therefore by OP and PQ in directions perpendicular to those lines. That is, the velocity of P in the simple harmonic motion is

PQ; which, when P passes through O, becomes V.

74. The acceleration of a point executing a simple harmonic motion is at any time simply proportional to the displacement from the middle point, but in opposite direction, or always towards the middle point. Its maximum value is that with which a velocity equal to that of the circular motion would be acquired in the time in which an arc equal to the radius is described.

For in the fig., the acceleration of Q (by § 36) is along Q0.

Q

Supposing, for a moment, QO to represent the magnitude of this acceleration, we may resolve it into QP, PO. The acceleration of Pis therefore represented on the same scale by PO. Its magnitude V PO Ꮴ

is therefore 20'2020 PO, which is proportional to PO, and

has at A its maximum value, an acceleration under which the

Displacement Qα T

where T is the time of describing the circle, or the period of the harmonic motion.

75. Any two simple harmonic motions in one line, and of one period, give, when compounded, a single simple harmonic motion;

S

R

P

C

of the same period; of amplitude equal to the diagonal of a parallelogram described on lengths equal to their amplitudes measured on lines meeting at an angle equal to their difference of epochs; and of epoch differing from their epochs by angles equal to those which this diagonal makes with the two sides of the parallelogram. Let P and P be two points executing simple harmonic motions of one period, and in one line B'BCAA'. Let Q and be the uniformly moving points in the relative circles. On CQ and CQ describe a parallelogram SQCQ; and through S draw SR perpendicular to BA' produced. We have PR CP (being projections of the equal and parallel lines Q'S, CQ, on CR). Hence CR CP+ CP' ; and therefore the point R executes the resultant of the motions P and P. But CS, the diagonal of the parallelogram, is constant

B

B'

(since the angular velocities of CQ and CQ' are equal, and therefore the angle OCO is constant), and revolves with the same angular velocity as CO or CQ; and therefore the resultant motion is simple harmonic, of amplitude CS, and of epoch exceeding that of the motion of P, and falling short of that of the motion of P, by the angles QCS and SCO respectively.

This geometrical construction has been usefully applied by the tidal committee of the British Association for a mechanical tideindicator (compare § 77 below). An arm CQ' turning round C carries an arm Q'S turning round Q. Toothed wheels, one of them fixed with its axis through C, and the others pivoted on a framework carried by CQ', are so arranged that Q'S turns very approximately at the rate of once round in 12 mean lunar hours, if CQ be turned uniformly at the rate of once round in 12 mean solar hours. Days and half-days are marked by a counter geared to CQ. The distance of S from a fixed line through C shows the deviation from mean sealevel due to the sum of mean solar and mean lunar tides for the time of day and year marked by CQ' and the counter.

76. The construction described in the preceding section exhibits the resultant of two simple harmonic motions, whether of the same period or not. [If they are very nearly, but not exactly, of the same period, the diagonal of the parallelogram will not be constant, but will diminish from a maximum value, the sum of the component amplitudes, which it has at the instant when the phases of the component motions agree; to a minimum, the difference of those amplitudes, which is its value when the phases differ by half a period. Its direction, which always must be nearer to the greater than to the less of the two radii constituting the sides of the parallelogram, will Oscillate on each side of the greater radius to a maximum deviation amounting on either side to the angle whose sine is the less radius divided by the greater, and reached when the less radius deviates by this together with a quarter circumference, from the greater. full period of this oscillation is the time in which either radius gains a full turn on the other. The resultant motion is therefore not simple harmonic, but is, as it were, simple harmonic with periodically increasing and diminishing amplitude, and with periodical acceleration and retardation of phase. This view is most appropriate for the case in which the periods of the two component motions are nearly equal, but the amplitude of one of them much greater than that of the other.

The

To find the amount of the maximum advance and maximum backwardness of phase, and when they are experienced, let CA be equal to the greater half-amplitude. From A as centre, with AB the less half-amplitude as radius, describe a circle. CB touching this circle represents the most deviated resultant. Hence CBA is a right angle; and

A

AB

sin BCA

CA

The angle BCA thus found is the amount by which the phase of the resultant motion is advanced or retarded relatively to that of the larger component; and the supplement of BCA is the difference of phase of the two components at the time of maximum advance or backwardness of the resultant.]

77. A most interesting application of this case of the composition of harmonic motions is to the lunar and solar tides; which, except in tidal rivers, or long channels or bays, follow each very nearly the simple harmonic law, and produce, as the actual result, a variation of level equal to the sum of variations that would be produced by the two causes separately.

The amount of the lunar tide in the equilibrium theory is about 1 times that of the solar. Hence the spring tides of this theory are 3.1, and the neap tides only 1'1, each reckoned in terms of the solar tide; and at spring and neap tides the hour of high water is that of the lunar tide alone. The greatest deviation of the actual tide from the phases (high, low, or mean water) of the lunar tide alone, is about 95 of a lunar hour, that is, '98 of a solar hour (being the same part of 12 lunar hours that 28° 26', or the angle whose sine is, is of 360°). This maximum deviation will be in advance or in arrear according as the crown of the solar tide precedes or follows the crown of the lunar tide; and it will be exactly reached when the interval of phase between the two component tides is 3'95 lunar hours. That is to say, there will be maximum advance of the time of high water approximately 4 days after, and maximum retardation the same number of days before, spring tides.

