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screen: the light by which the eye then sees the screen is, of course, undulatory

The phenomena in these exhausted tubes reveal to physical science a new world—a world where matter exists in a fourth state, where the corpuscular theory of light holds good, and where light does not always move in a straight line; but where we can never enter, and in which we must be content to observe and experiment from the outside.

II. "On a Machine for the Solution of Simultaneous Linear

Equations." By Sir WILLIAM THOMSON, LL.D., F.R.S.,
President of the Royal Society, Edinburgh. Received
August 30, 1878.

. . Bn be n bodies each supported on a fixed axis (in practice each is to be supported on knife-edges like the beam of a balance Let P1, P2, P31, Pni, n pulleys each pivoted on B.; P12, P2, P32,

B2;
P1, P2, P3,

Let B1, B2,

PX2

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Pni

B3;

C1, C2, C3, Cn, n cords passing over the pulleys;
Di, P11, P 127

, , P13, Pin, En, the course of Cı;
D, P1, P2, P3, P, E,

C.;

.

D, E, D2, E2, Dn, En, fixed points ; 11, 12, 13, ... In the lengths of the cords between D, E, and D, E, ... and D», En, along the courses stated above, when B1, B2,

. . B., are in particular positions which will be called their zero positions ;

lite, hten In+en, their lengths between the same fixed points, when B1, B2, . . . By are turned through angles X1, X2, from their zero positions ;

(11), (12), (13), . . . (ln),
(21), (22), (23), . . . (2n),
(31), (32), (33), (3n),

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... Pne from

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(21), &c., do not vary sensibly from the values which they have where X1, 22,

Xn, are each infinitely small. In practice it will be convenient to so place the axes of B1, B2, • . . Bn, and the mountings of the pulleys on B., B., . . . Bn, and the fixed points D., E1, D2, &c., that when X1, 22,

Xn are infinitely small, the straight parts of each cord and the lines of infinitesimal motion of the centres of the pulleys round which it passes are all parallel. Then ;(11), ž(21), ... (n) will be simply equal to the distances of the centres of the pulleys P11, P21, Pn, from the axis of Bi;

(12), (22) . . · }(n2) the distances of P12, P22, the axis of B2, and so on.

In practice the mounting of the pulleys are to be adjustable by proper geometrical slides, to allow any prescribed positive or negative value to be given to each of the quantities (11), (12), . . . (21), &c.

Suppose this to be done, and each of the bodies B1, B2, .. B, to be placed in its zero position and held there. Attach now the cords firmly to the fixed points D., D., . . . Du respectively; and passing them round their proper pulleys, bring them to the other fixed points E1, E2 En, and pass them through infinitely small smooth rings fixed at these points. Now hold the bodies B1, B2, . . . each fixed, and (in practice by weights hung on their ends, outside E1, E,, En) pull the cords through E1, E,, . . . En with any given tensions * T1, T2, :. Tn.

Let G1, G2, ... Gr be moments round the fixed axes of Bi, B2, Bn of the forces required to hold the bodies fixed when acted on by the cords thus stretched. The principle of “virtual velocities," just as it came from Lagrange (or the principle of “work"), gives immediately, in virtue of (I),

G=(11)T, +(21)T + +(nl)T,
G=(12)T. +(22)T + +(n2)T,

(II). Gx=(in)Ti+(2n)T,+ ... + (nn)T, Apply and keep applied to each of the bodies, B1, B, ... Bm (in practice by the weights of the pulleys, and by counter-pulling springs), such forces as shall have for their moments the values G1, G2 ... G., calculated from equations (II) with whatever values seem desirable for the tensions T1, T2 ... Tn. (In practice, the straight parts of the cords are to be approximately vertical, and the bodies B1, B2, are to be each balanced on its axis when the pulleys belonging to it are

* The idea of force here first introduced is not essential, indeed is not technically admissible to the purely kinematic and algebraic part of the subject proposed. But it is not merely an ideal kineinatic construction of the algebraic problem that is in. tended; and the design of a kinematic machine, for success in practice, essentially involves dynamical considerations. In the present case some of the most important of the purely algebraic questions concerned are very interestingly illustrated by these dynamical considerations.

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removed, and it is advisable to make the tensions each equal to half the weight of one of the pulleys with its adjustable frame.) The machine is now ready for use. To use it, pull the cords simultaneously or successively till lengths equal to e, en, ... en are passed through the rings E1, E2, ... En, respectively.

The pulls required to do this may be positive or negative; in practice, they will be infinitesimal, downward or upward pressures applied by hand to the stretching weights which (s) remain permanently hanging on the cords.

Observe the angles through which the bodies Bi, B2, . . . Bn are turned by this given movement of the cords. These angles are the required values of the unknown X1, X2, . . . Xn, satisfying the simultaneous equations (I).

