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\Large\bfseries
Commentatio mathematica, qua respondere tentatur
quaestioni ab Ill${}^{\mathrm{ma}}$ Academia Parisiensi
propositae:
\begin{quote}
\large
``Trouver quel doit \^{e}tre l'\'{e}tat calorifique d'un corps
solide homog\`{e}ne ind\'{e}feni pour qu'un syst\`{e}me de courbes
isothermes, \`{a} un instant donn\'{e}, restent isothermes
apr\`{e}s un temps quelconque, de telle sorte que la
temp\'{e}rature q'un point puisse s'exprimer en fonction du temps
et de deux autres variables ind\'{e}pendantes.''
\end{quote}.\\[12 pt]
Bernhard Riemann\\[12 pt]
[Bernhard Riemann's Gesammelte Mathematische Werke, ed.
Heinrich Weber, 2nd edition, Teubner 1892, pp.~391--404.]\\[24 pt]
\large\mdseries
Transcribed by D. R. Wilkins\\[12 pt]
Preliminary Version: December 1998\\
Corrected: April 2000
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\title{Commentatio mathematica, qua respondere tentatur
quaestioni ab Ill${}^{\mathrm{ma}}$ Academia Parisiensi
propositae:
\begin{quote}
\normalsize
``Trouver quel doit \^{e}tre l'\'{e}tat calorifique d'un corps
solide homog\`{e}ne ind\'{e}feni pour qu'un syst\`{e}me de courbes
isothermes, \`{a} un instant donn\'{e}, restent isothermes
apr\`{e}s un temps quelconque, de telle sorte que la
temp\'{e}rature q'un point puisse s'exprimer en fonction du temps
et de deux autres variables ind\'{e}pendantes.''
\end{quote}}
\author{Bernhard Riemann}
\date{[Bernhard Riemann's Gesammelte Mathematische Werke, ed.
Heinrich Weber, 2nd edition, Teubner 1892, pp.~391--404.]}
\maketitle


\begin{flushright}
Et his principiis via sternitur ad majora.
\end{flushright}

\medbreak

\centerline{1.}

\nobreak\medskip

Quaestionem ab ill${}^{\mathrm{ma}}$ Academia propositam ita
tractabimus, ut primum quaestionem generaliorem solvamus:
\begin{quote}
quales esse debeant proprietates corporis motum caloris
determinantes et distributio caloris, ut detur systema linearum
quae semper isothermae maneant,
\end{quote}
deinde
\begin{quote}
ex solutione generali hujus problematis eos casus seligamus, in
quibus proprietates illae evadant ubique eaedem, sive corpus sit
homogeneum.
\end{quote}

\medbreak

\centerline{\large\bfseries Pars prima.}

\nobreak\medskip

\centerline{2.}

\nobreak\medskip

Priorem quaestionem ut aggrediamur, considerandus est motus
caloris in corpore qualicunque.  Si $u$ denotat temperaturam
tempore $t$ in puncto $(x_1, x_2, x_3)$ aequationem generalem,
secundum quam haec functio~$u$ variatur, hujus esse formae
constat,
\begin{equation}
\begin{array}{rcl}
\displaystyle
      \mathbin{\phantom{+}}
      \frac{\displaystyle \partial
         \left(
            a_{1,1} \frac{\partial u}{\partial x_1}
          + a_{1,2} \frac{\partial u}{\partial x_2}
          + a_{1,3} \frac{\partial u}{\partial x_3}
         \right)}{\partial x_1} \\[12 pt]
\displaystyle
    + \frac{\displaystyle \partial
         \left(
            a_{2,1} \frac{\partial u}{\partial x_1}
          + a_{2,2} \frac{\partial u}{\partial x_2}
          + a_{2,3} \frac{\partial u}{\partial x_3}
         \right)}{\partial x_2} \\[12 pt]
\displaystyle
    + \frac{\displaystyle \partial
         \left(
            a_{3,1} \frac{\partial u}{\partial x_1}
          + a_{3,2} \frac{\partial u}{\partial x_2}
          + a_{3,3} \frac{\partial u}{\partial x_3}
         \right)}{\partial x_3}
   &=& \displaystyle h \frac{\partial u}{\partial t}.
\end{array}
\tag{I}
\end{equation}
Qua in aequatione quantitates~$a$ conductibilitates resultantes,
$h$~calorem specificum pro unitate voluminis, sive productum ex
calore specifico in densitatem designant et tanquam functiones pro
lubitu datae ipsarum $x_1, x_2, x_3$ spectantur.  Disquisitionem
nostram ad eum casum restringimus, in quo conductibilitas eadem
est in binis directionibus oppositis ideoque inter quantitates~$a$
relatio
\[ a_{\iota,\iota'} = a_{\iota', \iota} \]
intercedit.  Praeterea quum calor a loco calidiore in frigidiorem
migret necesse est, ut forma secundi gradus
\[ \left(
      \begin{array}{ccc}
         a_{1,1}, & a_{2,2}, & a_{3,3} \\
         a_{2,3}, & a_{3,1}, & a_{1,2}
      \end{array}
   \right) \]
sit positiva.

\medbreak

\centerline{3.}

\nobreak\medskip

Iam in aequatione (I) in locos coordinatarum rectangularium
$x_1, x_2, x_3$ tres variabiles independentes quaslibet novas
$s_1, s_2, s_3$ introducamus.

Haec transformatio aequationis (I) facillime  inde peti potest,
quod haec aequatio conditio est necessaria et sufficiens, ut,
designante $\delta u$ variationem quamcunque infinite parvam
ipsius~$u$, integrale
\begin{equation}
\delta \int \! \! \! \int \! \! \! \int \sum_{\iota ,\iota'} a_{\iota ,\iota'}
         \frac{\partial u}{\partial x_\iota}
         \frac{\partial u}{\partial x_{\iota'}}
         \, dx_1 \, dx_2 \, dx_3
      + \int \! \! \! \int \! \! \! \int 2h \, \frac{\partial u}{\partial t}
         \, \delta u \, dx_1 \, dx_2 \, dx_3
\tag{A}
\end{equation}
per corpus extensum, solum a valore variationis $\delta u$ in
superficie pendeat.  Introductis novis variabilibus haec
expressio (A) transibit in
\begin{equation}
\delta \int \! \! \! \int \! \! \! \int \sum_{\iota ,\iota'} b_{\iota ,\iota'}
         \frac{\partial u}{\partial s_\iota}
         \frac{\partial u}{\partial s_{\iota'}}
         \, ds_1 \, ds_2 \, ds_3
      + \int \! \! \! \int \! \! \! \int 2k \, \frac{\partial u}{\partial t}
         \, \delta u \, ds_1 \, ds_2 \, ds_3
\tag{B}
\end{equation}
posito brevitatis causa
\[ \frac{
      \displaystyle \sum_{\iota ,\iota'} a_{\iota ,\iota'}
            \frac{\partial s_\mu}{\partial x_\iota}
            \frac{\partial s_\nu}{\partial x_{\iota'}}}{
      \displaystyle \sum \pm
            \frac{\partial s_1}{\partial x_1}
            \frac{\partial s_2}{\partial x_2}
            \frac{\partial s_3}{\partial x_3}}
   = b_{\mu, \nu},\quad
   \frac{h}{
      \displaystyle \sum \pm
            \frac{\partial s_1}{\partial x_1}
            \frac{\partial s_2}{\partial x_2}
            \frac{\partial s_3}{\partial x_3}}
   = k.\]

