first order or of a higher order. And it will be seen that quantities of an order which we must retain in one part of the process may be rejected in another part. For instance: the elasticity which determines the motion of a small volume of air is not the absolute elasticity in that volume, but the difference between the elasticity in front and that in rear; and this difference is a small quantity of a higher order than the principal term; but the motion depends entirely on it, and it must be carefully retained. But the mass of matter in that volume, to be moved by the differential elasticity, is not the difference between two masses, but is the entire mass in the volume; the difference of the densities in front and in rear is unimportant, and may be wholly rejected. Thus it will be seen that continued attention is necessary for judging on the import of the terms which it is proposed to reject, and on their value as compared with those which it is proposed to retain.

35.

Investigation of the Elastic Force at any point of the disturbed Air.

Let x, y, z, be the ordinates of a particle in a tranquil state; x+X, y + Y, z + Z, the ordinates of the same particle in its disturbed state at the time t. Conceive seven neighbouring particles forming with the first, in the quiescent state, a rectangular parallelopiped; two bounding planes being defined by the ordinates x and x+h, two by the ordinates y and y+k, and two by the

8

ordinates z and z+l. Confining our attention for a moment to the four particles in a plane parallel to xy, with ordinate z; their original ordinates, and their disturbed ordinates at the time t, parallel to that plane, will be as follows:

And if, for the disturbed state, we subtract the ordinates of the 1st point from those of 2nd, 3rd, 4th, we have (see Figure 9),

Affecting this with the external operation

it becomes

=

1 d

dv. du = 4 ⋅ dv (4′ (v)}* — — {(4′ (v)}°. p′′ (v)

Or, neglecting (u), and restoring t and x,

When we are considering the state of a solitary wave at a great distance from the origin, v or (at — x), which (see Article 31) is limited within the value B, may be considered very small with regard to a; and in the factors of the small terms, for at +x or 2x + (at — x) we / may put 2x; and we have

+ 2 (p'′ (at − x)}". ¿′′ (at — x).

This degree of approximation will suffice for our

present purposes.

It appears from these that the projections on the plane xy, of the four points which were at the angles of a parallelogram, are now at the angles of a lozenge. The ordinates of 1st and 2nd points in the direction of x are not now equal, and those of 1st and 3rd in the direction of y are not now equal. Let the new distance from 1st point to 2nd be p, making the angle & with y; and that from 1st to 3rd be q, making the angle x with x. Then