and the rest over the several barriers. The coefficient of any is evidently the total flux across the corresponding barrier, in a motion of which is the velocity-potential. The values of 4 in the first and last terms of the equation are to be assigned in the manner indicated in Art. 56.

If

also be a cyclic function, having the cyclic constants *,, *, &c., then (22) becomes in the same way

Equations (30) and (31) together constitute Thomson's extension of Green's theorem.

67. If in (30) we put = 4, and suppose to be the velocitypotential of an incompressible fluid, we find

To interpret the last member of this formula we must recur to the artificial method of generating cyclic irrotational motion explained in Art. 61. The first term has already been interpreted as twice the work done by the impulsive pressure - po applied to every part of the original boundary of the fluid. Again, ρê, is the impulsive pressure applied, in the positive direction, to the infinitely thin massless membrane by which the place of the first barrier was in Art. 61 supposed to be occupied; so that the do

expression S pdo, denotes the work done by the impulsive

dn 1

forces applied to that membrane; and so on. Hence (32) expresses the fact that the energy of the motion is equal to the work done by the whole system of impulsive forces by which we may suppose it generated.

In applying (32) to the case where the fluid extends to infinity and is at rest there, we must replace the first term of the third member by

where the integration

extends over the internal boundary only;

or, when the total flux across this boundary is zero, by

CHAPTER IV.

MOTION OF A LIQUID IN TWO DIMENSIONS.

68. If the velocities u, v be functions of x, y only, whilst w is zero, the motion takes place in a series of planes parallel to xy, and is the same in each of those planes. The investigation of the motion of a liquid under these circumstances is characterized by certain analytical peculiarities; and the solutions of several problems of great interest are readily obtained.

Since the whole motion is known when we know that in the plane z = 0, we shall confine our attention to the motion which takes place in that plane. When we speak of points and lines drawn in that plane, we shall in general understand them to represent respectively the straight lines parallel to the axis of z, and the cylindrical surfaces having their generating lines parallel to the axis of 2, of which they are the traces.

By the flux across any curve we shall understand the volume of fluid which in unit time crosses that portion of the cylindrical surface having the curve as base, which is included between the planes z = 0, z= = 1.

69. Let A, P be any two points in the plane xy. The flux across any two lines joining AP is the same, provided they can be reconciled without passing out of the region occupied by the moving liquid; for otherwise the space included between these two lines would be gaining or losing fluid. Hence if A be fixed, and P variable, the flux across any line AP is a

function of the position of P. Let be the function; more precisely, let denote the flux across AP from left to right, as regards an observer placed on the curve, and looking along it from A in the direction of P. Analytically, if 1, m be the directioncosines of the normal (drawn to the right) to any element ds of the curve, we have

If the region occupied by the liquid be aperiphractic, is necessarily a single-valued function, but in periphractic regions the value of may depend on the nature of the path joining AP. For spaces of two dimensions however periphraxy and multiplecontinuity become the same thing, so that the properties of 4, when it is a many-valued function, in relation to the nature of the region occupied by the moving liquid, may be inferred from Arts. 55, 56, where we have discussed the same question with regard to p.

The cyclic constants of y, when the region is periphractic, are the values of the flux across the closed curves forming the several parts of the internal boundary.

A change, say from A to B, of the point from which is reckoned has merely the effect of adding a constant, viz. the flux across a line BA, to the value of ; so that we may, if we please, regard as indeterminate to the extent of an arbitrary constant.

If P move about in such a manner that the value of does not alter, it will trace out a curve such that no fluid anywhere crosses it, i.e. a stream-line. Hence the curves = const. are the stream-lines, and is called the 'stream-function.'

If P receive an infinitesimal displacement PQ (= dy) parallel to y, the increment of is the flux across PQ from left to right, i.e. dy = u. PQ, or

Again displacing P parallel to r, we find in the same way

(2).

(3).

The existence of a function related to u and v in this manner might also have been inferred from the form which the equation of continuity takes in this case, viz.

which is the analytical condition that udy - vdx should be an exact differential.

The foregoing considerations apply whether the motion be rotational or irrotational. The formulæ for the component angular velocities, given in Art. 38, become

70. In what follows we confine ourselves to the case of irrotational motion, which is, as we have already seen, characterized by the existence, in addition, of a velocity-potential 4, connected with u, v by the relations

and, since we are considering the motion of incompressible fluids only, satisfying the equation of continuity

The theory of the function 4, and the connection of its properties with the nature of the two-dimensional space through which the irrotational motion holds, may be readily inferred from the corresponding theorems in three dimensions proved in the last chapter. The alterations, both in the enunciation and in the proof, which are requisite to adapt these to the case of two dimensions are for the most part purely verbal. An exception, which we will briefly examine, occurs however in the case of the theorem of Art. 46 and of those which depend on it.