Quodsi formarum secundi gradus
\[ \mathrm{(1)}\qquad
   \left(
      \begin{array}{ccc}
         a_{1,1}, & a_{2,2}, & a_{3,3} \\
         a_{2,3}, & a_{3,1}, & a_{1,2}
      \end{array}
   \right) \qquad
   \mathrm{(2)}\qquad
   \left(
      \begin{array}{ccc}
         b_{1,1}, & b_{2,2}, & b_{3,3} \\
         b_{2,3}, & b_{3,1}, & b_{1,2}
      \end{array}
   \right) \]
determinantes sunt $A$, $B$ et formae adjunctae
\[ \mathrm{(3)}\qquad
   \left(
      \begin{array}{ccc}
         \alpha_{1,1}, & \alpha_{2,2}, & \alpha_{3,3} \\
         \alpha_{2,3}, & \alpha_{3,1}, & \alpha_{1,2}
      \end{array}
   \right) \qquad
   \mathrm{(4)}\qquad
   \left(
      \begin{array}{ccc}
         \beta_{1,1}, & \beta_{2,2}, & \beta_{3,3} \\
         \beta_{2,3}, & \beta_{3,1}, & \beta_{1,2}
      \end{array}
   \right) \]
invenietur
\[ A = B \sum \pm
            \frac{\partial s_1}{\partial x_1}
            \frac{\partial s_2}{\partial x_2}
            \frac{\partial s_3}{\partial x_3} \]
et
\[ \beta_{\mu, \nu} = \sum_{\iota ,\iota'} \alpha_{\iota ,\iota'}
            \frac{\partial x_\iota}{\partial s_\mu}
            \frac{\partial x_{\iota'}}{\partial s_\nu} \]
ideoque
\[ \sum_{\iota ,\iota'} \alpha_{\iota ,\iota'} \, dx_\iota \, dx_{\iota'}
   =  \sum_{\iota ,\iota'} \beta_{\iota ,\iota'} \, ds_\iota \, ds_{\iota'} \]
et
\[ \frac{h}{A} = \frac{k}{B}.\]

Unde facile perspicitur transformationem aequationis (I) reduci
posse ad transformationem expressionis
$\sum\displaystyle \alpha_{\iota, \iota'} \, dx_\iota \, dx_{\iota'}$.

Quae quum ita sint, problema nostrum generale hoc modo solvere
possumus, ut primum quaeramus, quales esse debeant functiones
$b_{\iota, \iota'}$ et $k$ ipsarum $s_1, s_2, s_3$, ut $u$ ab una
harum quantitatum non pendere possit.  Qua quaestione soluta
expressio $\sum \beta_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}$
formari poterit.  Tum ut, datis valoribus quantitatum
$a_{\iota, \iota'}$ et quantitatis $h$, inveniamus, num $u$
functio temporis et duarum tantum variabilium fieri possit et
quibusnam in casibus, quaerendum est, an expressio illa
$\sum \beta_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}$
in formam datam transformari possit; et hanc quaestionem infra
videbimus eadem fere methodo tractari posse, qua \emph{Gauss}
in theoria superficierum curvarum usus est.

\medbreak

\centerline{4.}

\nobreak\medskip

Primum igitur quaeramus, quales esse debeant functiones
$b_{\iota, \iota'}$ et $k$ ipsarum $s_1, s_2, s_3$, ut $u$ ab una
harum quantitatum non pendere possit.  Ut denotationem
simpliciorem reddamus, quantitates $s_1, s_2, s_3$ per
$\alpha, \beta, \gamma$ designemus et formam (2) per
\begin{equation}
   \left(
      \begin{array}{ccc}
         a, & b, & c \\
         a', & b', & c'
      \end{array}
   \right)
\tag{2}
\end{equation}
si $u$ a $\gamma$ non pendet, aequatio differentialis erit formae
\begin{equation}
      a \frac{\partial^2 u}{\partial \alpha^2}
    + 2 c' \frac{\partial^2 u}{\partial \alpha \, \partial \beta}
    + b \frac{\partial^2 u}{\partial \beta^2}
    + e \frac{\partial u}{\partial \alpha}
    + f \frac{\partial u}{\partial \beta}
    - k \frac{\partial u}{\partial t}
   = F = 0
\tag{II}
\end{equation}
posito
\[    \frac{\partial  a}{\partial \alpha}
    + \frac{\partial c'}{\partial \beta}
    + \frac{\partial b'}{\partial \gamma}
   = e,\quad
      \frac{\partial  b}{\partial \beta}
    + \frac{\partial c'}{\partial \alpha}
    + \frac{\partial a'}{\partial \gamma}
   = f.\]

Tribuendo ipsi $\gamma$ valores determinatos diversos ex
aequatione (II) inter sex quotientes differentiales ipsius $u$
obtinebuntur aequationes diversae, quarum coefficientes a
$\gamma$ non pendent.  Quodsi ex his aequationibus $m$ sunt a se
independentes
\[ F_1 = 0,\quad F_2 = 0,\ldots,\quad F_m = 0,\]
ita ut caeterae omnes ex iis sequantur, aequatio $F = 0$ necesse
est pro quovis ipsius $\gamma$ valore ex his $m$ aequationibus
fluat, unde $F$ formae esse debet
\[ c_1 F_1 + c_2 F_2 + \cdots + c_m F_m,\]
qua in expressione solae quantitates $c$ a $\gamma$ pendent.

Iam casus singulos, quando $m$ est $1$, $2$, $3$, $4$ paulo
accuratius examinemus simulque aequationes a $\gamma$
independentes, in quas aequatio $F = 0$ dissolvitur, in formas
simpliciores redigere curemus.

Casus primus $m = 1$.

