Paris, le 22 octobre 09
Observatoire De Paris
Cabinet Du Directeur

Monsieur et cher confrère,

M. Nordmann me communique la note ci-jointe.¹ Avant de partir, il m’avait exprimé le désir de qu’elle fût présentée à l’Académie par vous qui lui en avez suggéré l’idée. Je m’empresse de vous l’envoyer.

Veuillez agréer, Monsieur et cher confrère l’expression de mon entier dévouement et de mon profond respect

B. Baillaud

[deux figures]

\[\begin{array}{cc} d\varphi & Z \texttt{ fixe}\\ d\theta & \Omega \\ -d\phi & z \texttt{ mobile}\\ p=\theta' & O \Omega \\ q=\varphi' \sin \theta & OA \\ r=\varphi' \cos \theta - \psi' & Oz \end{array}\] \[2T=A[\theta'^2 + \varphi'^2 \sin^2\theta]+ C(\varphi'\cos\theta - \psi')^2\] \[2T=A\theta'^2 + \frac{A+C}{2}\varphi'^2 + \frac{C-A}{2}\varphi'^2 \cos2\theta -2C\varphi' \psi' \cos\theta + C\psi'^2\] \[\begin{array}{cc} \Theta = \frac{dT}{d\theta'} = A\theta'\texttt{;}& \Psi=\frac{dT}{d\psi'}=C\psi'-C\varphi'\cos\theta=-Cr \end{array}\] \[\phi=\frac{dT}{d\varphi'}=\frac{A+C}{2}\varphi'+\frac{C-A}{2}\varphi'\cos2\theta - C\psi'\cos\theta\]

\[\begin{array}{cc} 2T=\alpha x^2 + 2\beta x y +\gamma y^2 = x X + yY \texttt{;}&\mu=\beta^2 - \alpha \gamma\\ \mu x = - \gamma X+ \beta Y \texttt{;}&\mu y = \beta X - \alpha Y\\ X = \alpha x + \beta y \texttt{;}&Y = \beta x + \gamma y \end{array}\] \[2T = \frac{- \gamma X^2 + 2 \beta XY - \alpha Y^2}{\mu}\]

\[\mu = C^2 \cos^2 \theta - C [\frac{A+C}{2} + \frac{C-A}{2}\cos2\theta]\] \[\mu = \frac{C}{2} [(C-(A+C) + cos2\theta (C-(C-A)]=\frac{AC}{2} (cos2\theta -1)\] \[T = \frac{\Theta^2}{2A} - \frac{C\phi^2 + 2C \cos \theta \phi \psi + \psi^2 [\frac{A+C}{2} + \frac{C-A}{2} \cos2\theta]}{2 \mu}\]

\[\begin{array}{cccc} \Phi = G \texttt{;}& Cr = G \cos \theta \texttt{;}& \Theta = 0 & \texttt{pour} \end{array}\]

[équation illisible équivalente à la suivante]

\[T = G^2 \frac{C(1-\cos^2 \theta)^2 + A \cos^2 \theta \sin^2 \theta}{2 A C \sin^2 \theta} = G^2 \frac{C \sin^2 \theta + A \cos^2 \theta}{2 A C }\]

\[\begin{array}{cc} 2 T = A h^2 + C r^2 \texttt{;}& G^2 = A^2 h^2 + C^2 r^2 \end{array}\]

\[\begin{array}{ccc} A h = G \sin \theta \texttt{;}& C r = G \cos \theta \texttt{;}& 2 T = G^2 [\frac{sin^2 \theta}{A} + \frac{\cos^2 \theta}{C}] \end{array}\]

[ligne suivante en partie illisible] \[\begin{array}{ccc} \frac{dS}{d \theta} = f(\theta, \phi, \psi,T)\texttt{;}& \varphi + \int \frac{df}{d \phi} d \theta = const. \texttt{;}& \psi + \int \frac{df}{d \psi} d \theta = const. \end{array}\]

\[\int \frac{df}{dT} d \theta = t + const.\]

\[f = \sqrt{2 A (T + F )}\]

\[\begin{array}{cc} \int \frac{A d \theta}{f} = t + const. \texttt{;}& \varphi + \int \frac{A \frac{dF}{d \phi} d \theta}{f} = const. \end{array}\]

\[\begin{array}{ccc} \psi + \int \frac{A \frac{dF}{d \psi} d\theta}{f} \texttt{;}& \theta = const. \texttt{;}& \varphi + \frac{dF}{d \phi} t = const. \end{array}\]

\[\psi + \frac{dF}{d\psi} t = const.\]

\[C r = G \cos \theta_0\]

