increase in resistance, but also an action depending on the phase of the current. It was thought probable that, although the bismuth spiral had no real self-induction, yet it might have something equivalent. To test this, two equal resistances were prepared whose coefficients of self-induction could be varied at pleasure, for description of which see Addendum III. By no means could complete silence be obtained when the telephone and alternating current were employed; there was always a decided minimum noise. The minimum noise given by the telephone was certainly diminished by the introduction of suitable self-induction, the amount required diminishing with the strength of the field in which the bismuth was placed. But here, again, on attempting to measure I found difficulties; for the amount required varied from time to time, depending certainly on the state of the telephone (proved by tampering with the diaphragm), and possibly on the state of the ears, or on small peculiarities of the current. The resistance of the bismuth spiral was determined alone and in series with an ordinary resistance, the alternating current and telephone being of course employed; the results were the same in each case: if the bismuth had behaved as if it had self-induction, the results would have been different. The bismuth spiral was replaced by an ordinary resistance, and self-induction was introduced until the apparent change of resistance, measured by the telephone, was equal to that produced by the action of a magnetic field of certain strength on the bismuth: the minimun noise was very much greater than that observed when the bismuth was employed. Before making more experiments, I thought it advisable to make up some theory as a working hypothesis. In the first place, it is very probable that the increase of resistance with a constant current, produced by a strong magnetic field, is caused by, or accompanied by, a molecular or crystalline rearrangement of the bismuth. Again, it is not unlikely that the passage of a current along the bismuth may cause or require an additional rearrangement; and it is possible that an arrangement which has a certain resistance for one strength of current may not have the same resistance for another strength. It is not suggested that different strengths of current may be subject to different resistances, but that the arrangement for one strength of current may produce-the previous arrangement still obtaining-a different resistance for another strength of current. The changes in resistance would in this case be of a complicated character; I worked on the above assumption, assisted by mathematics, without obtaining any serviceable results. I made another attempt at a theory, which, with the help of sufficient hypotheses, gives results agreeing to some extent with the facts. For simplicity, initially, the bismuth spiral will not be considered as forming part of a Wheatstone bridge. It is very probable that when the bismuth wire is put in a magnetic field the ultimate particles, crystals or molecules, will become magnetized. The stronger the field the stronger will the magnetization be. If now an alternating current be sent along the wire, the magnetized particles will take up forced oscillations. = Let e e cos pt represent the E.M.F. acting at the extremity of a bismuth wire. For simplicity, let us confine our attention to one particle of bismuth. Let u= any displacement of the particle caused by the alternating current; then, approximately, the oscillation of the particle will be represented by ü+ku+n2u= E cos pt; where k depends on friction and damping, and n = (1) 2π T T being the time of oscillation of the particle. T will depend on the degree of magnetization of the particle, on the strength of the field, and on the ultimate structure of the bismuth. A solution of (1) is The movement of the particle will cause an E.M.F. along the wire; if the amplitude of the oscillation be small, the E.M.F. produced will vary approximately as du dt (5) From (5), du dt E cos a pk •pcos (pt+a). The p's cancel-this is an important part of the analysis— and we get du E cos a = dt If we assume that the value of pk is large in comparison with that of n2-p2, a will be small, and cos a will approximately equal unity. Let us suppose that n>p, so that a is positive. Let e the back E.M.F., due to the forced oscillation of the particle. Let Ka multiplier which will increase with the magnetization of the particle, i. e. with the strength of the field. We have To obtain a rough result we may put a=0, (7) Let F strength of magnetic field: for the sake of mathematical simplicity let us suppose that the magnetization of the bismuth particle varies as the square root of F. (8) This deduction diverges from the observed facts; its chief interest lies in the result that, so long as k is great compared n2-p2 2-p2, the frequency of the alternating current has p with little influence. The higher the value of k, the lower is the frequency for which (8) is still true. In the preceding it has been assumed that n>p; it seems impossible to obtain the actual value of n. In general, rapid oscillations are characteristic of small dimensions; and if the oscillations were due to mechanical causes, as an hypothesis I should feel inclined to take n large. The frequency of the oscillations of a magnetized particle could not depend entirely, however, on mechanical forces, for it would certainly depend on the strength of the field and the relative position of the particle with respect to the field. It may be noticed also that ra depends to some extent on p; and it will be shown later that, if the bismuth spiral formed one arm of a Wheatstone bridge, the positions of minimum noise would vary slightly with different frequencies; so that unless a perfectly harmonic alternating current was employed, the sound in the neighbourhood of the minimum position would differ in constitution from the sound corresponding to the original alternating current. The preceding treatment is very incomplete, a fuller examination is given below. The adjoining figure represents a Wheatstone bridge. The telephone is placed in BD, the bismuth spiral in CD. It will be assumed that the increase in resistance of the bismuth, when measured with a telephone, is altogether apparent. A B P Q T Let Y = true resistance of the bismuth spiral. Q P+Q Let us suppose the resistance of BD is so great that an inappreciable current goes through the telephone. Let E.M.F. at A=0. وو وو C=cos pt. Let i current along C D A. Let u=displacement of a bismuth particle, d2u du +ki dt2 dt +n2u=k2i.. (1) Let B-back E.M.F. due to motion of bismuth particles; since the back E.M.F. will increase with the number of particles, if we confine our attention to the same kind of wire, |