数理解析研究所講究録
第 1989 巻 2016 年 161-169

161

FLUID MOTION AND HEAT MOTION IN CLASSICAL FLUID DYNAMICS
流体数理冑典理論研究所増照茂

SHIGERU MASUDARESEARCH WORKSHOP ON CLASSICAL FLUID DYNAMICS,
EX. LONG-TERM RESEARCHER, INSTITUTE OF $MA’PHEMA’1^{\tau}ICAI_{I}$ SCIENCES, KYO ro UNIV.

ABSTRACT.
We discuss ,he historical $wav$ theories including the fluid dynamics, the heat theory, inthat of Fourier and Poisson. We think, from the heat theories, we have had themany mathernatically fruitful productions of derivatives from the heat theories, such as the$\dagger$$particular_{\backslash }$trigonometric series, the eigenvalue 1roblems and the $ral$ idly decrcaskg function.

Mathematics Subject$Classih^{\backslash }cati_{oI1}2010$: OIAxx: $76A02,$) $-02$$76?\vee’Ixx_{\dot{r}}7t\backslash$, 76-03, $33A15_{i}35Qxx$ 35-x :$35J10.$

Key words. Thc Navier-Stokes equations, fluid dynamics, fluid, hydrostaties, hydrodynamics, hydromechanics, thermodynamics, heat difusion equations, wave equations, mathematical history,
physico mathematics, trigonometric series, Fourier series.
$moeha\iota\backslash ics$

1. INTRODICTION
1,2,3

Fourier explains the motion of the hcat in the irxterior of solid. The diﬀerence is that
determines its increment of the temperature during an instant :Kdydz $d( \frac{dv}{dx})dt+Kanaz$ $a( \frac{d?J}{dy})dl+Kdxdyd(\frac{dt}{d\sim\sim})dt$$(d)_{F2_{cJ}^{r}}.

where,$\Rightarrow$$Kdxdyd_{\wedge} \prime(\frac{d^{2}.u}{dx^{2}}+\frac{d^{2}u}{dy^{2}}+\frac{d^{2}u\prime}{d_{\sim}^{2}7})dt$(1)\frac{du}{dt}=\frac{K}{C.D}(\frac{d^{2}u}{dx^{2}}+\frac{d^{2}u}{dy^{2}}+\frac{d^{2}u}{dz^{2}})$internal conductibility,
capacity,
density of the substance. ([3, pp.120-2]). We
think that Fourier’s deductive method is very diﬀuse style and simpler than Poisson’s inductive
method described over 10 pages in original [16], we shovx his poiat below in \S 3.1.
$C$

$D$

1.1. Poisson’s paradigm and singularity.
Poisson publishes the last books consist of three elements : [13, 14, 15, 16]. ([14, 15] are
the salne title and are divided into two volumes.) These are his paradigm of the mathematicalphysics through all his academic life, entitled a study of mathematical phisics. (Un Trait\’e dePhysiquematique.) In the rivalry to Euler, Lagrange, $La1^{1ace}$ , Fourier, Navier, et al.,$Matf\kappa’$)

Date: 2016/01/12.
lTranslation from Latin/French/German into Enghsh mine, except for Boltzmann.
establish time line of these contributor, we list for easy reference the year of their birth and death: Newton (1643-1727), Euler, $d’ Alernbelt(1717-83)$ , Lagrange(1736-1813), Laplace $(1749-1827),$ $Fourier(1768-$
1830), $p_{(\rangle}i\Re on(1781-1840)$ , Cauchy(1789-1857),, Riemann $(1826arrow 66)$ , Boltzmann(1844-1906),$2_{To}$$c\backslash$$(1707-8_{\backslash }{\})$.

$Diri く^{}\backslash ,h1et(1805arrow 59)$

Hilbcrt $(1862-1943)$ Schr\"odinger(1887-1961).
use
means our remark not original, when we want to avoid the confusions betweensic.
means our translation in citing the origin.
$3_{We}$$(\Leftarrow$ $)$$(J\})$$0\iota\iota r$

opinion and162
we think, he struggles to make his paradigm. On the other hand, as its proofs, there are somesingular but important sugestions such as:$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$rigorous sum instead of integral, (cf. \S 1.2)
critics to easy applying the rule comes from real to transcendental function, (cf. fig. 1)
conjecture on the defect of the proof in the eternity of exact diﬀerential,
contribution to the fluid dynamics, especially, to the Navier-Stokes equations, (cf. [7,pp. 261-271], Table 2, and 3)
deduction of another heat equation from the basically molecular analysis. (cf. \S 3 and \S
3.1)