2.1

78. We may consider next the case of equal amplitudes in the two given motions. If their periods are equal, their resultant is a simple harmonic motion, whose phase is at every instant the mean of their phases, and whose amplitude is equal to twice the amplitude of either multiplied by the cosine of half the difference of their phase. The resultant is of course nothing when their phases differ by half the period, and is a motion of double amplitude and of phase the same as theirs when they are of the same phase.

When their periods are very nearly, but not quite, equal (their amplitudes being still supposed equal), the motion passes very slowly from the former (zero, or no motion at all) to the latter, and back, in a time equal to that in which the faster has gone once oftener through its period than the slower has.

In practice we meet with many excellent examples of this case, which will, however, be more conveniently treated of when we come to apply kinetic principles to various subjects in practical mechanics, acoustics, and physical optics; such as the marching of troops over a suspension bridge, the sympathy of pendulums or tuning-forks, etc.

79. If any number of pulleys be so placed that a cord passing from a fixed point half round each of them has its free parts all in parallel lines, and if their centres be moved with simple harmonio

motions of any ranges and any periods in lines parallel to those lines, the unattached end of the cord moves with a complex harmonic motion equal to twice the sum of the given simple harmonic motions. This is the principle of Sir W. Thomson's tidepredicting machine, constructed by the British Association, and ordered to be placed in South Kensington Museum, availably for general use in calculating beforehand for any port or other place on the sea for which the simple harmonic constituents of the tide have been determined by the harmonic analysis' applied to previous observations'. We may exhibit, graphically, the various preceding cases of single or compound simple harmonic motions in one line by curves in which the abscissae represent intervals of time, and the ordinates the corresponding distances of the moving point from its mean position. In the case of a single simple harmonic motion, the corresponding curve would be that described by the point P in $ 70, if, while Q maintained its uniform circular motion, the circle were to move with uniform velocity in any direction perpendicular to OA. This construction gives the harmonic curve, or curve of sines, in which the ordinates are proportional to the sines of the abscissae, the straight line in which O moves being the axis of abscissae. It is the simplest possible form assumed by a vibrating string; and when it is assumed that at each instant the motion of every particle of the string is simple harmonic. When the harmonic motion is complex, but in one line, as is the case for any point in a violin-, harp-, or pianoforte-string (differing, as these do, from one another in their motions on account of the different modes of excitation used), a similar construction may be made. Investigation regarding complex harmonic functions has led to results of the highest importance, having their most general expression in Fourier's Theorem, to be presently enunciated. We give below a graphic representation of the composition of two simple harmonic motions in one line, of equal amplitudes and of periods which are as 1 : 2 and as 2 : 3, the epochs being each a quarter circumference. The horizontal line is the axis of abscissae of the curves; the vertical line to the left of each being the axis of ordinates. In the first case the slower motion goes through

See British Association Tidal Committee's Reports, 1868, 1872, 1875: or Lecture on Tides, by Sir W. Thomson (Collins, Glasgow, 1876).

one complete period, in the second it goes through two periods. These and similar cases when the periodic times are not commensurable, will be again treated of under Acoustics.

80. We have next to consider the composition of simple harmonic motions in different directions. In the first place, we see that any number of simple harmonic motions of one period, and of the same phase, superimposed, produce a single simple harmonic motion of the same phase. For, the displacement at any instant. being, according to the principle of the composition of motions, the geometrical resultant of the displacements due to the component motions separately, these component displacements in the case supposed, all vary in simple proportion to one another, and are 'in constant directions. Hence the resultant displacement will vary in simple proportion to each of them, and will be in a constant direction.

But if, while their periods are the same, the phases of the several component motions do not agree, the resultant motion will generally be elliptic, with equal areas described in equal times by the radiusvector from the centre; although in particular cases it may be uni form circular, or, on the other hand, rectilineal and simple harmonic.

81. To prove this, we may first consider the case, in which we have two equal simple harmonic motions given, and these in per pendicular lines, and differing in phase by a quarter period. Their resultant is a uniform circular motion. For, let BA, B'A' be their ranges; and from 0, their common middle point as centre, describe

B

P

A'

a circle through AA' BB'. The given motion of P in BA will be (§ 67) defined by the motion of a point Q round the circumference of this circle; and the same point, if moving in the direction indicated by the arrow, will give a simple harmonic motion of P, in B'A', a quarter of a period behind that of the motion of P in BA. But, since A'OA, QPO, and QPO are right angles, the figure QPOP is a parallelogram, and therefore Q is in the position of the displacement compounded of OP and OP. Hence two equal simple harmonic motions in perpendicular lines, of phases differing by a quarter period, are equivalent to a uniform circular motion of radius equal to the maximum displacement of either singly, and in the direc tion from the positive end of the range of the component in advance of the other towards the positive end of the range of this latter.

B'

82. Now, orthogonal projections of simple harmonic motions are clearly simple harmonic with unchanged phase. Hence, if we project the case of § 81 on any plane, we get motion in an ellipse, of which the projections of the two component ranges are conjugate diameters, and in which the radius-vector from the centre describes equal areas (being the projections of the areas described by the radius of the circle) in equal times. But the plane and position of