nents, c the point which travels uniformly in the auxiliary circle of the resultant. Then O A, O B, O c, are the three amplitudes, and o c, being the diagonal of a parallelogram of which OA, OB are the sides, may have any value intermediate between their sum and difference, according

FIG. 15.

to the magnitude of the angle A O B, which is the difference of phase (or difference of epoch) of the two components. When this angle is zero they have the same phase, and the resultant amplitude is the sum of the given amplitudes. When it is 180° they have oppo

site phases, and the resultant amplitude is the difference of the given amplitudes. In this case if the given amplitudes are equal the two components destroy each other, and the resultant is absolute rest..

The formula for the square of the amplitude of the resultant is evidently

o c2 = 0 A2 + O B2 + 2 O A. O B COS AO B,

A O B being the difference of phase of the two components, and O A, O B their amplitudes.

34. If the two component S.H. motions have not rigorously the same period, as hitherto supposed, the angular velocities of A and B in the two auxiliary circles will not be rigorously equal, and the angle AO B will change at a constant rate. We shall suppose this rate to

be slow in comparison with the angular velocities themselves, so that the angle A O B will only undergo a small change in each revolution. Then the path of c in one revolution will be nearly circular, and its velocity in this path sensibly uniform, so that the projection of its motion upon a given line will be very approximately simple harmonic; but the radius of the circle will gradually alter in successive revolutions, taking all values intermediate between the sum and difference of o A, O B. Remembering that the radius of the circle is the amplitude of the resultant, we see that the resultant of two S.H. motions of slightly unequal periods (both having the same line of motion) may be described as a S.H. motion with amplitude varying between the sum and the difference of the two given amplitudes.

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35. The variation of the square of the amplitude is simple harmonic. For if we drop a perpendicular C D on o A or o A produced, we have

o c2 = 0 A2 + A C2± 2 0 A. A D,

where all the quantities on the second side are constant except A D. The variation of o c2 is therefore the variation of 2 0 A. A D, and is proportional to AD. But the motion of c relative to OA is uniform circular motion round a, and the motion of D along the line oa is therefore simple harmonic motion with a as central point.

It follows that the mean value of the term ± 2 0 A. A D is zero; and therefore the mean value of oc2 is o A2+0 B2,

or the mean value of the square of the resultant amplitude is the sum of the squares of the component amplitudes.

36. These principles explain the throbbing character of the sound which is produced by the combination of two sounds differing slightly in pitch. The drum of the ear vibrating under their joint influence performs vibrations whose amplitudes vary from the sum to the difference of the amplitudes due to the two separate sounds. If the separate effects of the two sounds were exactly equal and were simple harmonic, there would be momentary silence at the instant when the phases became opposite. Two 'stopped' organ-pipes mounted side by side on the same wind-chest, and tuned as nearly as possible to unison, will often maintain this opposition of phase (and almost complete extinction of sound) for a considerable time.

The alternations of loudness produced by the cause here explained are called beats. Each beat indicates that one of the two sources has gained a complete vibration upon the other; and hence, if the number of vibrations made by one source is known, the number made by the other can be found by adding or subtracting the number of beats.

37. They also explain the phenomena of spring and neap tides.

Speaking broadly, the variation of tidal level at a given place is the sum of two S.H. variations, one depending on the moon and the other on the sun, the

former having the larger amplitude and a rather longer period. When the phases of these two S.H. variations concur, we have spring tides with amplitude equal to the sum of the lunar and solar amplitudes, and when the phases are opposite we have neap tides with amplitude equal to their difference,

COMPOSITION OF RECTANGULAR S.H. MOTIONS.

38. We have seen that the resultant of two s.H. motions of the same period and not in the same straight line is elliptic harmonic motion. We shall now investigate the form and position of the ellipse as affected by the amplitudes and epochs of the two components. We shall in the first instance, and throughout the greater part of our discussion, suppose the two components to be at right angles, this being the only case of practical importance.

Let the directions of the two components, for convenience of language, be called horizontal and vertical. Describe two concentric circles (Fig. 16) whose radii are the two amplitudes; then one component will be the horizontal motion of a point н travelling uniformly round one circle, and the other component will be the vertical motion of a point v travelling round the other with the same angular velocity. It is optional to regard the directions of revolution in the two circles as the same or opposite; we can, therefore, without loss of generality,

suppose them to be the same.

The angle Ho v between

the revolving radii will then be constant.

If we draw two tangents at the extremities of the horizontal diameter of the circle in which н moves, these will be the limits of the horizontal motion of the resultant; and in like manner two tangents at the extremities of the vertical diameter of the circle described by v will be the

limits of vertical motion. Hence the ellipse will be inscribed in the rectangle formed by these four tangents.

To determine the points of contact of the ellipse with the vertical sides of the rectangle, we must consider where v will be when H is at the extremities of the horizontal diameter. v will evidently be at the extremities of a diameter inclined to the horizontal at the given angle Hence we must draw this diameter of v's circle, and from its extremities draw horizontal lines one to the right and the other to the left to meet the vertical sides

HOV.