Calculations

[Ca. end of October, 1909]11endnote: 1 The three pages of calculations transcribed here are from a one-page letter from Benjamin Baillaud to Poincaré, 22.10.1910 (§ 3-2-3).

[deux figures]

dφZ fixedθΩ-dϕz mobilep=θOΩq=φsinθOAr=φcosθ-ψOz\begin{array}[]{cc}d\varphi&Z\texttt{ fixe}\\ d\theta&\Omega\\ -d\phi&z\texttt{ mobile}\\ p=\theta^{\prime}&O\Omega\\ q=\varphi^{\prime}\sin\theta&OA\\ r=\varphi^{\prime}\cos\theta-\psi^{\prime}&Oz\end{array}
2T=A[θ2+φ2sin2θ]+C(φcosθ-ψ)22T=A[\theta^{\prime 2}+\varphi^{\prime 2}\sin^{2}\theta]+C(\varphi^{\prime}% \cos\theta-\psi^{\prime})^{2}
2T=Aθ2+A+C2φ2+C-A2φ2cos2θ-2Cφψcosθ+Cψ22T=A\theta^{\prime 2}+\frac{A+C}{2}\varphi^{\prime 2}+\frac{C-A}{2}\varphi^{% \prime 2}\cos 2\theta-2C\varphi^{\prime}\psi^{\prime}\cos\theta+C\psi^{\prime 2}
Θ=dTdθ=Aθ;Ψ=dTdψ=Cψ-Cφcosθ=-Cr\begin{array}[]{cc}\Theta=\frac{dT}{d\theta^{\prime}}=A\theta^{\prime}\texttt{% ;}&\Psi=\frac{dT}{d\psi^{\prime}}=C\psi^{\prime}-C\varphi^{\prime}\cos\theta=-% Cr\end{array}
ϕ=dTdφ=A+C2φ+C-A2φcos2θ-Cψcosθ\phi=\frac{dT}{d\varphi^{\prime}}=\frac{A+C}{2}\varphi^{\prime}+\frac{C-A}{2}% \varphi^{\prime}\cos 2\theta-C\psi^{\prime}\cos\theta
2T=αx2+2βxy+γy2=xX+yY;μ=β2-αγμx=-γX+βY;μy=βX-αYX=αx+βy;Y=βx+γy\begin{array}[]{cc}2T=\alpha x^{2}+2\beta xy+\gamma y^{2}=xX+yY\texttt{;}&\mu=% \beta^{2}-\alpha\gamma\\ \mu x=-\gamma X+\beta Y\texttt{;}&\mu y=\beta X-\alpha Y\\ X=\alpha x+\beta y\texttt{;}&Y=\beta x+\gamma y\end{array}
2T=-γX2+2βXY-αY2μ2T=\frac{-\gamma X^{2}+2\beta XY-\alpha Y^{2}}{\mu}
μ=C2cos2θ-C[A+C2+C-A2cos2θ]\mu=C^{2}\cos^{2}\theta-C[\frac{A+C}{2}+\frac{C-A}{2}\cos 2\theta]
μ=C2[(C-(A+C)+cos2θ(C-(C-A)]=AC2(cos2θ-1)\mu=\frac{C}{2}[(C-(A+C)+cos2\theta(C-(C-A)]=\frac{AC}{2}(cos2\theta-1)
T=Θ22A-Cϕ2+2Ccosθϕψ+ψ2[A+C2+C-A2cos2θ]2μT=\frac{\Theta^{2}}{2A}-\frac{C\phi^{2}+2C\cos\theta\phi\psi+\psi^{2}[\frac{A+% C}{2}+\frac{C-A}{2}\cos 2\theta]}{2\mu}
