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We have tried several times to obtain indications of combination-tones when the primary notes were produced by tuning-forks. Two of König's large forks adjusted to 48 and 112 vibrations produced no effect when bowed simultaneously before the collector, and, as has been stated, smaller forks giving 256 and 320 vibrations have been placed inside the collector when sounding loudly. No effect whatever was produced, and there can be no doubt that if objective combination-tones are produced in such cases they are very much less intense than those generated by the siren.
Experiments have been made with reeds and with organpipes, but up to the present with uncertain results in the first case and negative results in the second. We hope to investigate these cases further.
We have made several attempts to detect combinationtones of higher orders, such as 2p+q and 2p-q, but without
We may in conclusion refer to some of the suggestions which have been made to account for the combination-tones by theories other than those of Helmholtz.
König's suggestion that they are the beat-tones of upper partials has been discussed and shown to be inadequate to explain the facts of observation.
Again, it has been argued that the summation-tone is the beat-note between the second partial of the higher note (the octave) and the beat-tone of the two primaries. It follows as a matter of algebra that such an explanation must always be numerically correct, for 2a-(a-b)=a+b, and our experiments throw no new light on the matter. It appears to us, however, that since propinquity between the sources of sound, causing a violent disturbance, is favourable to the production of combination-tones, while it is not necessary for the production of beats, the facts of experiment are in this case also in favour of von Helmholtz's views.
A still more subtle objection has been taken by Terquem (Annales d'Ecole Normale, 1870, p. 356). When two rows of holes are open in the siren, there may be occasions on which all the holes of both rows are opened simultaneously and others on which only one row is in action at one time. Terquem attempts to calculate the effects of irregularities such as these, but in the first place he specifically refrains from attacking the theory of Helmholtz; secondly, he does not apply calculation to the siren of Helmholtz; thirdly, he points out that the relatively large size of the holes in that instrument would reduce the effects he predicts; and, lastly, he admits that his results require confirmation by experiment. Putting these points aside, however, his theory leads to the conclusion that the two notes which we have been regarding as fundamental are reinforced harmonics in a series of which the fundamental note corresponds to the greatest common measure of these frequencies. Both the summation and the difference tone must be included in such a series; but Terquem's theory gives no reason why they should have such exceptional importance as experiment proves that they have. Lastly, as he expressly repudiates the idea that partials have an objective existence (loc. cit. p. 274), and includes the combination-tones in a series of partials, the experiments described by us must on this point be regarded as opposed to his views.
We think, then, that our experiments prove that von Helmholtz was correct in stating that the siren produces two objective notes the frequencies of which are respectively equal to the sum and difference of the frequencies of the fundamentals, and that our observations are also more or less opposed to the theories by which König, Appun, and Terquem have sought to account for the production of these notes.
We believe that the method we have devised is capable of greater sensitiveness. It can be extended by employing forks of different pitches, and it is quite possible that less massive forks may enable us to detect effects which have hitherto escaped us. We therefore refrain from any wide generalizations until a wider foundation of experiment has been laid.
P.S. Since the above was written Prof. S. P. Thompson has drawn our attention to a paper by O. Lummer, published in 1886 (Verh. phys. Gesell. Berlin, 1886, No. 9, p. 66), which had escaped our notice, as it is not abstracted in the Beiblätter. Herr Lummer obtained evidence of the objective character of the summation-tone by means of the microphone.
XXXIV. Energy Movements in the
ARADAY and Maxwell have shown that it is possible to look on the potential energy of electric separation as residing in the surrounding dielectric, and that each of the cells, bounded by the walls of a tube of force and two neighbouring equipotential surfaces, can be looked upon as containing a certain definite amount of energy.
This energy-distribution is not in general permanent, and can only be regarded as a step towards some simpler arrangement. Thus a positive and a negative electrified body suspended in space, and acted on only by the electrical forces between them, are never in equilibrium until they are in actual contact. The energy-distribution in the dielectric changes constantly as they approach. Poynting has shown† how energy is transferred from one point to another in an electromagnetic field; and we are quite accustomed to think of energy as flowing from dynamo to motor through the æther, or from primary to secondary in an alternating-current transformer.
In the following pages an attempt is made to deduce a few
*Communicated by the Physical Society: read March 8, 1895.
Phil. Mag. S. 5. Vol. 39. No. 239. April 1895. 2 B
of the consequences of Maxwell's suggestion with regard to energy-distribution. The changes needed in the theory in order to apply it to gravitation are also indicated.
2. If two equal charges of positive and negative electricity +M and M are separated by a distance l, the equation to the curve in which the equipotential surface V intersects the plane of the paper is
the origin being the point midway between the particles, and the X-axis being the line joining this point to the positive particle.
3. The number of tubes of force proceeding from a charge M has sometimes been taken as M but more generally as 4 M. For graphical methods the former plan seems most convenient, and in order to avoid confusion it is proposed that the resulting tubes should be called "tubes of polarization," while the smaller tubes retain the name "tubes of induction." The polarization, as defined by J. J. Thomson in his 'Recent Researches in Électricity and Magnetism,' is measured by the number of these polarization tubes which pass through a square centimetre perpendicular to their direction.
Using Maxwell's method of drawing the boundary lines between the tubes of polarization, the equation to the nth line in the case mentioned above will be
where the straight line drawn from the positive to the negative particle is called zero, and that drawn in the opposite direction M.
4. The point of intersection of the equipotential surface V, the line of force n, and the plane of the paper lies on the curves (1) and (2). If, then, 7 in these equations be regarded as a variable parameter, they will together represent the curve along which the corner of an energy-cell moves, when the two particles come together. To plot this curve, the equipotential surfaces and lines of force might be drawn for a number of different distances between the particles, and corresponding points of intersection joined.
The path of the energy-cell can be obtained with less labour from measures made on a single diagrain, drawn to represent the lines of force and equipotential surfaces about the two particles, when these are separated by a given distance. This is done by taking advantage of the following properties of these lines :
1. When the distance between the particles is changed, corresponding tubes of polarization in the two diagrams are similar to one another.
2. The equipotential surfaces in the two diagrams are also similar to one another. If, however, the distance between the particles changes from 1 to al, the equipotential surface V
V changes to a similar surface, on which the potential is a If, then, the nth line of force cuts the equipotential surface V at the point x, y in the first diagram, the nth line will cut the equipotential surface at the point ax, ay in the second diagram.
Thus, in the case where M=12 and 7=10 the coordinates of the following intersections, among others, were found by measurement on a carefully prepared diagram.
If the diagram were enlarged in the ratio of 4 to 1, the coordinates of the point corresponding to the intersection of 7 and 4 would be 4 and 4y or 21.4 and 9.12.
The potential at this point would be V'= 1.
Thus we have found the coordinates of the point of intersection of the line of force 7 and the equipotential surface 1, when the distance between the particles is 40 centim.
In the same way, and by means of the same original diagram, the point of intersection can be found for a number of other distances, and the path which it follows, when the two particles approach one another, can be plotted as in fig. 1.
In this figure the first of the numbers in brackets attached to each curve gives the number of the equipotential surface, and the second that of the intersecting line of force. The numbers along the curves show the distance of the particles from the centre of gravity, when the energy-cell is at the point marked on the curve.
It will be seen that the energy-cells move in more or less parabolic curves towards the centre of gravity of the two particles, and that during this process they are constantly approaching the two particles; so that, if these are not infinitely small, energy must be constantly passing into them from the æther.
Maxwell has shown that in all cases the number of energycells in the æther is twice the potential energy of the system. So that, if we suppose each cell to contain half a unit of