for convertible circular processes. At that time I derived it from the maxim that heat cannot of itself pass from a colder to a hotter body. I afterwards* derived the same equation in a very different way, namely from the law cited above, that the work which can be done by heat in an alteration of the arrangement of a body is proportional to the absolute temperature, in conjunction with the assumption that the heat actually present in a body is dependent on its temperature only, and not on the arrangement of its constituents. Therewith I considered the circumstance that in this way we could arrive at the already otherwise proved equation a main support of that law. Now the preceding analysis shows how that law, and with it the second axiom of the mechanical theory of heat, can be reduced to general mechanical principles.

XXI. Description of a Model of a Conoidal Cubic Surface called the "Cylindroid," which is presented in the Theory of the Geometrical freedom of a Rigid Body. By ROBERT STAWELL BALL, A.M., Professor of Applied Mathematics and Mechanism, Royal College of Science for Ireland+.

WE

WE become acquainted with the geometrical freedom which a rigid body enjoys by ascertaining the character of all the displacements which the nature of the restraints will permit the body to accept. If a displacement be infinitely small, it is produced by screwing the body along a certain screw. If a displacement have finite magnitude, it is produced by an infinite series of infinitely small screw displacements. For the analysis of geometrical freedom we shall only consider infinitely small screw displacements. This includes the initial stages of all displacements.

To analyze the geometrical restraints of a rigid body we proceed as follows. Take any line in space. Conceive this line to be the axis about which screws are successively formed of every pitch from ∞ to +∞. (The pitch of a screw is the distance its nut advances when turned through the angular unit.) We endeavour successively to displace the body about each of these screws, and record the particular screw or screws, if any, about which the restraints have permitted the body to receive a displacement. The same process is to be repeated for every other line in space. If it be found that the restraints have not permitted the body to receive any one of these displacements, then the body is rigidly fixed in space.

*Pogg. Ann. vol. cxvi. p. 73; Abhandlungen über die mechanischen Wärmetheorie, vol. i. p. 242.

Abstract of a paper read before Section A of the British Association at its Meeting at Edinburgh, August 1871. Communicated by the Author.

If, after all the screws have been tried, the body be found capable of displacement about one screw only, the body possesses the lowest degree of freedom. If one screw (A) be discovered, . and, the trials being continued, a second screw (B) be found, the remaining trials may be abridged by considering the information which the discovery of two screws affords.

The body may receive any displacement about one or both. of the two screws A and B. The composition of these displacements gives a resultant which could have been produced by displacement about a single screw. The locus of this single screw is the conoidal cubic surface which has been called the "cylindroid" (at the suggestion of Professor Cayley). The equation of the surface is

Any line (s) upon the surface is considered to be a screw, of which the pitch is

c+a cos 20,

where c is any constant, and is the angle between s and the line

y=0,
2=0.

The fundamental property of the cylindroid is thus stated. If any three screws of the surface be taken, and if a body be displaced by being screwed along each of these screws through a small angle proportional to the sine of the angle between the remaining screws, the body after the last displacement will occupy the same position that it did before the first.

The equation of the cylindroid is thus deduced. Take as axes of x and y two screws intersecting at right angles whose pitches are ca and c-a; a body is rotated about each of these screws through angles o cos 0, w sin O respectively. The corresponding translations are (c+a) w cos 0, and (c—a) w sin 0. The resultant of the translations may be resolved into two components, of which (c+a cos 20) ∞ is parallel to the resultant of the rotations, and aw sin 20 is perpendicular to the same line. The latter component has merely the effect of transferring in a normal plane the resultant of the rotations to a distance a sin 20, the resultant moving parallel to itself. The two original screwmovements are therefore compounded into a single screw whose pitch is c+a cos 20. The position of the screw is defined by the equations

=x tan 0,

z=a sin 20.

Eliminating, we have the equation of the surface.

The property of the surface is thus proved. Let 01, 02, 03 be the angles of three screws upon the surface, and w1, W2, Wz be the displacements about them. Each of these displacements may be resolved into screw-displacements about the screws of x and y. The conditions necessary and sufficient for the displacements to neutralize each other are

Thus each rotation is proportional to the sine of the angle between the other two screws.

For the complete determination of the cylindroid and the pitch of all its screws, we must have the quantities a and c. These quantities, as well as the position of the cylindroid in space, are completely determined when two screws of the system are known.

In the model of the cylindroid which is exhibited, the parameter a is 2.6 inches. The wires which correspond in the model with the generating lines of the surface represent the axes of the screws. The distribution of pitch upon the generating lines is shown by colouring a length of 26 x cos 20 inches upon each wire. The distinction between positive and negative pitches is indicated by colouring the former red and the latter black. This model is in the possession of the London Mathematical Society, 22 Albemarle Street.

It is remarkable that the addition of any constant (c) to all the pitches attributed in the model to the screws does not affect the fundamental property of the cylindroid.

