Henri Poincaré: calculs

$$V=S_{0}RMN\mathit{\hspace{1em}}\text{intér}\mathit{\hspace{1em}\hspace{1em}}V=R_{0}SMN\mathit{\hspace{1em}}\text{extér.}$$

$$V_1=S_1^0{R}_1{M}_1{N}_1;\int V\frac{d{V}_1}{dn}𝑑\omega =0;\int RMN{R}_1^{\prime }{M}_1{N}_1\frac{d\rho }{dn}𝑑\omega =0$$

$$\frac{d\rho }{dn}=\frac{1}{\alpha }=\frac{OP}{\rho }=\frac{\sqrt{{\mu }^2-{\nu }^2}A}{Q};OP=\frac{\sqrt{{\mu }^2-{\nu }^2}\rho A}{Q}=\frac{\sqrt{({\rho }^2-a^2)({\rho }^2-b^2)({\rho }^2-c^2)}}{\sqrt{({\rho }^2-{\mu }^2)({\rho }^2-{\nu }^2)}}$$

$$\mathrm{\ell }=\frac{1}{\sqrt{({\rho }^2-{\mu }^2)({\rho }^2-{\nu }^2)}}=\frac{OP}{\sqrt{({\rho }^2-a^2)({\rho }^2-b^2)({\rho }^2-c^2)}};\frac{d\rho }{dn}=\mathrm{\ell }iA$$

$$\int \mathrm{\ell }MN{M}_1{N}_1𝑑\omega =0;F=\sum K_i{M}_i{N}_i;\int \mathrm{\ell }{M}_j{N}_{j}F𝑑\omega$$

$$=\sum K_i\int \mathrm{\ell }{M}_i{N}_i{M}_j{N}_j𝑑\omega =K_j\int \mathrm{\ell }{M}_j^2{N}_j^2𝑑\omega$$

Problème de Dirichlet.

$$F=\sum KMN;\text{int.}V=\sum \frac{KRMN}{R_0};\text{ext.}V=\sum \frac{KSMN}{S_0}$$

Simple couche.

$$\frac{\zeta }{\mathrm{\ell }}=\sum C\cdot MN;\text{int.}V=\sum \frac{KRMN}{R_0};\text{ext.}V=\sum \frac{KSMN}{S_0}$$

$$\frac{dV}{dn}=\sum \mathrm{\ell }iA\frac{K{R}^{\prime }\cdot MN}{R}\mathit{\hspace{1em}}\text{int.}\mathit{\hspace{1em}\hspace{1em}}\text{ext.}\mathit{\hspace{1em}}\sum \frac{\mathrm{\ell }iAK{S}^{\prime }\cdot MN}{S}$$

$$\mathrm{\ell }iAK\left(\frac{R^{\prime }}{R}-S^{\prime }S\right)=4\pi \mathrm{\ell }C$$

$$A(R^{\prime }S-S^{\prime }R)=-i(2n+1);\frac{+K(2n+1)}{RS}=4\pi C$$

$$K=+\frac{4\pi CRS}{2n+1};V=+\sum \frac{4\pi C{S}_{0}RMN}{2n+1}$$

Couche ellipsoïdale;

$$\text{rapport}\epsilon ;\zeta =P{P}_1=\epsilon OP=$$

$$\frac{\epsilon \mathrm{\ell }}{\sqrt{({\rho }^2-a^2)({\rho }^2-b^2)({\rho }^2-c^2)}}=\frac{4}{3}\pi \frac{\epsilon \mathrm{\ell }}{T};T\text{vol. ellips.}$$

$$V=\frac{{(4\pi )}^2}{3}\frac{\epsilon }{T}S\mathit{\hspace{1em}}\text{ou}\mathit{\hspace{1em}}R=1.$$

Ellipsoïde plein.

$$\epsilon \frac{dV}{dx}$$

$$M{M}_1=\epsilon$$

$$MP=\zeta \mathit{\hspace{1em}\hspace{1em}}\text{angle}PM{M}_1=\theta .$$

$$\zeta =\epsilon \cos \theta .$$

$$M(\rho ,\mu ,\nu ,)Q\rho +d\rho ,\mu ,\nu$$

$$x,y,z\mathit{\hspace{1em}\hspace{1em}}x+dx\text{etc.}\mathit{\hspace{1em}}R=\sqrt{{\rho }^2-a^2}$$

