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wekuge, v. to murmur, cf. wiker. wer, n. a star ; egg (? a ball); golegole wer, the pupil of the eye. wer, n. appetite, cf. werer. weraki (J), n. a wig worn by men. wěrěm, n. child, children, family; the vertical fire-stick; sursur wěrěm,

a baby ; deg werem (Hunt), abortion ; werem pez (M), to have an abortion ; we do not know what deg means, pez may be

unripe (cf. upez, green coconut). werer, v. to be hungry; werer udili (), v. to starve. wererge, a. hungry (lit. in hunger). wěrěs, n. a conical basket used in catching tup. werir, n. the poles used for frightening the tup in order to catch

them. werkab, a. blessed. Mark, xi. 9, John, xx. 29. wertik (m), n. the Milky Way, cf. dad. werut, n. the tongue. wěsẽemur, see eseаmuda. wěser (J), v. to eat; n. a glutton (1). wěskip (J), n. a kind of yam. wěsor (L), n. turtle eggs. weswes, n. a branching coral (Madrepore). wětpur, n. a feast. weu, an exclamation of sorrow. wez, n. the croton. wez, n. grass or leaf tail, inserted in the belt when dancing. wi (u), n. a guest. wi, pron, they two. wiaba, pron. they, their. wiabielam, pron. through them. wiabim, pron. to them. wid (M) = wit, a. bad. wiker (s), v. to murmur, cf. wekuge. wit, a. bad, v. to do bad, cf. adud. witha (1), n. a bivalve shell, a kind of Tellina. wiu (J), n. a shell, a kind of Triton. wiwar, n. a stone used in sorcery for producing sickness, wonwon, n. a sea-urchin, Echinus. wonpis (J), n. a mask.

Zab (J), n. a war spear. zauber, n. a wave, breakers. zazer, zarazer, zager, a. white. zerem (m), n. a cross. zewar, n. a land crab. zi (J), n. the mangrove. ziai, n. the south west; ziai pek, S.S.W. ziau, n. the dura-mater. Paper is called ziau-wali or jiauwali, "jauali”

of service book, etc. (wali = calico, cloth), as it is the wali which resembles the dura-mater. Owing to their familiarity with corpses (making mummies, etc.), the natives were well acquainted with the parchment-like character of the outer

membranous covering of the brain. zigerziger, a. thorny, prickly. ziru, a. cold. ziz, n. a wound. zogo, n. a fetish, charm, an oracle (possibly also a totem, cf. agud);

hence, sacred, holy; zogo jiauali, Bible ; zogo meta, a church. zogoem, v. to sanctify. zogole, n. the temporary sacred men who divine with any particular

zogo, or the three chief men of the initiation ceremonies (used

in the Gospels for “ priest”). zogozogo, a. tabu, sacred, hence holy. . zole, n. a fire-charm in the shape of a roughly fashioned, crouching,

pregnant woman in stone. zor (1), dead coral. zorom, a. bright, shining, like reflection of sun on water; hence, glory. zoromzorom, a. shining. zup, n. a bamboo tobacco pipe. zug. zurkagem (m), a. round.

Part II. of this “Study' will be published in the next Volume of the Proceedings, and will contain : vii. Sketch of Saibai Grammar : ix. Specimens of the Saibai Language : x Saibai-English Vocabulary : XI. Sketch of the Daudai Grammar : XII. Specimens of the Daudai Language : Xu. Daudai-English Vocabulary : xir. Native Names of people and places in Torres Straits : xv. A List of Introduced and Adapted Words : xvi. Concluding Remarks.

XXXII.

ON A PROBLEM IN VORTEX MOTION. BY FRANCIS

ALEXANDER TARLETON, Sc.D., LL.D., F.T.C.D.

[Read DECEMBER 12, 1892.]

