Page:Mécanique céleste Vol 1.djvu/21

Rh described by the projection of the radii is a maximum, continues always parallel to itself, [187, &.C.] The principles of the living forces and of the areas, may be reduced to certain relations between the co-ordinates of the mutual distances of the bodies of the system, [189, &c.] Planes passing through each of the bodies of the system, parallel to the invariable plane drawn through the centre of gravity, possess similar properties, [189"] §22

Principle of the least action, [196']. Combined with that of the living forces, it gives the general equation of motion § 23

CHAPTER VI. ON THE LAWS OF MOTION OF A SYSTEM OF BODIES, IN ALL THE RELATIONS MATHEMATICALLY POSSIBLE BETWEEN THE FORCE AND VELOCITY 137

New principles which, in this general case, correspond to those of the preservation of the living forces, of the areas, of the motion of the centre of gravity, and of the least action. In a system which is not acted upon by any external force, we have, Jirst, the sum of the finite forces of the system, resolved parallel to any axis, is constant ; second, the sum of the finite forces, to turn the system about an axis, is constant ; third, the sum of the integrals of the finite forces of the system, multiplied respectively by the elements of their directions, is a minimum : these three sums are nothing in the state of equilibrium, [196'", &c.] § 24

CHAPTER VII. ON THE MOTIONS OF A SOLID BODY OF ANY FIGURE WHATEVER 144

Equations which determine the progressive and rotatory motion of the body, [214 — ^234]. § 25, 26

On the principal axes, [235]. In general a body has but one system of principal axes, [245"]. On the momentum of inertia, [245'"]. The greatest and least of these momenta appertain to the principal axes, [246"], and the least of all the momenta of inertia takes place with respect to one of the three principal axes which pass through the centre of gravity, [248']. Case in which the solid has an infinite number of principal axes, [250, &c.] § 27

Investigation of the momentary axis of rotation of the body, [254"] . The quantities which determine its position relative to the principal axes, give also the velocity of rotation, [260'*]. . . § 28

Equations which determine this position, and that of the principal axes, in functions of the time, [263, &.C,] Application to the case in which the rotatory motion arises from a force which does not pass through the centre of gravity. Formula to determine the distance from this centre to the direction of the primitive force [274]. Examples deduced from the planetary motions, particularly that of the earth [275*] ^29

On the oscillations of a body which turns nearly about one of its principal axes, [278]. The motion is stable about the principal axes, whose momenta of inertia are the greatest and the least ; but it is unstable about the third principal axis, [281'«] § 30

On the motion of a solid body, about a fixed axis, [287]. Determination of a simple pendulum, which oscillates in the same time as this body, [293'] § 31

CHAPTER VIII. ON THE MOTION OF FLUIDS. .... ,n^ Equations of the motions of fluids, [296] ; condition relative to their continuity [303'"]. . § 32

Transformation of these equations ; they are integrable when the density is any function of the pressure, and at the same time, the sum of the velocities parallel to three rectangular axes, each being multiplied by the element of its direction, is an exact variation, [304, &c.] Proof that this condition will be fulfilled at every instant of time, if it is so in any one instant, [316]. . § 33