Si $m = 1$, in aequatione (II) rationes coefficientum a $\gamma$
non pendebunt.  At introducendo in locum ipsius $\gamma$ novam
variabilem $\int k \, d\gamma$ semper effici potest, ut $k$ fiat
$= 1$, quo pacto coefficientes omnes a $\gamma$ evadent
independentes.  Porro introducendo in locos ipsarum
$\alpha$, $\beta$ novas variabiles semper effeci potest, ut $a$
et $b$ evanescant.  Hoc enim eveniet, si expressio
$b \, d\alpha^2 + 2 c' \, d\alpha \, d\beta + a \, d\beta^2$
(quae quadratum expressionis differentialis linearis esse nequit,
si (2) est forma positiva) in formam $m \, d\alpha' \, d\beta'$
redigitur et quantitates $\alpha'$, $\beta'$ tanquam variabiles
independentes sumuntur.

Aequatio igitur differentialis (II) hoc in casu in formam
\[ 2 c' \frac{\partial^2 u}{\partial \alpha \, \partial \beta}
      + e \frac{\partial u}{\partial \alpha}
      + f \frac{\partial u}{\partial \beta}
   = \frac{\partial u}{\partial t} \]
redigi potest et in forma (2) $a$, $b$ tum erunt $= 0$, $a'$ et
$b'$ functiones lineares ipsius $\gamma$, et $c'$ a $\gamma$
independens.  Caeterum patet temperaturam in hoc casu semper a
$\gamma$ independentem manere, si temperatura initialis sit
functio quaelibet solarum $\alpha$ et $\beta$.

Casus secundus, $m = 2$.

Sic aequatio (II) in duas aequationes a $\gamma$ independentes
discinditur, ope alterius
$\displaystyle \frac{\partial u}{\partial t}$
ex altera ejici potest.  Brevitatis causa haec ita exhibeatur
\begin{equation}
   \Delta u = 0,
\tag{1}
\end{equation}
illa
\begin{equation}
   \Lambda u = \frac{\partial u}{\partial t},
\tag{2}
\end{equation}
denotantibus $\Delta$ et $\Lambda$ expressiones characteristicas
ex $\partial_\alpha$ et $\partial_\beta$ conflatas.

Aequationem priorem facile perspicitur mutatis variabilibus
independentibus ita transformari posse ut sit $\Delta$
\begin{eqnarray*}
\mbox{vel}
   &=& \partial_\alpha \partial_\beta
         + e \partial_\alpha + f \partial_\beta \\
\mbox{vel}
   &=& \partial_\alpha^2
         + e \partial_\alpha + f \partial_\beta \\
\mbox{vel}
   &=& \partial_\alpha,
\end{eqnarray*}
valoribus $e = 0$, $f = 0$ non exclusis.

Quoniam sit
\[ 0  = \partial_t \Delta u = \Delta \partial_t u
      = \Delta \Lambda u,\]
ex his duabus aequationibus (1) et (2) sequitur
\begin{equation}
   \Delta \Lambda u = 0.
\tag{3}
\end{equation}

Iam duo distinguendi sunt casus, prout haec aequatio (3) vel ex
aequatione (1) fluat, ($\alpha$), sive sit
\[ \Delta \Lambda = \Theta \Delta \]
denotante $\Theta$ novam expressionem characteristicam, vel non
fluat, ($\beta$), novamque aequationem a $\Delta u$ independentem
sistat.

Casum priorem ($\alpha$) ut saltem pro una forma ipsius $\Delta$
perscrutemur, supponamus
\[ \Delta = \partial_\alpha \partial_\beta
      + e \partial_\alpha + f \partial_\beta.\]
Tum $\Delta \Lambda u$ ope aequationis $\Delta u = 0$ ad
expressionem reduci potest, quae solas derivationes secundam
alteram utram variabilem contineat et coefficientes omnes cifrae
aequales habere debeat.  Ponamus, quum terminis
$\partial_\alpha \partial_\beta$ continens ope aequationis
$\Delta u = 0$ ejeci possit,
\[ \Lambda
   = a \, \partial_\alpha^2 + b \, \partial_\beta^2
      + c \, \partial_\alpha + d \, \partial_\beta \]
formemusque expressionem
\[ \Delta \Lambda - \Lambda \Delta.\]
In hac expressione quum coefficientes ipsarum $\partial_\alpha^2$,
$\partial_\beta^2$ evanescere debeant, invenitur
$\displaystyle\frac{\partial a}{\partial \beta}  = 0$,
$\displaystyle\frac{\partial b}{\partial \alpha} = 0$,
unde si casus speciales $a = 0$, $b = 0$ excluduntur, mutatis
variabilibus independentibus effici potest, ut sit $a = b = 1$.
Tum autem invenitur ponendo coefficientes ipsarum
$\partial_\alpha^2$, $\partial_\beta^2$
in expressione reducta $\Delta \Lambda$ cifrae aequales
\[ \frac{\partial c}{\partial \beta}
   = 2 \frac{\partial e}{\partial \alpha},\quad
   \frac{\partial d}{\partial \alpha}
   = 2 \frac{\partial f}{\partial \beta},\]
unde poni potest
\begin{eqnarray*}
\Delta
   &=& \partial_\alpha \partial_\beta
      + \frac{\partial m}{\partial \beta}  \, \partial_\alpha
      + \frac{\partial n}{\partial \alpha} \, \partial_\beta \\
\Lambda
   &=& \partial_\alpha^2 + \partial_\beta^2
      + 2 \frac{\partial m}{\partial \alpha} \, \partial_\alpha
      + 2 \frac{\partial n}{\partial \beta}  \, \partial_\beta
\end{eqnarray*}
denotantibus $m$, $n$ functiones ipsarum $\alpha$, $\beta$, quae
jam duabus aequationibus differentialibus sufficere debent, ut
coefficientes ipsarum $\partial_\alpha$, $\partial_\beta$ in
expressione reducta $\Delta \Lambda$ evanescant.

Prorsus simili modo in reliquis casibus specialibus formae
simplicissimae ipsarum $\Delta$ et $\Lambda$ inveniuntur
conditioni
\[ \Delta \Lambda = \Theta \Delta \]
satisfacientes.  Sed huic disquisitioni prolixiori quam
difficiliori hic non immoramur.

