and therefore the whole excess of pressure urging the rhomb in the direction 2 is

Now II was found in Article 35 to be

HD (1-dr-dy_dz),

And p in this investigation is the same as l in Article

dX dY 35. Also, the retention of and

dx dy in the external

factor would retain terms of the next higher order, which in the formation of II we have rejected; and therefore they must not be kept here. Thus we obtain for excess of pressure in direction z,

HDhkl.

d dx dy dz
+ +
dz dx dy

And the mass to be moved is Dhkl.

dz

Hence, by the usual laws of mechanics (Article 21, note), all our densities and pressures being estimated by weights (Article 12), and omitting every consideration of temperature,

F

or, as a does not depend on t, and gH = a3,

2

These three equations express the relation of the motion of the air to the forces producing the motion, in all the directions in which motion can be conceived. They are therefore absolutely sufficient; and no other equation can be introduced, except as equivalent to or deduced from these three.

37. Introduction of the Characteristic Function F.

The solution of these equations is in many cases facilitated by the use of a very peculiar Characteristic Function, for which we shall always use the letter F F is a function of x, y, z, and t. We cannot in all cases find a form of F which shall correspond to an assumed form of solution; but, if we assume the principal characters of a form of F, we can in all cases find the differential equations leading to a solution; and by

careful choice of the form, we can usually find solutions possessing the characteristics that we desire.

The definition of the form of F is contained in these

which is the form of equation now to be used. The function x(t) can only produce, in the solution, a func

tion of t, which will vanish in forming

be omitted without loss of generality.

dF

dx'

&c.: it may

We should have arrived at the same final result if

we had proceeded in the last step from

ďY d'Z
or
dt dt

38. Inferences from the value of F when its form has been found.

In Article 35, the density of air at any point is

The existence of terms dependent on t only would imply some general and simultaneous alteration of the density of air in every part of the atmosphere. As

this is not consistent with our physical assumptions, we must always suppose F to be so taken that x(t) is not required. With this notice, we shall abandon that function, and we have

The motions of a particle are found by the first assumptions,

These expressions, as may be expected, go through

some changes in special applications.

39. Application to a plane wave of air.

The equation to a plane is,

Normal from origin of co-ordinates

=x.cos a+y.cos ẞ+z.cos y;

where a, B, y are constant angles, subject to the condition cos a+ cos' B+ cos' y = 1. It seems likely therefore that our object will be gained by supposing F to be