1.2. A comment on continuum by Duhamel.
Duhamel 1829 [1] points out the theory of continuum from the viewpoint of scientific history,
citing from the Poisson’s paper in the argument with Navier on the nonsense of Navier’s nullaction in nature.
Up to now, the reserchers have considered the corps of the nature as continue, it makes illusion to this regards, however, partly because this hypothesis
simplify the calcul, and partly because they think that it gives a suﬃcient approximation. Mr. Poisson think that this hypothesis isn’t never admissible, andjustify his opinion with following considerations.
$(\Leftarrow$ $)$In this state, the distance which separate the molecules mustbe such that this condition were replaced, in having regard totheir mutual attraction and the caloric repulsion which we takealso among the molecular actions. However the corps is hard or
something solid, the force which opposes the separation of their
parties is zero or doesn’t exist in the state of which we discuss.
It doesn’t begin the existence that when we seek to eﬀectuate
this separation, and when we change only a few distance of themolecules. Namely, if we explain this force with integral it gets, it will be soto as its value being zero in the natural state ofeven if after the variation of the molecular distances, so that, thecorps will opposite any resistance to the separatiopn of its parties; this is what will be nonsense. It results from here, that thesum which explain the total action of a series of disjoint molecules
can’t convert the sum instead of the definite integral ; this is whatholds in the nature of the function of distances which represent
the action of each molecule. The molecular force, of which wewill find the expression in the \S 1 of this Memoire, is calculated
according to this principle, and reduced at least in the simplest[1, pp.98-99] (trans. andform of which it were susceptible.
italics mine.)$a$$co7y_{J}s$

2. $c\circ NFUSI0NS$ AND UNIFY ON CONTINUUM THEORY

The hysico-mathematicians are must construct at first the physical structure, then allpies
the mathematical concept on it. The former is necessary to fit with the actual phenomena.
Arago 1829 $[?]$ seeks to separate these items to Navier 1829 [8] in the current of dispute with
Poisson and Arago. This is comes from the word what-Navier-called l’une sur l’autre, he fails
to explain exactly it, and since then, his theories and the equations are neglected up to the topof the 20th century. We consider that the confusions and unify are as follows :Poisson and Fourier discuss on the handling of the De Gua’s theorem into the transcendental equations. Without clear explanation, Fourier passed away in 1830. cf. (fig.1)$\bullet$

163
On the attraction and replusion of molecule, Navier depends on Fourier’s principle ofheat molecule. The then hysico-mathematiciams had little evaluated Navier until the
top of the 20th century. For formulation of heat motion in the fluid, Fourier cites not
Navier’s fluid equations, but
fluid equations.
The hydrodynamists like Navier, Poisson, Cauchy are propose the wave equations in theelasticity, and the last two hydrodynamists proposes the total equations in unity on thecontinuum.
On the formulation of heat motion in the fluid, Fourier had submitted this paper, however, until his death, he has not published it,which he seems to aim the unity ofhydro- and thermodynamics, however, he has given up it.
We show the diﬀerence between Poison and Fourier in applying the rule of De Gua, which is
reduced for real to the transcendental. Fourier shows only real case in $a=2,$ $b=1$ in Poisson’s.
$\bullet$$Euler^{\rangle}s$$\bullet$$\bullet$$i_{R}$

fig.1 Diﬀerence of applying the De Gua’s theorem into the transcendental
between Poisson and Fourier

3. THE HEAT AND FLUID THEORIES IN THE $19TH$ CENTURY
Poisson [11] traces Fourier’s work of heat theory, from the another point of view. Poissonemphasizes, in the head paragraph of his paper, that although he totally takes the diﬀerent
approaches to formulate the heat diﬀerential equations or to solove the various problems or todeduce the solutions from them, the results by Poisson are coincident with Fourier’s.
I will take care of, through this Memoire, to cite the principle result which
Mr. Fourier have obtained before me ; and I dare to say at first, in all theparticular problems which we have taken the one and the another for examples,
and which being naturally indicated in this material, the formulae of my Memoire
coincides with that this piece includes. However, just only that there common
between our two oeuvres ; because, it were to formulate the diﬀerential equationsof the motion of the heat, or it were to solve them and deduce the definitive$\dot{u}$164solution of each problem, I am using the entirely diﬀerent methods from that Mr.
Fourier is tracing. $[11, pp.1-2]$ (trans. and italics mine.)

