Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/161

 The terms which involve b and those which involve e must be separately zero, since they represent respectively the irrotational and the circuital parts of the equation. Thus, c satisfies the pair of equations

while b is to be determined from

A particular solution of the equations for c is easily seen to be

which represents a transverse plane wave propagated with velocity ✓(n/ρ). It can be shown that the general solution of the differential equations for c is formed of such waves as this, travelling in all directions, superposed on each

A particular solution of the equations for b is

which represents a longitudinal wave propagated with velocity

the general solution of the differential equation for b is formed by the superposition of such waves as this, travelling in all directions.

Poisson thus discovered that the waves in an elastic solid are of two kinds: those in c are transverse, and are propagated with velocity (n/ρ)$1⁄2$; while those in b are longitudinal, and are propagated with velocity {(k + $4⁄3$n)/ρ}$1⁄2$ The latter are waves of dilatation and condensation, like sound-waves; in the c-waves, on the other hand, the medium is not dilated or condensed, but