the heat derivable from the combustion of native sulphur, and of meteoric iron, every kind of motion (heat and light included) that takes place naturally, or that can be called into existence through man's directing powers on this earth, derives its mechanical energy either from the sun's heat or from motions and forces among bodies of the solar system.

In a speculation recently communicated to the Royal Society of Edinburgh, the author has shown that the sun's heat is probably* due to friction in the atmosphere between his surface and a vortex of vapours, fed externally by the evaporation of small planets, in a region of very high temperature round the sun, which they reach by gradual spiral paths, and falling in torrents of meteoric rain, down from the luminous atmosphere of intense resistance, to the sun's surface.

A continuation of the inquiry raises the question, from what source do the planets, large and small, derive the mechanical energy of their motions? This is a question, to the answering of which mechanical reasoning may legitimately be applied: for we know that from age to age the potential energy of the mutual gravitation of those bodies is gradually expended, half in augmenting their motions, and half in generating heat; and we may trace this kind of action either backwards or forwards; backwards for a million of million of years with as little presumption as forwards for a single day. If we trace them forwards, we find that the end of this world as a habitation for man, or for any living creature or plant at present existing in it, is mechanically inevitable; and if we trace them backwards according to the laws of matter and motion, certainly fulfilled in all the actions of nature which we have been allowed to observe, we find that a time must have been when the earth, with no sun to illuminate it, the other bodies known to us as planets, and the countless smaller planetary masses at present seen as the zodiacal light, must have been indefinitely remote from one another and from all other solids in space. All such conclusions are subject to limitations, as we do not know at what moment a creation of matter or energy may have given a beginning, beyond which mechanical speculations cannot lead us. If in purely mechanical science we are ever liable to forget this limitation, we ought to be reminded of it by con* [For correction of this conclusion see Appendix to Art. LXVI., pp. 25-27 above.]

sidering that purely mechanical reasoning shows a time when the earth must have been tenantless; and teaches us that our own bodies, as well as all living plants and animals, and all fossil organic remains, are organized forms of matter to which science. can point no antecedent except the Will of a Creator, a truth amply confirmed by the evidence of geological history. But if duly impressed with this limitation to the certainty of all speculations regarding the future and pre-historical periods of the past, we may legitimately push them into endless futurity, and we can be stopped by no barrier of past time, without ascertaining at some finite epoch a state of matter derivable from no antecedent by natural laws. Although we can conceive of such a state of all matter, or of the matter within any limited space, and have cases of it in the arbitrary distributions of temperature, prescribed at "initial” in the theory of the conduction of heat (see* Cambridge Mathematical Journal, Vol. IV. 1843, or Art. XI. of Vol. I. of Mathematical and Physical Papers), yet we have no indications whatever of natural instances of it, and in the present state of science we may look for mechanical antecedents to every natural state of matter which we either know or can conceive at any past epoch however remote.

It is by tracing backwards the motions which are at present observed, according to the known laws of motion and heat, with no limit as to time, that the author arrives at the conclusion that the bodies now constituting our solar system have been at infinitely greater distances from one another in space than they are now. He remarked that the nebular theory, as ordinarily stated, assuming as it does a previously gaseous state of matter, is not only untrue, but the reverse of the truth, according to the views now brought forward, since these show evaporation as a necessary consequence of heat generated by collisions and friction, and the general past and present tendency of matter is seen to be the conglomeration of solids and liquids accompanied by a gradual increase of the quantity of gaseous fluid occupying space.

Prof. Helmholtz, in a most interesting popular lecture on transformations of natural forces, delivered on the 7th of February last

* "Note on some points in the Theory of Heat," a short article in which it was shown how to test the age of a distribution of heat, by applying a certain criterion of convergence to its expression in the infinite series characteristic of the external circumstances of the body in which it is given.

at Königsberg, has estimated that, if the particles at present constituting the sun's mass have been drawn together by mutual gravitation, from a state of infinite diffusion, as supposed in the nebular theory, not however a gaseous state, as ordinarily supposed, but a state in which the particles exercise no mutual action except that of gravitation, the whole heat generated must have amounted to about 28,000,000 thermal units Centigrade per pound of the sun's mass. This estimate would not, as the author of the present paper shows, require any change, whether we assume, as the antecedent condition of the solar mass, a state of infinite diffusion, or a state of aggregation in solid masses of any dimensions small compared with his present dimensions, and separated from one another at comparatively great distances; provided always there has been no relative motion among them except what is generated by mutual gravitation. If, then, the whole mass of the sun has grown by the process which, according to the author's theory of solar heat (certain as regards a part, whether or not it may be sufficient to account for the whole of the radiation of solar heat), we know to be augmenting it at present, there must have been generated, in the whole process of conglomeration, the quantity of heat stated above; a quantity which amounts to about 20,000,000 times as much as is at present radiated off in one year. The author gave reasons for believing that this heat must have been nearly all radiated off immediately on being generated, and that enough of it has not been retained in the conglomerated mass to be the store from which the heat at present radiated is drawn*.

