Integrated equations of motion, expressing the fundamental modes of vibration; or of falling away from configuration of unstable equilibrium. Infinitely small disturbance from un stable equilibrium. Potential and Kinetic energies expressed as functions of time. Example of fundamental modes. be either stable or unstable, or neutral; but terms of higher orders in the expansion of V in ascending powers and products of the co-ordinates would have to be examined to test it; and if it were stable, the period of an infinitely small oscillation in the value of the corresponding co-ordinate or co-ordinates would be infinitely great. If any or all of a, ẞ, y, ... are negative, Vis not a minimum, and the equilibrium is (§ 292) essentially unstable. The form (7) for the solution, for each co-ordinate for which this is the case, becomes imaginary, and is to be changed into the exponential form, thus; for instance, let-a=p, a positive quantity. Thus 4 = Ce+t√p + Ke-t√p .(8), which (unless the disturbance is so adjusted as to make the arbitrary constant C vanish) indicates an unlimited increase in the deviation. This form of solution expresses the approximate law of falling away from a configuration of unstable equilibrium. In general, of course, the approximation becomes less and less accurate as the deviation increases. and, verifying the constancy of the sum of potential and kinetic energies, or T+V= (aA+BA" + etc.) ... .(11). One example for the present will suffice. Let a solid, immersed in an infinite liquid (§ 320), be prevented from any motion of rotation, and left only freedom to move parallel to a certain fixed plane, and let it be influenced by forces subject to the conservative law, which vanish in a particular position of equilibrium. Taking any point of reference in the body, choosing its position when the body is in equilibrium, as origin of rectangular co-ordinates OX, OY, and reckoning the potential energy from it, we shall have, as in general, 2T = Ax2 + Bỳ2 + 2Cxÿ; 2V = ax2 + by3 + 2cxy, fundamen the principles stated in § 320 above, allowing us to regard the Example of Ax2+By+2Cxy = const., and the ellipse or hyperbola, ax2 + by2+2cxy = const., and choosing these as oblique axes of co-ordinates (x,, y1), we 2T = Â ̧à ̧2 + B ̧ÿ ̧3, and 2V =α ̧x ̧2 + b12y,3. And, as A, B, are essentially positive, we may, to shorten our the normal expressions, according to the general forms shown The interpretation of the general solution is as follows: theorem of tal modes of small about a con of equi 338. If a conservative system is infinitely little displaced General from a configuration of stable equilibrium, it will ever after fundamen vibrate about this configuration, remaining infinitely near it; infinitely each particle of the system performing a motion which is com- motion posed of simple harmonic vibrations. If there are idegrees of figuration freedom to move, and we consider any system (§ 202) of gene- librium. ralized co-ordinates specifying its position at any time, the deviation of any one of these co-ordinates from its value for the configuration of equilibrium will vary according to a complex harmonic function (§ 68), composed of i simple harmonics generally of incommensurable periods, and therefore (§ 67) the whole motion of the system will not in general recur periodically through the same series of configurations. There are, however, i distinct displacements, generally quite determinate, which we shall call the normal displacements, fulfilling the condition, that Normal disif any one of them be produced alone, and the system then left from equito itself for an instant at rest, this displacement will diminish and increase periodically according to a simple harmonic func placements librium. Fundamen- tion of the time, and consequently every particle of the system vibration. will execute a simple harmonic movement in the same period. tal modes of Theorem of kinetic energy; This result, we shall see later (Vol. II.), includes cases in which there are an infinite number of degrees of freedom; as for instance a stretched cord; a mass of air in a closed vessel; waves in water, or oscillations of water in a vessel of limited extent, or of an elastic solid; and in these applications it gives the theory of the so-called "fundamental vibration," and successive "harmonics" of a cord or organ-pipe, and of all the different possible simple modes of vibration in the other cases. In all these cases it is convenient to give the name "fundamental mode” to any one of the possible simple harmonic vibrations, and not to restrict it to the gravest simple harmonic mode, as has been hitherto usual in respect to vibrating cords and organ-pipes. The whole kinetic energy of any complex motion of the system is [§ 337 (4)] equal to the sum of the kinetic energies of of potential the fundamental constituents; and [§ 337 (5)] the potential energy of any displacement is equal to the sum of the potential energies of its normal components. energy. Infinitesi mal motions in neighbourhood of Corresponding theorems of normal constituents and fundamental modes of motion, and the summation of their kinetic configura- and potential energies in complex motions and displacements, stable equi- hold for motion in the neighbourhood of a configuration of un tion of un librium. Case of equality among periods. Graphic representation. stable equilibrium. In this case, some or all of the constituent motions are fallings away from the position of equilibrium (according as the potential energies of the constituent normal vibrations are negative). 