3.1. The deduction of hrat equations by Poisson.
Poisson deduces his heat equations of the motion in interior of solid corps or liquid with thefunction , which depend on the distance between the two molecules.
(\S 44.) There is always the heat in motion in all the corps, even when of all their points isinvariable,
were each point would have a particular temperature,
were its would have all a same temperature.
However, the expression motion of the heat is taken here, in the another sense ; it signifies thevariation of temperature which holds from an instant to the other in a corps which is heated oris cooled ; and the velocity of this motion, in each point of the corps, is the primary diﬀerential
coeﬃcient of the temperature with respect to the time.
I will call the corps solid or liquid, homogeneous or heterogeneous, in which we are going
to consider the motion of the heat. Let$M$ a certain point of $A,$
and $m$ a particle of this corps, of insensible magnitude (no. 7),
and take the point $M.$
At the end of a certain time
designate with $x,$
, the three rectangular coordinates of $M,$
with the volume of $m,$
and with its density,
so that we have $m=v\rho$ . Let also, at the same instant, the temperature and as 4 the velocity
of motion of the heat which responds to the point $M.$
, dependent on an equation in the partial
The quantity will be a function of
diﬀerences with respect to these four variables, which it is the problem to form. If is a corps
solid, and which we make neglect its small dilations, positive or negative, products with thevariations of relative to time, the coordinates $x,$
, according to independent of , and wewill have simply, as
If in contrast, we have regard to small displacement of the point $M$ caused from these
dilations,or also, if is a fluid in which the integrality of temperature, or all other cause, hold tothe motions of its molecules,, will be the function of ; and then we will have with the known
then the coordinates
rules of the diﬀerentiation of functions made of functions,5$R$$\bullet$$\bullet$$A$$\bullet$$\bullet$$\bullet$$t_{:}$$\bullet$$\bullet$

$y,$

$z$

$v$

$\bullet$$\rho$

$u$

$t,$

$u$

$x,$

$y,$

$z$

$A$

$u$

$t$

$z$

$y,$

$= \frac{du}{dt}.$$\bullet$$A$$\bullet$

$x,$

$y,$

$t$

$z$

(1)

$[S= \frac{du}{dt}+\frac{dx}{dt}\frac{du}{dx}+\frac{dy}{dt}\frac{du}{dy}+\frac{dz}{dt}\frac{du}{dz}$;(2)where, expression in which, will be the components of the velocities at the point $M,$
parallel to the axes $x,$ $y,$ . (Below, original citation omitted.)(\S 45.) Let $M’$ a second point of $A$ very near to $M$ , and $m’$ a particle ofof insensible
magnitude, like $m$ which will contain $M’$ . At the end of time , we call $x’,$ $y’,$ , the coordinates
of $M’$ in relating to same axes with $x,$ $y,$ , and designate with $u’$ the temperature of $m’$ ; also
$4ddt,$$\frac{dx}{dt}$$\frac{dz}{dt}$

)

$z$

$A$

$t$

$z’$

$z$

let the distance $MM’.$
According to the general hypothesis on which the mathematical theory of the heat (no. 7) isbased, there will be a continuous exchange of heat between $m$ and $m’$ . I will represent with$r$$\delta$

We use , because, in origin, Poisson uses the vertical type
like the opened shape in upper of thenumerical letter 8, however, this exact type isn’t in our LaTex font system.
sic. The function is repeated.
$4(\Downarrow)$$s_{(\Downarrow)}$$tJ$$of\propto$

165
the augmentation of heat which will result then for $m$ during the instantpositive or negative, during this instant,$dt$, namely, the excessof the heat emitted from $m’$ and absorbed with $m,$over the heat emitted from $m$ and absorbed with $m’.$
It will be able to suppose this excess proportional to product $m7n’dt$ , or to $vv^{f}p\rho’dt$ , incalling and the volume and the density of , so that we would have $m’=v’\rho^{;}$ , as we havealready $m=v\rho$ . It will be zero in the case of $u’=u$ , and same sign with the diﬀerence $u^{f}-u,$$w\}zen$ it won’t be zero ; in the $vacuum_{3}$ it will comethe reverse ratio of the square of ; andgenerally its value will be the form$\bullet$$\bullet$$v’$$\uparrow n’$$\rho’$$i1’1$(2)$r$$\delta=\frac{vv’}{r^{2}}R(u’-u)dt$(3),, whichwhere, in designating with $R$ a positive coeﬃcient, in whiclx we contain the factor
will decrease very rapidly for the values increasing with , which will be also able to depend on
materials and the temperatures of $m$ and $m’$ , and will vary with the direction of $MM’$ , when
the absorption of the heat won’t be the same in all direction around of $M.$
$p\rho’$$r$

