7-1-18. Calculs

V=S0RMNintér  V=R0SMNextér.\displaystyle V=S_{0}RMN\quad\text{intér}\qquad V=R_{0}SMN\quad\text{extér.}
V1=S10R1M1N1;VdV1dn𝑑ω=0;RMNR1M1N1dρdn𝑑ω=0\displaystyle V_{1}=S^{0}_{1}R_{1}M_{1}N_{1};\quad\int V\frac{dV_{1}}{dn}d% \omega=0;\int RMNR^{\prime}_{1}M_{1}N_{1}\frac{d\rho}{dn}d\omega=0
dρdn=1α=OPρ=μ2-ν2AQ;OP=μ2-ν2ρAQ=(ρ2-a2)(ρ2-b2)(ρ2-c2)(ρ2-μ2)(ρ2-ν2)\displaystyle\frac{d\rho}{dn}=\frac{1}{\alpha}=\frac{OP}{\rho}=\frac{\sqrt{\mu% ^{2}-\nu^{2}}A}{Q};\quad 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})}}
=1(ρ2-μ2)(ρ2-ν2)=OP(ρ2-a2)(ρ2-b2)(ρ2-c2);dρdn=iA\displaystyle\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})}};\quad\frac{d\rho}% {dn}=\ell iA
MNM1N1𝑑ω=0;F=KiMiNi;MjNjF𝑑ω\displaystyle\int\ell MNM_{1}N_{1}d\omega=0;\quad F=\sum K_{i}M_{i}N_{i};\quad% \int\ell M_{j}N_{j}Fd\omega
=KiMiNiMjNj𝑑ω=KjMj2Nj2𝑑ω\displaystyle=\sum K_{i}\int\ell M_{i}N_{i}M_{j}N_{j}d\omega=K_{j}\int\ell M^{% 2}_{j}N^{2}_{j}d\omega

Problème de Dirichlet.11endnote: 1 Ces calculs sont datés par leur support : une lettre de G. Cres à Poincaré, 25.01.1912.

F=KMN;int.V=KRMNR0;ext.V=KSMNS0F=\sum KMN;\quad\text{int.}\;V=\sum\frac{KRMN}{R_{0}};\quad\text{ext.}\;V=\sum% \frac{KSMN}{S_{0}}

Simple couche.

ζ=CMN;int.V=KRMNR0;ext.V=KSMNS0\displaystyle\frac{\zeta}{\ell}=\sum C\cdot MN;\quad\text{int.}\;V=\sum\frac{% KRMN}{R_{0}};\quad\text{ext.}\;V=\sum\frac{KSMN}{S_{0}}
dVdn=iAKRMNRint.  ext.iAKSMNS\displaystyle\frac{dV}{dn}=\sum\ell iA\frac{KR^{\prime}\cdot MN}{R}\quad\text{% int.}\qquad\text{ext.}\quad\sum\frac{\ell iAKS^{\prime}\cdot MN}{S}
iAK(RR-SS)=4πC\displaystyle\ell iAK\left(\frac{R^{\prime}}{R}-{S^{\prime}}{S}\right)=4\pi\ell C
A(RS-SR)=-i(2n+1);+K(2n+1)RS=4πC\displaystyle A(R^{\prime}S-S^{\prime}R)=-i(2n+1);\quad\frac{+K(2n+1)}{RS}=4\pi C
K=+4πCRS2n+1;V=+4πCS0RMN2n+1\displaystyle K=+\frac{4\pi CRS}{2n+1};\quad V=+\sum\frac{4\pi CS_{0}RMN}{2n+1}

Couche ellipsoïdale;

rapportε;ζ=PP1=εOP=\displaystyle\text{rapport}\;\varepsilon;\quad\zeta=PP_{1}=\varepsilon OP=
ε(ρ2-a2)(ρ2-b2)(ρ2-c2)=43πεT;Tvol. ellips.\displaystyle\frac{\varepsilon\ell}{\sqrt{(\rho^{2}-a^{2})(\rho^{2}-b^{2})(% \rho^{2}-c^{2})}}=\frac{4}{3}\pi\frac{\varepsilon\ell}{T};\;T\;\text{vol. % ellips.}
V=(4π)23εTSouR=1.\displaystyle V=\frac{(4\pi)^{2}}{3}\frac{\varepsilon}{T}S\quad\text{ou}\quad R% =1.

Ellipsoïde plein.