IF a perfect liquid, extending in all directions to infinity, be in motion under the influence of two parallel rectilinear vortices of small section and infinite length whose strengths are equal and opposite, the motion will take place in planes perpendicular to the axes of the vortices; and the lines of flow in one of these planes will (as is well known) be a system of coaxal circles having as their limiting points the intersections of the plane with the vortex axes. From hence it has been concluded that, if a single rectilinear vortex of small section exist in a liquid bounded by a parallel circular cylinder, the motion of the liquid will be that due to the given vortex A, together with another parallel vortex of equal and opposite strength, passing through the point which is the inverse with respect to the circular section of the cylinder of the intersection of A with the plane of this section.

When the liquid whose motion is under consideration is contained within the rigid cylinder, the inference above is correct; but it seems to have hitherto escaped the attention of mathematicians that, in the case of a liquid extending to infinity, and bounded internally by a rigid cylinder, the conditions of the problem are not satisfied by a motion such as has been described. In order to see this, let us suppose that, at first, a single rectilinear vortex exists in an infinite liquid unbounded in every direction. The current function due to the vortex will then be given by the equation y=-m logr, where om is the strength of the vortex, and r the distance of any point in the plane of motion from the point A in which it is met by the vortex axis. Let us now suppose that a circular cylinder of liquid parallel to the vortex becomes rigidified, and let us inquire how the motion of the remaining liquid is thereby affected. In the first place, the current function in the new state of motion must be constant at all points of the circle 8 bounding the section of the cylinder in the plane of motion. This. condition is fulfilled by supposing the motion to be the same as that due to the effect of the given vortex along with another of strength - am situated at the point B, which is the inverse or electrical image of A in the circle S. The current function at any point P is then

BP

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and this is constant when P is on the circle.

Up to the present time it seems to have been supposed that the condition above is sufficient. This, however, is not the case; for we have no warrant for asserting that there can be only one function of the co-ordinates which differs by a constant from m log AP at each point of s, is constant at infinity, and satisfies Laplace's equation throughout the field of motion. On the other hand, there is another condition which the motion must satisfy, which apparently has been hitherto overlooked.

When a cylinder of liquid is rigidified, the only immediate effect on the surrounding liquid is an impulsive pressure at each point of the cylindrical boundary. Such a pressure will produce an additional velocity potential in the moving liquid ; and the essential character of a velocity potential so produced is that it should have only one value at each point of the field-in other words, be acyclic throughout the field, though not necessarily a single valued function. Hence it follows that the circulation in every circuit throughout the field outside the cylinder must remain unaltered by its rigidification.

, BP The function m logo, when taken as the current function, does not satisfy the condition now mentioned, and hence cannot be the current function. In fact, in order to get the motion of the liquid outside the cylinder, we must suppose this motion to be that due to the original vortex at A, along with a vortex of equal and opposite strength at B, and a third vortex, of the same strength as that of the original one, placed at the centre C of the circle 8.

The current function is then given by the equation

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and all the conditions of the question are satisfied.

It is easy to see now that we have obtained the only possible correct solution of the problem.

In fact, we know by Green's theorem that, disregarding a constant, there can be only one function of the co-ordinates which satisfies the equation vap = 0 and is acyclic throughout the field, and whose differential coefficient along the normal is given at each point of the circle S, and also at infinity. Now the velocity potential correspond

BP ing to the current function m log o satisfies all the specified conditions, and is therefore the only function which does. Hence the state of motion supposed is the only possible state of motion satisfying the conditions of the problem.

The case of a circular cylinder has been considered at length ; but it is obvious that whatever be the form of the curve bounding the section of the cylinder in the plane of motion, the current function must be constant along this curve, and such that the corresponding circulation in every circuit throughout the field of motion is the same as that due to the actually existing vortex which causes the motion.

By these two conditions the motion of the liquid is determined in all cases in which it is due to a rectilinear vortex of infinite length and small section, situated in a liquid extending to infinity, and containing a fixed rigid cylinder of infinite length parallel to the vortex axis.

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