158 On a Dynamical Theory of 



coefficients of the corresponding variations in the triple integrals, 

 we get 



_ 



d? C dy ' dz 1 



., , ~ _ _ 77 _ 



dt? dx dy ' 



which are the equations of propagation, giving the expression 

 for the accelerating force parallel to each axis of co-ordinates. 



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 consider 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 x' y, the 

 displacements will be functions of z' only ; and if i// be any 

 function of the displacements, we shall have, by formulae (d'), 



d\L d\L dz' dil d\L d\L , d\L d\L 



lT = -r,-r=-r, COS a", -f- = - COS 0", -* = - COS y", 



dx dz due dz dy dz dz dz 



so that the equations (5) may be written 



(% z dZ dY 



= #, cos /3 - b z r cos 7 ", 



dt~ dz dz 



z z 



= a? Tr cosy - c 1 , cos a , 

 df dz dz 



. z z 



= b z -V7 cos a - a z - cos /3 ; 

 d& dz dz 



and when we combine these with the following, 



