and the Laws of Plane Waves propagated through them. 123 



d-^ _ A dm 



'df~d^' 



and, by differentiating with respect to (.?■, y,z), we obtain 



rf-fl) A fd-(o d'u) d'^a)\ ,„„, 



7¥=7[d?+df^l?)' (26) 



which determines tlie cubical compression as a function of (,r, y, z, t). 



The equations (25) and (26) are applicable to the propagation of sound in 

 gases, and to the longitudinal vibrations of elastic soUds ; but there will be an 

 essential difference between the two cases. If the gas were freed from the 

 action of external forces, and unconfined at its limits, its molecules would sepa- 

 rate, and a displacement would develope no force tending to restore them to 



their original positions ; hence, ( -j— Bia j would become positive, which is in- 

 consistent with stable equilibrium, and the velocity of wave propagation 

 ( ^ —] would become imaginary. In order, therefore, that the equiUbrium of 



the gas be stable, we must suppose a constant pressure exerted at the limits, 

 sufficient to keep the molecules together, and restore them to their original 

 positions, if displaced. No such condition is requisite in the elastic solid, for 



-J— CO) I will be negative, without the assistance of forces appUed at the bound- 



ing surface. 



SECTION lU. LAWS OF TRANSVEESE VIBRATIONS. 



In order to obtain the form of the fimction V pecuUar to transverse vibra- 

 tions, we must suppose that, if a plane be drawn through the body in any 

 direction, the molecular forces exerted on any element of the plane will be 



altogether tangential. Hence we obtain 



b2 



