7. The second motion.-The second motion, a rotational motion having zero normal velocity at the boundary, is got by an imaging of the vortex distribution; and the utility of the periodic conformal transformation consists in the fact that it makes this imaging process possible.

The specification of the motion may be by a streamfunction, or by functions (2, v2) equal to dy2/dn, -dy/d, the corresponding velocity components in the z plane being u2/h, v2/h.

It is convenient, for a moment, to think of 2 as the stream-function of a motion in the plane, of which (u, v) would be the velocity. If such a motion, unhampered by any rigid boundary, were due to a line-vortex representing a circulation m, situated at =', and periodically repeated at ±rλ, (r=1, 2, 3,...), it is known that the corresponding stream-function would be

When the line n=0 is a rigid boundary an image must be introduced at the point ", where " is the complex conjugate to ', and the stream-function is

m

2π

π

log | sin(-5)/sin (5-5)].

Instead of single vortices a continuous periodic distribution of vorticity may be postulated over the whole area between 7=0 and 7=t, the former line being still a rigid boundary; and if m be replaced by o(') dS', where ds" is an element of area and a density of distribution, the stream-function is

1

π

-Jo (5) log sin (5-5) sin (-) ds', (15)

2π

the area integral being taken over a rectangle of length t

so, if the circulation round the contour of any area in the plane is to be equal to 2 times the corresponding area in the z plane, it is necessary that

o(5') = 2w{h(5')}2.

* Proc. Royal Irish Academy, l. c. § 13.

(16)

and this specifies a motion in the plane which has vorticity o
at all points for which t>n> 0, and has the curve corre-
sponding to 7=0 as a fixed boundary.
The corresponding u, v functions are

It is to be noticed that if is inside the area of integration the subjects of integration in (17), (18), and (19) have infinities at '=; these infinities, however, are not sufficiently powerful to make the integrals divergent.

8. These formulæ may be checked by noting directly what conditions ut, vt must satisfy if they are to represent the kind of motion in the z plane which has been described in the previous article.

§ and ʼn are curvilinear coordinates in the z plane, and the corresponding velocity components are U=u/h and V=vr/h. The boundary condition V=0 or v=0, when =0, is clearly

satisfied. The equations of continuity and of vorticity are

when is inside the area of integration.

=

2∞ {h()}2, . (20)

The testing of these equalities involves the differentiation of the integrals of formulæ (18) and (19), and this cannot be done by the ordinary rule of differentiation under the sign of integration, since that would yield semi-convergent integrals. It is, however, easy to apply the method of differentiation explained in the Cambridge Tract on Volume and Surface Integrals used in Physics,' articles 21 and 23, and it is then readily verified that u and v satisfy both conditions.

A single compact formula giving both u, and v, is

π

π

ve + iu, = { [" ( ^ {(4 (5') } 3 [cot (5 — 5')—cot 7 (5—5')] d§'dn'. (21)

vt

λ

0 0

The integral on the right-hand side has the appearance of being a function of the complex variable; but this appearance is deceptive, for if v and u were conjugate functions there would be no vorticity *.

9. The next step that suggests itself is a passage to limits fort infinitely great. This is feasible in the case of vt, but the integral representing u proves to be divergent.

It will be shown that this can be remedied by adding to us before passage to limit, a suitable function of t which does not involve or n. An addition to u means simply the superposition of an irrotational motion with circulation round the fixed boundary. This will serve to cancel an undesired circulation at infinity in the motion defined by ut and vt.

No corresponding addition to v need be or could be made.

It being necessary to consider not only the convergence of the u and v integrals but also the forms to which these, regarded as functions of and 7, tend for n very great and positive, it is important to notice two expansions of the function which appears in square brackets in formula (21).

On this point compare the writer's note "On Functionality of a Complex Variable" in the 'Mathematical Gazette,' early in 1918.

= 4i [+Σ exp (2) cosh2 {n—i—'))], (22)

and if n>n'>0

λ

cot (5—5′)—cot (5—5'')

λ

λ

=-4i Σ sinh (28π

2 Sπ

exp

λ

{i(§—§')—n},. (23)

where s=1, 2, 3, ....

The formulæ can be verified by noticing that each side of each equality is equivalent to

For n' great the most important terms of {h(')} are

+ 2ks exp (2) cos (+-) +

πη
λ

λ

+73-70)+..., (24)

the terms decreasing by successive negative powers of exp (2πn'); it is to be noted that the functions of which multiply these exponentials are sums which may contain constant as well as harmonic terms.

In studying the form of the subject of integration in formula (21), for great values of n', with a view to examining the divergence for to, the product of the series (22) and (24) may be used. For this purpose a term of the product may be ignored if its integral with respect to ' through a range λ is zero, or if it contains as factor the exponential of a negative multiple of n'. By one or other of these tests every term is negligible except one, namely,

which contributes to the integral, at its upper limit (after integration with respect to '),

(i×3λ2/2π) exp (4πt/λ).

Hence the subtraction of (iwλ/2π) exp (4πt/λ) from the right-hand side of formula (21) gives an expression which has a definite limit for to. This justifies the definition

π

Σπ

+SC^{M(E)) { cot (5-5)-cot (5—5") } de d']. (25)

10. Limiting form of the second motion at infinity.—The formula (25) defines a motion which has all the characteristics required for the second motion, with the, as yet, possible exception of tending to the proper form at infinity. It is now necessary to inquire what are the limiting forms to which u and v2, functions of & and 7, tend with indefinite increase of n.

For this purpose the series-expansions of formulæ (22) and (23) may be used, each within the appropriate range of n'. If [22], [23] be used as abbreviations for the expressions on the right-hand side of these formulæ,

As the integral of formula (25) is absolutely convergent in respect of the infinity of the subject of integration at, it is safe to use the series [22] and [23] right up to the critical value n'n which separates the ranges within which they are respectively valid. For {h()} the series of formula (24) is again employed.

In taking the term-by-term products of the two series which are multiplied under the sign of integration, any resulting term may be passed over whose integral with respect to over a range λ is zero. Thus a term of the the type

cos { (2πm/X)(§' + a)} cos {(2πn/λ) (E' +B)}

need not be considered unless m=n. Further, when only an approximation for n great is desired, an estimate of the importance of an exponential in n' and 7 is to be made on the hypothesis that ʼn is very great but that ŋ' is of