IX. Magnetic-Elongation and Magnetic-Twist Cycles. IN measuring the successive changes of length of iron and nickel wires subjected to cyclic magnetizations, Mr. Nagaoka has worked out a problem of extreme difficulty. Not only is the quantity to be measured excessively minute, but, unless very refined precautions are taken, it is certain to be disturbed beyond recognition by inevitable temperature-changes. With admirable experimental skill, Mr. Nagaoka has applied the principle of the gridiron-pendulum; and the remarkable smoothness of the curves which embody his instructive results is an evidence of the perfection of the compensation. Joule's original experiments, and all similar experiments of later date, demonstrated the existence of what might be called a residual elongation when the magnetizing force was removed. Mr. Nagaoka has now given us the whole history of the magneticelongation cycle, and has done for this very much more difficult enquiry what Warburg and Ewing did for the magnetization cycle. It may not be without interest to compare Mr. Nagaoka's results with what I believe to be closely related results obtained by myself several years ago. As long ago as 1858 Wiedemann discovered † that, when a current passes along a longitudinally magnetized iron wire, the wire twists. In 1883 I observed the same phenomenon with nickel wire similarly treated, and more recently also with cobalt‡. This phenomenon is, for present purposes, conveniently called the Wiedemann Effect. Maxwell suggested that it could be explained in terms of the Joule Effect, as we may also conveniently term the phenomenon which has been engaging Mr. Nagaoka's attention. In my papers on the subject I have brought this probable explanation prominently forward. The general features of the Wiedemann Effect in the three magnetic metals were just what was to be expected if Maxwell's explanation were admitted. Not only so, but certain of these features suggested corresponding characteristics in the Joule Effectcharacteristics which Mr. Bidwell has subsequently observed §. In my paper of 1891 (Trans. R. S. E. vol. xxxvi.) I have given the magnetic-twist cycles both for iron and nickel. While a steady current was passing along the wire the longi * Communicated by the Author. † See Wiedemann's Electricität, Bd. iii. See Trans. Roy. Soc. Edin. vols. xxxii., xxxv., and xxxvi. $ See Proceedings Roy. Soc. vol. i. (1892), p. 495. tudinal field acting on the wire was gradually altered between limits H, and at suitable intervals observations of the twist were made. With small range of field the hysteresis curve obtained by plotting twist against field was very similar to the well-known hysteresis curve of magnetization. But with limiting fields stronger than the field which produced the maximum twist, the hysteresis curve crossed itself twice and formed three loops. A typical example is shown in the figure, the magnetic-twist cycle being the Z-shaped graph with the thick lines. In the magnetic-elongation cycle, the change of sign of the magnetizing force does not produce a change of sign in the elongation. On the other hand, in the magnetic-twist cycle, as the magnetizing force passes through zero from positive to negative, the twist tends to do the same, though laggingly. Suppose, now, that we change the sign of the twist throughout one-half of the cycle. In other words, take the reflexion of the one-half of the graph in the horizontal axis, as indicated by the thin lines in the figure. Then round off the sharp angles by the dotted lines, and we get a curve similar to Mr. Nagaoka's magnetic-elongation curve, fig. 2, Plate II. Or, taking Mr. Nagaoka's curves of fig. 4, Plate II., we may reflect one half of each in the horizontal axis, join the parts so as to have continuous flowing lines, and thus get curves identical in form with those of the magnetic-twist cycle (see plate iii. of my paper already referred to). The magnetic-elongation curves obtained for nickel are all of the simpler two-looped form, the reason being that there is no maximum contraction for nickel. But with high ranges of field, the magnetic-twist curve for nickel is exactly similar to the same for iron. The reason is simply that with nickel there is a field of maximum twist for each value of current passing along the wire, although there is no field of maximum elongation. The explanation of this is given in my paper of 1888 (Trans. R. S. E. vol. xxxv.). It is sufficient to note that the twist, under a given combination of circular and longitudinal magnetizing forces, depends not only upon the elongations but also upon some function of these forces which changes sign with each, and to which the existence of the maximum twist is largely or altogether due. For even in the case of iron, which has a maximum elongation, the maximum twist occurs in quite a different field. Indeed the field of maximum twist varies with the value of the current along the wire. Meanwhile the broad character of the hysteresis in the magnetic-elongation cycle, as established by Mr. Nagaoka's delicate experiments, agrees perfectly with what might be inferred from the character of the hysteresis in the magnetictwist cycle-a phenomenon whose experimental study is one of the simplest in the whole subject of magnetic strains. Edinburgh University, IF October 28, 1893. X. On the Law of Distribution of Energy. By S. H. BURBURY, F.R.S.* F there be in any space a great number of mutually acting molecules, Boltzmann's law of distribution of energy requires that the number per unit of volume of molecules whose coordinates and momenta lie between assigned limits varies as ex+) in the known notation. The proofs of that law given by Boltzmann and Watson are based on the hypothesis that, from the instant when mutual action commences between two molecules to the instant when it ceases, no action takes place between either of them and any third molecule, or, as it is called, the encounters are binary. I propose to consider the more general case when, for instance, no group of molecules is ever free from the action of other parts of the system. PART I. (1) The following is a known proposition in the theory of Least Squares. Let the chance of a certain magnitude lying between a and x+dx be f(x)dx, f(x) being a function which vanishes *Communicated by the Author. 144 Mr. S. H. Burbury on the for infinite values of a, and is not altered by changing the sign of x, so that So f(x) da = 1. 81 dx Let there be N, a very great number, of such magnitudes, each independent of all the others, denoted by 1, 2 ・ ・ ・ XÑ. Let it be required to find the chance that ... shall lie between c and c+de, and let this chance be F(c)dc. Then by the known method we find + f(x)dx √√√_ay(a)da (which is zero as stated) N A V W ta fd Law of Distribution of Energy. 145 changing the It is then shown that X=1 if 0=0, X<1 if 0 0; and therefore in forming XN, N being very great, we may neglect powers of 0 above 02, and so e be F(c)de. ero as stated) which is proportional to 02 2x2 and the constant C is found from the condition that n (2) That proposition has been extended as follows. Let X1, X2... x be n mutually dependent magnitudes, and let the chance that they shall simultaneously have the values denoted by ... x1+dx1, &c., be f(x1... x)dx1... dx n where f(x) is a function which vanishes for infinite values of the variables ... x, and which is not altered when they all change sign together. Let us call the simultaneous occurrence of x1. x an association. The limits for each x shall be +. Then we have ... n SS... f(x1 ... xn)x12dxı ... dx = x2, ... n Let there now be N such associations, N being very large, and let each be independent of all the others. The variables in the first association shall for distinction be called 11, those in the second association 21, X22 Phil. Mag. S. 5. Vol. 37. No. 224. Jan. 1894. |