and the Laws of Plane Waves propagated through them. 103 



The equations of motion derived from (4) will be, 



(5) 



These equations contain the laws of propagation of every variety of wave 

 not depending on external forces ; and as the function has not been restricted 

 by any hypothesis as to the law of molecular action, they will be the dynamical 

 equations of propagation of sound in air, water, solids, and of light, if we adopt 

 the undulatory hypothesis. In all these cases, the difficulty is to ascertain the 

 correct form of V peculiar to the particular case ; the form being found, the 

 coefficients must be determined by experiment for each body : so far as theory 

 is concerned, the mechanical classification of bodies would be complete, if the 

 form of V were known for all. 



The conditions to be satisfied at the limits will be different, according as 

 the surface is fixed, free, acted on by special forces, or in contact mth other 

 bodies. As there is no difficulty in forming the equations for any of these 

 cases, I shall here give only the conditions at the limiting surface in contact 

 with other bodies. 



Leti^(.i", y, z) = <) be the equation of the surface passing through the posi- 

 tions of rest of the molecules which at any instant form the actual limiting 

 surface /(.r, y, z, t) ~ 0, hence 



dF ^ dF , dF , ^ 

 ^dx + ^dy+-^dz = 0; 



and if \, f^, v, be the angles which determine the position of the normal, we 

 shall have 