Φ=G;Cr=Gcosθ;Θ=0 pourT=G22Ccos2θ+cos2θ(Asin2θ+Ccos2θAC(1-cos2θ)T=G2C(1-cos2θ)2+Acos2θsin2θ2ACsin2θ=G2Csin2θ+Acos2θ2AC2T=Ah2+Cr2;G2=A2h2+C2r2Ah=Gsinθ;Cr=Gcosθ;2T=G2[sin2θA+cos2θC]dSdθ=f(θ,ϕ,ψ,T):φ+dfdϕ𝑑θ= const. |ψ+dfdψ𝑑θ=const.\begin{gathered}\Phi=G;\qquad Cr=G\cos\theta;\qquad\Theta=0\text{ pour}\\ T=G^{2}\frac{2C\cos^{2}\theta+\cos^{2}\theta(A\sin^{2}\theta+C\cos^{2}\theta}{% AC(1-\cos^{2}\theta)}\\ T=G^{2}\frac{C(1-\cos^{2}\theta)^{2}+A\cos^{2}\theta\sin^{2}\theta}{2AC\sin^{2% }\theta}=G^{2}\frac{C\sin^{2}\theta+A\cos^{2}\theta}{2AC}\\ 2T=Ah^{2}+Cr^{2};\qquad G^{2}=A^{2}h^{2}+C^{2}r^{2}\\ Ah=G\sin\theta;\qquad Cr=G\cos\theta;\qquad 2T=G^{2}\left[\frac{sin^{2}\theta}% {A}+\frac{\cos^{2}\theta}{C}\right]\\ \frac{dS}{d\theta}=f(\theta,\phi,\psi,T):\varphi+\int\frac{df}{d\phi}d\theta=% \text{ const. }|\psi+\int\frac{df}{d\psi}d\theta=\text{const.}\end{gathered}
dfdT𝑑θ=t+const.\int\frac{df}{dT}d\theta=t+const.
f=2A(T+F)f=\sqrt{2A(T+F)}
Adθf=t+const.;φ+AdFdϕdθf=const.\begin{array}[]{cc}\int\frac{Ad\theta}{f}=t+const.\texttt{;}&\varphi+\int\frac% {A\frac{dF}{d\phi}d\theta}{f}=const.\end{array}
ψ+AdFdψdθf;θ= const. ;φ+dFdϕt= const.\begin{array}[]{ccc}\psi+\int\frac{A\frac{dF}{d\psi}d\theta}{f};\qquad\theta=% \text{ const. };\qquad\varphi+\frac{dF}{d\phi}t=\text{ const.}\end{array}
ψ+dFdψt=const.\psi+\frac{dF}{d\psi}t=const.
Cr=Gcosθ0Cr=G\cos\theta_{0}
TG2=Csin2θ0+Acos2θ02AC;Θ22AG2=Csin2θ0+Acos2θ02AC-C(1-cosθcosθ0)2+Acos2θ0sin2θ2ACsin2θ\frac{T}{G^{2}}=\frac{C\sin^{2}\theta_{0}+A\cos^{2}\theta_{0}}{2AC};\quad\frac% {\Theta^{2}}{2AG^{2}}=\frac{C\sin^{2}\theta_{0}+A\cos^{2}\theta_{0}}{2AC}-% \frac{C(1-\cos\theta\cos\theta_{0})^{2}+A\cos^{2}\theta_{0}\sin^{2}\theta}{2AC% \sin^{2}\theta}
(1-cosθcosθ0)2-sin2θsin2θ0=0(1-\cos\theta\cos\theta_{0})^{2}-\sin^{2}\theta\sin^{2}\theta_{0}=0
[1-cos(θ+θ0)][1-cos(θ-θ0)][1-\cos(\theta+\theta_{0})][1-\cos(\theta-\theta_{0})]

