and the Laws of Plane Waves propagated through them. 105 



equations, which denote that the vibrating molecules at the bounding surface 

 may be considered as belonging to either body ; they are three in number, 



^o = C V»-ih, to = ^«- (7) 



Equations (6) and (7) contain the laws of reflexion and refraction of vibra- 

 tions for all bodies, and are completely determinate when the form of Fis 

 given for each of the bodies in contact. 



Equations (5), (6), (7), are necessary and sufficient to determine the propa- 

 gation and reflexion of waves, so far as they are connected with each other • 

 and no mechanical theory of vibrations is correct which does not exhibit thi«' 

 connexion, or which assumes such laws of reflexion and refraction as contra- 

 dict the laws of propagation ; the connexion between these laws is no proof of 

 the truth of any theory, but the want of a connexion would be a proof of the 

 inconsistency of a theory. 



The reduction of the general equations by the omission of the external 

 forces may require some explanation. There are two kinds of waves, as I have 

 stated in making the reduction, one only of which is the subject of our present 

 inquiry. Fluid bodies, such as the atmosphere and the ocean, can propagate tidal 

 waves depending on external forces ; they are also capable of propagating waves 

 of sound which depend directly on the molecular forces ; solid bodies cln only 

 propagate the second species of waves. In considering this kind of vibration, 

 we neglect the external forces, as they are of so much less intensity than the' 

 molecular forces, that they produce no effect on the motion ; but if we suppose 

 the motion to cease, and inquire into the state of equilibrium of the body, we 

 should then use the general formula, which includes external forces ; and, even 

 in the case of motion, all the coefficients must be considered as variable, in con- 

 sequence of the position of constrained equilibrium to which the body would 

 return, if the motion ceased. 



It is evident from an inspection of equations (5) and (6) that the difieren- 

 tial coefficients of the function F, with respect to (a„ o,, &c.), occupy an impor- 

 tant position in the theory of elastic media. Hitherto, I have only given to them 

 a mathematical definition, I now proceed to explain their physical meaning, by 

 the consideration of an elementary parallelepiped of the body. If we conceive a 

 plane drawn in any direction in the interior of the body, and consider the parts 



VOL. XXIL p 



