nomena of magnetism has been denied; but when I find it affirmed by such a number of witnesses, comprising Morichini, Carpi, Ridolfi, Gmelin, Davy, Playfair, Barlocci, Zantedeschi, Christie, Somerville, Baumgartner, and the Messrs. Knox, I cannot believe that they were all deceived. The statement of Playfair, as given by Sir David Brewster, is too strong, striking and circumstantial, to admit the suspicion of mistake. The advantage to be derived from admitting that the electric fluid is a compound of elementary forces, which, according to the circumstances of its excitation, may vary in its constitution, is that our explanations of phænomena are thereby disentangled from a multiplicity of embarrassments occasioned by the supposed identity of the various forms of electricity; and the endeavour to reduce irreconcileable effects under the operation of a single cause is no longer necessary. These laboured efforts to sustain the doctrine of identity, have had their influence in causing the contradictory speculations promulgated concerning electricity in connexion with matter. Were there as much truth as boldness in them, we should by this time have attained a thorough knowledge of the internal structure of matter; the very shape of atoms, and their individual constitution and properties, must have been ascertained. By one philosopher we are informed that atoms are spherical; others find that electricity is an integrant part of their composition, and even declare that without it the atoms could not exist. Again, we are informed that each atom has two electrical poles. This is utterly denied by another authority; the argument adduced is, that polarity is incompatible with sphericity, all the points in the surface of a sphere being symmetrically placed with relation to the centre. The idea of polarity is also denied by others, who have discovered that some atoms are positively and some negatively electrical throughout their mass. To this is also added, that the electricity proper to each atom is disguised by an electrical atmosphere which surrounds it. Some philosophers declare that the polar electricities of atoms are not of equal intensity, one always predominating: this is most emphatically denied by others, who conceive that the admission of equality is indispensable. It is not to be forgotten also, that some will admit but one electric fluid; others must have two, or they explain nothing; but others, again, explain all the phænomena without any electric fluid at all. The list of contradictory speculations should be considered in connexion with this very extraordinary fact, that the fundamental principle of one theory of galvanism is, that water is a conductor of electricity; but in the rival one it is assumed to be an insulator; and without the admission of one or other of these conflicting positions neither theory can stand its ground. In the observations thus made, it is as far from being my wish, as it is beyond my capability, to depreciate the labours of the illustrious persons who have erected splendid specimens of art on the foundations just described. They used these foundations as they found them; and if they be not sound and permanent, we have only to lament that so much skill, ingenuity and industry, were not bestowed on a more solid basis. 11 Clare Street, Dublin. [To be continued.] XIX. Remarks on the Researches of Dr. Goodman " On the Identity of the Existences or Forces, Light, Heat, Electricity and Magnetism." By Dr. TYNDALL*. THE HE December Number of the Philosophical Magazine contains an abstract of a paper bearing the above title, and recently read before the Royal Society by its Secretary Mr. Bell. Dr. Goodman finds, that on suspending a magnetized sewingneedle within the helix of a galvanometer, and covering the instrument with a glass shade, when the sun is permitted to shine strongly upon the instrument an electric current is developed in the helix, which is indicated by its action upon the magnetized needle. The direction of the current varies with the portion of the instrument shone upon, and in some cases a permanent deflection of 10 or 12 degrees was obtained even when the two ends of the galvanometer wire were disunited. 'During the course of the experiments the circuit was established by means of a connecting wire between the mercury cups, and the circuit was again and again completed, and as frequently broken, without any deviation occurring in any of the results." The remarks made further on will, perhaps, excuse me to Dr. Goodman if I propose the following modifications of his experiment: first, to remove the magnetized sewing-needle and put an unmagnetized one in its place; secondly, to remove the steel needles altogether and substitute in their stead one of copper or of wood; thirdly, to remove the helix also, and leave the wooden or copper needle and dial-plate alone within the shade. If, on submitting the apparatus to the conditions described, in none of these proposed cases an action quite the same as that exhibited by the magnetized needle can be observed, then is the discovery a most surpri sing one, and the momentous conclusion drawn from it in some measure justified. I believe, however, that it has been the lot of many experimenters to deal with phænomena similar to those * Communicated by the Author. described by Dr. Goodman, in cases where there was no possibility of an electric current being formed. During the inquiry on diamagnetism and magnecrystallic action, an account of which appears in the September Number of this Magazine for 1851, the torsion balance there described was placed before a window through which the sun shone during the forenoon. In experimenting with the spheres of bismuth, I was often perplexed and baffled by the contradictory results obtained at different hours of the same day. With the spheres of calcareous spar, where the diamagnetic action was weak, the discrepancies were still more striking. Once while gazing puzzled at the clear ball of spar resting on the torsion balance, my attention was attracted to the bright spot of sunlight formed by the convergence of the beams which traversed the spar, and the thought immediately occurred to me that