When a rigid body is found capable of being displaced about a pair of screws, it is necessarily capable of being displaced about every screw on the cylindroid determined by that pair.

The theorem of the cylindroid includes as particular cases the well-known rules for the composition of two displacements parallel to given lines, or of two small rotations about intersecting axes. If the parameter a be zero, the cylindroid reduces to a plane, and the pitches of all the screws become equal. If the arbitrary constant (c) which expresses the pitch be infinite, we have the theorem for displacements; and if the pitch be zero we have the theorem for rotations. As far as the composition of two displacements is concerned, the plane can only be regarded as a degraded form of the cylindroid, from which the most essential feature has disappeared.

Royal College of Science,

July 1871.

XXII. On the steady Flow of a Liquid. By HENRY MOSELEY, M.A., D.C.L., Canon of Bristol, F.R.S., Corresponding Member of the Institute of France, &c.*

THE

HE hydraulic experiments of M. Darcy, continued after his death by M. Bazin, were made with remarkable industry and scientific skill, and on a large scale. In the first series+ circular pipes were used, and they were placed horizontally. In the second the channels were rectangular, open and closed, and they were sloped. The experiments of the first series, which are those referred to in the following paper, were made with pipes of different materials, in different states of roughness or smoothness of internal surface, and of different diameters, from inch to 20 inches. Their lengths were generally 120 yards, but some of them 60 yards; and the water was made to traverse them with velocities varying from 1 inch per second to 20 feet.

All the necessary precautions were taken to determine the mean diameters of these pipes, and to measure the water discharged from them. To feed them, it was received from the reservoirs at Chaillot into a cylindrical vessel 28 metres high, in which it could be made to stand at any required height by opening more or less a cock in the supply-pipe. It passed from the bottom of this reservoir by means of a horizontal pipe 300 metres long, into a great horizontal cylinder, I metre in diameter and 3 metres long, to one end of which horizontal cylinder were adjusted the pipes to be experimented upon. This cylinder was crossed internally by an iron diaphragm pierced with small holes, through which the water was made to pass that its vis viva might (as far as possible) be destroyed before it entered the pipes experimented upon. Pressure-gauges were fixed at four different points of each pipe-the first being placed near the end by which the water escaped from the pipe into the reservoir of efflux, the second at 50 metres from it, the third at 100 metres from the first, the fourth near the point of entrance of the water from the horizontal cylinder into the pipe at 47 metres from the third, and the fifth in the horizontal pipe. They were water-gauges.

The first and third gauges being 100 metres apart, the difference of the heights of the water in these gauges showed, when the flow of the water had become steady, the head of water necessary to overcome the resistances opposed to the flow of that portion of the water in the pipe which intervened between these * Communicated by the Author.

Recherches Expérimentales relatives au mouvement de l'eau dans les Tuyaux. Paris: Bachelier, 1857.

Recherches Hydrauliques, par M. Darcy. Continuées par M. Bazin. Paris: Dunot, 1865.

two gauges. This head of water is that designated in the following paper by the symbol h.

Besides determining the efflux under different conditions, M. Darcy determined also the velocities of the water at different distances from the axis of certain of the pipes on which he experimented; and with reference to the theory of the flow of liquids, this was the most interesting feature of his experiments. He effected it by means of the instrument well known as Pitot's tube, into the construction and use of which he introduced some admirable improvements, for the particulars of which the reader is referred to his work. The results he arrived at are stated at

length in Table I. of the following paper.

A film in a liquid flowing through a pipe, in the sense in which the word is used in the following paper, is a continuous portion of the liquid, every molecule in which flows with the same velocity. A filament is an exceedingly narrow film. To the surface of the pipe a film of the liquid is supposed to adhere and to remain at rest. The film adjacent to it moves over this fixed film, the third over the second, the fourth over the third, &c. with continually increasing velocities; the film nearer to the surface moving always slower than that more remote. This is proved by the experiments of MM. Darcy and Bazin.

The resistance opposed by the surface of the pipe to the flow of the liquid immediately in contact with it is represented by the formula

where P is the resistance per unit of surface, V the velocity of the flow, A, a constant, and μ, a term as to which there is a difference of opinion whether it is constant or increases with the velocity.

This formula is founded on experiment*. The first term in it is considered to represent that part of the resistance which is due to the adherence of the liquid to the surface of the pipe, and which is of the nature of that which in solid bodies is opposed to shearing.

The second term is understood to represent the resistance caused by the impacts of the molecules of the flowing liquid on those of the film of liquid fixed to the surface of the pipe, and on the eminences of the solid surface of the pipe which project through that film.

Let the steady flow of a liquid in a horizontal circular pipe of uniform dimensions and roughness of surface be supposed to be maintained by the pressure of the liquid in a reservoir whose surface is always on the same level; let

* See Poncelet, Introduction à la Mécanique Industrielle, art. 387, 388.