$$\cos \theta =\frac{dx}{d\rho }\frac{d\rho }{dn}=\frac{x\rho }{\sqrt{{\rho }^2-a^2}}\frac{d\rho }{dn};RMN=x\sqrt{(a^2-b^2)(a^2-c^2)}$$

$$\cos \theta =\frac{x\rho }{\sqrt{{\rho }^2-a^2}}\frac{d\rho }{dn}=\mathrm{\ell }iA\frac{x\rho }{\sqrt{{\rho }^2-a^2}}=\mathrm{\ell }x\sqrt{({\rho }^2-b^2)({\rho }^2-c^2)}$$

$$\zeta =\mathrm{\ell }\epsilon RMN\sqrt{\frac{({\rho }^2-b^2)({\rho }^2-c^2)}{(a^2-b^2)(a^2-c^2)}}=\frac{\mathrm{\ell }\epsilon T}{\left(\frac{4}{3}\pi \right)}\frac{MN}{\sqrt{(a^2-b^2)(a^2-c^2)}}$$

$$\frac{dV}{dx}=\sqrt{({\rho }^2-b^2)({\rho }^2-c^2)}\frac{4}{3}\pi \frac{RMN{S}_0}{\sqrt{(a^2-b^2)(a^2-c^2)}}$$

$$\mathrm{△}u=0,S:u=Q;M-Q=\varpi ;S:\varpi =0,\mathrm{△}\varpi =-\mathrm{△}Q=f$$

$$\int \left[\sum {\left(\frac{d\varpi }{dx}\right)}^2+2\varpi f\right]𝑑\tau =J;\int \left[\sum \frac{d\varpi }{dx}\frac{d\delta \varpi }{dx}+f\delta \varpi \right]𝑑\tau =0$$

$$\int \delta \varpi \frac{d\varpi }{dn}𝑑\omega -\int \delta \varpi (\mathrm{△}\varpi -f)𝑑\tau =0;\varpi =\sum A_i{\psi }_i$$

1° $\psi$ continue sur $R$ et sur son bord ainsi que ses dérivées principales. $m,n,p=0,1,2$
2° Sur le bord $\psi =0$; $\zeta$
3° $\mathrm{\cdots }$ $\zeta$ représ. par séries, toutes les fois que $\zeta$ continue ainsi que ses dérivées princip.
4° ${\zeta }_m=0$ entraîne $A_1=A_2=\mathrm{\cdots }=A_m=0$ si ${\zeta }_m=\sum A_i{\psi }_i$

On fera ${\psi }_i=F\begin{array}{c}\hfill \sin \hfill \\ \hfill \cos \hfill \end{array}mx\begin{array}{c}\hfill \sin \hfill \\ \hfill \cos \hfill \end{array}ny\begin{array}{c}\hfill \sin \hfill \\ \hfill \cos \hfill \end{array}pz$; $F=0$.

$$\int \left({\varpi }_m\mathrm{△}{\zeta }_m-f{\zeta }_m\right)𝑑\tau =0;\int \left[\sum \frac{d{\varpi }_m}{dx}\frac{d{\zeta }_m}{dx}+f{\zeta }_m\right]𝑑\tau =0$$

$$\int {\varpi }_m\frac{d{\zeta }_m}{dn}𝑑\omega -\int \left({\varpi }_m\mathrm{△}{\zeta }_m-f{\zeta }_m\right)𝑑\tau =0$$

$J_p$ $={\displaystyle \int \left[\sum {\left(\frac{d{\varpi }_p}{dx}\right)}^2+2f{\varpi }_p\right]𝑑\tau }\mathit{\hspace{1em}\hspace{1em}}(1)\mathit{\hspace{1em}\hspace{1em}}pq$

$J_q$ $={\displaystyle \int \left[\sum {\left(\frac{d{\varpi }_q}{dx}\right)}^2+2f{\varpi }_q\right]𝑑\tau }\mathit{\hspace{1em}\hspace{1em}}(-1)$

$Q$ $={\displaystyle \int \left[\sum \frac{d{\varpi }_q}{dx}\frac{d({\varpi }_q-{\varpi }_p}{dx}+f({\varpi }_q-{\varpi }_p)\right]𝑑\tau }\mathit{\hspace{1em}\hspace{1em}}(2)$

$$J_p-J_q=\int \sum {\left[\frac{d({\varpi }_q-{\varpi }_p}{dx}\right]}^{2}d\tau$$

AD 3p. Collection particulière, Paris 75017.

Time-stamp: "19.03.2015 01:51"