3-49-1. Gaëtan Blum to H. Poincaré

Paris, le 3 janvier 191011endnote: 1 Gaëtan Blum was the assistant secretary of the Astronomical Society of France, and in this capacity, he invited Poincaré to a meeting of the council of the society. This mundane correspondance is of particular interest, in that Poincaré used its blank pages to investigate the stability of rotating fluid masses.

Société astronomique de France

Fondée en 1887, reconnue d’utilité publique en 1897

Monsieur et Cher Collègue,

J’ai l’honneur de vous informer que l’Ordre du Jour de la Séance du Conseil du 5 Janvier, 8h1/4\nicefrac{{1}}{{4}}, sera le suivant :

«   Examen de modifications à apporter au projet de statuts votés par le Conseil dans sa dernière réunion   ».

Veuillez agréer, Monsieur et Cher Collègue, mes bien sincères salutations.

L’un des Secrétaires Adjoints,

Blum


axes fixes; axes mobiles; sphère fictive ax2=1\sum ax^{2}=1

λ\lambda, μ\mu, ν\nu, ρ\rho ori. axes mobiles axes fixes;22endnote: 2 rayé: sphère fictive

λ1\lambda_{1}, μ1\mu_{1}, ν1\nu_{1}, ρ1\rho_{1} ori. sphère fictive axes mobiles; (sphère fictive mob. [???])

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pp, qq, rr; p1p_{1}, q1q_{1}, r1r_{1} diff. entre les deux cas proj. sur axes mobiles.

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ddtdTdλ-dTdλ+dUdλ=kλ;ddtdTdλ-dTdλ1=k1λ1;λ2=λ12=1\displaystyle\frac{d}{dt}\frac{dT}{d\lambda^{\prime}}-\frac{dT}{d\lambda}+% \frac{dU}{d\lambda}=k\lambda;\frac{d}{dt}\frac{dT}{d\lambda^{\prime}}-\frac{dT% }{d\lambda_{1}}=k_{1}\lambda_{1};\sum\lambda^{2}=\sum\lambda_{1}^{2}=1
12p=-μλ+λμ-ρν+νρ12p1=-μ1λ1+λ1μ1-ρ1ν1+ν1ρ1)  pourquoi à l’envers\displaystyle\left.\begin{aligned} \frac{1}{2}p&=-\mu\lambda^{\prime}+\lambda% \mu^{\prime}-\rho\nu^{\prime}+\nu\rho^{\prime}\\ \frac{1}{2}p_{1}&=-\mu_{1}\lambda_{1}^{\prime}+\lambda_{1}\mu_{1}^{\prime}-% \rho_{1}\nu_{1}^{\prime}+\nu_{1}\rho_{1}^{\prime}\end{aligned}\right)\qquad% \text{pourquoi à l'envers}

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T=T2+T1+T0;T2=12Ap2;T0=12A1p12\displaystyle T=T_{2}+T_{1}+T_{0};T_{2}=\frac{1}{2}\sum Ap^{2};T_{0}=\frac{1}{% 2}\sum A_{1}p_{1}^{2}
T1=Pp=A1pp1;A1=A1\displaystyle T_{1}=\sum Pp=\sum A_{1}^{\prime}pp_{1};A_{1}=A_{1}^{\prime}

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ddtdTdp+rdTdq-qdTdr=-L;ddtdTdp1-r1dTdq1+q1dTdr1=0\frac{d}{dt}\frac{dT}{dp}+r\frac{dT}{dq}-q\frac{dT}{dr}=-L;\frac{d}{dt}\frac{% dT}{dp_{1}}-r_{1}\frac{dT}{dq_{1}}+q_{1}\frac{dT}{dr_{1}}=0

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Xi(f)\displaystyle X_{i}(f) =Xiμdfdxμ;XiXk-XkXi=ciksXs\displaystyle=\sum X_{i}^{\mu}\frac{df}{dx_{\mu}};\quad X_{i}X_{k}-X_{k}X_{i}=% \sum c_{iks}X_{s}
X(f)\displaystyle X(f) =zdfdy-ydfdz;XY-YX=Z\displaystyle=z\frac{df}{dy}-y\frac{df}{dz};\quad XY-YX=Z
Y(f)\displaystyle Y(f) =xdfdz-zdfdx;YZ-ZY=X\displaystyle=x\frac{df}{dz}-z\frac{df}{dx};\quad YZ-ZY=X
Z(f)\displaystyle Z(f) =ydfdx-xdfdy;ZX-XZ=Y\displaystyle=y\frac{df}{dx}-x\frac{df}{dy};\quad ZX-XZ=Y