In the supposition the most general,
of $M$ and $M’$ ; so that we will have$R$will be hence a function of$r,$$u_{\grave{ノ}}u$, and the coordinates$R=\Phi\langle r, u, u’, x, y, z, x’, y’, z$(Below, original citation omitted.)(\S 46.) The total augmentation of heat of $m$ during the instant will be the sum of values of, extended to all the point $M’$ of which the distance at the point $M$ is smaller. I will$vdt$
indicate a such sum with the characteristic . The factor
being common to all the value of, their sum will be$dt$$thax\backslash l$$\delta$$\Sigma$$\delta$$vdt \sum\frac{R}{r^{2}}(u’-u)v’$

.

(4)

as

However, during the instance , the temperature of $m$ augments with $dt$ ; if hence, we callits specific heat, $cvtJdt$ will be also its augmentation of heat during this instant ; hence insuppressing the common factor $vdt$ , we will have
$dt$

$c$

(3)

$c$

as

$= \sum\frac{R}{r^{2}}(u^{f}-u)v’$.

(5)

for the equation of motion of the heat equally applicable to asolid and to a liquid, in
substituting the convenient expression with as.
(Below, original citation omitted.)(\S 47.) Of the point $M$ as
and a radius equal to the linear unit, we describe a sphericalsurface ; were $ds$ the diﬀerential element of this surfaee, to which gets, the radius of which thedirection is that of $MM’$ , we will have $dv’=r^{2}drds$ ; and according to the value of the sum, the equation (5) will turn out$cox\cdot ps$$cex\iota ter$$\sum$(4)$c \frac{du}{dt}=\int\int R(u’-u)drds$;(6)

We put here, for abridgement, , instead of ; however, weremember that this diﬀerential
coeﬃcient needs to be taken with relation toto all this that depend ; so that it needs to, of the point $M$ will vary with
with the formula (2), when the coordinates
replacethe time.
The limit relative to of the integral contains in this equation (6) won’t be the same, according
to the distance of the point $M$ to the surface of will surpass or $wiX$ be shorter than thissmall segment. In this chapter we will suppose that this were the primary case which holds ;the integral relative to will come to be hence taken from $r=0$ to $r=l$ , in all the direction$\frac{du}{dt}$

$tJ$

$w$

$t_{\theta t1}d$$x,$$\frac{dc\iota}{dt}$

$y,$

$z$

$r$

$A$

$r$

$l$

166
around $M$ ; we will be able hence to describe the equation (6) under the form
(5)

$c\frac{du}{dt}=\int_{0}^{\ell}[\int R(u’-u)ds]dr$(7);where, the integral in respecting to $ds$ will come to be extended to all the element $ds$ from thespherical surface, and with the reduction in series, we will obtain easily the approximate value.

, the angles which the segment $MM’$ makes
For these things, I designate with
with the parallels to the axes
, traced through the point $M$ . Because of $MM’=r$ , thenit will result(\S 48.)$\alpha,$$x,$$\beta,$$\gamma$

$z$

$y,$

$x’-x=r\cos\alpha, y’-y=r\cos\beta, z’-z=r\cos\gamma$ ;and, according to the theory of Taylor, we will have$u’-u=$$\frac{du}{dx}r\cos\alpha+\frac{du}{dy}r\cos\beta+\frac{du}{dz}r\cos\gamma$$1 d^{2}u 2 1d^{2}u 2 1d^{2}u 2$$+ \overline{2}\overline{dx^{2^{r}}} \cos 2\alpha+\overline{2}\overline{dy^{2^{r}}} \cos 2\beta+\overline{2}\overline{dz^{2}}r \cos^{2}\gamma$$+$$\frac{d^{2}u}{dxdy}r^{2}cos$$a$$\cos\beta+\frac{d^{2}u}{dxd\approx}r^{2}\cos$a$\cos\gamma+\frac{d^{2}u}{dydz}r^{2}\cos\beta\cos\gamma$

If we develop similarly in accordance with the power and the products of $u’-u,$ $x’-x_{\rangle}y’-$$z’-z$ , we will have also
$R$

$y,$

$R=V+( \frac{dR}{du})(u’-u)+(\frac{dR}{dx})(x’-x)+(\frac{dR}{dy})(y’-y)+(\frac{dR}{dz’})(z’-z)+\cdots$;

where, the parentheses indicating here that it needs to put $u’=u,$ $x’=x,$ $y’=y,$ $z’=z$
according to the diﬀerentiation which supposes invariable, and designating this which comesat the same time from the function of the (no. 45), so that we have
$V$

$r$

$\Phi$$V=\Phi(r, u, u, x, y\}z, x, y, z)$.

