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intensities at corresponding points have been made with reflecting surfaces of rectangular form or consisting of either two or three elements in the same plane for various angles of incidence; the results show that the expression for the illumination at any point of the diffraction pattern contains a factor proportional to the square of the cosine of the obliquity at such point.

The experiments and observations described in the note were carried out in the Palit Laboratory of Physics. The writer hopes to carry out further work on the subject of oblique diffraction by various forms of aperture, and particularly in regard to the positions of the points of maximum intensity in the pattern, which would no doubt differ from those given by the usual formulæ owing to the asymmetry of the illumination curves.


8th June, 1917.

XII. On the Two-Dimensional Motion of Infinite Liquid produced by the Translation or Rotation of a Contained Solid. By J. G. LEATHEM, M.A., D.Sc., Fellow of St. John's College, Cambridge


ERIODIC conformal transformations-that is, transformations by which doubly connected regions in the plane of a variable z=x+iy, externally unbounded and bounded internally by a closed polygon or curve, may be represented conformally and repeatedly upon successive semi-infinite strips of width λ in the half-plane ŋ>0 of a variable +in-have been studied by the present writer in a previous paper †. It has there been shown how the knowledge of such a transformation for any particular curve makes possible the specification of a field of circulatory liquid flow (with or without logarithmic singularities) round a fixed solid body bounded by the curve.

It is now proposed to show that such knowledge makes possible also the determination of the field of irrotational motion due to any translation or rotation of the same solid in surrounding infinite liquid.

Communicated by the Author.

J. G. Leathem, "On Periodic Conformal Curve-Factors and CornerFactors," Proc. Royal Irish Academy, vol. xxxiii. Sec. A, August 1916.

2. The geometrical, or (z, Č), relation may be either in the form


or in the differential form

dz = C(5)ds,


where is a periodic curve-factor of linear period λ and angular period 27. With this angular period it is necessary that both () and ƒ() should, for n great and positive, tend to infinity like exp (2πn/λ). In fact, () is expan

sible in the form

z = f($),
· ·



(C) = exp(-2πi/λ). Σcexp (2πsič/λ), (3) where s=0, 2, 3, 4..., and the coefficients may be complex. The periodicity of z makes it necessary + that c1 = 0. From integration of (3) it follows that

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exp(-6) 23


* L. c. § 5.

† L. c. § 4.

z=ƒ($)=c+(iλ/2π) exp(− 2πiɣ/λ). Σ{cs/(1—s)} exp (2πsiğ|λ), (4)

where c is another complex constant ‡.

It is also to be noticed that, if | C(5)| = h, and if c=к, exp (iv),


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h3= exp(+17) [ko2 + exp (− 4") 2x JK: 05 ( 1 + 73-70)







+ exp(-8) { 2xx, cos (+1670) + x2)}...]. (5)




the terms containing ascending integral powers of

exp (−2πη/λ).



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The problem of obtaining a transformation of the type of formula (4), so that a given closed curve shall correspond to n=0, is the same as that of the parametric representation of the given curve by a formula of the type

x+iy=mexp(−ip)+μ+ Σμ exp (isp),

where is a real parameter, m and the u's are complex constants, and s takes positive integral values.

It is to be noted that a formula of this type need not represent a curve free from nodes unless the constants are suitably restricted.

It is understood that the boundary in the z plane is the locus corresponding to n=0.

3. Field of flow due to translation of the boundary.—If the boundary have a velocity V in a direction making an angle μ with the axis of x, the superposition on the whole system of such uniform velocity as brings the boundary to rest gives an irrotational fluid motion which has zero normal velocity at the boundary and tends, for z infinite, to flow V in the direction μ+. If this motion have velocity potential and stream-function, so defined that the velocity is the upward gradient of p, and if w=+iy, w must tend, for z infinite, to the form

or, in terms of Y,

Now if

− V2 exp (−iμ) + const.,

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- Vo(i/2T) expi(-μ-2π/λ).

w=-(V/π) sin (2π/λ—+μ),

this tends, for n∞, to the form (7); and as
¥=− (Vкλ/π) cos (2π§/λ—¥+μ) sinh (2πŋ/X),

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it is clear that is zero along the boundary n=0. The corresponding form of shows that there is no circulation round the boundary; and w is free from infinities in the relevant region.

