Motion of Solid and Fluid Bodies. 173 
It is important to observe that the directions of the wave and ray, being both 
given, (by the radius vector and perpendicular on tangent plane of the surface of 
wave-slowness ), completely determine the direction of molecular vibration in any 
given case. If the direction of the wave alone be given, the problem is indeter- 
minate ; for three parallel tangent planes may be drawn to the wave surface, each 
tangent plane being accompanied by its own direction of molecular vibration, and 
the three directions of vibrations being at right angles to each other: but if the 
direction of the ray be also given, there will be nothing indeterminate, for this 
will show which of the three tangent planes we must select, and, consequently, 
which of the three directions of molecular vibration. 
We now come to the conditions which must be satisfied at the limits; the 
facilities for obtaining which constitute the great advantage of the method of the 
Mecanique Analytique, as compared with any other method in Mechanics. The 
principal cases of limits which occur are the following: Ist. The limits may be 
completely fixed; 2nd. The body may have particular forces (2, ©, ¥) applied 
throughout the whole extent of the limiting surface: 3rd. The limits may be 
perfectly free; 4th. The limits may be the surfaces separating different kinds of 
bodies, so that the vibration passes from one into the other, changing the laws of 
its propagation at the limits, and requiring particular conditions to be satisfied at 
the separating surfaces. 
Ist. If the external surface of the body be completely fixed, so that its 
equation is independent of the time, then there are two cases to be considered : 
the molecules constituting the external surface may be absolutely fixed, so that 
no motion whatever is possible at the limits ; or the external surface may be fixed, 
so that motion perpendicular to it is impossible; but there may be motion along 
the surface itself. In the first case we have no condition at the limits; for the 
condition to be satisfied at the limits being, 
0 = A=§S((Neosr + Joos + w~cosy )eE + (Acosu + 5cosy + JcosA )oy 
+ (jcosy + ~cosdA + dcosp ee )w (32) 
is identically true, because 
NOY, Cy) = CE (0); 
But in the second case (which occurs, for example, in hydrodynamics), the 
variations at the limiting surfaces are not zero, but only subjected to the restric- 
