know that the resultant motion must be in some closed

curve, the locus is, in general, an ellipse.

Describe a rectangle whose sides are the double amplitudes of the two given components. Then all the ellipses obtained by giving different values to ò can be inscribed in this rectangle, since the extreme values of x are always ±a, and the extreme values of y are ±6.

It will be found upon examination that the direction in which the moving point travels round the ellipse depends upon the value of ò, and is reversed as through the values o and . We shall examine the two

cases

passes

75 d=37, for which we have obtained in last

2

2

Hence in one quarter period the motion is from the point x=a, y=o to the point x=0, y=b. The direction of revolution is therefore from the positive axis of x to the positive axis of y'.

Hence the direction of revolution is from the positive axis of x to the negative axis of y; which is opposite to the direction of revolution in the preceding case.

45. Next, let us suppose the periods of the two mutually perpendicular components to be only approximately equal. Then the resultant motion at any moment will be approximately one of the ellipses represented by equation (7), but will gradually change, and thus, instead of the motion repeating itself in a fixed ellipse, it will approximate in succession to all the ellipses which equation (7) can be made to represent by giving every possible value to ò. The first figure in Plate III. shows the trace left, by a point describing these approximate ellipses, upon a sheet of paper travelling uniformly past it.

the

If ò is increasing, the x vibrations are gaining upon y vibrations. For the phase of the x vibrations is , and the phase of the y vibrations is 6-d; but if d increases with (and at a much slower rate) the increment of 6-8 in any time is less than the increment of 0; that is to say, the phase of the y vibrations increases more

slowly than the phase of the x vibrations. The opposite will be the case if d is decreasing.

45. * To investigate the resultant of two s.H. motions along lines inclined at an angle other than a right angle, we have only to suppose the axes of co-ordinates in the preceding sections 43-45 to be oblique. The algebraic work is unchanged, and the interpretation of the results presents no difficulty.

CHAPTER IV.

WAVES.

46. LET a a1, bb1, cc1, d d1, &c. (Fig. 20), be equal parallel and equi-distant straight lines, such that their extremities a, b, c, d, &c., are in one straight line, and their other extremities therefore also in a straight line. Let there be a number of particles, one in each of the lines a a1, b b1, &c., executing simple harmonic vibrations

in them, so that a and a1, are the extreme positions of the first particle, b and b1, of the second particle, and so on. Let the periods of their vibrations be all equal, but let the second particle be a little later in its phases than the first, the third a little later than the second, and so on, the differences of phase from particle to particle being all equal. Then we shall have a simple harmonic undulation-in other words, a series of simple harmonic waves -traversing the particles.

In the figure, the first particle is supposed to be at the middle of its descent from a to a1, and the seventh particle to be at the middle of its ascent from g1 to g. The wave is travelling from left to right.

47. Let the common difference of phase from particle

to particle be Þ

of the common period, þ being a large

integer, then the difference of phase between the first particle and the p+1th, or between the second particle and the p2th, or between the rth particle and the p+ rth will amount to one period. But a difference of phase of one period means identity of phase. Hence the phase of the first particle at any moment is the same as that of the p+1th, also of the 2p+1th, and the 3p+1th; the phase of the second particle is the same as that of the p+2th, the 2p+2th, the 3p+2th, &c., and in general the phase of the rth particle is the same as that of the p+rth, the 2p+rth, the 3p+rth, and so on.

The first particles will form one complete wave, the next particles another complete wave, which at any given moment will be precisely similar to the first; the next particles will form another similar wave, and so on. Any p successive particles will form one complete wave; and if we include p+1 particles, the phases of the first and last will be identical. The distance from the first particle to the p+1th (in other words, p times the common distance a b or b c or cd) is called the length of the waves, or the wave-length, and is usually denoted by