If the density of the sphere is variable, the only difficulty is that of finding the particular integral f. As an example of variable density, take the case of a sphere in which the density varies as the square of the distance from a diametral plane. Here and the general solution involves zonal harmonics (Pn). Hence, inside, 10. The gravitation potential of a tore has been recently considered by Mr. Dyson* by means of special functions and Inside the tore -20-4πσ with σ constant. A Hence : Outside, Inside, $ = √(C-c)ΣAP, cos nv, p' = —πσа2S2/(С−c)2+√(C−c)ΣBQ, cos nr. f must be expanded in a series of the form Writing P for R, and Q, for T in § 3, we find But n n A,(PQ-PQ) = Q2F-Q2F B(PQ-PQ) = PF-PF PhQn-PnQn S - Inserting the values of F, and F, we get 16 / 2 a2C { (n2 - } ) -σι 3 The external and internal potentials are thus completely found. "The Potential of an Anchor Ring," Phil. Trans. 1893, pp. 43, 1041. LVI. On the Energy of the Amperian Molecule. By A. P. CHATTOCK, Professor of Physics, University College, Bristol, and F. B. FAWCETT, Associate of University College, Bristol*. THE THE following experiments were undertaken for the purpose of determining, if possible, whether the molecular currents of Ampère are accompanied by motions of the molecules themselves. They were suggested during an attempt by one of us to express certain physical properties of solids in terms of the ionic charges of their molecules. The theory put forward by Weber, that Amperian molecules may be rotating charged carriers of electricity, lends itself at first sight to the view that the molecules of all matter, whether in the electrolytic form or not, carry upon them the charges they possess when in the condition of ions. It is only necessary to suppose that the molecules of a magnetic substance rotate with these charges in virtue of their heat motions, to account for the permanence of their magnetic moments at any given temperature. Upon this supposition, if a bar of iron is saturated in a magnetic field, and this field is suddenly strengthened, the effect upon the iron will be two-fold. There will be a sudden decrease in the magnetic moments of the molecules corresponding to a decrease in their rates of rotation, and therefore to a cooling of the iron as a whole; and this will be followed by a slow return to their original condition as the iron receives heat from surrounding objects. The result of Ewing's work on iron subjected to intense fields has been to show that no certain alteration in the value of I (the magnetic moment per cub. centim. of the iron) can be detected after saturation within the wide limits of fieldstrength which he employed. Upon the present hypothesis there should be no permanent alteration; and even the temporary fall of I on the first application of the field would be far too small to detect; its value being about 5 × 10-11I when the magnetizing force is raised to 40,000 after saturation (see below). But though the alteration of I is inappreciable, the accompanying fall of temperature is not; and we therefore decided to look for it. We were, moreover, encouraged to make the experiment by the publication of an interesting paper on the subject of ionic charges and their consequences by Dr. F. * Communicated by the Authors. † Phil. Mag. Dec. 1892, p. 480. Phil. Mag. S. 5. Vol. 38. No. 234. Nov. 1894. 2 K Richarz*, in which (p. 410) the author suggests rotating ions as the cause of molecular magnetism, and then gives quantitative support to his view by showing that the saturation values of I for magnetic metals are of the same order of magnitude as those calculated from reasonable assumptions as to the period of rotation of the molecules. Magnitude of the Effect sought. For simplicity of calculation, consider the molecules to have the form of thin rings of diameter 8(=10-8), rotating about their principal axes, carrying charges 9(=3x 10-22 E.M.) and numbering n (=1025) to a cubic centimetre of iron. Let the magnetic axes of these rings be parallel and similarly directed; i.e. let the iron be saturated and of magnetic moment I per cub. centim. It is easy to see that if V stands for the linear velocity of any point on a ring along its circumference, I 4 πδ 4I V: = = πδ 4 4' (i.) Suppose now that the field by which the molecules are held in position is suddenly increased by an amount H. The number of lines of force which will enter each ring per centim. Η πε H8 of its circumference is and the momentum imparted thereby to the matter of the ring is consequently From this it follows that if v is the change produced by this process in the original velocity (V) of a ring of mass m, HS 49. Now the ratio is the fraction by which the magnetism in the iron decreases when H is increased. If we put H=40,000 (Ewing's maximum value about) and I=1500, v comes out to be 5 x 10-11, as stated above. V From (i.) and (ii.) it is also easy to calculate the loss of heat in the iron due to any increase of H. Loss of kinetic energy of the } \ rings in ergs per cub. centim. = nm V2 — 1 nm (V—v)2, =nmV v-nm v2 ; and since v is negligible compared with V, the second term * Wied. Ann. lii. p. 385 (1894). vanishes. Hence, as nm Vv=IH from (i.) and (ii.), it follows where J is Joule's equivalent. For an increase of H of 250 at saturation this would mean a fall of temperature of about one hundredth of a degree C. This is a well-known expression for magnetic-field energy ; and though obtained in this somewhat roundabout manner, it is of course quite independent of any hypothesis as to the source of the energy being in the motion of the molecules or elsewhere. The above equations simply suggest a mechanical explanation, in the case of magnetic molecules, for the fact that, when energy is put into surrounding space by starting any current A (say the source of H in the above), it is less when A is alone than it is if a second current B (the molecular current) is already flowing in its neighbourhood; the difference being wholly derived from the source of B (which, according to the present hypothesis, is the kinetic energy of the molecule) provided В does not change in strength during the process. If it does-in other words, if the second term in the above expression does not vanish-part of the extra field-energy is derived from the source of A, and calculation becomes practically impossible. Description of Apparatus. After several unsuccessful attempts to obtain reliable values, the arrangement shown in fig. 1 was adopted. AA are the upper portions of the coils of a large electromagnet. BB are small auxiliary coils for the purpose of producing an alteration |