and the Laws of Plane Waves propagated through them. 109 



bodies, as it enables us to pass directly from the resultant forces of the mole- 

 cules to the form of the function which determines the laws of propagation 

 and reflexion. Introducing these forces into the conditions at the limits (6), 

 we obtain 



dF dF clF _ dF ,jy,dF dF 



^"dx +^°^ dy^ " dz -'" d^'^^'^dy'^^'^li' 



n> dF dF dF_ dF dF dF 



"' dx ^ ^'' dy ^ ^ °' dz ~ ^^'" Tv^^'^'dbj^ ^"^ ~dz- 



If the axis of z be made to coincide with the normal, we shall have 



dx ' dy ' dz 



Hence, 



P' _ p" t' — t" 



-f 03 — -t 03 1 SO — So I 



Qo3 = 0^3. V'o — Vo'. (11) 



Tit jnif yl yll 



■iios — ■''■03 I io — So • 



These equations at the limits are the mathematical statement of two facts, oi' 

 which one is mechanical and the other geometrical. 



1. That the forces, normal and tangential (arising from molecular action), 

 acting upon an element of the bounding surface, must be equal and opposite for 

 the two bodies in contact. 



2. That the motion of the particles in the bounding surface may be con- 

 sidered as common to both bodies. 



I shall now return to the general equation (4); let the function Fbe di- 

 vided into homogeneous parts, so that 



F=0o + 9!>] + 02 + ^3 + &C., 



where <^o is constant, (/>, of the first degree with respect to (a,, a,), and so of the 

 others ; then, neglecting (/>o and terms higher than the second, we obtain 



