not more closely still, the mechanical equivalent of a unit of heat; that is, the number of foot-pounds of mechanical energy which must be expended in order to raise the temperature of one pound of water by one degree. For Fahrenheit's degree, that quantity is 772 footpounds: for the Centigrade degree, x 772=1389-6 foot-pounds & (Phil. Trans., 1850). This, the most important numerical constant in molecular physics, has been styled by other writers on the subject "Joule's Equivalent," in order that the name of its discoverer may be perpetuated by connection with the most imperishable of memorials a truth. Mr. Joule, at the same time, proved by experiment the law which had previously been only a matter of speculative theory with others: that not only heat and motive power, but all other kinds of physical energy, such as chemical action, electricity, and magnetism, are convertible and equivalent; that is to say, that any one of those kinds of energy may, by its expenditure, be made the means of developing any other in certain definite proportions. Meanwhile, partly through a theoretical anticipation of this law, and partly through the influence of the hypothesis of molecular motions as applied to heat, the formation of a systematic theory of the relations between heat and motive power advanced. Messrs. Helmholtz and Waterston may be referred to as having aided that progress. The investigations of the Count de Pambour on the theory of the steam engine, although not involving the discovery of any principle in thermodynamics properly speaking, were conducive to the progress of that science by pointing out the proper mode of applying mechanical principles to the expansive action of an elastic fluid.

The general equation of thermodynamics, which expresses the relations between heat and mechanical energy under all circumstances, was arrived at independently, and by different methods, in 1849, by Professor Clausius and the Author of this work; and published by the former in Poggendorff's Annalen, and communicated by the latter to the Royal Society of Edinburgh in February, 1850. (Edin. Trans., 1850). The consequences of that equation have since been developed, and applied to scientific and practical questions in a series of papers which have appeared in Poggendorff's Annalen; the Philosophical Magazine since 1850; the Edinburgh Philosophical Journal for 1849 and 1855; the Transactions of the Royal Society of Edinburgh, since 1850, Vol. xx.; and the Philosophical Transactions for 1854 and 1859.

*Professor William Thomson, adopting the true theory of heat, in 1850, not only solved some new problems in thermodynamics, and devised and carried out, jointly with Mr. Joule, some most important experiments; but he extended analogous principles to electricity and magnetism, and thereby created what may justly be

Now Lord Kelvin.

styled a new science. His papers have appeared in the Transac tions of the Royal Society of Edinburgh for 1851, and subsequently in the Philosophical Magazine since 1851, and the Philosophical Transactions since 1854. Numerical data, without which the theoretical researches before referred to would have been fruitless, were furnished by the experiments of Dulong, and MM. Bravais, Martins, Moll, Van Beek, and others, on the velocity of sound; by those of M. Rudberg, on the expansion of gases; by the experiments, almost unparalleled for extent and precision, of M. Regnault, on the properties of gases and vapours, made at the expense of the French Government, and published in the Proceedings and Memoirs of the Academy of Sciences, from 1847 to 1854; and by the joint experiments of Messrs. Joule and Thomson, on the thermic effects of currents of elastic fluids, made at the expense of the Royal Society, and published in the Philosophical Transactions for 1854. Amongst later experimental researches may be specially mentioned those of Messrs. Fairbairn and Tate on the density of steam, and those of M. G. A. Hirn on vapours, and on the disappearance of heat in steam engines.

HYPOTHESIS OF MOLECULAR VORTICES.-In thermodynamics as well as in other branches of molecular physics, the laws of phenomena have to a certain extent been anticipated, and their investigation facilitated, by the aid of hypotheses as to occult molecular structures and motions with which such phenomena are assumed to be connected. The hypothesis which has answered that purpose in the case of thermodynamics, is called that of "molecular vortices," or otherwise, the "centrifugal theory of elasticity." (On this subject, see the Edinburgh Philosophical Journal, 1849; Edinburgh Transactions, vol. xx.; and Philosophical Magazine, passim, especially for December, 1851, and November and December, 1855.)

SCIENCE OF ENERGETICS.-Although the mechanical hypothesis just mentioned may be useful and interesting as a means of anticipating laws, and connecting the science of thermodynamics with that of ordinary mechanics, still it is to be remembered that the science of thermodynamics is by no means dependent for its certainty on that or any other hypothesis, having been now reduced to a system of principles, or general facts, expressing strictly the results of experiment as to the relations between heat and motive power. In this point of view the laws of thermodynamics may be regarded as particular cases of more general laws, applicable to all such states of matter as constitute Energy, or the capacity to perform work, which more general laws form the basis of the science of energetics,-a science comprehending, as special branches, the theories of all physical phenomena.*

Proceedings of the Philosophical Society of Glasgow, 1853; Edinburgh Philosophical Journal, 1855.