That the present solar radiation is supplied chiefly from a store of heat contained in the mass, whether created there or generated mechanically by the impact of meteors which have fallen in during remote periods of past time, appears very improbable [most probable].

On the contrary, there must in all probability be some [is certainly no] agency continually supplying [any approach to a

* [Reasons which soon led me to a reversal of this conclusion were given in an article "On the Age of the Sun's Heat," published in Macmillan's Magazine for March 1862, and re-published as Appendix (E) to Thomson and Tait's Natural Philosophy, Part II. Second Edition (1883). See also Appendix to "Mechanical Energies of the Solar System (Article LXVI. pp. 25-27 above) where will be found justification of the amendments on the text of the present page, now inserted in square brackets. W. T. May 27, 1883.]

sufficiency of] heat to compensate the loss constantly experienced by radiation from the sun; and that agency, as the author has shown elsewhere, can be no other than the mechanical action of masses coming from a state of very rapid motion round the sun to rest on his surface.

The author showed how a system of solid bodies, large and small, initially at rest and at great distances from one another, may, by their mutual gravitations, and by the resistance their motions must experience in the gaseous atmosphere, evaporated from them by the heat of their collisions after a vast period of time, come into a state of motion, heat, and light, analogous to the present condition of our solar system and the stars. The origin of rotatory motion is explained by showing that different systems starting from rest will influence one another so as to acquire contrary rotatory motions, without any aggregate of rotatory momentum being acquired by the whole. Any system or group beginning to concentrate round one principal mass, after having thus acquired a momentum of rotatory motion, will acquire from it, in a certain stage of advancement, just such approximately circular motions as those of the planets, the particles of the zodiacal light and the satellites of our solar system, and such rotatory motions as the central and other masses are known to have, all chiefly in one direction.

In considering the question whether all the heat and motion existing in matter have their origin in that action by which their amount is at present being increased, it is shown that, unless their entire actual energy [kinetic energy] exceeds a certain definite limit, namely, the value of the whole potential energy of gravitation that would be spent in drawing all the particles of matter from a state of infinite diffusion into their present positions, it is quite possible they may be so produced; or that the potential energy of gravitation may be in reality the ultimate created antecedent of all the motion, heat, and light at present in the universe.

ART. LXX. ELEMENTARY DEMONSTRATIONS OF PROPOSITIONS IN THE THEORY OF MAGNETIC FORCE.

[Phil. Mag., April, 1855.

ELECTROSTATICS AND MAGNETISM, Article XXXVII.]

ART. LXXI. ON THE MAGNETIC MEDIUM AND ON THE EFFECTS
OF COMPRESSION.

[Phil. Mag., April and Dec. 1855, and Jan. 1856.
ELECTROSTATICS AND MAGNETISM, Article XXXVIII.]

ART. LXXII. COMPENDIUM OF THE FOURIER MATHEMATICS FOR THE CONDUCTION OF HEAT IN SOLIDS, AND THE MATHEMATICALLY ALLIED PHYSICAL SUBJECTS OF DIFFUSION OF FLUIDS AND TRANSMISSION OF ELECTRIC SIGNALS THROUGH SUBMARINE CABLES.

[From the Encyclopædia Britannica, new edition (1880), being Appendix and § 82 of Article "Heat," and Quarterly Journal of Mathematics, Vol. I., March, 1856.]

[THE most important part of this article (the system of formulas I. to XVII.) is taken from a private mathematical memorandum-book, under dates September and October, 1850. Though first published in 1880, it is reproduced in this place because it constituted part of the substance of the paper on the conduction of electricity and heat referred to at the commencement of Art. LXXIII. below, as intended for communication to the Royal Society, and because the substance of it may be found useful for mathematical readers of Article LXXIII. and others which follow it on the "Theory of the Electric Telegraph."]

Let v be the temperature at any point P specified by §, n, १ according to any system of three sets of plane or curved orthogonal surfaces used for coordinates. Let λdę, μdŋ, vd be the lengths of the edges of the infinitesimal rectangular parallelepiped having P for its centre, and its sides parts of the six surfaces § – Įd§, § + 1 d§, n − 1 dn, n+1 dn, (-1 dc, & + Id}.