28 Mr. Mac Cuntacu on the dynamical Theory of 
But as the variations &, 6, 8¢ are arbitrary and independent, this equation can- 
not hold unless the double integrals, which relate to the limits of the system, 
reduce themselves to zero, leaving the equality to subsist, independently of the 
variations, between the triple integrals alone. Equating, therefore, the coeffi- 
cients of the corresponding variations in the triple integrals, we get 
ae ° de ° dat a 
i ES » ak 
dé Ya ee dy 
which are the equations of propagation, giving the expression for the accelerating 
force parallel to each axis of coordinates. 
When there is a single medium extending indefinitely on all sides, the 
conditions relative to the limits are of no importance, and we have only to con- 
sider the equations (5), from which we shall now deduce the laws by which a 
system of plane waves is propagated. 
Supposing the waves to be parallel to the plane of 2’ y’, the displacements 
will be functions of 2’ only, and if y be any function of the displacements, we 
shall have, by formule (d’), 
dy _dydz' _ dy , ay _ dy , ay _ dy 
— cosa , 
dx dz dz dz 
so that the equations (5) may be written 
@é _ dz ; » dY 7 
ip ° qe co B —b qe C81» 
dn 2 dx eer naz Beal! 
gpa ag Ty eae S ; 
ae 
dy dx 
— > = & —, cosa’ — a? —, cos B’ 5 
dt dz’ dz‘ ; 
and when we combine these with the following, 
