and theLaios of Plane Waves propagated through them. 115 



There are, therefore, three possible directions of molecular vibration for a 

 given direction of wave plane ; and there will be three parallel waves moving 

 with velocities determined by the magnitude of the axes of the ellipsoid, the 

 direction of vibration in each wave being parallel to one of the axe^ The 

 theorem involved in equation (16) was first proved by M. CAUcirr,* for bodies 

 whose molecules act by attractions and repulsions in the line joining them. It 

 is here extended to every kind of molecular action, and shown to be a funda- 

 mental property in molecular dynamics. It may be worth while to examine 

 the reason of its truth. It arises from the homogeneity of equations (14) 

 resulting from the absence of external forces. The right hand member of each of 

 the three equations of motion consists of eighteen terms; there will, therefore, 

 be in all fifty-four terms ; if each of these were supposed to have different and 

 independent coefiicients, equations (16) would cease to be true, and the coeffi- 

 cients of (cos a, cos |8, cos 7) would be nine distinct quantities, so that tlie 

 theorem which represents the direction and magnitude of the molecular vibra- 

 tion by means of an ellipsoid whose coefficients are functions of the direction 

 of the wave, would no longer be applicable. Equations (14) contain only 

 thirty-six distinct constants, and this reduction in the number of constants arises 

 from the assumption that the virtual moments of the system may be repre- 

 sented by jj'jc Ft/.i'f/ycZ* ; which is equivalent to assuming that the virtual mo- 

 ments of the molecular forces applied at any point may be represented by the 

 variation of a single function : 



o*- 



Q^ + Q'cq' -1- &C. = c r. 



The cubic equation whose roots are the squares of the reciprocals of the 

 axes of the ellipsoid (17) is 



(-P'-s){Q'-s){E-s)-F'-{F-s)-G"{Q'-s)-H"(R'-s)+2F'G'H'=0; 



where s—ev-. Hence, if ( P, Q, B, F, G, H) denote the same functions of (.r, y, z) 

 that {P, Q', R', F', G', H') are of (/, m, n), it may be shown, in a manner similar 



Exercices de Mathematiques, torn. v. p. 32. 

 q2 