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dxμdv=ηρXρμ;δxμ=ωρXρμ\displaystyle\frac{dx_{\mu}}{dv}=\sum\eta_{\rho}X_{\rho}^{\mu};\delta x_{\mu}=% \sum\omega_{\rho}X_{\rho}^{\mu}
(dTdx-dUdx)δx=Ωiωi\displaystyle\sum\left(\frac{dT}{dx}-\frac{dU}{dx}\right)\delta x=\sum\Omega_{% i}\omega_{i}
ddtdTdηs=cskidTdηiηk+Ωs\displaystyle\frac{d}{dt}\frac{dT}{d\eta_{s}}=\sum c_{ski}\frac{dT}{d\eta_{i}}% \eta_{k}+\Omega_{s}

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rot. pp, qq, rr, rot. p1p_{1}, q1q_{1}, r1r_{1}; une [???] à l’autre.

signif. des c¯\underline{c}; inversion; signif. des Ω\Omega.

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Eq. des Aires.

Vecteur des moments dTdp\frac{dT}{dp}, dTdq\frac{dT}{dq}, dTdr\frac{dT}{dr}; vitesse relative de ce

ddtdTdp\displaystyle\frac{d}{dt}\frac{dT}{dp} +rdTdq-qdTdr\displaystyle+r\frac{dT}{dq}-q\frac{dT}{dr} =-L\displaystyle=-L
vitesse vitesse vitesse
relative d’entraînement absolue.

Eq. de Helmholtz

u=r1bay-q1caz+ry-qz\displaystyle u=r_{1}\sqrt{b}{a}y-q_{1}\sqrt{c}{a}z+ry-qz
dudy-dvdz=r1(ba+ab)+2r  abc=1\displaystyle\frac{du}{dy}-\frac{dv}{dz}=r_{1}\left(\sqrt{\frac{b}{a}}+\sqrt{% \frac{a}{b}}\right)+2r\qquad abc=1

integr. de Helm. grand cercle =rab(dudy-dvdz)=r1(1a+1b)+2rab=\frac{r}{\sqrt{ab}}\left(\frac{du}{dy}-\frac{dv}{dz}\right)=r_{1}\left(\frac{% 1}{a}+\frac{1}{b}\right)+\frac{2r}{\sqrt{ab}}

fact. const. près 4d15abc\frac{4d}{15\sqrt{abc}}; A1r1+A1r=dTdp1A_{1}r_{1}+A_{1}^{\prime}r=\frac{dT}{dp_{1}}

Vecteur de Helmholtz dTdp1\frac{dT}{dp_{1}} dTdq1\frac{dT}{dq_{1}} dTdr1\frac{dT}{dr_{1}}

invar. lié à la sph. fictive

dTdp1\displaystyle\frac{dT}{dp_{1}} =r1dTdq1-q1dTdr1\displaystyle=r_{1}\frac{dT}{dq_{1}}-q_{1}\frac{dT}{dr_{1}}
vitesse vitesse
absolue d’entraînement

ellipticity 126000.365\frac{1}{26000.365} 1300\frac{1}{300} prév. 3 [???]

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Mouv. simples

ax2=1,aux=0\displaystyle\sum ax^{2}=1,\sum aux=0 dudx=0;x=xa,y=yb,z=zc\displaystyle\qquad\sum\frac{du}{dx}=0;x^{\prime}=x\sqrt{a},y^{\prime}=y\sqrt{% b},z^{\prime}=z\sqrt{c}
u=rbay-qcaz;\displaystyle u=r\sqrt{\frac{b}{a}}y-q\sqrt{\frac{c}{a}}z; dudy-dvdx=r(ba+ab)\displaystyle\qquad\frac{du}{dy}-\frac{dv}{dx}=r\left(\sqrt{\frac{b}{a}}+\sqrt% {\frac{a}{b}}\right)
v=pcbz-rabx;\displaystyle v=p\sqrt{\frac{c}{b}}z-r\sqrt{\frac{a}{b}}x; dvdz-dwdy=p(cb+bc)\displaystyle\qquad\frac{dv}{dz}-\frac{dw}{dy}=p\left(\sqrt{\frac{c}{b}}+\sqrt% {\frac{b}{c}}\right) (1)
w=qacx-pbcy;\displaystyle w=q\sqrt{\frac{a}{c}}x-p\sqrt{\frac{b}{c}}y; dwdx-dudz=q(ac+ca)\displaystyle\qquad\frac{dw}{dx}-\frac{du}{dz}=q\left(\sqrt{\frac{a}{c}}+\sqrt% {\frac{c}{a}}\right)

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mouv. simples se combinent comme rotations, invariant a(x1-x2)2\sum a(x_{1}-x_{2})^{2}

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Conserv. des mouv. simples.

