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

1

sine is is of 360°). This maximum deviation will be in advance 2.1'

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 4 days after, and maximum retardation the same number of days before, spring tides.

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. 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 § 66, 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 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 uniform 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 perpendicular lines, and differing in phase by a quarter period. Their resultant is a uniform circular motion. For, let BA, B'A' be their

Α'

P

ranges; and from O, their common middle point as centre, describe 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

B'

O P

in the direction indicated by the arrow, will B 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 QP'O are right angles, the figure QP'OP 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 direction 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.

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 the circle of which this projection is taken may clearly be found so as to fulfil the condition of having the projections of the ranges coincident with any two given mutually bisecting lines. Hence any two given simple harmonic motions, equal or unequal in range, and oblique or at right angles to one another in direction, provided only they differ by a quarter period in phase, produce elliptic motion, having their ranges for conjugate axes, and describing, by the radius-vector from the centre, equal areas in equal times.

83. Returning to the composition of any number of equal simple harmonic motions in lines in all directions and of all phases: each component simple harmonic motion may be determinately resolved into two in the same line, differing in phase by a quarter period, and one of them having any given epoch. We may therefore reduce the given motions to two sets, differing in phase by a quarter period, those of one set agreeing in phase with any one of the given, or with any other simple harmonic motion we please to choose (i. e. having their epoch anything we please).

All of each set may (§ 75) be compounded into one simple harmonic motion of the same phase, of determinate amplitude, in a determinate line; and thus the whole system is reduced to two simple fully-determined harmonic motions differing from one another in phase by a quarter period.

Now the resultant of two simple harmonic motions, one a quarter of a period in advance of the other, in different lines, has been

proved (§ 82) to be motion in an ellipse of which the ranges of the component motions are conjugate axes, and in which equal areas are described by the radius-vector from the centre in equal times. Hence the proposition of § 80.

84. We must next take the case of the composition of simple harmonic motions of different kinds and in different lines. In general, whether these lines be in one plane or not, the line of motion returns into itself if the periods are commensurable; and if not, not. This is evident without proof.

Also we see generally that the composition of any number of simple harmonic motions in any directions and of any periods, may be effected by compounding, according to previously explained methods, their resolved parts in each of any three rectangular directions, and then compounding the final resultants in these directions.

85. By far the most interesting case, and by far the simplest, is that of two simple harmonic motions of any periods, whose directions must of course be in one plane.

Mechanical methods of obtaining such combinations will be afterwards described, as well as cases of their occurrence in Optics and Acoustics.

We may suppose, for simplicity, the two component motions to take place in perpendicular directions. Also, it is easy to see that we can only have a reëntering curve when their periods are commensurable. The following figures represent the paths produced by the combination of simple harmonic motions of equal amplitude in two rectangular directions, the periods of the components being as 1: 2, and the epochs differing successively by 0, 1, 2, etc., sixteenths of a circumference.

In the case of epochs equal, or differing by a multiple of, the curve is a portion of a parabola, and is gone over twice in opposite directions by the moving point in each complete period.

If the periods be not exactly as 1:2 the form of the path produced by the combination changes gradually from one to another of the series above figured; and goes through all its changes in the time in which one of the components gains a complete vibration on the other.

86. Another very important case is that of two groups of two simple harmonic motions in one plane, such that the resultant of each group is uniform circular motion.

If their periods are equal, we have a case belonging to those already treated (§ 80), and conclude that the resultant is, in general, motion in an ellipse, equal areas being described in equal times about the centre. As particular cases we may have simple harmonic, or uniform circular, motion.

If the circular motions are in the same direction, the resultant is evidently circular motion in the same direction. This is the case of the motion of S in § 75, and requires no further comment, as its amplitude, epoch, etc., are seen at once from the figure.

87. If the radii of the component motions are equal, and the periods very nearly equal, but the motions in opposite directions, we have cases of great importance in modern physics, one of which is figured below (in general, a non-reëntrant curve).

This is intimately connected with the explanation of two sets of important phenomena,—the rotation of the plane of polarization of light, by quartz and certain fluids on the one hand, and by transparent bodies under magnetic forces on the other. It is a case of the hypotrochoid, and its corresponding mode of description will be described in § 104. It may be exhibited experimentally as the path of a pendulum, hung so as to be free to move in any vertical plane