If we suppose a second vibration to be communicated to v, in the same line, and of the same duration, but whether of the same extent or not does not matter, the compound vibration is only equivalent to another simple vibration. Let a circle move with q, and in that circle let a point (a) revolve uniformly, and let R v be perpendicular to o A. Then, while r vibrates about o, V performs a vibration in the same time relatively to r; or a spectator who does not see the motion of r, will see no motion in v except a vibration about P. Now it is easily shown that R not only describes a circle about q, but also actually describes either a circle in apace, about the centre o, or an ellipse, in the manner presently explained. And v, vibrating about r, which itself vibrates about o, does, if these vibrations be of the same duration, nothing but vibrate about o. Mathematically, this is easily obtained as follows :—Let the angles A 0 Q and a q R (Q a being parallel to o A) be at some one moment a and $, and let o q =a, q =b, and let the time be measured from the instant at which the angles are a and /3. Then we have x=a cos (nt +a) +6 cos (n t 0), the sign + being used when the circular vibrations are in the same. — when they are in opposite, directions. This is equivalent to x cos (nt + A), provided / and A be found from /cosA=acosa+bcosft ,/sinX=asina--1-bsin/3; and the joint vibration is one of the excursion 1, and such that the angle is A when the angles of the component vibrations are a and B. It is easy to show in like manner that any number of vibrations whatsoever, made in the same times and in the same lines, are not distinguishable from one single vibration, of the same duration and in the same line.
Again, it is easily shown that a vibration which is represented in direction and excursion by the diagonal of a parallelogram is the com pound effect of two vibrations of the same duration, represented in direction and excursion by the two sides of the parallelogram, if the particles of the component vibrations begin to describe the sides at the same instant as the particle of the resultant vibration begins to de scribe the diagonal ; and the same thing may be shown of the diagonal of a parallelopiped and its three sides. Hence any number of vibrations of equal times about any lines drawn through one point may each be decomposed into three in the direction of three given axes passing through that point, and those in the several axes may be compounded together into one. The student who appreciates the similarity of tho laws by which velocities, pressures, and rotations are compounded and decomposed, will see that to the list must be added vibrations. But the only vibrations which bear the application of these rules are those of equal duration.
Let us now suppose that any number of vibrations of equal times, and about the mine point, are reduced to three, in the directions of three axes of x, y, and e. When a cos t represents the distance of a
vibrating particle from its centre of vibration, let the angle t be called the phase of the vibration. If the three vibrations be always in the same phase, the diagonal of the parallelopiped described on the three excursions represents the direction and excursion of the resulting vibration, which is simple and rectilinear. But if the simultaneous phases be not the same, so that x= a cos (at + a), y = b cos (at z= c cos (nt+7), represent the simultaneous distances in the three vibrations, and also the co-ordinates of a point which is affected by them all, the particle, thus triply vibrating, does not move in a straight line, but in an ellipse. Let us consider two vibrations in a plane, and let A a and R b be their double excursions about the common centre 0. The axes in the figure are drawn at right angles, but any angle will do equally well. Draw the parallelogram w x Y z, which always con tains the particle, and suppose that P and v are contemporaneous positions in the two vibrations, whence N is one of the positions of the particle. Through N can be drawn two ellipses. having the centre 0, and touching all tho four sides of the parallelogram IV x Y Z. The particle must describe one or other of these ellipses : one when P and v are both leaving the centre or both returning to it; the other when one is leaving the centre and one returning to it. In the figure, and supposing c v e to be the direction of motion, v is leaving, and r returning to, the centre. And if c za c be the circle described about thin ellipse, and K L M be always perpendicular to c c, the law of the motion of the particle L is that sr moves uniformly round the circle, or K moves through a simple vibration. This is exactly the law of motion shown by Newton to obtain when the particle L is attracted towards o by a force which varies as its distance from o; and mechanical con siderations might easily be used to establish the whole theorem. If the vibrations be thus compounded for each pair of axes, three ellipses are obtained on the three co-ordinate planes, which are the projections of the ellipse which the particle describes in space.
We may attempt to compound two different vibrations on the same line, that is, two vibrations of different durations. if in the first figure we suppose the angular velocity of n round q to be different from that of Q round o, we ace that n describes a trochoidal curve, and supposing such a curve to be described by uniform circular motions, the motion of the projection of It upon the line of vibration will show the effect of the two vibrations. Some simple instances may be readily obtained from the diagrams in the article cited; but an attempt at a description of the multifarious effects of even two vibrations would baffle all human power of classification.