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glass with the diameter of pencil of parallel rays 'emerging from the eye-piece, when a point of light is placed at a great distance in front of the object-glass), as a particular case. 287. Let
number of attracting or repelling masses, or perfectly smooth elastic objects, be fixed in space. Let two stations, 0 and 0, be chosen. Let a shot be fired with a stated velocity, V, from 0, in such a direction as to pass through O. There may clearly be more than one natural path by which this may be done; but, generally speaking, when one such path is chosen, no other, not sensibly diverging from it, can be found; and any infinitely small deviation in the line of fire from 0, will cause the bullet to pass infinitely near to, but not through O. Now let a circle, with infinitely small radius r, be described round 0 as centre, in a plane perpendicular to the line of fire from this point, and let—all with infinitely nearly the same velocity, but fulfilling the condition that the sum of the potential and kinetic energies is the same as that of the shot from 0-bullets be fired from all points of this circle, all directed infinitely nearly parallel to the line of fire from 0, but each precisely so as to pass through O. Let a target be held at an infinitely small distance, d', beyond O, in a plane perpendicular to the line of the shot reaching it from 0, The bullets fired from the cirumference of the circle round 0, will, after passing through O, strike this target in the circumference of an exceedingly small ellipse, each with a velocity (corresponding of course to its position, under the law of energy) differing infinitely little from V", the common velocity with which they pass through O. Let now a circle, equal to the former, be described round 0, in the plane perpendicular to the central path through 0, and let bullets be fired from points in its circumference, each with the proper velocity, and in such a direction infinitely nearly parallel to the central path as to make it pass through 0. These bullets, if a target is held to
V receive them perpendicularly at a distance a = a'
beyond 0, will strike it along the circumference of an ellipse equal to the former and placed in a corresponding position; and the points struck by the individual bullets will correspond in the manner explained below. Let P and P' be points of the first and second circles, and Q and & the points on the first and second targets which bullets from them strike; then if P be in a plane containing the central path through O, and the position which would take if its ellipse were made circular by a pure strain ($ 159); Q and are similarly situated on the two ellipses.
288. The most obvious optical application of this remarkable result is, that in the use of any optical apparatus whatever, if the eye and the object be interchanged without altering the position of the instrument, the magnifying power is unaltered. This is easily understood when, as in an ordinary telescope, microscope, or opera-glass (Galilean telescope), the instrument is symmetrical about an axis, and
is curiously contradictory of the common idea that a telescope diminishes' when looked through the wrong way, which no doubt is true if the telescope is simply reversed about the middle of its length, eye and object remaining fixed. But if the telescope be removed from the eye till its eye-piece is close to the object, the part of the object seen will be seen enlarged to the same extent as when viewed with the telescope held in the usual manner. This is easily verified by looking from a distance of a few yards, in through the object-glass of an opera-glass, at the eye of another person holding it to his eye in the usual way.
The more general application may be illustrated thus :—Let the points, 0, 0 (the centres of the two circles described in the preceding enunciation), be the optic centres of the eyes of two persons looking at one another through any set of lenses, prisms, or transparent media arranged in any way between them. If their pupils are of equal sizes in reality, they will be seen as similar ellipses of equal apparent dimensions by the two observers. Here the imagined particles of light, projected from the circumference of the pupil of either eye, are substituted for the projectiles from the circumference of either circle, and the retina of the other eye takes the place of the target receiving them, in the general kinetic statement.
289. If instead of one free particle we have a conservative system of
any number of mutually influencing free particles, the same statement may be applied with reference to the initial position of one of the particles and the final position of another, or with reference to the initial positions, or to the final positions of two of the particles. It thus serves to show how the influence of an infinitely small change in one of those positions, on the direction of the other particle passing through the other position, is related to the influence on the direction of the former particle passing through the former position produced by an infinitely small change in the latter position, and is of immense use in physical astronomy. A corresponding statement, in terms of generalized co-ordinates, may of course be adapted to a system of rigid bodies or particles connected in any way. All such statements are included in the following very general proposition :
The rate of increase of the component momentum relative to any one of the co-ordinates, per unit of increase of any other coordinate, is equal to the rate of increase of the component momentum relative to the latter per unit increase or diminution of the former co-ordinate, according as the two co-ordinates chosen belong to one configuration of the system, or one of them belongs to the initial configuration and the other to the final.