(8)(Below, original citation omitted.)(\S 49.) (General equation of the motion of heat) 6
In this hypothesis, we will stop the development of $R$ at the terms dependent on the squareof exclusively. By reason of the system of $R$ in respect to and $u’,$ and $x’,$ and $y’,$ and
$r$

$z’$

$u$

, and of this one which

$V$

$x$

$z$

$y$

represents, we have evidently$( \frac{dR}{du})=\frac{1}{2}\frac{dV}{du}, (\frac{dR}{d\alpha})=\frac{1}{2}\frac{dV}{dx}, (\frac{dR}{dy})=\frac{1}{2}\frac{dV}{dy}, (\frac{dR}{dz’})=\frac{1}{2}\frac{dV}{dz}$;then, it will result hence$R=V+ \frac{1}{2}\frac{dV}{du}(u’-u)+\frac{1}{2}\frac{dV}{dx}(x’-x)+\frac{1}{2}\frac{dV}{dy}(y’-y)+\frac{1}{2}\frac{dV}{dz}(z’-z)$;and of this value jointed to that of $u’-u$ , we will conclude$H_{2}=$$+$$\frac{1}{2}[V\frac{d^{2}u}{dx^{2}}+(\frac{dV}{du}\frac{du}{dx}+\frac{dV}{dx})\frac{du}{dx}]\int c\circ s^{2}\alpha ds+\frac{1}{2}[V\frac{d^{2}u}{dy^{2}}+(\frac{dV}{du}\frac{du}{dy}+\frac{dV}{dy})\frac{du}{dy}]\int\cos^{2}\beta ds$$\frac{1}{2}[V\frac{d^{2}u}{dz^{2}}+(\frac{dV}{du}\frac{du}{dz}+\frac{dV}{dz})\frac{du}{dz}]\int\cos^{2}\gamma ds,$

This article is the most frequently referred from other article, such as 52, 58, 64, 68, 70, 76, 85, 89, 117,
119, 120, 137, 162. (These are the article numbers, referred to the no. 49, and in the bold numbers, the anotherequations are expressed.)$6(\Downarrow)$

167
or more simply$H_{2}= \frac{1}{2}[V\frac{d^{2}u}{dx^{2}}+\frac{dV}{dx}\frac{du}{dx}]\prime\cos^{2}\alpha ds+\frac{1}{2}[V\frac{d^{2}u}{dy^{2}}+\frac{dV}{dy}\frac{du}{dy}]\prime\cos^{2}\beta ds$$+ \frac{1}{2}[V\frac{d^{2}u}{dz^{2}}+\frac{dV}{dz}\frac{du}{dz}]\prime\cos^{2}\gamma ds$

the partial diﬀerences 7 of with espect toof these three coordinates, and without varying
We have additionally
$V$

$x,$

$1’$

$y,$

;
$z$

, being taken in considering$u$

as a function$r.$$\int\cos^{2}\alpha ds=\int\cos^{2}\beta ds=\prime\cos^{2}\gamma ds.$

Moreover, if we callthe angle which
the plane of the segment $MM’$ and of a parallel$M$
to the axis of traced through the point , with a fixed plane traced through this parallel, we
will have $ds=\sin\alpha d\alpha d\psi$ ; and the integral relative to $ds$ will come to be extended to all thespherical surface, to which this element belongs, then it will result$ma1_{\iota^{r}}es$$\psi$$x$$\int\cos^{2}\alpha ds=\int_{0}^{\prime/r}\cos^{2}\alpha\sin\alpha d\alpha\int_{0}^{2pr}d\psi=\frac{4\pi}{3}.$in reducing the value of $\int R(u’-u)$ at the primary telm
equation (7) will come to be$8Hence_{\}}$$c \frac{du}{dt}$$=$$H_{2}r^{2}$of the series ?? , the
$($

$)$

$\frac{2\pi}{3}(\frac{d^{2}u}{dx^{2}}I_{0}^{l_{Vr^{2}dr}}+\frac{du}{dx}\int_{0}^{l}\frac{dV}{dx}r^{2}dr)+\frac{2rr}{3}(\frac{d^{2}u}{dy^{2}}\int_{\zeta)}^{l}Vr^{2}dr+\frac{du}{dy}\int_{0}^{l}\frac{dV}{dy}r^{2}dr)$$+ \frac{2\pi}{3}(\frac{d^{2}u}{dz^{2}}\prime_{0^{l}}Vr^{2}dr+\frac{du}{dz}\int_{0}^{l}\frac{dV}{d_{\tilde{\rho}}}r^{2}dr)$.

