strikes the target, with the phosphorescence on the glass screen accompanying molecular impacts. Focus of Heat of Molecular Impact. The author finally describes an apparatus in which he shows that great heat is evolved when the concentrated focus of rays from a nearly hemispherical aluminium cup is deflected sideways by a magnet to the walls of the glass tube. By using a somewhat larger hemisphere and allowing the negative focus to fall on a strip of platinum foil, the heat rises to the melting point of platinum. An Ultra-gaseous State of Matter. The paper concludes with some theoretical speculations on the state in which the matter exists in these highly exhausted vessels. The modern idea of the gaseous state is based upon the supposition that a given space contains millions of millions of molecules in rapid movement in all directions, each having millions of encounters in a second. In such a case, the length of the mean free path of the molecules is exceedingly small as compared with the dimensions of the vessel, and the properties which constitute the ordinary gaseous state of matter, which depend upon constant collisions, are observed. But by great rarefaction the free path is made so long that the hits in a given time may be disregarded in comparison to the misses, in which case the average molecule is allowed to obey its own motions or laws without interference; and if the mean free path is comparable to the dimensions of the vessel, the properties which constitute gaseity are reduced to a minimum, and the matter becomes exalted to an ultragaseous state, in which the very decided but hitherto masked properties now under investigation come into play. Rays of Molecular Light. In speaking of a ray of molecular light, the author has been guided more by a desire for conciseness of expression than by a wish to advance a novel theory. But he believes that the comparison, under these special circumstances, is strictly correct, and that he is as well entitled to speak of a ray of molecular or emissive light when its presence is detected only by the light evolved when it falls on a suitable screen, as he is to speak of a sunbeam in a darkened room as a ray of vibratory or ordinary light when its presence is to be seen only by interposing an opaque body in its path. In each case the invisible line of force is spoken of as a ray of light, and if custom has sanctioned this as applied to the undulatory theory, it cannot be wrong to apply the expression to emissive light. The term emissive light must, however, be restricted to the rays between the negative pole and the luminous 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. ... Let B1, B2, . . . 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). 11, 12, 13, . . . In the lengths of the cords between D1, E1, and D, E, . . . and Dn, En, along the courses stated above, when B1, B2, . . . Bu, are in particular positions which will be called their zero positions; liten, l1⁄2te2, . . . ln+en, their lengths between the same fixed points, when B1, B2, . . . B, are turned through angles x1, x2, from their zero positions; (21), &c., do not vary sensibly from the values which they have where x1, x2, an, are each infinitely small. In practice it will be convenient to so place the axes of B1, B2, . . . B2, and the mountings of the pulleys on B1, B2, Bn, and the fixed points D1, E1, D2, &c., that when 1, X2, .., 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 1(11), 1(21), . . . 1(n) will be simply equal to the distances of the centres of the pulleys P11, P21, . . . Pn, from the axis of B1; (12), (22). (n2) the distances of P12, P22, Pr2 from 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, . . . B2 to be placed in its zero position and held there. Attach now the cords firmly to the fixed points D1, D2, . . . D, 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, E2, . . . En) pull the cords through E1, E2, . . . En with any given tensions* T1, T2, . . Tn. Let G1, G2... G be moments round the fixed axes of B1, B2, . . B 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), ... ... +(nl) Tn In ] G„=(ln)T1+(2n)T2+ ... + (nn) Tn, (II). Apply and keep applied to each of the bodies, B1, B2 . . . B2 (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 . . . Gя, 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 kinematic construction of the algebraic problem that is intended; 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. 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 e1, e2,... 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 (§) remain permanently hanging on the cords. n Observe the angles through which the bodies B1, B2, . . . B, 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) suffices 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 approxima tions. 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. I. "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 August 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 experimental 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 downhill 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 |