Natural-Philosophical Collections. 4g2 
* 
NATURAL-PHILOSOPHICAL COLLECTIONS. 
Extract from the Analysis of the Labours of the Royal Academy of Sciences 
during the year 1828; by Baron FouRIER. 
At the sitting of the 23d March 1828, M. Poinsot presented to the Academy 
a very important notice respecting the Theory and the precise determination of 
the Invariable Plane of the Area in the System of the Universe. 
It is known that, in the motion of a system of bodies which react in a given 
manner upon each other, if there be projected upon a plane the area traced around 
a fixed point or focus, by the vector radii drawn from this focus to all the equal 
particles of the system, the sum of these projected areze remains constant, not- 
withstanding the variations which each of them undergoes, from the connection 
and reciprocal action of the different bodies; and it is in this that the so well- 
known principle of the preservation of the aree consists. 
In thus considering the areze upon the different planes which may be drawn 
through the same focus, it is found that there exists a plane distinguished from 
all the others by the following property: If the areew be projected upon one of 
the planes perpendicular to this unique plane which we have said to be distinct 
from all the rest, the sum of the projections is always null. It is thus found, 
that the projection of the aree upon the single plane in question is the greatest 
possible. This property had already been remarked by geometricians ; they had 
selected this plane for simplifying their calculations in determining the motion 
of certain systems, and, for example, in the case of the motion of a solid body 
turning freely around its centre of gravity. ‘The author adds, it is this plane 
which M. Laplace has considered in our planetary system, and to which he has 
given the name of Invariable Plane. He has endeavoured to determine the po- 
sition which this plane ought to have had at the commencement of 1750, and 
his formule have given for that period the inclination of the plane to the ecliptic 
equal to 1°.7689 and 114°.3979 for the longitude of its ascending node. M. 
Poinsot remarked that this great geometrician, in establishing his analysis, only 
considered the aree described around the sun by the different planets considered 
as so many points of which each is charged with the entire mass of the planet 
and of its satellites. Now, it is well known, that M. Poinsot has discovered a 
new theory of the motions and are, in which these kinds of quantities are. with 
him only the geometrical expression of the couples or forces of rotation, which are 
at present exercised in the system. All geometricians are acquainted with these 
beautiful and ingenious researches which have contributed to the improvement 
of statics, and which at the same time possess the advantage of being clear and 
profound. : Rts 
He now concludes from his theory of motions that the truly invariable plane 
is nothing else than that of the area which would result from arew simultaneous- 
ly described. by the particles of the system, were all these arew composed toge« 
ther in the manner of simple forces applied upon a point, and that, consequently, 
to determine the true invariable plane, it is necessary to combine together, not 
only the aree which M. Laplace has considered, but also other aree which his 
analysis does not comprehend, viz.. those which result from the particular mo- 
tions of the satellites around their principal planets, and those which arise from 
the rotation of all these bodies and of the sun itself upon their respective axes. 
M. Poinsot remarked, that the plane of this resultant area is the only one of 
which it can be affirmed that it remains motionless in the heavens, or that it al- 
ways remains parallel to itself, whatever may be the changes which the succes- 
sion of ages may induce in the motions, the figure and the mutual position of the 
dierent celestial bodies. He adds, that if only a part ofthese simultaneous are 
be considered together, it cannot in that case be said that the partial area which 
results from them is invariable in magnitude and in its position in space ; whence 