(9)

The function be\’ing zero for all the value of longer than , we will be able to nvw extend
the integral relative to beyond this limit, and if we want to be until $r=\infty$ . If we put also
$V$

$l$

$r$

$r$

$\frac{2\pi}{3}\int_{0}^{\infty}Vr^{2}dr\equiv k$

where,$k$will be a mction of$f\iota$$u,$$\frac{2\pi}{3}\int_{0}^{\infty}\frac{dV}{dx}r^{2}dr=\frac{dkarrow}{dx},$

$x,$

$y,$

$z$

,

(10)

, and we will have$\frac{2_{7}r}{3}\int_{0}^{\infty}\frac{dV}{dy}r^{2}dr=\frac{dk}{dy},$$\frac{2\pi}{3}\int_{0}^{\infty}\frac{dV}{dz}r^{2}dr=\frac{dk}{dz}$;in consequence, the general equation of the motion of the heat will come to be finally 9(7)

When all the point of$A$$c \frac{du}{dt}=\frac{d.k\frac{du}{dx}}{dx}+\frac{d.k\frac{du}{dy}}{dy}+\frac{d.k\frac{d\prime u}{dz}}{dz}$gets to a stationary state, we will have

.

(21)

$\frac{du}{dt}=0$, an then it will result$(i$$\frac{d.k\frac{du}{dx}}{dx}+\frac{d.k\frac{d\uparrow x}{dy}}{dy}+\frac{d.k\frac{du}{dz}}{dx}=0,$$\fbox{Error::0x0000}r_{(\Downarrow)}$$8(\Downarrow)$id.
According to [9, p.41, no.277],$\int\cos^{m}x\sin xdx=-\frac{\cos^{rn+1}x}{m+1}.$$9(\Downarrow)$

The expression (9) is reduced into
$c \frac{du}{dt}=(\frac{d^{2}u}{dx^{2}}k+\frac{du}{dx}\frac{dk}{dx})+(\frac{d^{2}u}{dy^{2}}k+\frac{du}{dy}\frac{dk}{dy})+(\frac{d^{2}u}{dz^{2}}k+\frac{du}{dz}\frac{dk}{dz})$$\Rightarrow$

(11).

168
for the equation relative to this stationary state.
(\S 50.) The equation (11) coincides with that which I found in years ago for the case of a
, however, in never supposing hence that the quantity depended on theheterogeneous$corps^{1\fbox{Error::0x0000}}$$k$temperature
If is a corps heterogeneous,
will depend only on
and the equation (11) will be changed as follows : 11
$u.$

$A$

$\bullet$

$k$

$u,$

$\bullet$(8)$c \frac{du}{dt}=k(\frac{d^{2}u}{dx^{2}}+\frac{d^{2}u}{dy^{2}}+\frac{d^{2}u}{dz^{2}})+\frac{dk}{du}(\frac{du^{2}}{dx^{2}}+\frac{du^{2}}{dy^{2}}+\frac{du^{2}}{dz^{2}})$In supposing that this quantity
(9)

$k$

.

(12)

were independent of , we could have the equation$u$$c \frac{du}{dt}=k(\frac{d^{2}u}{dx^{2}}+\frac{d^{2}u}{dy^{2}}+\frac{d^{2}u}{dz^{2}})$

,

(13)

12

which we give it ordinarily, and which is reduced, in the case of the stationary state, to an
equation independent of two quantities and , viz.,
$c$

$k$

$\frac{d^{2}u}{dx^{2}}+\frac{d^{2}u}{dy^{2}}+\frac{d^{2}u}{dz^{2}}=0$

.

(14)

13

Poisson puts also the another heat equations :(1)

where,$u$is the heat,$\frac{du}{dt}=a^{2}(\frac{d^{2}u}{dx^{2}}+\frac{d^{2}u}{dy^{2}}+\frac{d^{2}u}{dz^{2}})$

$k$

and

$c$

,

$\frac{k}{c}=a^{2}$

,

(15)

are the conductibility and the specific heat of the material.

