Memoirs of the Institute of France. 597 



Memoir on the Equation the roots of which are the principal moments 

 of inertia of a solid body, and on some other Equations of the 

 same kind. By the same Author. 



M. CAUCHY remarks that it has not hitherto been shewn, that the 

 roots of the above equation arc possible, except by indirect methods, 

 such as, for instance, the reduction of the cubic equation to a qua- 

 dratic, by the transformation of coordinates in space : he then pro- 

 poses a direct demonstration, depending on some theorems enun- 

 ciated but not demonstrated in this paper. 



Memoir on the Movement of a System of Molecules which attract or 

 repel each other at very small distances, and on the Theory of 

 Light. By the same Author. 



THE author states, that the equations of motion of such a system of 

 molecules may be integrated by the methods given by himself in 

 the 19th Number of the Journal de VEcole Poly technique, and that 

 these integrals have led him to form the following conclusions : 



1st. If in any system of molecules the electricity of the system be 

 equal in every direction, and a vibratory motion be produced at any 

 point, two spherical undulations, of constant, but unequal velocities, 

 will be propagated from that point. The first of these vibrations 

 will cease, when the initial dilatation of the volume ceases; if, then, 

 we suppose the vibrations to have been originally parallel to a given 

 plane, they will continue so. 



2d. If the system be such that the elasticity continues the same 

 about an axis parallel to a given line, in all directions perpendicular 

 to the axis, the equations of motion will contain several coefficients 

 depending on the nature of the system ; and a relation may be esta- 

 blished between their coefficients, such that the propagation of a 

 vibration first produced at any point may give rise to three undula- 

 tions, each of which coincides with a surface of the second degree. 

 Furthermore, omitting that undulation which disappears with the 

 dilatation of the volume, when the elasticity again becomes equal in 

 every direction, the surfaces of the two remaining undulations will 

 be reduced to a system comprising a sphere and a spheroid, of which 

 the axis of revolution coincides with the diameter of the sphere. 



The author observes, that the remarkable coincidence of this 

 result with the theorem of Huyghens on the double refraction of 

 light in crystals having a single axis, is deserving of our attention ; 

 and we may hence be induced to conclude, that the equations of the 

 motion of light coincide with those which express the movement of 

 a system of particles very little disturbed from the position of equi- 

 librium. 



VOL. I. MAY, 1831. 2 R 



