5 and 6. The corresponding greatest distances are, in (I), between 2 and 3; in (II), between 1 and 2; in (III), between 4 and 5. The remaining particles likewise exhibit differences of relative distance in the three cases. Thus, the positions of greatest shortening and greatest lengthening occupy dif ferent situations in the wave, and the intermediate variations between them proceed according to different laws, when the modes of particle-vibration are different. The more particles we lay down in their proper positions in a and a', the less abrupt will be the changes of distance between neighbouring particles. By indefinitely increasing the number of vibrating particles, we should ultimately arrive at a state of things in which perfectly continuous changes of shortening and lengthening intervened between the positions of maximum shortening and maximum lengthening in the same wave. 17. Let us now replace our row of indefinitely numerous, but mutually unconnected, particles by the slenderest filament of some material whose parts (like those of an elastic string) admit of being compressed or dilated at pleasure. When any portion of the filament is shortened, a larger quantity of material is forced into the space which was before occupied by a smaller quantity. The matter within this space is therefore more tightly packed, more dense, than it was, i.e. a process of condensation has occurred. On the other hand, when a portion of the filament is lengthened, a smaller quantity is made to occupy the space before occupied by a larger quantity. Here the matter is more loosely packed, more rare, than it was, i.e. a process of rarefaction has taken place. Let us now suppose the particles of the filament to be thrown into successive vibrations in the manner already so fully explained. Alternate states of condensation and rarefaction will then travel along the filament. It will be convenient to call these states 'pulses '-of condensation or rarefaction as the case may be. A pulse of condensation and a pulse of rarefaction together make up a complete wave. 18. The degree of condensation, or rarefaction, existing at any given point of a wave has been shown to depend on the mode in which the particles of the filament vibrate. It is therefore desirable to have some simple method, appealing directly to the eye, of exhibiting the law of any assigned mode of vibration. We may arrive at such a method by the following considerations. When a line of particles vibrate longitudinally, they give rise to alternate pulses of condensation and rarefaction; when transversely,' they produce alternate protuberances on opposite sides of the line of particles in their positions of rest. Nevertheless, if the vibrations in the two cases are identical in all respects save direction alone, the distance which, at any moment, separates an assigned particle from its position of rest will be the same whether the vibrations are longitudinal or transverse. It is therefore only necessary to construct the wave corresponding to any system of transverse vibrations, in the way shown in § 12, in order to obtain the means of fixing the position of an assigned particle at any given moment for the same system of vibration executed longitudinally. Let AB and CD, Fig. 16, be lines of particles executing vibrations transverse to AB and along CD respectively. Let a and b be corresponding particles in their positions of rest. Draw the transverse wave for any given instant of time: the particle originally at a will now be at a', and that originally at b will be at b', if bb' be made equal to aa'. Fig.16. -B C... D By performing the same process for different instants, we can find as many corresponding positions of the longitudinally-vibrating particle as we please. It is true that we learn nothing new by this, since we cannot construct the wave-curve without knowing beforehand the mode of the particle's vibration [§ 12]. Still, when we are dealing with longitudinal particle-vibrations, and require to know the law of the variation of condensation and rarefaction at different points of a single wave, it is convenient to have a picture of the mode of vibration by which, as we know [§ 16], that law is determined. Such a picture we have in the form of the wave produced by the same mode of vibration when executed transversely. Let us call the wave so related to a given wave of condensation and rarefaction its associated wave. 19. Before leaving this portion of the subject, it will be advisable to draw the associated wave for that particular mode of longitudinal vibration in which each particle moves as if it were the extremity of a pendulum traversing a path which is very short compared to the pendulum's length. The meaning of this limitation will be easily seen from Fig. 17. Fig.17. B Α D Let O be the fixed point of suspension; OA the pendulum in its vertical position; AB a portion of a circle with centre O and radius OA; a, b, c, d, points on this circle; AD a horizontal straight line through A; aa', bb', cc', dd' verticals though a, b, c, d, respectively. If the pendulum is placed in the position Oa, and left to itself, it will reach an equally inclined position on the other side of OA, i.e. will swing through twice the angle aOA before it turns back again. Similarly, if started at Ob it will swing through twice the angle bOA; if at Oc, through |