453 



where p, 6, ^ are the polar co-ordinates of the second particle 

 with regard to the first. 



The tendency of this force will evidently be to change the 

 relative position of the two molecules. From these principles 

 the author has deduced, by the method of Lagrange, the 

 equations of equilibrium or motion of any body, homogeneous 

 or heterogeneous, whose particles satisfy the hypothesis of 

 independent action. These are, in general, partial differen- 

 tial equations of the second order. If the body be homoge- 

 neous, the coefficients in these equations will become constant, 

 and the differential coefficients of the first order will dis- 

 appear. 



The author finds that in the case of a homogeneous solid 

 the number of distinct coefficients which these equations con- 

 tain will be fifty-four, namely, eighteen for each equation. 

 The equations of motion are in this case of the form 



d^K , d'^^ d^-^ d?l 



dSj^ 

 dx^ 



dx^ 



d-K (Pn fJ'^T 



+ 2Di — I- + &c. + 2£i -^ + &c. + 2F,4-r^ &c. 

 dydz dydz dydz 



^ = &c^ = &c 

 df ^^- df ^^' 



The coefficients Ai, Bi, &c. being all independent, their num- 

 ber will plainly be as above stated. 



The author has integrated these equations for the case of 

 plane waves and rectilinear vibrations. He finds that for each 

 direction of wave plane there are three directions of vibration. 

 These directions are not, however, at right angles, nor are 

 they necessarily all real. 



