From these experiments it is established that the value of the Ay increase AVY in the specific heat of diamond has diminished at a red heat and upwards to a white heat, until it has become but the seventeenth part of what it was for the temperature-interval 0° to 100°. This increase is now of the same magnitude as the Ay value of in the case of those elements which obey Dulong AT and Petit's law. Further remarks and deductions will be made for graphite and diamond together, inasmuch as a value for the specific heat of graphite has been obtained almost identical with that obtained for diamond. The curve marked diamond in the Table (Plate VII.) exhibits graphically the results of the foregoing experiments; the plain line shows the results actually observed; the notched line is interpolated from these. [To be continued.] XXII. A Statical Theorem. By LORD RAYLEIGH, M.A., F.R.S. [Continued from vol. xlviii. p. 456.] SINCE INCE the publication of the paper in the December Number of the Philosophical Magazine, entitled "A Statical Theorem," I have made some tolerably careful experimental measurements in illustration of one of the results there given, which are perhaps worth recording. The "system" consisted of a strip of plate glass 2 feet long, 1 inch broad, and about inch in thickness, supported horizontally at its ends on two very narrow ledges. In the first experiment two points, A and B, were marked upon it, A near the centre, and B about 5 inches therefrom, for which the truth of the theorem was to be tested. When a weight W is suspended at A, the deflection in a vertical direction at B should be the same as is observed at A when W is attached at B. The weight was suspended from a hook whose pointed extre mity rested on the upper surface of the bar at the marked points. In this way there was no uncertainty as to the exact point at which the weight was applied. The measurements of deflection were made with a micrometer-screw reading to the ten-thousandth of an inch, the contact of the rounded extremity of the screw with its image in the upper surface of the glass being observed with a magnifier. The reading in each position was repeated four times, with the following results. Case 1. W hung at A. Deflection observed at B:- Accordingly the deflection at A due to W at B was 1400. The difference of the two deflections, amounting to only about per cent., is quite as small as could be expected, and is almost within the limits of experimental error. In the second experiment the test was more severe, B being replaced by another point B' 7 inches distant from A, instead of only 5 inches. The deflections in the two cases here came out identical and equal to 993 divisions, W being the same as before. With W at A, the deflection at A was about 1700; and with W at B', the deflection at B' was 760. The theorem here verified might sometimes be useful in determining the curve of deflection of a bar when loaded at any point A. Instead of observing the deflection at a number of points P, it might be simpler to measure the deflections at the fixed point A, while the load is shifted to the various points P. For the benefit of those whose minds rebel against the vagueness of generalized coordinates, a more special proof of the theoretical result may here be given. The equation of equilibrium of a bar (whose section is not necessarily uniform) is * Thomson and Tait's 'Natural Philosophy,' § 617. 184 Lord Rayleigh on a Statical Theorem. way mity rested on the upper surface of the bar at the marked points. In this there was no uncertainty as to the exact point at which the weight was applied. The measurements of deflection were made with a micrometer-screw reading to the ten-thousandth of an inch, the contact of the rounded extremity of the screw with its image in the upper surface of the glass being observed with a magnifier. The reading in each position was repeated four times, with the following results. Case 1. W hung at A. Deflection observed at B:- W off. 1473 1473 1476 1474 W on. 79 82 79 76 Mean 1474 Mean.. 79 The deflection at B due to W at A was therefore 1395. Case 2. W hung at B. Deflection observed at A: W.on. W off. 1447 50 47 45 45 1449 1445 1446 Mean.. 1447 Mean .. 47 Accordingly the deflection at A due to W at B was 1400. The difference of the two deflections, amounting to only about per cent., is quite as small as could be expected, and is almost within the limits of experimental error. In the second experiment the test was more severe, B being replaced by another point B' 7 inches distant from A, instead of only 5 inches. The deflections in the two cases here came out identical and equal to 993 divisions, W being the same as before. With W at A, the deflection at A was about 1700; and with W at B', the deflection at B' was 760. The theorem here verified might sometimes be useful in determining the curve of deflection of a bar when loaded at any point A. Instead of observing the deflection at a number of points P, it might be simpler to measure the deflections at the fixed point A, while the load is shifted to the various points P. For the benefit of those whose minds rebel against the vagueness of generalized coordinates, a more special proof of the theoretical result may here be given. The equation of equilibrium of a bar (whose section is not necessarily uniform) is B dx2 d2 day Y* (1 dx2 * Thomson and Tait's 'Natural Philosophy,' § 617. (A) in which for the present application Y denotes the impressed force, not including the weight of the bar itself, and y is the vertical displacement due to Y. Let y, y' denote two sets of displacements corresponding to the forces Y, Y'. Then d2 dzy d2 S{v ± (82%) -v 2 (B2)} dr={{yY—yY'}de, (B) B dx dx2 dx2 where the integration extends over the whole length of the bar. Now, integrating by parts, day d2 d2 d dy dy' day S +dx2 dxe da, dx dx in which the integrated terms always vanish in virtue of the terdzy day minal conditions. In the present case, for example, y, y', 'd vanish at each extremity. Thus the left-hand member of (B) vanishes, and we derive 2 S{y'Y―yY'}dx=0.. (C) Let us now suppose that Y vanishes at all points of the bar except in the neighbourhood of A, and also that Y' vanishes except in the neighbourhood of B. Then from (C), y'1SYdx=y1SY'dx; d2y Terling Place, Witham, January 16, 1875. or if SYdx=SY'da, Y'A =YB, as was to be proved. A similar method is applicable to all such cases. I may here add that, corresponding to each of the statical propositions in my former paper, there are others relating to initial motions in which impulses and velocities take the place of forces and displacements. Thus, to take an example from electricity, if A and B represent two circuits, the sudden generation of a given current in one of them gives rise to an electromotive impulse in the other, which is the same, whether it be A or B in which the current is generated. Or, to express what is really the same thing in another way, the ratio of the currents in ÁÅ and B due to an electromotive impulse in B is the negative of the ratio of the impulses in B and A necessary in order to prevent These statements are not the development of a current in B. affected by the presence of other circuits, C, D, &c., in which induced currents are at the same time excited. (D) XXIII. Studies on Magnetism. By E. BoUTY, Professor of [Concluded from p. 98.] III. ON THE BREAKING OF MAGNETIZED NEEDLES. IT has long been known that when a magnet is broken the fragments possess magnetic properties; but I do not think that up to the present any one has set himself to determine the laws which govern the formation of breakage-magnets. In the act of fracture of a magnet we shall distinguish the fact of the separation of the parts (with its consequences, such as would be presented in the case of the simple disjunction of the same parts juxtaposed*, not welded, in the primitive magnet) from the mechanical fact of the breaking. I purpose, in the first place, to ascertain if this mechanical fact modifies in any way the magnetic state of the fragments. The following are the experiments I have made on this subject. 1. The proper effect of the Rupture. A regular magnetized needle is obtained by passing a freshly tempered steel needle through a spiral traversed by a current. If we break this needle in the middle, two cases may present themselves 1 1st. If the needle is tempered hard enough to break between the fingers like glass, the two halves will be magnets possessing the same magnetic moment, as was to be expected by reason of symmetry. 2nd. If the needle is tempered soft, so as to bend several times in opposite directions before breaking, the two halves possess unequal magnetic moments, in an apparently arbitrary manner. In the first place this difference must be accounted for. For that purpose I take a needle slightly tempered and regular. I grasp it by the middle between two plates of lead so that one of the halves remains immovable during the breaking, while the other, seized with the hand, is submitted to flexions in opposite directions till rupture takes place. It is found that the half which was submitted to the flexions possesses a lower magnetic moment than that of the half which was nipped, and so much the more as the breaking was more difficult. * When two pieces of steel are united by two equal plane faces, the parts opposed to each other are in reality separated by a lamina of air, the thickness of which is very great in proportion to the distance of two magnetic molecules. The thing in question here is perfect juxtaposition, such as exists between the different portions of a colierent solid. ་ If a needle slightly tempered be grasped on both sides of and very near its middle by means of two pincers, so that only a very thin section on each side of the plane of separation takes part in the flexions which precede rupture, the two halves of the needle present very nearly equal magnetic moments. Therefore the difference above found is due to the flexions which precede the fracture of needles that are tempered soft. It has moreover long been known that mechanical actions of this sort, when subsequent to magnetization, diminish the magnetic moment of the needles submitted to them. In needles that break like glass, the mechanical act of breaking concerns only an infinitely thin layer of molecules on each side of the plane of separation; it must be presumed that the effect of such an action is infinitesimal. I have ascertained that, in the case of hard-tempering, the magnetic moment of a fragment depends neither on the number nor the mode of the breakings by which it has been detached from the mother needle-which would be very difficult to account for if any peculiar appreciable influence were exerted by the act of breaking. In a word, in none of my experiments have I found a weakening of the magnetic moment that could be attributed to such an influence. But proofs still more conclusive will present themselves in the course of this Part. For all the following experiments I employed only needles of hard and nearly invariable temper-obtained by heating a rectilinear steel wire, of greater length than the needle required, in a gas-flame supplied with air from a bellows, and dipping it when bright-red hot in an earthen pan full of water-the ends being then detached, so as to reserve only the middle portion of the wire, the temper of which is very regular. The length of the needles obtained was, at the most, 150 millims. 2. Saturated cylindrical Needles broken perpendicularly to the axis. In his 'Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism' (Nottingham, 1828), Green deduced from the hypothesis of coercive force the following formula, which gives the magnetic moment y of a needle of length x magnetized to saturation*: 2 e B-e-B y= Aa2 eß + e-B (1) * Beer (Elektrostatik) has demonstrated that Green's formula applies to a needle placed in a magnetizing spiral, provided that the turns are of large dimensions in proportion to the diameter of the needle. In that case A is proportional to the magnetizing force f. |