making any assumption not warranted by experimental evidence, and that there is, as yet, no experimental evidence to shew whether the electric current is really a current of a material substance, or a double current, or whether its velocity is great or small as measured in feet per second. A knowledge of these things would amount to at least the beginnings of a complete dynamical theory of electricity, in which we should regard electrical action, not, as in this treatise, as a phenomenon due to an unknown cause, subject only to the general laws of dynamics, but as the result of known motions of known portions of matter, in which not only the total effects and final results, but the whole intermediate mechanism and details of the motion, are taken as the objects of study. 575.] The experimental investigation of the second term of Xme, is more difficult, as it involves the observation of the effect of forces on a body in rapid motion. Fig. 34. The apparatus shewn in Fig. 34, which I had constructed in 1861, is intended to test the existence of a force of this kind. 575.] EXPERIMENT OF ROTATION. 203 The electromagnet A is capable of rotating about the horizontal axis BB', within a ring which itself revolves about a vertical axis. Let A, B, C be the moments of inertia of the electromagnet about the axis of the coil, the horizontal axis BB, and a third axis CC' respectively. Let be the angle which CC' makes with the vertical, the azimuth of the axis BB', and a variable on which the motion of electricity in the coil depends. Then the kinetic energy of the electromagnet may be written 2 T = A ¿2 sin2 0 + B 02 + C¿2 cos2 0 + E († sin 0 + y)2, where E is a quantity which may be called the moment of inertia of the electricity in the coil. If is the moment of the impressed force tending to increase 0, we have, by the equations of dynamics, — {(4— C) ¿2 sin 0 cos 0 + E + cos 0 (¿ sin 0 +↓)}. By making, the impressed force tending to increase y, equal to zero, we obtain $ sin 0+↓ = Y, a constant, which we may consider as representing the strength of the current in the coil. If C is somewhat greater than A, will be zero, and the equi librium about the axis BB will be stable when sin 0 = Εγ This value of depends on that of y, the electric current, and is positive or negative according to the direction of the current. The current is passed through the coil by its bearings at B and B', which are connected with the battery by means of springs rubbing on metal rings placed on the vertical axis. To determine the value of 0, a disk of paper is placed at C, divided by a diameter parallel to BB' into two parts, one of which is painted red and the other green. When the instrument is in motion a red circle is seen at C when is positive, the radius of which indicates roughly the value of 0. When is negative, a green circle is seen at C. By means of nuts working on screws attached to the electromagnet, the axis CC' is adjusted to be a principal axis having its moment of inertia just exceeding that round the axis A, so as to make the instrument very sensible to the action of the force if it exists. The chief difficulty in the experiments arose from the disturbing action of the earth's magnetic force, which caused the electromagnet to act like a dip-needle. The results obtained were on this account very rough, but no evidence of any change in could be obtained even when an iron core was inserted in the coil, so as to make it a powerful electromagnet. If, therefore, a magnet contains matter in rapid rotation, the angular momentum of this rotation must be very small compared with any quantities which we can measure, and we have as yet no evidence of the existence of the terms Tme derived from their mechanical action. 576.] Let us next consider the forces acting on the currents of electricity, that is, the electromotive forces. Let Y be the effective electromotive force due to induction, the electromotive force which must act on the circuit from without to balance it is Y' - Y, and, by Lagrange's equation, Hence, Since there are no terms in T involving the coordinate y, the second term is zero, and Y is reduced to its first term. electromotive force cannot exist in a system at rest, and with constant currents. Again, if we divide Y into three parts, Y, Y, and Yme, corresponding to the three parts of T, we find that, since Tm does not contain, Y = 0. We also find d dr.. dt dy Here is a linear function of the currents, and this part of dy the electromotive force is equal to the rate of change of this function. This is the electromotive force of induction discovered by Faraday. We shall consider it more at length afterwards. 577.] From the part of T, depending on velocities multiplied by d dTme currents, we find dt dy dTme Ym = me Now If, therefore, any terms of Tme have an actual existence, it would be possible to produce an electromotive force independently of all existing currents by simply altering the velocities of the conductors. is a linear function of the velocities of the conductors. dy 577.] ELECTROMOTIVE FORCE. 205 For instance, in the case of the suspended coil at Art. 559, if, when the coil is at rest, we suddenly set it in rotation about the vertical axis, an electromotive force would be called into action proportional to the acceleration of this motion. It would vanish when the motion became uniform, and be reversed when the motion was retarded. Now few scientific observations can be made with greater precision than that which determines the existence or non-existence of a current by means of a galvanometer. The delicacy of this method far exceeds that of most of the arrangements for measuring the mechanical force acting on a body. If, therefore, any currents could be produced in this way they would be detected, even if they were very feeble. They would be distinguished from ordinary currents of induction by the following characteristics. (1) They would depend entirely on the motions of the conductors, and in no degree on the strength of currents or magnetic forces already in the field. (2) They would depend not on the absolute velocities of the conductors, but on their accelerations, and on squares and products of velocities, and they would change sign when the acceleration becomes a retardation, though the absolute velocity is the same. Now in all the cases actually observed, the induced currents depend altogether on the strength and the variation of currents in the field, and cannot be excited in a field devoid of magnetic force and of currents. In so far as they depend on the motion of conductors, they depend on the absolute velocity, and not on the change of velocity of these motions. We have thus three methods of detecting the existence of the terms of the form Tme, none of which have hitherto led to any positive result. I have pointed them out with the greater care because it appears to me important that we should attain the greatest amount of certitude within our reach on a point bearing so strongly on the true theory of electricity. Since, however, no evidence has yet been obtained of such terms, I shall now proceed on the assumption that they do not exist, or at least that they produce no sensible effect, an assumption which will considerably simplify our dynamical theory. We shall have occasion, however, in discussing the relation of magnetism to light, to shew that the motion which constitutes light may enter as a factor into terms involving the motion which constitutes magnetism. CHAPTER VII. THEORY OF ELECTRIC CIRCUITS. 578.] WE may now confine our attention to that part of the kinetic energy of the system which depends on squares and products of the strengths of the electric currents. We may call this the Electrokinetic Energy of the system. The part depending on the motion of the conductors belongs to ordinary dynamics, and we have shewn that the part depending on products of velocities and currents does not exist. Let A1, A2, &c. denote the different conducting circuits. Let their form and relative position be expressed in terms of the variables x1, x2, &c., the number of which is equal to the number of degrees of freedom of the mechanical system. We shall call these the Geometrical Variables. Let y1 denote the quantity of electricity which has crossed a given section of the conductor A, since the beginning of the time t. The strength of the current will be denoted by 1, the fluxion of this quantity. We shall call y1 the actual current, and y, the integral current. There is one variable of this kind for each circuit in the system. Let T denote the electrokinetic energy of the system. It is a homogeneous function of the second degree with respect to the strengths of the currents, and is of the form where the coefficients L, M, &c. are functions of the geometrical variables 1, 2, &c. The electrical variables 31, 32 do not enter into the expression. We may call L1, L2, &c. the electric moments of inertia of the circuits 41, 42, &c., and two circuits 4, and 42. 12 M12 the electric product of inertia of the |