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 periods 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 adding their components in each of any three rectangular directions. The final resultant motion is thus fully expressed by formulae giving the rectangular co-ordinates as 'complex harmonic functions' of the time.

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 I : 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 pairs of simple harmonic motions in one plane, such that the resultant of each pair 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 through its point of suspension, and containing in its bob a fly-wheel in rapid rotation.

88. [Before leaving for a time the subject of the composition of harmonic motions, we must enunciate Fourier's Theorem, which is not only one of the most beautiful results of modern analysis, but may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. To mention only sonorous vibrations, the propagation of electric signals along a telegraph wire, and the conduction of heat by the earth's crust, as subjects in their generality intractable without it, is to give but a feeble idea of its importance. Unfortunately it is impossible to give a satisfactory proof of it without introducing some rather troublesome analysis, which is foreign to the purpose of so elementary a treatise as the present.

The following seems to be the most intelligible form in which it can be presented to the general reader :

THEOREM.-A complex harmonic function, with a constant tèrm added, is the proper expression, in mathematical language, for any arbitrary periodic function; and consequently can express any function whatever between definite values of the variable

89. Any arbitrary periodic function whatever being given, the amplitudes and epochs of the terms of a complex harmonic function, which shall be equal to it for every value of the independent variable, may be investigated by the method of indeterminate coefficients." Such an investigation is sufficient as a solution of the problem,—to find a complex harmonic function expressing a given arbitrary periodic function,-when once we are assured that the problem is possible; and when we have this assurance, it proves that the reso lution is determinate; that is to say, that no other complex harmonic function than the one we have found can satisfy the conditions.]

90. We now pass to the consideration of the displacement of a rigid body or group of points whose relative positions are unalterable. The simplest case we can consider is that of the motion of a plane figure in its own plane, and this, as far as kinematics is concerned, is entirely summed up in the result of the next section.

91. If a plane figure be displaced in any way in its own plane, there is always (with an exception treated in § 93) one point of it common to any two positions; that is, it may be moved from any one position to any other by rotation in its own plane about one point held fixed.

To prove this, let A, B be any two points of the plane figure in a first position, A', B' the position of the same two after a displacement. The lines AA', BB' will not be parallel, except in one case to be presently considered. Hence the line equidistant from A and A will meet that equidistant from B and B in some point O. Join OA, OB, OA', OB. Then, evidently, because OA'=OA, OB'=OB, and A'B' = AB, the triangles OA'B' and OAB are equal and similar. Hence O is similarly situated with regard to A'B' and AB, and is therefore one and the same point of the plane figure in its two positions. If, for the sake of illustration, B we actually trace the angle OAB upon the plane, it becomes OA'B' in the second posi tion of the figure.

A

92. If from the equal angles A'OB', AOB of these similar triangles we take the common part A'OB, we have the remaining angles AOA', BOB' equal, and each of them is clearly equal to the angle through which B the figure must have turned round the point to bring it from the first to the second position. The preceding simple construction therefore enables us not only to demonstrate the general proposition (§ 91), but also to determine from the two positions of one line AB, A'B' of the figure the common centre and the amount of the angle of rotation.

93. The lines equidistant from A and A', and from B. and B, are parallel if AB is parallel to A'B'; and therefore the construction

fails, the point O being infinitely distant, and the theorem becomes nugatory. In this case the motion is in fact a simple translation of the figure in its own plane without rotation-since as AB is parallel and equal

to A'B', we have AA' parallel and equal to BB'; and instead of there being one point of the figure common to both positions, the lines joining the successive positions of every point in the figure are equal and parallel.

94. It is not necessary to suppose the figure to be a mere flat disc or plane-for the preceding statements apply to any one of a set of parallel planes in a rigid body, moving in any way subject to the condition that the points of any one plane in it remain always in a fixed plane in space.

95. There is yet a case in which the construction in § 91 is nugatory-that is when AA' is parallel

to BB, but AB intersects A'B'. In A

this case, however, it is easy to see at once that this point of intersection is the point required, although the former 4 method would not have enabled us to find it,

B

B

B

96. Very many interesting applications of this principle may be made, of which, however, few belong strictly to our subject, and we shall therefore give only an example or two. Thus we know that if a line of given length AB move with its extremities always in two fixed lines OA, OB, any point in it as P describes an ellipse. (This is proved. in § 101 below.) It is required to find the direction of motion of P at any instant, i.e. to draw a tangent to the ellipse. BA will pass to its next position by rotating about the point Q; found by the method of § 91 by drawing perpendiculars to ОA and OṚ at A and B. Hence P for the instant revolves about Q, and thus its direction of motion, or the tangent to the ellipse, is perpendicular to QP. Also AB in its motion always touches a curve (called in geometry its envelop); and the same principle enables us to find the point of the envelop which lies in AB, for the motion of that point must évidently be ultimately (that is for a very smal displacement) along AB, and the only point which so moves is the intersection of AB, .with the perpendicular to it from Q Thus our construction would enable us to trace the envelop by points.

A