INTRODUCTION.

OF MACHINES IN GENERAL

SECTION 1.-Of Resistance and Work.

1. The Action of a Machine is to produce Motion against Resistance. For example, if the machine is one for lifting solid bodies, such as a crane, or fluid bodies, such as a pump, its action is to produce upward motion of the lifted body against the resistance arising from gravity; that is, against its own weight: if the machine is one for propulsion, such as a locomotive engine, its action is to produce horizontal or inclined motion of a load against the resistance arising from friction, or from friction and gravity combined: if it is one for shaping materials, such as a planing machine, its action is to produce relative motion of the tool and of the piece of material shaped by it, against the resistance which that material offers to having part of its surface removed; and so of other machines.

2. Work. (A. M., 513.)—The action of a machine is measured, or expressed as a definite quantity, by multiplying the motion which it produces into the resistance, or force directly opposed to that motion, which it overcomes; the product resulting from that multiplication being called wORK.

In Britain, the distances moved through by pieces of mechanism are usually expressed in feet; the resistances overcome, in pounds avoirdupois; and quantities of work, found by multiplying distances in feet by resistances in pounds, are said to consist of so many foot-pounds. Thus the work done in lifting a weight of one pound, through a height of one foot, is one foot-pound; the work done in lifting a weight of twenty pounds, through a height of one hundred feet, is 20 x 100 = 2,000 foot-pounds.

In France, distances are expressed in mètres, resistances overcome in kilogrammes, and quantities of work in what are called kilogrammètres, one kilogrammètre being the work performed in lifting a weight of one kilogramme through a height of one mètre.

The following are the proportions amongst those units of distance, resistance, and work, with their logarithms :-

B

One mètre

One foot

One kilogramme

One lb. avoirdupois

='453593 kilogramme,.

1*484011

I-656666

One kilogrammètre =723308 foot-pounds, ...........0859323 One foot-pound 0138254 kilogrammètres,.....I'140677

3. The Bate of Work of a machine means, the quantity of work which it performs in some given interval of time, such as a second, a minute, or an hour (A. M., 661). It may be expressed in units of work (such as foot-pounds) per second, per minute, or per hour, as the case may be; but there is a peculiar unit of power appropriated to its expression, called a HORSE-POWER, which is, in Britain, 550 foot-pounds per second,

or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour.

This is also called an actual or real horse-power, to distinguish it from a nominal horse-power, the meaning of which will afterwards be explained. It is greater than the performance of any ordinary horse, its name having a conventional value attached to it.

In France, the term FORCE DE CHEVAL, or CHEVAL-VAPEUR, is applied to the following rate of work :

being about one-seventieth part less than the British horse-power. 4. Velocity. If the velocity of the motion which a machine causes to be performed against a given resistance be given, then the product of that velocity into the resistance obviously gives the rate of work, or effective power. If the velocity is given in feet per second, and the resistance in pounds, then their product is the rate of work in foot-pounds per second, and so of minutes, or hours, or other units of time.

It is usually most convenient, for purposes of calculation, to express the velocities of the parts of machines either in feet per second or in feet per minute. For certain dynamical calculations to be afterwards referred to, the second is the more convenient unit of time: in stating the performance of machines for practical purposes, the minute is the unit most commonly employed.

The units of time being the same in all civilized countries, the proportions amongst their units of velocity are the same with those amongst their linear measures.

5. Work in Terms of Angular Motion. (A. M., 593.)—When a resisting force opposes the motion of a part of a machine which moves round a fixed axis, such as a wheel, an axle, or a crank, the product of the amount of that resistance into its leverage (that is, the perpendicular distance of the line along which it acts from the fixed axis) is called the moment, or statical moment, of the resistance. If the resistance is expressed in pounds, and its leverage in feet, then its moment is expressed in terms of a measure which may be called a foot-pound, but which, nevertheless, is a quantity of an entirely different kind from a foot-pound of work.

Suppose now that the body to whose motion the resistance is opposed turns through any number of revolutions, or parts of a revolution; and let T denote the angle through which it turns, expressed in revolutions, and parts of a revolution; also, let

denote, as is customary, the ratio of the circumference of a circle to its radius. Then the distance through which the given resistance is overcome is expressed by

the leverage × 2 = × T ;

that is, by the product of the circumference of a circle whose radius is the leverage, into the number of turns and fractions of a turn made by the rotating body.

The distance thus found being multiplied by the resistance overcome, gives the work performed; that is to say,

The work performed

= the resistance the leverage × 2 × T.