268 Analysis of the Mecanique Celeste ofM. La Place. 



could take place in every possible mathematical relation 

 between the velocity and force. He shows that there exist 

 in this general case, principles analogous to those of the 

 conservation of the lining forces, of the areas, of the move- 

 ment of the ecntre of gravity, and of the least action in na- 

 ture. Ke draws from these* results the conditions which es- 

 sentially distinguish the state of motion from that of equili- 

 brium. — These very remarkable connections are entirely new. 

 The laws of the motions of transposition and rotation of 

 solid bodies are afterwards developed with the greatest extent. 

 The author here demonstrates the properties of the principal 

 axes, and their use in the determination of the momenta 

 inertia? : he searches for the place of the points which re- 

 main immoveable during the instantaneous movement of the 

 body ; and he is led in a very simple manner to observe, that 

 these points are situated upon a straight line, whence he 

 infers, that every movement of rotation, of whatever kind 

 it may be, is nothing else than a movement of rotation 

 around a straight hue fixed during an instant, and variable 

 from one instant to another, a property which has procured 

 it the name of instantaneous axis of rotation. The author 

 applies these principle's to the case where the movement of 

 the body is owing to a primitive impulsion which does not 

 pass by its centre of gravity : he shows how we may deter- 

 mine the discance of the centre of gravity from this impul- 

 sion, when tji J circumstance* of the movement of the body 

 arc known, and he gives an example of it drawn from the 

 movement of the carJi. 



He after urrds considers the oscillations of a body which 

 turns very nearly round one of its principal axes. He de- 

 monstrates that this movement is stable around the two 

 principal axes, the momenta inertice of which are the greatest 

 and the smallest, and that it is' not around the third princi- 

 pal axis; so that this last motion may be sensibly affected 

 by the slightest cause. He afterwards integrates the equa- 

 tions which determine the movement of rotation in the hy- 

 pothesis of the very small oscillations. Finally, he examine? 

 ihe movement of a body subjected to turn around a fixed 

 axis) and supposing this body animated by gravity alone, 



he 