The actual construction of a practically useful machine for calculating as many as eight or ten or more of unknowns from the same number of linear equations does not promise to be either difficult or over-elaborate. A fair approximation being found by a first application of the machine, a very moderate amount of straightforward arithmetical work (aided very advantageously by Crelle's multiplication tables) soffices to calculate the residual errors, and allow the machines (with the setting of the pulleys unchanged) to be re-applied to calculate the corrections (which may be treated decimally, for convenience): thus, 100 times the amount of the correction on each of the original unknowns, to be made the new unknowns, if the magnitudes thus falling to be dealt with are convenient for the machine. There is, of course, no limit to the accuracy thus obtainable by successive approximations. The exceeding easiness of each application of the machine promises well for its real usefulness, whether for cases in which a single application suffices, or for others in which the requisite accuracy is reached after two, three, or more of successive approximations.

December 12, 1878.

W. SPOTTISWOODE, M.A., D.C.L., President, in the Chair.

Dr. Philipp Hermann Sprengel was admitted into the Society.

The Presents received were laid on the table, and thanks ordered for them.

The following Papers were read :

1. “On the Flow of Water in Uniform Régime in Rivers and

other Open Channels.” By JAMES THOMSON, LL.D., D.Sc., F.R.S., and F.R.S.E., Professor of Civil Engineering and Mechanics in the University of Glasgow. Received Au

gust 15, 1878. In respect to the mode of flow of water in rivers, a supposition which has been very perplexing in attempts to form a rational theory for its explanation, has during many years past, during at least a great part of the present century, been put forward as a result from experi. mental observations on the flow of water in various rivers, and in artificially constructed channels. It was, I presume, put forward in the earlier times only as a vague and doubtful supposition; but, in later times it has, in virtue of more numerous and more elaborately conducted experimental observations, advanced to the rank of a confirmed supposition, or even of an experimentally established fact. This experimentally derived and gradually growing supposition was perplexing, because it was in conflict with a very generally adopted theory of the flow of water in rivers which appeared to be well founded and well reasoned out.

That commonly received theory, which for brevity we may call the laminar theory, was one in which the frictional resistance applied by the bottom or bed of the river against the forward motion of the water was recognized as the main or the only important drag hindering the water, in its downbill course under the influence of gravity, from advancing with a continually increasing velocity; and in which it was assumed that if the entire current is imagined as divided into numerous layers approximately horizontal across the stream, or else trough-shaped so as to have a general conformity with the bed of the river, each of these layers should be imagined as flowing forward quicker than the one next below it, with such a differential motion as would generate through fluid friction or viscosity, or perhaps jointly with that, also through some slight commingling of the waters of contiguous layers, the tangential drag which would just suffice to prevent further acceleration of any layer relatively to the one next below it. Under this prevailing view it came to be supposed that for points at various depths along any vertical line imagined as extending from the surface of a river to the bottom, the velocity of the water passing that line would diminish for every portion of the descent from the surface to the bottom.

The experimentally derived and perplexing supposition for which no tenable theory appears to have been proposed, though the want of such a theory has been extensively felt as leaving the science of the flow of water in rivers in a state of general bewilderment, is, that inconsistently with the imagination of the water's motion conceived under the laminar theory, the forward velocity of the water in rivers is, in actual fact, sometimes or usually not greatest at the surface with gradual abatement from the surface to the bottom; but that when the different forward velocities are compared which are met with at successive points along a vertical line traversing the water from the surface to the bottom, it may often be found that the velocity increases with descent from the surface downwards through some part of the whole depth, until a place of maximum velocity is reached, beyond which the velocity diminishes with further descent towards the resisting bottom.

That the superficial stratum of water flowing downhill under the influence of the earth's attraction should not have its forward velocity continually accelerated until, by its moving quicker than the bed of water on which it lies, a frictional drag would be communicated to it from below, by that supporting bed of water, sufficient to hold it back against further acceleration, has appeared very paradoxical. In various cases, during a long period of time, the alleged result appeared so incredible that the experimental evidence was doubted, or was dismissed as untrustworthy. In some cases the phenomenon was admitted as a fact, but was attributed to a frictional drag or resistance applied to the surface of the water by the superincumbent air, even in case of the air being at rest with the water flowing below, or more strongly so when the wind might be blowing contrary to the motion of the river.

Omitting to touch on the experimental results, and the opinions of various investigators in the older times, as I have not had sufficient opportunity to scrutinise them in detail, I have to refer to the investigations conducted at about the year 1850 by Ellet on the Mississippi and Ohio Rivers.* He was led to the conclusiont from his own experiments on the Mississippi, that the mean velocity of that river (or at least the mean velocity of the great body of its current, as the part near the bottom or bed of the river had not been definitely included in his researches) instead of being less, is in fact greater than the mean surface velocity. He attributed this phenomenon, which he regarded as indubitably proved, and which if true niust certainly be very remarkable, to a frictional drag or resistance, against the forward inotion, applied to the surface of the water by the atmosphere in contact with the surface. Like suppositions had previously been made by some observers and theoretical investigators in Europe, as may be gathered from D’Aubuisson “ Traité d'Hydraulique,” 2nd edition, 1840, p. 176, and from other sources of information.

Ellet on the “ Mississippi and Ohio Rivers.” Philadelphia : 1853. This is a republished edition of a Report to the American War Department by Ellet on his investigations, which were made under authority of an Act of Congress.

+ Pages 37 and 38 of the book referred to in the preceding note.

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