Th. de Helmholtz. (udx+vdy+wdz)=\int(udx+vdy+wdz)= fonc. lin. de x𝑑x\int xdx, y𝑑x\int ydx, x𝑑y\int xdy; etc y𝑑x=-x𝑑y\int ydx=-\int xdy

Récipr. dudy-dvdx=\frac{du}{dy}-\frac{dv}{dx}= const. — sol. (1); sol. unique, car si deux sol. u-u=dφdxu-u^{\prime}=\frac{d\varphi}{dx}, v-v=dφdyv-v^{\prime}=\frac{d\varphi}{dy}, w-w=dφdzw-w^{\prime}=\frac{d\varphi}{dz}; φ=0\triangle\varphi=0, dφdu=0\frac{d\varphi}{du}=0

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x=x+udtx^{\prime}=x+udt; x𝑑x\int x^{\prime}dx^{\prime}, y𝑑x\int y^{\prime}dx^{\prime}, …fonc. lin. de x𝑑x\int xdx etc.

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Extens. à l’ellipsoïde mobile volume constant.

dudy-dvdx=const.,u=rbay-qcaz+dφdx\displaystyle\frac{du}{dy}-\frac{dv}{dx}=\text{const.},u=r\sqrt{\frac{b}{a}}y-% q\sqrt{\frac{c}{a}}z+\frac{d\varphi}{dx}
ax2=1,ax2=1+2ψdt;a(x+udt)2=1+2ψdt\displaystyle\sum ax^{2}=1,\sum ax^{2}=1+2\psi dt;\sum a(x+udt)^{2}=1+2\psi dt
aux=ψ;axdφdx=ψ;φ=0,vol.=const.\displaystyle\sum aux=\psi;\sum ax\frac{d\varphi}{dx}=\psi;\triangle\varphi=0,% \text{vol.}=\text{const.}

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Force vive. ellip. immobile.

TT polyn. 2d degré p,q,r¯\underline{p,q,r}; symétrie pas de terme en pqpq; si r¯=0\underline{r}=0, changer qq en -q-q et yy en -y-y, c’est changer uu et ww en -u-u et -w-w.

q=r=0;d=1,p=1;u=0;2T=(v2+w2)𝑑x𝑑y𝑑z\displaystyle q=r=0;d=1,p=1;u=0;2T=\int(v^{2}+w^{2})dxdydz
=1abc(v2b+w2c)𝑑x𝑑y𝑑z=1abc(y2c+x2b)𝑑x𝑑y𝑑z\displaystyle=\frac{1}{\sqrt{abc}}\int\left(\frac{v^{\prime 2}}{b}+\frac{w^{% \prime 2}}{c}\right)dx^{\prime}dy^{\prime}dz^{\prime}=\frac{1}{\sqrt{abc}}\int% \left(\frac{y^{\prime 2}}{c}+\frac{x^{\prime 2}}{b}\right)dx^{\prime}dy^{% \prime}dz^{\prime}
y2𝑑x𝑑y𝑑z=4π15;2T=4π151abc(1b+1c)\displaystyle\int y^{\prime 2}dx^{\prime}dy^{\prime}dz^{\prime}=\frac{4\pi}{15% };2T=\frac{4\pi}{15}\frac{1}{\sqrt{abc}}\left(\frac{1}{b}+\frac{1}{c}\right)
T=12Ap2;A=4π15dabc(1b+1c);ordabc=1.\displaystyle T=\frac{1}{2}\sum Ap^{2};A=\frac{4\pi}{15}\frac{d}{\sqrt{abc}}% \left(\frac{1}{b}+\frac{1}{c}\right);\text{ord}\;abc=1.
(vz-wy)𝑑x𝑑y𝑑z=1abcvz-wybc𝑑x𝑑y𝑑z=8π15pdabc1bc\displaystyle\int(vz-wy)dxdydz=\frac{1}{\sqrt{abc}}\int\frac{v^{\prime}z^{% \prime}-w^{\prime}y^{\prime}}{\sqrt{bc}}dx^{\prime}dy^{\prime}dz^{\prime}=% \frac{8\pi}{15}\frac{pd}{\sqrt{abc}}\frac{1}{\sqrt{bc}}
Ap,Bq,Cr;A=8π15dabc1bc;A[???]=A\displaystyle A^{\prime}p,B^{\prime}q,C^{\prime}r;A^{\prime}=\frac{8\pi}{15}% \frac{d}{\sqrt{abc}}\frac{1}{\sqrt{bc};}A^{\prime}[???]=A

ALS 1p. Collection particulière, Paris 75017.

Time-stamp: "13.08.2022 12:35"

Notes

  • 1 Gaëtan Blum was the assistant secretary of the Astronomical Society of France, and in this capacity, he invited Poincaré to a meeting of the council of the society. This mundane correspondance is of particular interest, in that Poincaré used its blank pages to investigate the stability of rotating fluid masses.
  • 2 rayé: sphère fictive