4. CONCLUSIONS
Fourier doesn’t show the pricise deduction of the heat equation (1), while Poisson takes 9
pages to descrive it from \S 44 to \S 50. The diﬀefrence between Fourier and Poisson is the common
kernel function of molecular distance, which Poisson considers and manipulates in both fluid
motion and heat motion. Additionally, we concludes as follows :He presents the two constant theory’, which we assert, (id. cf. [7]) as visible in the
Navier-Stokes equations in 1831. After this, Stokes follows Poisson’s equation in 1849,
and Prandtl [18, 19] declears these equations as the ‘Navier-Stokes equations’ in the topof the twenty century.
He presents the heat kernel function in [16], which is equivalent, however more complex
than the fluid kernel function with the distance of fluid molecule : $f(r)$ .
$\langle$$\bullet$$\bullet$$\bullet$He proposes the alternative method of the definite integral, 14 instead of making theuniversal method of it, since by Euler, Lagrange and Laplace.

He shows the heat equation by deducing precisely, although Fourier’s series is the first,
however, its introduction isn’t deducing such as
or without demonstration.
Although his approach dues to the rivarly to the $Fourier’s$ theory, it brings up the derivative productions of the another solutions or thinking in making many breakthroughs
to $Fourier^{)}s$ method.

$\bullet$$Poisson^{\rangle}s$$\bullet$. Journal de l’\’Ecole Polytechnique, 19e cahier, page 87.

Poisson [11], [15, p. 677].
The second term of the right hand-side of the equation (12) is for $k=k(u)$ , in sic.
The equation (13) means, wheremeaning the Laplacian.
This function satisfying the equation (14) is called harmonic function. Poisson doesn’t mention theharmonic function, however, Poincar\’e [17, p.237] calls it so.
cf. [10], [15, pp.347-367].
$10_{sic}$$11(\Downarrow)$$12(\Downarrow)$$13(\Downarrow)$$14(\Downarrow)$$c \frac{du}{d{\}}=k\Delta u$$u$$\Delta$$(\Downarrow$

$)$

169
5. EPILOGUE
Poisson [16, pp.411-415] expects the earth warming before the Industorial Revolution up
to 17 years after. According to his speculation, in using this average rate of the increment per ayear is $0.22^{o}C$ , then we can estimate with this incretnent rate up to this year 2015, just at the
COP21, the temperature rises between 198 years, $2.447^{c)}C$ as follows :$1|i$$\frac{11.980-11.730}{17\frac{7}{1-}}\cross 198=\frac{0.22}{17.58333}\cross 198=2.477^{0}C.$This is what
the world.

$is$called the reason of the consensus about the increment of the earth$warmi_{1}\mathfrak{B}$in