Caeterum patet in hoc casu temperaturam semper a $\gamma$
independentem manere, si temperatura initialis est functio
quaelibet ipsarum $\alpha$ et $\beta$ aequationi
$\Delta u = 0$ satisfaciens, sequitur enim ex aequationibus
\begin{eqnarray*}
\Delta  u &=& 0 \\
\Lambda u &=& \frac{\partial u}{\partial t}
\end{eqnarray*}
$\displaystyle
0  = \Theta \Delta u = \Delta \Lambda u = \Delta \partial_t u
   = \frac{\partial \Delta u}{\partial t}$
et proin aequatio $\Delta u = 0$ subsistere pergit, si initio
valet et functio~$u$ secundum aequationem
$\displaystyle \Lambda u = \frac{\partial u}{\partial t}$
variatur.  Tum autem satisfit legi motus caloris sive aequationi
$F = 0$.

\medbreak

\centerline{5.}

\nobreak\medskip

Restat casus specialis alter ($\beta$) quando
$\Delta \Lambda u = 0$ a $\Delta u = 0$
est independens.  Ut simul et casus sequentes $m = 3$, $m = 4$
amplectemur, suppositionem generaliorem examinemus,
praeter aequationem $\Delta u = 0$ haberi aequationem
differentialem quamlibet linearem $\Theta u = 0$, ipsum
$\displaystyle\frac{\partial u}{\partial t}$
non continentem et a $\Delta u = 0$ independentem.

Si $\Delta$ est formae
$\partial_\alpha \partial_\beta + e \partial_\alpha + f \partial_\beta$,
ope aequationis $\Delta u = 0$ expressio $\Theta$ a
derivationibus secundum ambas variabiles liberari potest.

Iam duo distinguendi sunt casus.

Si ex expressione $\Theta$ omnes quotientes differentiales
secundum alteram utram variabilem ex.\ gr.\ secundum $\beta$
simul excidunt, obtinetur aequatio differentialis solos
quotientes differentiales secundum $\alpha$ continens formae
\begin{equation}
   \sum_\nu a_\nu \frac{\partial^\nu u}{\partial \alpha^\nu} = 0,
\tag{1}
\end{equation}
sin minus, semper elici poterit aequatio differentialis formae
\begin{equation}
   \sum_\nu a_\nu \frac{\partial^\nu u}{\partial t^\nu} = 0
\tag{2}
\end{equation}
sive solos quotientes differentiales secundum $t$ continens.

Nam in hoc casu expressiones
$\Lambda u, \Lambda^2 u, \Lambda^3 u,\ldots$,
quibus quotientes differentiales ipsius $u$ secundum $t$ aequales
sunt, ope aequationium $\Delta u = 0$, $\Theta u = 0$ semper ita
transformari possunt, ut solos quotientes differentiales secundum
alteram utram variabilem contineant eosque non altiores quam
$\Theta u$.  Quorum numerus quum sit finitus, eliminando
aequationem formae (2) obtineri posse manifestum est.
Coefficientes $a_\nu$ utriusque aequationis sunt functiones
ipsarum $\alpha$, $\beta$.

Observare conveniet, alteram utram harum aequationum semper
valere etiamsi $\Delta$ non st formae
$\partial_\alpha \partial_\beta + e \partial_\alpha + f \partial_\beta$.
Casus specialis, quando
$\Delta = \partial_\alpha^2 + e \partial_\alpha + f \partial_\beta$
ad utrumque casum referri potest, quum ope aequationis
$\Delta u = 0$ tum ex $\Theta u$, tum ex $\Lambda u$ omnes
derivationes secundam $\beta$ ejici possint, quo facto aequatio
utriusque formae facile obtinetur.  Si $f = 0$, hic casus sicuti
casus $\Delta = \partial_\alpha$ ad casum priorem referendus est.

Iam casum posteriorem accuratius perscrutemur.

Solutionem generalem aequationis
\[ \sum_\nu a_\nu \frac{\partial^\nu u}{\partial t^\nu} = 0 \]
e teminis formae $f(t) e^{\lambda t}$ conflatam esse constat,
denotante $f(t)$ functionem integram ipsius $t$ et $\lambda$
quantitatem a $t$ non pendentem, facileque perspicitur hos
terminos singulos aequationi (I) satisfacere debere.  Iam
demonstrabimus fieri non posse, ut sit $\lambda$ functio ipsarum
$x_1$,~$x_2$,~$x_3$.

Sit $kt^n$ terminus summus functionis $f(t)$ distinguanturque duo
casus.

1${}^{\mathrm{o}}$.
Quando $\lambda$ aut realis est aut formae $\mu + \nu i$ et
$\mu$,~$\nu$ functiones unius variabilis realis $\alpha$ ipsarum
$x_1$,~$x_2$,~$x_3$, substituendo $u = f(t) e^{\lambda t}$ in
parte laeva aequationis (I) coefficiens ipsius
$t^{n+2} e^{\lambda t}$ invenitur
\[ = k \left( \frac{\partial \lambda}{\partial \alpha} \right)^2
         \sum_{\iota, \iota'} a_{\iota, \iota'}
            \frac{\partial \alpha}{\partial x_\iota}
            \frac{\partial \alpha}{\partial x_{\iota'}}.\]

Sed haec quantitas evanescere nequit, nisi
\[    \frac{\partial \alpha}{\partial x_1}
   =  \frac{\partial \alpha}{\partial x_2}
   =  \frac{\partial \alpha}{\partial x_3}
   =  0 \]
sive $\alpha = \mbox{const.}$, quum forma
\[ \left(
      \begin{array}{ccc}
         a_{1,1}, & a_{2,2}, & a_{3,3} \\
         a_{2,3}, & a_{3,1}, & a_{1,2}
      \end{array}
   \right) \]
ut supra monuimus, sit forma positiva.

2${}^{\mathrm{o}}$.
Quando $\lambda$ est formae $\mu + \nu i$ et $\mu$,~$\nu$ sunt
functiones independentes ipsarum $x_1$,~$x_2$,~$x_3$, quantitates
$\mu + \nu i$ et $\mu - \nu i$ pro variabilibus independentibus
$\alpha$ et $\beta$ sumi poterunt continebitque ipsum $u$ praeter
terminum $f(t) e^{\alpha t}$ etiam terminum complexum conjugatum
$\varphi(t) e^{\beta t}$.  Quodsi
\[ \Delta u
   =     a \frac{\partial^2 u}{\partial \alpha^2}
       + b \frac{\partial^2 u}{\partial \alpha \, \partial \beta}
       + c \frac{\partial^2 u}{\partial \beta^2}
       + e \frac{\partial u}{\partial \alpha}
       + f \frac{\partial u}{\partial \beta} \]
est, ex aequatione $\Delta u = 0$ substituendo
$u = f(t) e^{\alpha t}$ et aequando coefficientem ipsius
$t^{n+2} e^{\alpha t}$ cifrae, obtinetur $\alpha = 0$ et perinde
$c = 0$ substituendo $u = \varphi(t) e^{\beta t}$.  Unde ope
aequationis $\Delta u = 0$ aequatio
$\displaystyle \Lambda u = \frac{\partial u}{\partial t}$
ita transformari potest, ut solos quotientes differentiales
secundum alteram utram variabilem contineat.  Sed substituendo
\[ u = f(t) e^{\alpha t},\quad u = \varphi(t) e^{\beta t} \]
coefficiens summi cujusque horum quotientium differentialium
invenitur $= 0$, unde et hi quotientes differentiales ex
aequatione
$\displaystyle \Lambda u = \frac{\partial u}{\partial t}$
omnes excidere debent, q.~e.~a., quum $u$ ex hyp.\ non sit
constans.

