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 term 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 co-efficients.' 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 resolution 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

B'

is therefore one and the same point of the plane figure in its two positions. If, for the sake of illustration, we actually trace the angle OAB upon the plane, it becomes OA'B' in the second position of the figure.

92. If from the equal angles A'OB', AOB of these similar triangles we take the com- 0, mon part A'OB, we have the remaining angles AOA', BOB' equal, and each of them is clearly equal to the angle through which the figure must have turned round the point O to bring it from the first to the second position.

B

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

B

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.

A'

В

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

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

B

by the method of § 91 by drawing perpendiculars to OA and OB 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 evidently be ultimately (that is for a very small 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.

P

A

0

97. Again, suppose ABDC to be a jointed frame, AB having a reciprocating motion about A, and by a link BD turning CD in the same plane about C. Determine the relation between the angular velocities of AB and CD in any position. Evidently the instantaneous direction of motion of B is transverse to AB, and of D transverse to CDhence if AB, CD produced meet

in O, the motion of BD is for an instant as if it turned about O. From this it may easily be seen that if the angular velocity of AB AB OD be w, that of CD is ω. A similar process is of course OB CD

applicable to any combination of machinery, and we shall find it very convenient when we come to apply the principle of work in various problems of Mechanics.

Thus in any Lever, turning in the plane of its arms-the rate of motion of any point is proportional to its distance from the fulcrum, and its direction of motion at any instant perpendicular to the line joining it with the fulcrum. This is of course true of the particular form of lever called the Wheel and Axle.

98. Since, in general, any movement of a plane figure in its plane may be considered as a rotation about one point, it is evident that two such rotations may, in general, be compounded into one; and therefore, of course, the same may be done with any number of rotations. Thus let A and B be the points of the figure about which in succession the rotations are to take place. By_rotation about A, B is brought say to B, and by a rotation about B, A is brought to A'. The construction of § 91 gives us at once the point 0 and the amount of rotation about it which singly gives the same effect as those about A and B in succession. But there is one case of

B'

exception, viz. when the rotations about A and B are of equal amount and in opposite directions. In this case A'B' is evidently parallel to AB, and therefore the compound result is a translation only. That is, if a body revolve in succession through equal angles, but in opposite directions, about two parallel axes, it finally takes a position to which it could have been brought by a simple translation perpendicular to the lines of the body in its initial or final position, which were successively made axes of rotation; and inclined to their plane at an angle equal to half the supplement of the common angle of rotation.

99. Hence to compound into an equivalent rotation a rotation and a translation, the latter being effected parallel to the plane of the former, we may decompose the translation into two rotations of equal amounts and opposite directions, compound one of them with the given rotation by § 98, and then compound the other with the resultant rotation by the same process. Or we may adopt the following far simpler method:-Let

OA be the translation common to B' all points in the plane, and let BOC be the angle of rotation about O, BO being drawn so that

C'

A

C

B

OA bisects the exterior angle COB'. Evidently there is a point B' in BO produced, such that B'C', the space through which the rotation carries it, is equal and opposite to OA. This point retains its former position after the performance of the compound operation; so that a rotation and a translation in one plane can be compounded into an equal rotation about a different axis.

100. Any motion whatever of a plane figure in its own plane might be produced by the rolling of a curve fixed to the figure upon a curve fixed in the plane.

For we may consider the whole motion as made up of successive elementary displacements, each of which corresponds, as we have

02

Ost

요

03

04

seen, to an elementary rotation about some point in the plane. Let 0,, 0, 0, etc., be the successive points of the figure about which the rotations take place, 01, 02, 03, etc., the positions of these points on the plane when each is the instantaneous centre of rotation. Then the figure rotates about O1 (or o1, which coincides with it) till O2 coincides with o2, then about the latter till O coincides with 03, and so on. Hence, if we join 01, 02, 03, etc., in the plane of the figure, and 01, 02, 0, etc., in the fixed plane, the motion will be the same as if the polygon 01020, etc., rolled upon the fixed polygon 0,0,0,, etc. By supposing the successive displacements small enough, the sides

2

01

02

of these polygons gradually diminish, and the polygons finally become continuous curves. Hence the theorem.

From this it immediately follows, that any displacement of a rigid solid, which is in directions wholly perpendicular to a fixed line, may be produced by the rolling of a cylinder fixed in the solid on another cylinder fixed in space, the axes of the cylinders being parallel to the fixed line.

101. As an interesting example of this theorem, let us recur to the case of § 96:—A circle may evidently be circumscribed about OBQA; and it must be of invariable magnitude, since in it a chord of given length AB subtends a given angle O at the circumference. Also OQ is a diameter of this circle, and is therefore constant. Hence, as Q is momentarily at rest, the motion of the circle circumscribing OBQA is one of internal rolling on a circle of double its diameter. Hence if a circle roll internally on another of twice its diameter any point in its circumference describes a diameter of the fixed circle, any other point in its plane an ellipse. This is precisely the same proposition as that of § 86, although the ways of arriving at it are very different.

102. We may easily employ this result, to give the proof, promised in § 96, that the point P of AB describes an ellipse. Thus let OA, OB be the fixed lines, in which the extremities of AB move. Draw the circle AOBD, circumscribing AOB, and let CD be the diameter of this circle which passes through P. While the two points A and B of this circle move along OA and OB, the points C and D must, because of the invariability of the angles BOD,

D

B

E

F

AOC, move along straight lines OC, OD, and these are evidently at right angles. Hence the path of P may be considered as that of a point in a line whose ends move on two mutually perpendicular lines. Let E be the centre of the circle; join OE, and produce it to meet, in F, the line FPG drawn through P parallel to DO. Then evidently EF=EP, hence F describes a circle about 0. Also FP: FG:: 2FE: FO, or PG is a constant submultiple of FG; and therefore the locus of P is an ellipse whose major axis is a diameter of the circular path of F. Its semi-axes are DP along OC, and PC along OD. 103. When a circle rolls upon a straight line, a point in its circumference describes a Cycloid, an internal point describes a Prolate Cycloid, an external point a Curtate Cycloid. The two latter varieties are sometimes called Trochoids.

G

The general form of these curves will be seen in the succeeding figures; and in what follows we shall confine our remarks to the cycloid itself, as it is of greater consequence than the others. The