(simple) irreconcileable non-evanescible circuits, and no more, can be drawn in it, is said to be 'n-ply-connected.'

The shaded portion of Fig. 3, Art. 40, is a triply-connected space of two dimensions.

It is shewn in note (B) that the above definition of an n-plyconnected space is self-consistent. In such simple cases as n = 2, n=3, this is sufficiently obvious without demonstration.

54. Let us suppose, now, that we have an n-ply-connected region, with n-1 simple independent non-evanescible circuits drawn in it. It is possible to draw a barrier meeting any one of these circuits in one point only, and not meeting any of the n − 2 remaining circuits*. A barrier drawn in this manner does not destroy the continuity of the region, for the interrupted circuit remains as a path leading round from one side of the barrier to the other. The order of connection of the region is however reduced by unity; for every circuit drawn in the modified region must be reconcileable with one or more of the n 2 circuits not met by the barrier.

n

A second barrier, drawn in the same manner, will reduce the order of connection again by one, and so on; so that by drawing 1 barriers we can reduce the region to a simply-connected one. A simply-connected region is divided by a barrier into two separate parts; for otherwise it would be possible to pass from a point on one side the barrier to an adjacent point on the other side by a path lying wholly within the region, which path would in the original region form a non-evanescible circuit.

Hence in an n-ply-connected region it is possible to draw n -1 barriers, and no more, without destroying the continuity of the region. We might, if we had so chosen, have taken this property as the definition of an n-ply-connected space. We leave it as an exercise for the student to prove that this definition is free from ambiguity, and that it is equivalent to the former one.

Irrotational Motion in Multiply-connected Spaces.

55. The circulation is the same in any two reconcileable circuits ABCA, A'B'C'A' drawn in a region occupied by fluid moving

* In simple cases this is obvious. For a general proof see note (B).

irrotationally. For the two circuits may be connected by a continuous surface lying wholly within the region; and if we apply the theorem of Art. 40 to this surface, we have, remembering the rule as to the direction of integration round the boundary,

or

I(ABCA)+I(ACBA)=0,

I (A BCA) = I (A'B'C'A').

If a circuit ABCA be reconcileable with two or more circuits A'B'CA', A"B"C"A", &c., combined, we can connect all these circuits by a continuous surface which lies wholly within the region, and of which they form the complete boundary. Hence

or

I(ABCA)+I(A'C'BA)+I(A'C” B'A')+ &c.

=

I(ABCA)=I(A'B'C' A)+I(4′′ B* C" A') + &c. ;

i.e. the circulation in any circuit is equal to the sum of the circulations in the several members of any set of circuits with which it is reconcileable.

Let the order of connection of the region be n+1, so that n independent simple non-evanescible circuits a,, a,,...a can be drawn in it; and let the circulations in these be к,, *,,..., respectively. The sign of any will of course depend on the direction of integration round the corresponding circuit; let the direction in which is estimated be called the positive direction in the circuit. The value of the circulation in any other circuit can now be found at once. For the given circuit is necessarily reconcileable with some combination of the circuits a,, a,,...a; say with a1 taken P1 times, a, taken P2 times and so on, where of course any p is negative when the corresponding circuit is taken in the negative direction. The required circulation then is

Since any two paths joining two points A, B of the region together form a circuit, it follows that the values of the flow in the two paths differ by a quantity of the form (14), where, of course, in particular cases some or all of the p's may be zero.

56. Let be the flow from a fixed point A to a variable point P, viz.

$ = ["(udx+vdy+wdz).....

.(15).

So long as the path of integration from A to P is not specified, & is indeterminate to the extent of a quantity of the form (14).

If however n barriers be drawn in the manner explained in Art. 54, so as to reduce the region to a simply-connected one, and if the path of integration in (15) be restricted to lie within the region as thus modified (i.e. it is not to cross any of the barriers), then becomes a single-valued function, as in Art. 42. It is continuous throughout the modified region, but its values at two adjacent points on opposite sides of a barrier differ by ±. To derive the value of when the integration is taken along any path in the unmodified region we must add the quantity (14), where any p denotes the number of times this path crosses the corresponding barrier. A crossing in the positive direction of the circuits interrupted by the barrier is here counted as positive, a crossing in the opposite direction as negative.

By displacing P through an infinitely short space parallel to each of the co-ordinate axes in succession, we find

so that o satisfies the definition of a velocity-potential, Art. 22. It is now however a many-valued or cyclic function; i.e. it is not possible to assign to every point of the original region a definite value of 4, such values forming a continuous system. On the contrary, whenever P describes in the region a non-evanescible circuit, & will not, in general, return to its original value, but will differ from it by a quantity of the form (14). The quantities K1, K,... are called by Thomson the 'cyclic constants' of p.

n

57. The foregoing theory is illustrated by Ex. 4, Art. 35. The formula there given make the velocity infinite at points on the axis of z, which must therefore be excluded from the region to which our theorems apply. This region becomes thereby doublyconnected, for we can connect any two points A, B of it by two

irreconcileable paths passing on opposite sides of the axis, e.g. ACB, ADB in the figure. The portion of the plane zx for which Fig. 4.

0

2π

x is positive may be taken as a barrier, and the region is thus made simply-connected. The circulation in any circuit meeting this barrier once only, e.g. in ACBDA, is . rde, or 2p. That in any circuit not meeting the barrier is zero. In the modified region may be put equal to a single-valued function, viz. μe, but its value on the positive side of the barrier is zero, that at an adjacent point on the negative side is 2πμ.

More complex illustrations of irrotational motion in multiplynected spaces will present themselves in the next chapter.

58. Before proceeding further we may briefly indicate a somewhat different method of presenting the above theory.

Starting from the existence of a velocity-potential as the characteristic of the class of motions which we wish to study, and adopting the second definition of an n+1-ply-connected region, given in Art. 54, we remark that in a simply-connected region every equipotential surface must either be a closed surface, or else form a barrier dividing the region into two separate parts. Hence, supposing the whole system of such surfaces drawn, we see that if a closed curve cross any given equipotential surface once it must cross it again, and in the opposite direction. Hence, corresponding to any element of the curve, included between two consecutive equipotential surfaces, we have a second element such that the flow along it, being equal to the difference between the corresponding values of 4, is equal and opposite to that along the former; so that the circulation in the whole circuit is zero.

If however the region be multiply-connected, an equipotential surface may form a barrier without dividing it into two separate parts. Let as many such surfaces be drawn as it is possible to draw

The number of

without destroying the continuity of the region. these cannot, by definition, be greater than n. Every other equipotential surface which is not closed will be reconcileable (in an obvious sense) with one or more of these barriers. A curve drawn from one side of one of these barriers round to the other, without meeting any of the remaining barriers, will cross every surface reconcileable with it an odd number of times, and every other surface an even number of times. Hence, the circulation in the circuit thus formed will not vanish, and 4 will be a cyclic function. In the method adopted above we have based the whole theory on the equations

and have deduced the existence and properties of the velocitypotential in the various cases as necessary consequences of these. In fact, Arts. 41, 42, and 53-56, may be regarded as a treatise on the integration of this system of differential equations.

The integration of (16), when we have, on the right-hand side, instead of zero known functions of x, y, z, will be treated in Chapter VI.

59. If the density of the fluid be either constant or a function of the pressure only, and if the external impressed forces have a single-valued potential, the cyclic constants of p do not alter with the time. For if 4, be the initial value of the velocity-potential, we have, Art. 23,

Under the circumstances stated x is a single-valued function, and

the cyclic constants of are the same as those of .. In other words the circulations in the several circuits of the region occupied by the fluid are constant.