In casu igitur posteriori functio $u$ componitur e numero finito
terminorum formae $f(t) e^{\lambda t}$ in quibus $\lambda$ est
constans et $f(t)$ functio integra ipsius~$t$.

In casu priori quando habetur aequatio formae
\begin{equation}
   \sum_\nu a_\nu \frac{\partial^\nu u}{\partial \alpha^\nu} = 0,
\tag{1}
\end{equation}
functio $u$ erit formae
\[ u = \sum_\nu q_\nu p_\nu,\]
denotantibus $p_1, p_2,\ldots$ solutiones particulares
aequationis (1) et $q_1, q_2,\ldots$ constantes arbitrarias sive
functiones solarum $\beta$ et $t$.  Quodsi haec expressio in
aequatione
\[ \Lambda u = \frac{\partial u}{\partial t} \]
substituitur, obtinetur aequatio formae
\[ \sum PQ = 0,\]
in qua quantitates $Q$ sunt quotientes differentiales ipsarum $q$
ideoque functiones solarum $\beta$ et $t$, quantitates $P$ autem
functiones solarum $\alpha$ et $\beta$.  At tali aequationi supra
vidimus, si ex $n$ terminis componatur, subjacere $\mu$
aequationes lineares inter functiones $Q$ et $n - \mu$
aequationes inter functiones $P$, quarum coefficientes sint
functiones solius~$\beta$, denotante $\mu$ quempiam numerorum
$0,1,2,\ldots, n$.  Obtinebuntur igitur expressiones ipsarum
$\displaystyle \frac{\partial q}{\partial t}$
per quotientes differentiales ipsarum $q$ secundum $\beta$ ab
ipsa $\alpha$ liberae.

Iam casus singulos problematis nostri ad hunc casum pertinentes
perlustremus.

Quando $m = 2$ et $\Delta$ est formae
$\partial_\alpha \partial_\beta + e \partial_\alpha + f \partial_\beta$,
aequatio reducta $\Delta \Lambda u = 0$, si a quotientibus
differentialibus secundum $\beta$ libera evadit, formam induet:
\[ \frac{\partial^3 u}{\partial \alpha^3}
       + r \frac{\partial^2 u}{\partial \alpha^2}
       + s \frac{\partial u}{\partial \alpha}
   = 0,\]
unde $u$ erit formae
\[ ap + bq + c,\]
denotantibus $a$,~$b$,~$c$ functiones solarum $\beta$ et $t$, $p$
et $q$ autem functiones solarum $\alpha$ et $\beta$.  Iam in
locum ipsius $\alpha$ variabilis independens $q$ introduci
potest.  Quo pacto obtinetur
\[ u = a p + b \alpha + c,\]
ubi jam sola $p$ est functio ambarum variabilium $\alpha$ et
$\beta$.  Substituendo hanc expressionem in aequationibus
\[ \Delta u  = 0,\quad
   \Lambda u = \frac{\partial u}{\partial t} \]
coefficientium formae facile eruuntur.

Restat casus quando jam una aequationeum, in quas aequatio
$F = 0$ discinditur, formam (1) habet, ideoque formam
\[       r \frac{\partial^2 u}{\partial \alpha^2}
       + s \frac{\partial u}{\partial \alpha}
   = 0.\]
Tum erit $u = ap + b$, denotantibus $a$ et $b$ functiones solarum
$\beta$ et $t$ et $p$ functionem solarum $\alpha$ et $\beta$.  Si
in locum ipsius $\alpha$ variabilis independens $p$ introducitur,
prodibit
\[ u = a \alpha + b,\quad
  \frac{\partial^2 u}{\partial \alpha^2} = 0.\]

Invenimus igitur, si $m$ sit $= 2$ sive aequatio $F = 0$ in duas
aequationes
\begin{eqnarray*}
\Delta u  &=& 0 \\
\Lambda u &=& \frac{\partial u}{\partial t}
\end{eqnarray*}
dissolvatur, esse aut $\Delta \Lambda = \Theta \Delta$, aut
functionem $u$ composituam esse e numero finito terminorum formae
$f(t) e^{\lambda t}$, in quibus $\lambda$ constans et $f(t)$
functio integra ipsius $t$ est, aut formam induere
\[    \varphi(\beta, t) \chi(\alpha, \beta)
    + \alpha \varphi_1(\beta, t) + \varphi_2(\beta, t),\]
sit $m = 3$, functionem $u$ aut esse e numero finito terminorum
$f(t) e^{\lambda t}$ conflatam aut formae
\[ \varphi(\beta, t) \alpha + \varphi_1(\beta, t).\]

Casus denique $m = 4$ nullo negotio penitus absolvi potest.

Si enim praeter aequationem
$\displaystyle \Lambda u = \frac{\partial u}{\partial t}$
habentur tres aequationes inter
\[ \frac{\partial^2 u}{\partial \alpha^2},\quad
   \frac{\partial^2 u}{\partial \alpha \, \partial \beta},\quad
   \frac{\partial^2 u}{\partial \beta^2},\quad
   \frac{\partial u}{\partial \alpha},\quad
   \frac{\partial u}{\partial \beta},\]
aut prodibit aequatio formae
\[    r \frac{\partial u}{\partial \alpha}
    + s \frac{\partial u}{\partial \beta}
   = 0 \]
et proin variabiles independentes ita eligere licebit, ut $u$
fiat functio unius tantum variabilis, aut
\[ \frac{\partial^2 u}{\partial \alpha^2},\quad
   \frac{\partial^2 u}{\partial \alpha \, \partial \beta},\quad
   \frac{\partial^2 u}{\partial \beta^2},\]
ideoque etiam $\Lambda u$, $\Lambda^2 u$, $\Lambda^3 u$ per
$\displaystyle \frac{\partial u}{\partial \alpha}$,
$\displaystyle \frac{\partial u}{\partial \beta}$
exprimi poterunt.  Tum autem emerget aequatio formae
\[    a \frac{\partial^3 u}{\partial t^3}
    + b \frac{\partial^2 u}{\partial t^2}
    + c \frac{\partial u}{\partial t}
   = 0,\]
unde $u$ habebit formam
\[ p e^{\lambda t} + q e^{\mu t}
   + r \mathop{\mathrm{vel}} (p + qt) e^{\lambda t} + r \]
constatque per praecedentia $\lambda$ et $\mu$ esse constantes.