this little "fire-place " might be sufficient to create currents of air strong enough to cause the anomalies observed. The light was shut out, and thus the source of my perplexity was effectually cut away. The air-currents, however, were far more owing to the warming of the glass cover of the instrument than to the convergence of the sun's rays; during the whole inquiry I was obliged to experiment every sunshiny forenoon with closed shutters. On mentioning this fact to Prof. Magnus, he informed me that during his investigation on thermo-electric, currents, a report of which appears in the present Number of the Philosophical Magazine, he was obliged to protect his galvanometer from the action of the sun, as the unequal heating of its glass shade rendered his astatic needles quite unsteady. It is almost incredible how slight a difference of temperature is sufficient to create these currents, and thus disturb the action of a finely suspended needle. M. Kohlrausch, whose refined experiments I had the pleasure of witnessing for several successive days last spring, has been obliged to construct a table for the express purpose of making allowance for the little whirlwinds which sometimes establish themselves in his electrometer. In his instrument, a needle of silver wire is suspended from a glass fibre of extreme tenuity, the needle being protected by a vessel of brass with a glass cover. The days on which we experimented were cold ones; and as long as a good fire was kept in the stove which heated the room, the experiments were satisfactory; but as soon as the fire became low, and radiation set in strongly against the cold window-panes before which the electrometer was placed, the action of the air within the brass vessel upon the needle was at once exhibited, and increased to such a degree, with the decreasing temperature, that further experiment had to be abandoned. M. Kohlrausch has mapped these little currents with great care. They resemble, to compare small things with great, the cyclones of Colonel Reid. To preserve equability of temperature, a screen of pasteboard was often found serviceable. Now that a needle suspended from a silken fibre sixteen inches long, covered with a glass shade and placed in strong sunlight, which are the conditions of Dr. Goodman's experiments, should also be influenced by air-currents, is exceedingly probable, and that a permanent current of electricity should circulate in a helix of covered copper wire with its two ends disconnected being exceedingly improbable, it appears worth the trouble to subject the matter to one or more of the three tests which have been above proposed. Queenwood College, XX. Homogeneous Functions, and their Index Symbol*. By ROBERT CARMICHAEL, A.B., Trinity College, Dublin. IN a valuable memoir published in the Philosophical Transactions for the year 1844, Professor Boole, by the aid of two fundamental principles, has given general methods of solution for certain extensive classes of linear differential equations. It is the chief object of the present paper to show that, by a generalization of those principles and a suitable development of the consequences of the higher principles, we can obtain similar general methods of solution of corresponding classes of partial differential equations. The solutions of such partial differential equations will be found to be unaffected by the number of independent variables which the equations may contain; but more especial reference is made to those in most common occurrence, containing but two independent variables, x and y. In the course of the investigation, extensions of many familiar and elementary theorems are furnished, which seem to possess much practical utility. From the spontaneity with which they evolve themselves, and the facility with which they admit of. employment, they appear to open a large and fruitful field for future speculation. By an application of the general principles to the subject of *The principal portion of the first seven articles in this paper has been already published in the Cambridge and Dublin Mathematical Journal, November 1851. In the Number of the same Journal for February 1851, will be found an interesting and masterly paper by Mr. Sylvester, "On certain general Properties of Homogeneous Functions." The same symbol is there applied to equations in finite terms, with a view to the subjects of surfaces and the linear solution of systems of indeterminate equations. In the present paper the symbol is applied to the subjects of partial differential equations and multiple definite integrals. + Communicated by the Author. Phil. Mag. S. 4. Vol. 3. No. 16. Feb. 1852. K Multiple Definite Integrals, it will be seen that valuable results can be obtained, and some examples are furnished. It will be observed that these general principles present the means of extending all multiple definite integrals, in which the variables enter, as complicated functions, in the indices of known quantities unconnected with the limits. This seems to be an important step, but an adequate development of its consequences would much exceed the limits of the present paper. The instrument employed is the symbol which occurs in the well-known theorem of homogeneous functions. The relation which the result of the operation of this symbol upon any homogeneous function bears to the degree of the function, seems to give ground for the appellation, Index Symbol. In conclusion, the writer begs to express in the most ample manner his acknowledgements to the distinguished mathematician above named. 1. In general, if um=f(x, y, z, &c.) be a homogeneous function of the mth degree between the n independent variables x, y, z, &c., cum dum or, putting the operating symbol we have d d d x +y +2 + &c. = f(x1).x d which is an extension of the theorem 2m=f(m).xm, or f(D).em®=f(m).em® (1) the first fundamental principle employed by Professor Boole. In fact, am is a particular homogeneous function of the mth d degree, and a is the first term of V. dx 2. Now if U be any mixed rational function of x, y, z, &c., it can, in general, be put under the form U = uo+u2+ u2+ &c. +Um; and we have a theorem for mixed rational functions corresponding |