X,Y,ZX,Y,Z fixes
x,y,zx,y,z mobiles

φOZ(angle deOMzavecyOz)ψOX(angle deOPavecOz)χOZ(angle dePOMavecPOZ)ωOX(angle deOPavecOZ)θOZ(angle dePOZavecYOZ)\begin{array}[]{ccc}\varphi&OZ&(\text{angle de}OM^{\prime}z\text{avec}yOz)\\ \psi&OX&(\text{angle de}OP^{\prime}\text{avec}Oz)\\ \chi&OZ&(\text{angle de}P^{\prime}OM^{\prime}\text{avec}P^{\prime}OZ)\\ \omega&OX&(\text{angle de}OP^{\prime}\text{avec}OZ)\\ \theta&OZ&(\text{angle de}POZ\text{avec}YOZ)\end{array}

OMOM^{\prime} lié aux axes mobiles ; deux systèmes d’axes coïncident.

OP=OZ=OzOP^{\prime}=OZ=Oz

1ère rotation OPOP^{\prime} reste fixe.

Oz=OP=OZ;ZPOMzOXOz=OP^{\prime}=OZ;\qquad ZP^{\prime}OM^{\prime}z\perp OX

2e rotation OPOP^{\prime} fixe

OP=OZ;Oz𝚊𝚗𝚐𝚕𝚎ψ𝚊𝚟𝚎𝚌OPcet angle ne changera plus ;ZPOMzOX\begin{array}[]{ccc}OP^{\prime}=OZ\texttt{;}&Oz\texttt{angle}\psi\texttt{avec}% OP^{\prime}\texttt{cet angle ne changera plus ;}&ZP^{\prime}OM^{\prime}z\perp OX% \end{array}

3e rotation OPOP^{\prime} peu importe POZP^{\prime}OZ futur(?) lié.

OP=OZ;POZ𝚏𝚞𝚝𝚞𝚛OX\begin{array}[]{cc}OP^{\prime}=OZ\texttt{;}&P^{\prime}OZ\texttt{futur}\perp OX% \end{array}

4e rotation OP’ et P’OZ futur liés axes mobiles
OP’ angle ω\omega avec OZ changera plus ; P’OZ futur [illisible] à poser par OP’ et OZ

5e rotation OPOP^{\prime} et POZP^{\prime}OZ futur liés
POZP^{\prime}OZ futur fait angle θ\theta avec ?OZ?OZ

dφd\varphi autour de OzOz ; dψd\psi perp. à POzMP^{\prime}OzM^{\prime}
dχd\chi autour de OPOP^{\prime} ; dωd\omega perp. à POZP^{\prime}OZ ; dθd\theta autour de OZOZ

ldχdt=hdφdt=A-CArhl\frac{d\chi}{dt}=h\frac{d\varphi}{dt}=\frac{A-C}{A}rh
lG=(A-C)hr;A2h2+C2r2=G2;Ah2+Cr2=2T\begin{array}[]{ccc}lG=(A-C)hr\texttt{;}&A^{2}h^{2}+C^{2}r^{2}=G^{2}\texttt{;}% &Ah^{2}+Cr^{2}=2T\end{array}
(λ+iμ+jν+kρ)=ekφ2(Cψ+isψ)ekχ2(Cω+isω)ekθ2(\lambda+i\mu+j\nu+k\rho)=e^{k\frac{\varphi}{2}}(C\psi+is\psi)e^{k\frac{\chi}{% 2}}(C\omega+is\omega)e^{k\frac{\theta}{2}}
ik=-ki=-j;ekφ2i=ie-kφ2\begin{array}[]{cc}ik=-ki=-j\texttt{;}&e^{k\frac{\varphi}{2}}i=ie^{-k\frac{% \varphi}{2}}\end{array}
CψCωek2(φ+χ+θ)+isψCωek2(χ+θ-φ)+isωCψek2(θ-φ-χ)-sωsψek2(θ-φ-χ)C\psi C\omega e^{\frac{k}{2}(\varphi+\chi+\theta)}+is\psi C\omega e^{\frac{k}{% 2}(\chi+\theta-\varphi)}+is\omega C\psi e^{\frac{k}{2}(\theta-\varphi-\chi)}-s% \omega s\psi e^{\frac{k}{2}(\theta-\varphi-\chi)}
λ=CψCωcosφ+χ+θ2-sψsωcosθ+φ-χ2\lambda=C\psi C\omega\cos\frac{\varphi+\chi+\theta}{2}-s\psi s\omega\cos\frac{% \theta+\varphi-\chi}{2}
μ=sψCωcosχ+θ-φ2+Cψsωcosθ-χ-φ2\mu=s\psi C\omega\cos\frac{\chi+\theta-\varphi}{2}+C\psi s\omega\cos\frac{% \theta-\chi-\varphi}{2}
ν=sψCωsinφ-χ-θ2+Cψsωsinφ+χ-θ2\nu=s\psi C\omega\sin\frac{\varphi-\chi-\theta}{2}+C\psi s\omega\sin\frac{% \varphi+\chi-\theta}{2}
ρ=CψCωsinφ+χ+θ2-sψsωsinθ+φ-χ2\rho=C\psi C\omega\sin\frac{\varphi+\chi+\theta}{2}-s\psi s\omega\sin\frac{% \theta+\varphi-\chi}{2}

AD 3p. Collection particulière, Paris 75017.

Time-stamp: "22.06.2022 12:00"

Notes

  • 1 The three pages of calculations transcribed here are from a one-page letter from Benjamin Baillaud to Poincaré, 22.10.1910 (§ 3-2-3).