339. If, as may be in particular cases, the periods of the vibrations for two or more of the normal displacements are equal, any displacement compounded of them will also fulfil the condition of being a normal displacement. And if the system be displaced according to any one such normal displacement, and projected with velocity corresponding to another, it will execute a movement, the resultant of two simple harmonic movements in equal periods. The graphic representation of the variation of the corresponding co-ordinates of the system, laid down as two rectangular co-ordinates in a plane diagram, will consequently (§ 65) be a circle or an ellipse; which will therefore, representa of course, be the form of the orbit of any particle of the system Graphic which has a distinct direction of motion, for two of the displace- tion. ments in question. But it must be remembered that some of the principal parts [as for instance the body supported on the fixed axis, in the illustration of § 319, Example (C)] may have only one degree of freedom; or even that each part of the system may have only one degree of freedom, as for instance if the system is composed of a set of particles each constrained to remain on a given line, or of rigid bodies on fixed axes, mutually influencing one another by elastic cords or otherwise. In such a case as the last, no particle of the system can move otherwise than in one line; and the ellipse, circle, or other graphical representation of the composition of the harmonic motions of the system, is merely an aid to comprehension, and is not the orbit of a motion actually taking place in any part of the system. systems. 340. In nature, as has been said above (§ 278), every system uninfluenced by matter external to it is conservative, when the ultimate molecular motions constituting heat, light, and magnetism, and the potential energy of chemical affinities, are taken into account along with the palpable motions and measurable forces. But (§ 275) practically we are obliged to Dissipative admit forces of friction, and resistances of the other classes there enumerated, as causing losses of energy, to be reckoned, in abstract dynamics, without regard to the equivalents of heat or other molecular actions which they generate. Hence when such resistances are to be taken into account, forces opposed to the motions of various parts of a system must be introduced into the equations. According to the approximate knowledge which we have from experiment, these forces are independent of the velocities when due to the friction of solids: but are simply proportional to the velocities when due to fluid viscosity directly, or to electric or magnetic influences; with corrections depending on varying temperature, and on the varying configuration of the system. In consequence of the last-mentioned cause, the resistance of a real liquid (which is always more or less viscous) against a body moving rapidly enough through it, to leave a great deal of irregular motion, in the shape of "eddies," in its wake, seems, when the motion of the solid has resistance been kept long enough uniform, to be nearly in proportion to Views of Stokes on to a solid moving through a liquid. bable law. the square of the velocity; although, as Stokes has shown, at the lowest speeds the resistance is probably in simple proportion to the velocity, and for all speeds, after long enough time of one speed, may, it is probable, be approximately expressed as Stokes' pro- the sum of two terms, one simply as the velocity, and the other as the square of the velocity. If a solid is started from rest in an incompressible fluid, the initial law of resistance is no doubt simple proportionality to velocity, (however great, if suddenly enough given;) until by the gradual growth of eddies the resistance is increased gradually till it comes to fulfil Stokes' law. Friction of solids. 341. The effect of friction of solids rubbing against one another is simply to render impossible the infinitely small vibrations with which we are now particularly concerned; and to allow any system in which it is present, to rest balanced when displaced, within certain finite limits, from a configuration of frictionless equilibrium. In mechanics it is easy to estimate its effects with sufficient accuracy when any practical case of finite oscillations is in question. But the other classes of dissipative agencies give rise to resistances simply as the velocities, Resistances without the corrections referred to, when the motions are invarying as velocities. finitely small; and can never balance the system in a con figuration deviating to any extent, however small, from a configuration of equilibrium. In the theory of infinitely small vibrations, they are to be taken into account by adding to the expressions for the generalized components of force, proper (§ 343 a, below) linear functions of the generalized velocities, which gives us equations still remarkably amenable to rigorous mathematical treatment. The result of the integration for the case of a single degree of freedom is very simple; and it is of extreme importance, both for the explanation of many natural phenomena, and for use in a large variety of experimental investigations in Natural Philosophy. Partial conclusions from it are as follows: If the resistance per unit velocity is less than a certain critical value, in any particular case, the motion is a simple |