εdVdx\varepsilon\frac{dV}{dx}
MM1=ε\displaystyle MM_{1}=\varepsilon
MP=ζ  anglePMM1=θ.\displaystyle MP=\zeta\qquad\text{angle}\;PMM_{1}=\theta.
ζ=εcosθ.\displaystyle\zeta=\varepsilon\cos\theta.
M(ρ,μ,ν,)Qρ+dρ,μ,ν\displaystyle M\;(\rho,\mu,\nu,)\;Q\;\rho+d\rho,\mu,\nu
x,y,z  x+dxetc.R=ρ2-a2\displaystyle x,y,z\qquad x+dx\;\text{etc.}\quad R=\sqrt{\rho^{2}-a^{2}}
cosθ=dxdρdρdn=xρρ2-a2dρdn;RMN=x(a2-b2)(a2-c2)\displaystyle\cos\theta=\frac{dx}{d\rho}\frac{d\rho}{dn}=\frac{x\rho}{\sqrt{% \rho^{2}-a^{2}}}\frac{d\rho}{dn};\quad RMN=x\sqrt{(a^{2}-b^{2})(a^{2}-c^{2})}
cosθ=xρρ2-a2dρdn=iAxρρ2-a2=x(ρ2-b2)(ρ2-c2)\displaystyle\cos\theta=\frac{x\rho}{\sqrt{\rho^{2}-a^{2}}}\frac{d\rho}{dn}=% \ell iA\frac{x\rho}{\sqrt{\rho^{2}-a^{2}}}=\ell x\sqrt{(\rho^{2}-b^{2})(\rho^{% 2}-c^{2})}
ζ=εRMN(ρ2-b2)(ρ2-c2)(a2-b2)(a2-c2)=εT(43π)MN(a2-b2)(a2-c2)\displaystyle\zeta=\ell\varepsilon RMN\sqrt{\frac{(\rho^{2}-b^{2})(\rho^{2}-c^% {2})}{(a^{2}-b^{2})(a^{2}-c^{2})}}=\frac{\ell\varepsilon T}{\left(\frac{4}{3}% \pi\right)}\frac{MN}{\sqrt{(a^{2}-b^{2})(a^{2}-c^{2})}}
dVdx=(ρ2-b2)(ρ2-c2)43πRMNS0(a2-b2)(a2-c2)\displaystyle\frac{dV}{dx}=\sqrt{(\rho^{2}-b^{2})(\rho^{2}-c^{2})}\frac{4}{3}% \pi\frac{RMNS_{0}}{\sqrt{(a^{2}-b^{2})(a^{2}-c^{2})}}
u=0,S:u=Q;M-Q=ϖ;S:ϖ=0,ϖ=-Q=f\displaystyle\triangle u=0,\quad S:u=Q;\quad M-Q=\varpi;\quad S:\varpi=0,\quad% \triangle\varpi=-\triangle Q=f
[(dϖdx)2+2ϖf]𝑑τ=J;[dϖdxdδϖdx+fδϖ]𝑑τ=0\displaystyle\int\left[\sum\left(\frac{d\varpi}{dx}\right)^{2}+2\varpi f\right% ]d\tau=J;\quad\int\left[\sum\frac{d\varpi}{dx}\frac{d\delta\varpi}{dx}+f\delta% \varpi\right]d\tau=0
δϖdϖdn𝑑ω-δϖ(ϖ-f)𝑑τ=0;ϖ=Aiψi\displaystyle\int\delta\varpi\frac{d\varpi}{dn}d\omega-\int\delta\varpi(% \triangle\varpi-f)d\tau=0;\quad\varpi=\sum A_{i}\psi_{i}

ψ\psi continue sur RR et sur son bord ainsi que ses dérivées principales. m,n,p=0,1,2m,n,p=0,1,2
2° Sur le bord ψ=0\psi=0; ζ\zeta
\cdots ζ\zeta représ. par séries, toutes les fois que ζ\zeta continue ainsi que ses dérivées princip.
ζm=0\zeta_{m}=0 entraîne A1=A2==Am=0A_{1}=A_{2}=\cdots=A_{m}=0 si ζm=Aiψi\zeta_{m}=\sum A_{i}\psi_{i}

On fera ψi=Fsincosmxsincosnysincospz\psi_{i}=F\begin{smallmatrix}\sin\\ \cos\end{smallmatrix}mx\begin{smallmatrix}\sin\\ \cos\end{smallmatrix}ny\begin{smallmatrix}\sin\\ \cos\end{smallmatrix}pz; F=0F=0.

(ϖmζm-fζm)𝑑τ=0;[dϖmdxdζmdx+fζm]𝑑τ=0\displaystyle\int\left(\varpi_{m}\triangle\zeta_{m}-f\zeta_{m}\right)d\tau=0;% \quad\int\left[\sum\frac{d\varpi_{m}}{dx}\frac{d\zeta_{m}}{dx}+f\zeta_{m}% \right]d\tau=0
ϖmdζmdn𝑑ω-(ϖmζm-fζm)𝑑τ=0\displaystyle\int\varpi_{m}\frac{d\zeta_{m}}{dn}d\omega-\int\left(\varpi_{m}% \triangle\zeta_{m}-f\zeta_{m}\right)d\tau=0
Jp\displaystyle J_{p} =[(dϖpdx)2+2fϖp]𝑑τ  (1)  pq\displaystyle=\int\left[\sum\left(\frac{d\varpi_{p}}{dx}\right)^{2}+2f\varpi_{% p}\right]d\tau\qquad(1)\qquad pq
Jq\displaystyle J_{q} =[(dϖqdx)2+2fϖq]𝑑τ  (-1)\displaystyle=\int\left[\sum\left(\frac{d\varpi_{q}}{dx}\right)^{2}+2f\varpi_{% q}\right]d\tau\qquad(-1)
Q\displaystyle Q =[dϖqdxd(ϖq-ϖpdx+f(ϖq-ϖp)]𝑑τ  (2)\displaystyle=\int\left[\sum\frac{d\varpi_{q}}{dx}\frac{d(\varpi_{q}-\varpi_{p% }}{dx}+f(\varpi_{q}-\varpi_{p})\right]d\tau\qquad(2)
Jp-Jq=[d(ϖq-ϖpdx]2dτJ_{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: " 5.05.2019 01:04"

Notes

  • 1 Ces calculs sont datés par leur support : une lettre de G. Cres à Poincaré, 25.01.1912.