and the Laws of Plane Waves propagated through them. 101 



The kind of motion which it is the object of this paper to investigate is of 

 the kind commonly called small oscillations ; and for this kind of motion it is 

 not necessary to use the most general equations, or to consider the unknown 

 quantities of the problem as functions of (.;■, y, z, t). We may use, instead of 

 {x, y, z), the co-ordinates {a, b, c) of the position of rest of the molecules. In 

 fact, any differential coefficient of a function <p, taken with respect to (a, b, c), 

 will be expressed by the equation, 



d<f) _ d(j) d.c d(l) dji d^ dz 

 da dx da dy da dz da ' 



but, since ,f = a + ^, y = [, + }j, z - c + ^, we obtain 



f?f _ , rf| (ly__dr} dz _ rff 



da da ' da ~ da ' da ~ da ' 



Hence, neglecting quantities of the second order, we find 



d(j) _ f/0 

 da dx ' 



and similarly for the other differential coefficients. 



In the remaining part of this memoir (unless the contrary be expressed), I 

 shall, therefore, consider {x, y, z) as the co-ordinates of the position of equili- 

 brium of the molecules, and (?, »;, f ) as the small displacements of the molecule ; 

 the element of the mass will be expressed by the equation dm = alxdydz, 

 where e denotes the density, not considered as a function of the time, since 

 {dxdydz) denotes the original element of the volume. 



Two kinds of waves can pass through such a body as water ; one, a surface 

 wave, depending on the action of gravity for its propagation ; the other, such- 

 a wave as propagates sound, and does not directly depend on external forces. 

 This latter is the kind of wave described in this paper. The equations pecu- 

 liar to it will be found by omitting (X, F, Z) from the general equations; but 

 though the external forces are not explicit in the formulaj, yet as they affect 

 the density and structure of the body differently at different points, though they 

 do not directly affect the wave, we must introduce them implicitly by rendering 

 e and the coefficients of V functions of (X, Y, Z), and therefore of [x, y, z) 