Iam sumta $p$ pro variabili independente $\alpha$ et substitutis
his expressionibus in aequatione
$\displaystyle \Lambda u = \frac{\partial u}{\partial t}$
invenitur fieri non posse, ut $q$ sit functio ipsius $\alpha$,
siquidem $\lambda$ et $\mu$ sint inaequales.  Ergo $p$ et $q$
vice variabilium independentium fungi possunt.  Praeterea ex
aequatione
$\displaystyle \Lambda u = \frac{\partial u}{\partial t}$
invenitur $r = \mbox{const.}$

In hoc igitur casu $u$ aut est functio ipsius $t$ et unius tantum
variabilis, aut alteram utram formarum
\[ \alpha e^{\lambda t} + \beta e^{\mu t} + \mbox{const.},\quad
   (\alpha + \beta t) e^{\lambda t} + \mbox{const.} \]
induet, valore $\mu = 0$ non excluso.

Postquam formae quas functio $u$ induere potest inventae sunt,
aequationes $F_\nu = 0$, quas brevitati consulentes perscribere
noluimus, facillimae sunt formatu.  Unde in singulis quibusque
casibus et forma
\[ \left(
      \begin{array}{ccc}
         b_{1,1}, & b_{2,2}, & b_{3,3} \\
         b_{2,3}, & b_{3,1}, & b_{1,2}
      \end{array}
   \right) \]
et forma adjuncta
\[ \left(
      \begin{array}{ccc}
         \beta_{1,1}, & \beta_{2,2}, & \beta_{3,3} \\
         \beta_{2,3}, & \beta_{3,1}, & \beta_{1,2}
      \end{array}
   \right) \]
innotescet.  Si jam in expressionibus
$\sum \beta_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}$
in locos quantitatum $s_1$,~$s_2$,~$s_3$ functiones quaelibet
ipsarum $x_1$,~$x_2$,~$x_3$ substituuntur, manifesto obtinebuntur
casus omnes, in quibus $u$ functio temporis et duarum tantum
varaibilium fieri possit.  Unde quaestio prior soluta erit.

Superest ut quaeramus, quando expressio
$\sum \beta_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}$
in formam datum
$\sum \alpha_{\iota, \iota'} \, dx_\iota \, dx_{\iota'}$
transformari possit.

\medbreak

\centerline{\large\bfseries Pars secunda.}

\begin{center}
{\itshape
De transformatione expressionis
$\sum \beta_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}$
in formam datum
$\sum \alpha_{\iota, \iota'} \, dx_\iota \, dx_{\iota'}$.}
\end{center}

\medskip

Quum quaestio ab Ill${}^{\mathrm{ma}}$ Academia ad corpora
homogenea restricta sit, in quibus conductibilitates resultantes
sint constantes, evolvamus primum conditiones, ut expressio
$\sum \beta_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}$,
aequando quantitates $s$ functionibus ipsarum~$x$, in formam
$\sum \alpha_{\iota, \iota'} \, dx_\iota \, dx_{\iota'}$,
constantibus coefficientibus $a_{\iota, \iota'}$ affectam
transformari possit.  Deinde de transformatione in formam
quamlibet datam pauca adjeciemus.

Expressionem
$\sum \alpha_{\iota, \iota'} \, dx_\iota \, dx_{\iota'}$,
si est, id quod supponimus, forma positive ipsarum $dx$, semper
in formam $\sum\limits_\iota dx_\iota^2$ redigi posse constat.
Unde si
$\sum \beta_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}$
in formam
$\sum \alpha_{\iota, \iota'} \, dx_\iota \, dx_{\iota'}$
transformari potest, redigi etiam pot\-est in formam
$\sum\limits_\iota dx_\iota^2$ et vice versa.  Quaeramus igitur,
quando in formam $\sum\limits_\iota dx_\iota^2$ transformari
possit.

Sit determinans
$\sum \pm b_{1,1} b_{2,2} \ldots b_{n,n} = B$
et determinantes partiales $= \beta_{\iota, \iota'}$; quo pacto
erit
$\sum\limits_\iota \beta_{\iota, \iota'} b_{\iota, \iota'} = B$
et
$\sum\limits_\iota \beta_{\iota, \iota'} b_{\iota, \iota''} = 0$,
si $i' \neq i''$.

Si
$\sum\limits_{\iota, \iota'} \beta_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}
   = \sum\limits_\iota dx_\iota^2$
pro valoribus quibuslibet ipsarum $dx$, substituendo $d + \delta$
pro $d$ invenitur etiam
$\sum\limits_{\iota, \iota'} b_{\iota, \iota'} \, ds_\iota \, \delta s_{\iota'}
   = \sum dx_\iota \, \delta x_\iota$
pro valoribus quibuslibet ipsarum $dx$ et $\delta x$.

Hinc si quantitates $ds_\iota$ per $dx_\iota$ et quantitates
$\delta x_\iota$ per quantitates $\delta s_\iota$ exprimuntur,
sequitur
\begin{equation}
\frac{\partial x_{\nu'}}{\partial s_\nu}
   = \sum_\iota b_{\nu, \iota}
         \frac{\partial s_\iota}{\partial x_{\nu'}}
\tag{1}
\end{equation}
et proinde
\begin{equation}
\frac{\partial s_\iota}{\partial x_{\nu'}}
   = \sum_\nu \frac{\beta_{\nu, \iota}}{B}
         \frac{\partial x_{\nu'}}{\partial s_\nu}.
\tag{2}
\end{equation}
Unde porro deducitur, quoniam sit
\[ \sum_\nu
      \frac{\partial s_\iota}{\partial x_\nu}
      \frac{\partial x_\nu}{\partial s_\iota}
   = 1
     \quad\mbox{et}\quad
   \sum_\nu
      \frac{\partial s_\iota}{\partial x_\nu}
      \frac{\partial x_\nu}{\partial s_{\iota'}}
   = 0, \mbox{ si $\iota \neq \iota'$},\]
\begin{equation}
   \sum_\nu
      \frac{\partial x_\nu}{\partial s_\iota}
      \frac{\partial x_\nu}{\partial s_{\iota'}}
   = b_{\iota, \iota'},\qquad
   \mathrm{(4)}\qquad
   \sum_\nu
      \frac{\partial s_\iota}{\partial x_\nu}
      \frac{\partial s_{\iota'}}{\partial x_\nu}
   = \frac{\beta_{\iota, \iota'}}{B}
\tag{3}
\end{equation}
et differentiando formulam (3)
\begin{equation}
   \sum_\nu
         \frac{\partial^2 x_\nu}{\partial s_\iota \, \partial s_{\iota''}}
         \frac{\partial x_\nu}{\partial s_{\iota'}}
      + \sum_\nu
         \frac{\partial^2 x_\nu}{\partial s_{\iota'} \, \partial s_{\iota''}}
         \frac{\partial x_\nu}{\partial s_\iota}
   =  \frac{\partial b_{\iota, \iota'}}{\partial s_{\iota''}}.
\tag{4}
\end{equation}