290. If a conservative system is infinitely little displaced from a configuration of stable equilibrium, it will ever after vibrate about this configuration, remaining infinitely near it; each particle of the system performing a motion which is composed of simple harmonic vibrations. If there are i degrees of freedom to move, and we consider any system of generalized 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 ($ 88), composed in general of i simple harmonics of incommensurable periods, and therefore ($ 85) the whole motion of the system will not recur periodically through the same series of configurations. There are in general, however, i distinct determinate displacements, which we shall call the normal displacements, fulfilling the condition, that if any one of them be produced alone, and the system then left to itself for an instant at rest, this displacement will diminish and increase periodically according to a simple harmonic function of the time, and consequently every particle of the system will execute a simple harmonic movement in the same period. This result, we shall see later, 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 in a vessel of water of limited extent, or in an elastic solid; and in these applications it gives the theory of the so-called 'fundamental vibration, and successive 'harmonics' of the cord, and of all the different possible simple modes of vibration in the other cases.
291. 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 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 ( 82) be a circle or an ellipse; which will therefore, of course, be the form of the orbit of any particle of the system which has a distinct direction of motion, for two of the displacements in question. But it must be remembered that some of the principal parts 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 not a representation of any motion actually taking place in any part of the system.
292. In nature, as has been said above ($ 250), 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 ($ 247)
practically we are obliged to 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; and 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 very rapidly through it, and leaving a great deal of irregular motion, such as eddies,' in its wake, seems to be nearly in proportion to 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 may, it is probable, be approximately expressed as the sum of two terms, one simply as the velocity, and the other as the square of the velocity.
293. The effect of friction of solids rubbing one against 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, without the corrections referred to, when the motions are infinitely small, and can never balance the system in a configuration deviating to any extent, however small, from a configuration of equilibrium without friction. 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, terms consisting of the generalized velocities each multiplied by a constant, 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, in the first place, stated in general terms, are as follows:
294. If the resistance is less than a certain limit, in any particular case, the motion is a simple harmonic oscillation, with amplitude decreasing by equal proportions in equal successive intervals of time. But if the resistance exceeds this limit, the system, when displaced from its position of equilibrium and left to itself, returns gradually
towards its position of equilibrium, never oscillating through it to the other side, and only reaching it after an infinite time.
In the unresisted motion, let na be the rate of acceleration, when the displacement is unity ; so that ($ 74) we have T=27: and let the rate of retardation due to the resistance corresponding to unit velocity be kThen the motion is of the oscillatory or non-oscillatory class according as k < 2n or k > 2 n. In the first case, the period of the oscillation is increased, by the resistance, from T to T.
(4k2_n2) 1: and the rate at which the Napierian logarithm of the amplitude diminishes per unit of time is lk.
295. An indirect but very simple proof of this important proposition may be obtained by means of elementary mathematics as follows:--A point describes a logarithmic spiral with uniform angular velocity about the pole-find the acceleration.
Since the angular velocity of SP and the inclination of this line to the tangent are each constant, the linear velocity of P is as SP. Take
Р a length PT, equal to n SP, to
T represent it. Then the hodograph, the locus of p, where Sp is parallel and equal to PT, is evidently another logarithmic spiral similar to
S the former, and described with the
uniform angular velocity. Hence ($$ 35, 49) pt, the acceleration required, is equal to'n Sp, and makes with Sp an angle Spt equal to SPT. Hence, if Pu be drawn parallel and equal to pt, and uv parallel to PT, the whole acceleration pt or Pu may be resolved into Pv and vu. Now Pvu is an isosceles triangle, whose base angles (v, u) are each equal to the constant angle of the spiral. Hence Pv and vu bear constant ratios to Pu, and therefore to SP and to PT respectively.
The acceleration, therefore, is composed of a central attractive part proportional to the distance, and a tangential retarding part proportional to the velocity.
And, if the resolved part of P's motion parallel to any line in the plane of the spiral be considered, it is obvious that in it also the acceleration will consist of two parts—one directed towards a point in the line (the projection of the pole of the spiral), and proportional to the distance from it; the other proportional to the velocity, but retarding the motion.