extremity of the line towards this point. In fig. 35, if q, is a rotational vector due to one magnetic field, and q, is a similar vector at the same point, due to a second field superposed on the first, is the diagonal of the parallelogram whose adjacent sides are q, and 92. This diagonal is determinate in direction and magnitude, and may therefore be taken to represent a rotational vector of the same kind as q, or q2. It represents, in fact, a rotational motion which is the resultant of the separate motions 9, and 92. FIG. 35 Very various proofs of this kinematical theorem are to be found in elementary textbooks. The reader may most easily satisfy himself of its correctness by considering any one point of a rigid body, and following its displacement as the body is rotated through small angles about the two axes successively; the small angles being proportional respectively to q, and 92. If the two vectors a, and o, are coincident in direction, the diagonal is equal in magnitude to their sum; if on the other hand they are in exactly opposite directions, is equal to their difference. Hence the following simple rule: Where lines of force in the same direction are superposed upon one another, the rotatory motion is strengthened; where oppositely directed lines are superposed, the motion is diminished. If the lines of force belonging to the two original systems are inclined at an angle to one another, they must be compounded by the parallelogram of rotatory vectors, which leads to the rule for the combination of fields of force so frequently applied in Chapter V. 124. Conception of a permanent motion.-If each line of force is really an axis about which a rotatory motion is taking place, there might seem to be some difficulty in explaining the fact that this motion continues undiminished so long as the corresponding magnetic force retains its value unaltered. For even bar magnets, with free ends from which lines of force are given off, lose their magnetism but slowly, while in closed magnetic circuits (such as that of a horseshoe magnet with a keeper) the intensity of magnetisation may remain almost constant for years. Here then rotatory motion must be maintained without appreciable loss of velocity. This would appear to contradict our hypothesis, since every observable motion of the kind which we postulate must ultimately be completely annihilated by friction, unless some means were from time to time used to excite it afresh. This, however, constitutes no real ground of objection against the existence of motion in the field, provided the special character of that motion is taken into account. It is a matter of every-day experience that the kinetic energy of finite bodies in motion becomes transformed by frictional resistances into energy of other forms, the most important of which is heat, and so that the velocity on which the kinetic energy depends is correspondingly diminished. In the mechanical theory of heat, thermal phenomena are referred to motions of the smallest particles, or molecules, of matter. But though we may constantly observe the transformation of the energy of motion of appreciable masses into heat by friction, no similar transformation takes place in the world of molecules; we may always conceive of the molecular movements as unaccompanied by friction without finding anything in experience to contradict our supposition. The rotatory movements of MAXWELL are supposed to actuate the smallest particles, which we may suppose to be affected with permanent motions about axes, frictional influences being absent. RICHARZ, taking as his starting point the phenomena of electrical atomic charges, has recently succeeded in explaining the magnetic properties of iron by means of such rotatory motions. The diffiiculty of conceiving of the existence of such infinitely enduring motions is not greater than that inherent in other conceptions of which we frequently make use. For example certain properties of crystals, of unannealed glass, and of melted lavas are referred to conditions of stress which have persisted in these bodies for thousands of years, though one might think that the molecular motions constituting heat must have long ago equalised the stresses in question. Astronomical observation furnishes examples of bodies whose motions endure through ages. Axial movements, such as we here suppose to take place in the field, have certain properties in relation to stability, as may be seen from the simple case of a spinning-top (law of free axes). Motions of this kind, in which the same succession of events occurs again and again-cyclical motions as they were appropriately termed by HELMHOLTZ-confer on the bodies which they actuate a certain stability which would not exist in the absence of such motions. Lord KELVIN has illustrated this by means of ingenious models, and has shown that a system of gyrostats may possess properties identical with those of an elastic solid. 'Permanent motions as apparent substances' was the title of an address which HELMHOLTZ was to have given at the Scientific Congress at Nuremberg in 1893, though unfortunately the congress never was held. Amongst his scientific remains only the introduction was discovered, though the author was fortunate enough to learn in conversation that the subject of the address was the persistent character of certain cyclic motions,' whereof vortex motion had already been investigated by HELMHOLTZ himself. See also under (C) and (D). So far we have failed to find any means of satisfying ourselves directly of the existence of elementary rotations in magnetic fields; the motions in question belong to the class called hidden motions' (verborgenen Bewegungen) by HELMHOLTZ, and play a prominent part in the Prinzipien der Mechanik' of HEINRICH HERTZ. In the second part of the book a complete exposition will be given of the highly suggestive kinetic theory of electro-magnetic and electro-dynamic phenomena, which in this work is only introduced by way of illustration. B.-Mechanical interpretation of the dynamical properties of magnetic lines of force. If a magnetic field behaves like a region filled with a rotatory motion of its minutest parts, we must attempt to discover how the dynamical properties of such a motion can account for the pressures and tensions in the field. To this end we must ascribe to the constituent motions a certain amount of energy-that is to say, we must suppose the thing which moves to possess inertia. This brings us to the consideration of MAXWELL's vortex theory. The assumption of Maxwellian molecular vortices has led to values for the density of the medium in motion which agree very closely with those deduced from other observations (GRÄTZ). We must here content ourselves with showing how the centrifugal tendency in a rotatory motion accounts qualitatively for the observed mechanical forces in the field, the rigorous investigation of MAXWELL (1861) being reserved for the second part of the book. 125. Pressure and tension in a rotating element. That a rotatory motion gives rise to a tension and consequent tendency to shorten along the axis, accompanied by a pressure, or tendency to lateral expansion, in directions perpendicular to the axis, may be exemplified by means of a model, which we may take to represent an element of a tube of force (O. J. LODGE). On the under face of the horizontal base-board C of a centrifugal apparatus there is a grooved pulley r which can turn freely about a vertical axis, and may be set in rapid rotation by means of the connecting band L, which couples it to a large driving wheel lying to the right, outside the range of the figure. On the axis of the rotating pulley an elastic cylindrical drum (shaded in the figure) is fastened by means of a thumbscrew F. This consists of two circular wooden discs D, D, above and below, over which the thick-walled rubber tube G is passed, and bound round with wire at S1, S2. In the upper disc are two holes closed by corks K, K, while to the lower one is attached the spindle A, which can turn freely in a hole drilled through the fixed support H. Weights may be hung from the lower end e of the axle A. Through the holes in the upper disc liquid is poured into the drum, mercury being used if the drum is very thick-walled, though otherwise, especially when the drum is of greater size (12 cm. high, 10 cm. in diameter), glycerine will be better. S2 If the cylindrical element G, filled with fluid (fig. 36), be set in rapid rotation about its axis, its elastic walls become bulged out laterally. At the same time the length of the drum will be diminished, as if its two ends attracted one another. The tendency to contract is so great that a considerable load may be attached to e, and will be lifted up. There is also a pressure exerted on the walls of the vessel, in directions perpendicular to the axis. If we imagine a large number of such rotating cylinders to be closely packed together, they will mutually hinder one another from undergoing a change of shape, and we shall have a mechanical representation of the tension along the lines of force and the pressure across them. II FIG. 36 This model furnishes an instructive example of the way in which a distribution of motion may give rise to apparent actions at a distance, the motions themselves being hidden from our observation. It might be called a gyrostatic model of a tube of force. The kinetic energy of its rotation is at each instant proportional to the angular velocity q (see also below, under D), and |