(uX +vY+wZ)dx dy , · · (17)
at

ді
i.e. molar energy is subject to change from two sources, the
first being the influence of external forces, the second being
the possible transformation of molecular energy E into molar
energy K or vice versâ. Let us denote the rate of change of

K
K due to the second source of variation by d'Krat; we have
ДЕ DK

- SMS[p&?a+pml +p8*c+pA+PŠĘB+p&nC]dx dy dz.(18) at

at
Let us put
3p=pP+pm? +p52.

• (19)
F=(p-pšaha +(p-pm?)6+(p-p52)c-onĞA-PSEB-PENC
=-ρξ? (a- ,θ) -ρη (1-3θ)-ρζ? (e-) - ρηζA - ρξξB-ρξης

} (20) =(p-p82)(a-10)+(p-on?)(1-10)+(p--P52)(c-30)

-ρηζA-ρξξB-ρξηC.
Equation (18) may then be written

ӘE
OK

SSS (F – po)da dy dz ; (21)
at at
the molecular energy E is therefore subject to change from
two sources : first, the work of the ordinary average pressure,
and second, the effect of a disturbance giving rise to tangential
pressures and to inequality of the norinal ones, F being the
rate (per unit of volume and time) at which molecular energy
is generated by the effect of the disturbance.
Put in (7)

Q=(u + 5)2 ; Q=(v+n)(w+$); . (22)

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.

S

.

a

.

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.

4p 8t

if the disturbance is not very violent, we shall find

8
(p-p?)= -2p(a–}0).

(23)
8

(png)=pa, st

(24)
and other equations which may be written down from
symmetry. Hence

1
F= {(p-p.) 2 + (p—pm?)? +(p-p52)

+2(95)2 + 2(255)2 + 2(87)}. • (25)
From this it appears that the direction of that transformation
of energy the rate of which is F depends on the nature of the
mutual action between molecules. Molar energy will become
converted into molecular energy if molecular interaction is
such as to tend to diminish the absolute values of the terms
p-?, ons and similar terms ; i.e. such as to tend gradually
to calm the disturhance existing : F is then always a positive
quantity and the opposite transformation of molecular energy
into molar energy is impossible. That this is precisely what
is realised in all fluids in Nature, as attested by the phe-
nomenon of viscosity, cannot, however, be deduced from
Kinematical Theory. We have ascertained, as it were, the
path of change of the molar energy, but we are unable to say
why one of the two possible directions of change is always
selected.
If now we suppose the following equations to be given :-
p-=2u(a-10),

(26)
ρηζ=-μΑ,

. (27) (the symbol je denoting a constant, the coefficient of viscosity), and four other equations, to be written down from symmetry, we shall find it possible to complete our solution. Maxwell deduced these equations from his well-known assumption as to the law of force between molecules; Poisson, Sir G. G. Stokes, and others have given them in the ordinary Theory of

Viscosity. It follows, then, that 1 F=={(P-pĚ) ? +}(p-pma)2 + 3(p-p52)2 +(995) + (p58)2 + (67)"},

}'

? ? =r{2(a–}0)2 +2(6-10)2 +2(0-10)? + A2+B+C2} 1

{f(pm –P52)2 + (982 – pĚ2)2 + ][ - pm?)2 + (ongga + (p5€)2 + (p&n)2} =r{3(6—c)2 +3 (c-a)? +3(a−b)2 + A2 + B2 +C2},

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2

(28)

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M

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