REFERENCES[1] J.M.C. Duhaniel, (Book review) M\’en oire sur let le Mouvement des COmps elabtiques ; par M. Poiss
Bulletin des sciences athematiques, astromatiqucs, physiques et chimiques, 11(1829), 98-111. (The title
number : No.35. )[2] J.-B.-J. Fourier,analytique de la chaleur. Deuxl\’erne \’Edition, Paris, 1822. This is available by
G.Darbonx $[?]$ [Tome Premier] with colxunents).
Th\’eone
Fourier,[3] J.-B.-J.
chaleur.,,anatytique
de
la
1822.
http://gallica.bnf. $fr/ark:/12148/bpt6kl0455e8v$ ( Remark. This is the original book version of BnF :que $Natior/,ale$ France, namely, it seems the original book of [2] edited by G.Darboux. After p.32,
there are $4_{1}>ag(ae$ of pp.23-26,
are followed after p.26. Total pages are 639 plus the 2 pages of
figures, while the book [2] is consisted of 563 pages. )[4] J.-B.-.t. Fourier, Memoire d’Analyse sur le mouvement de la chageur dans les Fluedes, pp.507-514, Extrait desnotes manuscntes
es par l’auteur, pp.515-530 M\’emoires de l’Acade’mie royale des Siences, 12(1833),
607-530. (Lu: $4/sept/1820$ . $)arrow http$ ://gallica.bnf.fr/ark: $/12148/bpt6k3227s/\mathfrak{B}20(Da1^{\backslash }bo\mathfrak{U}x2[?], I)p.59_{\backslash }*614$ )S. Masuda, Histoncal development of classical fluid dynamics, Scholars’ Press, 2 く}14.
[6] S. Masuda, Histo7vcal development of classecal physico-mathematics, Scholars’ Pxess, 2014.
[7] S. Masuda, Confusion and Unity \’in Handling of Heat Mohon cxrnd Fluid Motnon in the 19th Century, RIMS
K\^oky\^uroku 1946, Kyoto zmiv. 2015/4.
[8J C.L.M.H. Navier, Lettre de M.Nnvier \‘a M.Arago, Annales \‘ae chinie et de hysique, 39(1829), 99-107. (Thisis followed by Note du Redacteur, 107-110.)[9] B. O. Peirce, A Short Table of Integruls, 4th ed., Brain Publishing, 1956, (Jap.)[10] S.D. Poisson, M\’emoire sur les inkg7ales definies, (1813), J. \’Ecole Polytoch., Calxier 16, 9(1813), 216-246.
httl $://$ gallica. $bnf.fr/ark:/12148/$}}$pt6k4336720/f220$[11] S.D. Poisson, M\’emoire sur la Distribution de la Chaleur dans les Corps solides, J. \’Ecole Royale Polytech.,Cahier 19, 12(1823), 1-144. (Lu: $i\}1/el\acute{e}c/1821.$ ) $arrow http://$ gallica. $bnf.h/wk:/121\mathfrak{W}/bpt6k433675h$[12] S.D. Poisson, Memoire sur les equations
de l’\’equilibre et du mouvement des corps sodades \’elasttques
et des fluides, (1829), J. \’Ecole Royale Polytech., 13(1831), 1-174. (Lu : $12/oct/1829$ .
[13] S.D. Poisson, Nouvelle the’orie de l.action capillaire, Bachelier P\’ere et Fils, Paris,
http: //gallica. bnf. $fi\cdot/ark:/12148/bpt6kll03201$[14J S.D. Poisson, ?
de M\’echanique, second edition, (1) Bachelier, $Imp_{l}imeur$-Libraire, Paris, 1833.
http: //gallica. bnf. &/ark: $/12148/bpt6k9605452x$[15] S.D. Poisson, TVaik de)rimeur-Libraire, Paris, 1833.
, second edition, (2), Bachelier,
http: //gallica. bnf fr/ark: $/12148/$ })pt6k9605438n[16] S.I). Poisson, The’ orie
de la chaleur, Bachelier P\’ere at Fils, Paris, 1835. $arrow http://www.e-$$\gamma$$(\rangle n,$$n\backslash$$Tl\kappa’ori$$\langle$$Pa1\dot{Y}S$$Bibhoth_{\grave{\mathcal{L}}}$$h_{oWeve1_{\backslash P^{33ff}}}.\cdot$$conse\gamma\eta\acute{k}$$[0^{r}]$$f^{i}$$arrow$$g\’{e} r\iota\acute{e}rulex$$)$$183\downarrow.$$\triangleright aik’$$arrow$$arrow$$\ddagger mr$$M\’{e} chart_{\fbox{Error::0x0000}}i,qt \iota e$$arrow$$math_{\mathcal{L}’}matr.que$$r\delta xa.ch/doi/10.3931/e-ral\cdot a-16666$[17] E. Poincar\’e, Th\’eorie analytique de la propagation, de la chaleur. Le\caons professe’es pendant le premier sementre 1899-1894, Georges Carr\’e, Paris. $arrow http://$ gallica. $bnf.fr/ark:/12148/bpt6\mathfrak{B}50070_{\sim}^{9}f$[18J L.Prandtl,
Fl\"ussigkeitsbewegung bei sehr kleiner Rpibung, in III. Internationaler Mathematiker-Kongressin Heidelberg vom 8. bis 13.
1904. Verhundlungen, A. Krazer ed.) 18&91, Leibzig, 1905. Also: Ludwig
Prandtl, Gesammelte Abhundlungen ur Mecchanik. Hydro-und Aerodynamik, vols 3(1961): G\"ottingen. vo12,575-584. (reacl 1904.)[19] L. Pra idtl, Fundamentals of hydrcy and $aeromechan’ics$ , McGrawhill, 1934. (Based on lectures of L.Prandtl(1929 by O.G.Tietjens, translated to English by L.Rosenhead. 1934.
Remark. Lu: accepted date, (ex. Lu: $12/oct/1829,$ in the bibliographies of French M\’emoire.
$\dot{\zeta}\dot{1}ber$$Au\mathfrak{W}^{st}$$\langle$$\approx$$)$$\rangle$$)$$15_{(\Downarrow)}$

As we know, the$h_{\lambda}du$

torial Revolution was occurred at about in 1830 in England.