Motion of Solid and Fluid Bodies. 169 
cop = Sag t bag + Gat 2( Bagh + *aaty* ae 
+ agit net oT +2(8 ang tag dydz * +0 ae 
+ Big bgt gat? Braga + gags + oN 
Te eget ge t hay t?(doaniy + hades + eas) 
+n otha et 2 (wags + amas + * aay) 
+g tat Ate (waee + Aan +h Laas 
To these equations of motion we must add the equation of continuity, 
dé (dé d dy dé 
atiat a + a)=° (27) 
and we shall have four equations to re the four quantities (& y, % ¢) as 
functions of (, y, z,¢), « being the original density, and ’ the variable density 
at any point. The general integral of these equations is hardly to be expected 
in the present state of the integral calculus ; but there exists a particular integral, 
which represents the case of plane waves, and which leads to many important re- 
sults. Let 
E=cosa.f(d), y=cosB. flo), C=cosy.f(o), gp=le+my+nz — vi, 
where (a, 8, y) denote the direction of vibration of molecules, (/, m, 7) the direc- 
tion of normal to wave-plane, and v the velocity of waves. The question to be 
determined is the following: given (/, m, 2), to determine, by substituting (&, 1, ¢) 
in the equations of motion, whether it is possible to find real values of (a, 8, y, 7), 
for which such a direction of wave-plane is possible. 
Substituting the values of (& , ¢) in the equations (26), we obtain the fol- 
lowing equations of condition : 
ev’.cosa = p’cosa + n’cosB + G’cosy, 
ev’.cosB = Q’cosB + F’cosy + Hn’cosa, (28) 
ev.cosy = R’cosy + G’cosa + F’cosp; 
2A 2 ~ 