Hence formula (8) specifies that irrotational motion past the fixed boundary whose limit form, at indefinitely great distance, is the assigned uniform flow.

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4. The impulse of the motion due to translation.--Though modern speculation tends to regard wave-motion as the preponderating factor in suction and other inertia phenomena of floating bodies, it can hardly be doubted that, in the case of a submarine at least, the ordinary inertia coefficients measure approximately the resistance to quick changes of velocity. Thus the evaluation of the impulse (X, Y) of the combined motion of solid and fluid, when the solid has translatory motion, is of interest.

If an approximation to w for || great, closer than that afforded by formula (6), be

·V≈ exp (−iμ)+C+D/z, .


(6 a)

where C and D are complex constants, it is known * that X+iY=−2πpD,

where p is the density of the liquid, and it is supposed that the mean density of the solid is also p.

If formula (a) be expressed in terms of by means of formula (4), it yields the approximation

w= — Vxo (24) exp i (Yo—μ— 2 ) + C'




+ { V«;( 1 ) exp i(ve−μ) + D ( 37 ) exp (−ive) } exp(2xit



where C' is a constant. On comparison of this with formula (8), it appears that the coefficient of exp (2πiğ/λ) must equal Vo(iλ/2π) exp i (μ— Yo). Hence

D= − (Va2/4π2){κ2 exp (iμ) — к。«1⁄2 exp i(yo+Y1⁄2−μ)}, and therefore

X+iY=(pVx2/2π){ |co|2 exp (iu)-coc2 exp (-iu)}. . (9)

5. Field of flow due to rotation of the boundary.-When the boundary has a motion of rotation, the specification of the liquid motion presents greater difficulty; the outline of the procedure is as follows.

One motion is known which satisfies the proper condition at the moving boundary-namely, a rotation of the whole liquid, as if rigid, with the same angular velocity as the boundary. This may be called the first motion. It is not the required motion because it is rotational, and because it has infinite velocity at infinity.

Another motion, which will be called the second motion, can be specified. This also is rotational, having the same vorticity at every point as the first motion; but at the boundary its normal velocity is zero. It has infinite velocity

at infinity.

If the second motion be subtracted from the first motion, the result is an irrotational motion whose normal velocity at the boundary is the same as that of the boundary itself. This may be called the difference motion. If the velocities of the first and second motions tend to equality at infinity in such manner that the difference motion tends to zero at infinity and has no circulation round the solid, the difference

* J. G. Leathem, "Some Applications of Conformal Transformation to Problems in Hydrodynamics," Phil. Trans. Roy. Soc., A, vol. ccxv. 1915, § 17.

motion satisfies all the requirements of the problem, and is the motion due to the rotation of the solid.

6. The first motion.-The first motion will be taken to be a rotation, as if rigid, with angular velocity w, about the point = exp (y). There is no loss of generality in this choice, as the substitution of another centre of rotation can always be effected by superposing a motion due to translation of the boundary as explained in article 3.

The first motion may be specified by its stream function 1, namely

†1 = — 1W | 2 — K exp (iy) | 2, which is also expressible as a function of § and ŋ. specification is by u1, v1, where


v1 =-d41/d§,

(11) It is to

these also being regarded as functions of § and ŋ. be noted that u1 and v1 are not velocity components, but that the velocity components in the directions corresponding respectively to and n increasing are u/h and v1/h. From formula (4) it follows that


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(10) Another

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70) + ... }, (12)

+ 3 *ose

sin (+-%) + ... }, (14)

the terms being arranged in descending order of importance.

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