Iam ex his ipsarum
\[ \frac{\partial b_{\iota,  \iota' }}{\partial s_{\iota''}},\quad
   \frac{\partial b_{\iota,  \iota''}}{\partial s_{\iota'}},\quad
   \frac{\partial b_{\iota', \iota''}}{\partial s_\iota},\]
expressionibus eruitur
\begin{equation}
   2 \sum_\nu
         \frac{\partial^2 x_\nu}{\partial s_{\iota'} \, \partial s_{\iota''}}
         \frac{\partial x_\nu}{\partial s_\iota}
   =  \frac{\partial b_{\iota,  \iota' }}{\partial s_{\iota''}}
    + \frac{\partial b_{\iota,  \iota''}}{\partial s_{\iota'}}
    - \frac{\partial b_{\iota', \iota''}}{\partial s_\iota}
\tag{5}
\end{equation}
et si haec quantitas per $p_{\iota, \iota', \iota''}$ designatur
\begin{equation}
   2 \frac{\partial^2 x_\nu}{\partial s_{\iota'} \, \partial s_{\iota''}}
   = \sum_\iota \frac{\partial s_\iota}{\partial x_\nu}
         p_{\iota, \iota', \iota''}.
\tag{6}
\end{equation}
Quantitatibus $p_{\iota, \iota', \iota''}$ iterum differentiatis
obtinetur
\[    \frac{\partial p_{\iota, \iota', \iota'' }}{\partial s_{\iota'''}}
    - \frac{\partial p_{\iota, \iota', \iota'''}}{\partial s_{\iota'' }}
   =  2 \sum_\nu 
         \frac{\partial^2 x_\nu}{\partial s_{\iota'} \, \partial s_{\iota''}}
         \frac{\partial^2 x_\nu}{\partial s_\iota \, \partial s_{\iota'''}}
    - 2 \sum_\nu
         \frac{\partial^2 x_\nu}{\partial s_{\iota'} \, \partial s_{\iota'''}}
         \frac{\partial^2 x_\nu}{\partial s_\iota \, \partial s_{\iota''}},\]
unde tandem prodit, substitutis valoribus modo inventis (6) et (4)
\begin{equation}
\begin{array}{rl}
   \displaystyle \mathbin{\phantom{+}}
      \frac{\partial^2 b_{\iota, \iota''}}{
         \partial s_{\iota'} \, \partial s_{\iota'''}}
    + \frac{\partial^2 b_{\iota', \iota'''}}{
         \partial s_\iota \, \partial s_{\iota''}}
    - \frac{\partial^2 b_{\iota, \iota'''}}{
         \partial s_{\iota'} \, \partial s_{\iota''}}
    - \frac{\partial^2 b_{\iota', \iota''}}{
         \partial s_\iota \, \partial s_{\iota'''}} \\
   \displaystyle
    + {\textstyle\frac{1}{2}} \sum_{\nu, \nu'}
         (    p_{\nu, \iota', \iota'''} p_{\nu', \iota,  \iota''}
            - p_{\nu, \iota,  \iota'''} p_{\nu', \iota', \iota''} )
         \frac{\beta_{\nu,\nu'}}{B} &= 0.
\end{array}
\tag{I}
\end{equation}

Hujus modi igitur aequationibus functiones $b$ satisfaciant
necesse est, quando
$\sum b_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}$
in formam $\sum\limits_\iota dx_\iota^2$ transformari potest:
partes laevas harum aequationum designabimus per
\[ (\iota \iota', \iota'' \iota''').\]

Ut indoles harum aequationum melius perspiciatur, formetur
expressio
\[    \delta \delta
      \sum b_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}
    - 2 d \delta
      \sum b_{\iota, \iota'} \, ds_\iota \, \delta s_{\iota'}
    + d d
      \sum b_{\iota, \iota'} \, \delta s_\iota \, \delta s_{\iota'} \]
determinatis variationibus secundi ordinis $d^2$, $d \delta$,
$\delta^2$ ita, ut sit
\[    \delta'
      \sum b_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}
    - \delta
      \sum b_{\iota, \iota'} \, ds_\iota \, \delta' s_{\iota'}
    - d
      \sum b_{\iota, \iota'} \, \delta s_\iota \, \delta' s_{\iota'}
   = 0,\]
\[    \delta'
      \sum b_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}
    - 2 d
      \sum b_{\iota, \iota'} \, ds_\iota \, \delta' s_{\iota'}
   = 0,\]
\[    \delta'
      \sum b_{\iota, \iota'} \, \delta s_\iota \, \delta s_{\iota'}
    - 2 \delta
      \sum b_{\iota, \iota'} \, \delta s_\iota \, \delta' s_{\iota'}
   = 0,\]
denotante $\delta'$ variationem quamcunque.  Quo pacto haec
expressio invenietur
\begin{equation}
   = (\iota \iota', \iota'' \iota''')
         (    ds_\iota      \, \delta s_{\iota'}
            - ds_{\iota'}   \, \delta s_\iota )
         (    ds_{\iota''}  \, \delta s_{\iota'''}
            - ds_{\iota'''} \, \delta s_{\iota''} ).
\tag{II}
\end{equation}

Iam ex hac formatione hujus expressionis sponte patet, mutatis
variabilibus independentibus transmutari eam in expressionem a
nova forma ipsius
$\sum b_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}$
eadem lege dependentem.  At si quantitates $b$ sunt constantes,
omnes coefficientes expressionis (II) cifrae aequales evadunt.
Unde si
$\sum b_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}$
in expressionem similem constantibus coefficientibus affectam
transformari potest, expressio (II) identice evanescent necesse
est.