\[\begin{array}{cc} \frac{T}{G^2} = \frac{C \sin^2 \theta_0 + A \cos^2 \theta_0}{2 A C}\texttt{;}& \frac{\Theta^2}{2 A G^2} = \frac{C \sin^2 \theta_0 + A \cos^2 \theta_0}{2 A C} - \frac{C ( 1 - \cos \theta \cos \theta_0 )^2 + A \cos^2 \theta_0 \sin^2 \theta}{2 A C \sin^2 \theta} \end{array}\]

\[(1- \cos \theta \cos \theta_0 )^2 - \sin^2 \theta \sin^2 \theta_0 = 0\]

\[[1 - \cos (\theta + \theta_0 )] [1 - \cos (\theta -\theta_0)]\]

\(X,Y, Z\) fixes
\(x,y,z\) mobiles
\[\begin{array}{ccc} \varphi & OZ & (\texttt{angle de} OM'z \texttt{avec} yOz)\\ \psi & OX & (\texttt{angle de} OP' \texttt{avec} Oz )\\ \chi & OZ & (\texttt{angle de} P'OM' \texttt{avec} P'OZ)\\ \omega & OX & (\texttt{angle de} OP' \texttt{avec} OZ )\\ \theta & OZ & (\texttt{angle de} POZ \texttt{avec} YOZ ) \end{array}\]

\(OM'\) lié aux axes mobiles ; deux systèmes d’axes coïncident. \[OP'=OZ=Oz\]

1 rotation \(OP'\) reste fixe.
\[\begin{array}{cc} Oz=OP'=OZ \texttt{;}& ZP'OM'z \perp OX \end{array}\]

2 rotation \(OP'\) fixe
\[\begin{array}{ccc} OP'=OZ \texttt{;}& Oz \texttt{angle} \psi \texttt{avec} OP'\texttt{cet angle ne changera plus ;}& ZP'OM'z \perp OX \end{array}\]

3 rotation \(OP'\) peu importe \(P'OZ\) futur(?) lié.
\[\begin{array}{cc} OP'=OZ \texttt{;}& P'OZ \texttt{futur} \perp OX \end{array}\]

4 rotation OP’ et P’OZ futur liés axes mobiles
OP’ angle \(\omega\) avec OZ changera plus ; P’OZ futur [illisible] à poser par OP’ et OZ
5 rotation \(OP'\) et \(P'OZ\) futur liés
\(P'OZ\) futur fait angle \(\theta\) avec \(?OZ\)
\(d\varphi\) autour de \(Oz\) ; \(d\psi\) perp. à \(P'OzM'\)
\(d\chi\) autour de \(OP'\) ; \(d\omega\) perp. à \(P'OZ\) ; \(d\theta\) autour de \(OZ\)
\[l \frac{d\chi}{dt} = h \frac{d\varphi}{dt} = \frac{A-C}{A} rh\] \[\begin{array}{ccc} lG = (A-C) hr \texttt{;}& A^2 h^2 + C^2 r^2 = G^2 \texttt{;}& Ah^2+Cr^2 = 2T \end{array}\]

\[(\lambda + i \mu + j \nu + k \rho ) = e^{k \frac{\varphi}{2}} (C \psi + i s \psi) e^{k \frac{\chi}{2}} (C \omega + i s \omega) e^{k \frac{\theta}{2}}\] \[\begin{array}{cc} ik = - ki = -j \texttt{;}& e^{k \frac{\varphi}{2}} i = i e^{-k \frac{\varphi}{2}} \end{array}\] \[C \psi C \omega e^{\frac{k}{2} (\varphi + \chi + \theta)} + i s \psi C \omega e^{\frac{k}{2} (\chi + \theta - \varphi)} + i s \omega C \psi e^{\frac{k}{2} (\theta - \varphi -\chi)} - s \omega s \psi e^{\frac{k}{2} (\theta - \varphi - \chi)}\]

\[\lambda = C \psi C \omega \cos \frac{\varphi + \chi + \theta}{2} - s \psi s \omega \cos \frac{\theta + \varphi - \chi}{2}\] \[\mu = s \psi C \omega \cos \frac{\chi + \theta - \varphi}{2} + C \psi s \omega \cos \frac{\theta - \chi -\varphi}{2}\] \[\nu = s \psi C \omega \sin \frac{\varphi - \chi - \theta}{2} + C \psi s \omega \sin \frac{\varphi + \chi - \theta}{2}\] \[\rho = C \psi C \omega \sin \frac{\varphi + \chi + \theta}{2} - s \psi s \omega \sin \frac{\theta + \varphi - \chi}{2}\]