Perinde patet, si expressio (II) non evanescat, expressionem
\begin{equation}
    - {\textstyle\frac{1}{2}}
      \frac{\displaystyle
      \sum (\iota \iota', \iota'' \iota''')
         (    ds_\iota      \, \delta s_{\iota'}
            - ds_{\iota'}   \, \delta s_\iota )
         (    ds_{\iota''}  \, \delta s_{\iota'''}
            - ds_{\iota'''} \, \delta s_{\iota''} )}{
      \displaystyle
      \sum b_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}
      \sum b_{\iota, \iota'} \, \delta s_\iota \, \delta s_{\iota'}
       - \left(
            \sum b_{\iota, \iota'} \, ds_\iota \, \delta s_{\iota'}
         \right)^2}
\tag{III}
\end{equation}
mutatis variabilibus independentibus non mutari, insuperque
immutatam maneri, si in locos variationum
$ds_\iota$, $\delta s_\iota$ expressiones ipsarum lineares
quaelibet independentes
$\alpha \, ds_\iota + \beta  \, \delta s_\iota$,
$\gamma \, ds_\iota + \delta \, \delta s_\iota$
substituantur.  Valores autem maximi et minimi hujus functionis
(III) ipsarum
$ds_\iota$, $\delta s_\iota$
neque a forma expressionis
$\sum \beta_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}$
neque a valoribus variationum
$ds_\iota$, $\delta s_\iota$
pendebunt, unde ex his valoribus dignosci poterit, an duae
hujusmodi expressiones in se transformari possint.

Disquisitiones haece interpretione quadam geometrica illustrari
possunt, quae quamquam conceptibus inusitatis nitatur, tamen
obiter eam addigitavisse juvabit.

Expressio
$\sqrt{ \sum_{\iota, \iota'} b_{\iota, \iota'} \, ds_\iota \, ds_{\iota'} }$
spectari potest tanquam elementum lineare in spatio generaliore
$n$ dimensionum nostrum intuitum transcendente.  Quodsi in hoc
spatio a puncto $(s_1, s_2,\ldots, s_n)$ ducantur omnes lineae
brevissimae, in quarum elementis initialibus variationes ipsarum
$s$ sunt ut
\[ \alpha \, ds_1 + \beta \delta s_1 :
   \alpha \, ds_2 + \beta \delta s_2 :
   \, \ldots \, :
   \alpha \, ds_n + \beta \delta s_n,\]
denotantibus $\alpha$ et $\beta$ quantitates quaslibet, hae
lineae superficiem constituent, quam in spatium vulgare nostro
intuitui subjectum evolvere licet.  Quo pacto expressio (III)
erit mensura curvaturae hujus superficiei in puncto
$(s_1, s_2,\ldots, s_n)$.

Si jam ad casum $n = 3$ redimus, expressio (II) est forma secundi
gradus ipsarum
\[ ds_2 \, \delta s_3 - ds_3 \, \delta s_2,\quad
   ds_3 \, \delta s_1 - ds_1 \, \delta s_3,\quad
   ds_1 \, \delta s_2 - ds_2 \, \delta s_1,\]
unde in hoc casu sex obtinemus aequationes, quibus functiones $b$
satisfacere debent, ut
$\sum \beta_{\iota, \iota'} \, ds_\iota \, ds_{\iota'}$
in formam constantibus coefficientibus gaudentem transformari
possit.  Nec difficile, ope notionum modo traditarum, est
demonstratu, has sex conditiones, ut hoc fieri possit, sufficere.
Observandum tamen est ternas tantum esse a se independentes.

\nobreak\medskip

\centerline{\vrule height 0.2pt width 72pt}

\medbreak

Iam ut quaestionem ab Ill${}^{\mathrm{ma}}$ Academia propositam
persolvamus, in his sex aequationibus formae functionum~$b$,
methodo supra exposita inventae, sunt substituendae, quo pacto
omnes casus invenientur, in quibus temperatura $u$ in corporis
homogeneis functio temporis et duarum tantum variabilium fieri
possit.

Sed angustia temporis non permisit hos calculos perscribere.
Contenti igitur esse debemus, postquam methodos quibus usi sumus
exposuimus, solutiones singulas quaestionis propositae
enumerasse.

Si brevitatis causa casum simplicissimum, quando temperatura~$u$
secundum legem
\begin{equation}
      \frac{\partial^2 u}{\partial x_1^2}
    + \frac{\partial^2 u}{\partial x_2^2}
    + \frac{\partial^2 u}{\partial x_3^2}
   = a a \frac{\partial u}{\partial t}
\tag{I}
\end{equation}
variatur, solum respicimus, ad quem casus reliquos facile reduci
posse constat: casus $m = 1$ tum tantum evenire potest, quando
$u$ est constans aut in lineis rectis parallelis, aut in circulis
helicibusve, ita ut coordinatis rectangularibus
$z$, $r \cos \varphi$, $r \sin \varphi$ rite electis, poni possit
$\alpha = r$, $\beta = z + \varphi \mathbin{.} \mbox{const.}$

Casus $m = 2$ locum inveniet si
$u = f(\alpha) + \varphi(\beta)$,
casus $m = 3$ si
$u = \alpha e^{\lambda t} + f(\beta)$,
denotante $\lambda$ constantem realem, casus denique $m = 4$, ut
jam supra invenimus, si $u$ est aut
$= \alpha e^{\lambda t} + \beta e^{\mu t} + \mbox{const.}$,
aut
$= (\alpha + \beta t) e^{\lambda t} + \mbox{const.}$,
aut $= f(\alpha)$.

Iam ut formae functionis $u$ penitus innotescant, annotari tantum
opus est, temperaturam $u$, nisi sit formae
$\alpha e^{\lambda t}$, tum tantum functionem temporis et unius
variabilis esse posse, quando sit constans aut in planis
parallelis, aut in cylindris eadem axi gaudentibus, aut in
sphaeris concentricis.  Si $u$ est formae $\alpha e^{\lambda t}$,
ex aequatione differentiali (I) sequitur
\[    \frac{\partial^2 \alpha}{\partial x_1^2}
    + \frac{\partial^2 \alpha}{\partial x_2^2}
    + \frac{\partial^2 \alpha}{\partial x_3^2}
   = \lambda a a \alpha \]
et perinde in casu quarto substituendo valores ipsius $u$ in
aequatione differentiali (I), functiones $\alpha$ et $\beta$
facile determinantur, dummodo animadvertas, in hoc casu
$\alpha e^{\lambda t}$ et $\beta e^{\mu t}$ esse posse
quantitates complexas conjugatas.

\end{document}