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. Deter. mine 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 W. 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 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 exception, viz. when the rotations about A and B are of equal Bamount and in opposite directions. In this case A'B' is evidently parallel and equal to AB, and therefore the com pound result is a translation only. Thus we see that if a body revolve in succes» sion through equal angles, but in oppo site directions, about two parallel axes, it finally takes a position to which it could have been brought by a simple translation per pendicular 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.

A

B

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

B

OA bisects the exterior angle COB. Evidently there is a point B' in BQ 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 seen, to an elementary rotation about some point in the plane. Let O,, O2, O2, etc., be the successive points of the figure about which the rotations take place,·01, Og »› etc., the positions of these points on the plane when each is the instantaneous centre of rotation. Then the figure rotates about O, (or o, which coincides with it) till O, coincides with o,, then about the latter till O, coincides with o,, and so on. Hence, if

8

03

as

we join 0,, 0,, O,, etc., in the plane of the figure, and o,,,,,, etc., in the fixed plane, the motion will be the same as if the polygon' 0,0,0,, etc., rolled upon the fixed polygon 0,0,0,, etc. By supposing the successive displacements small enough, the sides 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 Qis 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,

B

D

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 O. 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 next section contains a simple investigation of those properties of the cycloid which are most useful in our subject.

P

P

B

P

C

104. Let ÅB be a diameter of the generating (or rolling) circle, BC the line on which it rolls. The points A and B describe similar and equal cycloids, of which AQC and BS are portions. If PQR be any subsequent position of the generating circle, and S the new positions of A and B, QPS is of course a right angle. If, therefore, QR be drawn parallel tó PS, PR is a diameter of the rolling circle, and R lies in a straight line AH drawn parallel to BC. Thus AR = BP. Produce QR to T, making RT QR PS. Evidently the curve AT, which is the locus of 7, is similar and equal to BS, and is therefore a cycloid similar and 4 equal to AC.. But QR is perpendicular to PQ, and is therefore the instantaneous direction' of motion of Q, or is the tangent to the cycloid AQC. Similarly, PS is perpendicular to the therefore TQ is perpendicular to AT at T. the evolute of AT, and arc AQ=QT=2QR.

=

105. When a circle rolls upon another circle, the curve described by a point in its circumference is called an Epicycloid, or a Hypocycloid, as the rolling circle is without, or within the fixed circle; and when the tracing-point is not in the circumference, we have Epitrochoids and Hypotrochoids. Of the latter classes we have already met with examples (SS 87, 101), and others will be presently mentioned. Of the former we have, in the first of the appended figures, the case of a circle rolling externally on another of equal size. The curve in this case is called the Cardioid.

In the second figure a circle rolls ex

R

cycloid BS at S, and Hence (§ 22) AQC is

P

P

ternally on another of twice its radius. The epicycloid so described is of importance in optics, and will, with others, be referred to when we consider the subject of Caustics by reflexion.

P

In the third figure we have a hypocycloid traced by the rolling of one circle internally on another of four times its radius.

The curve of § 87 is a hypotrochoid described by a point in the plane of a circle which rolls internally on another of rather more than twice its diameter, the tracing-point passing through the centre of the fixed circle. Had the diameters of the circles been exactly as I : 2, § 101 shows us that this curve

would have been reduced to a single straight line.

106. If a rigid body move in any way whatever, subject only to the condition that one of its points remains fixed, there is always (without exception) one line of it through this point common to the body in any two positions.

Consider a spherical surface within the body, with its centre at the fixed point C. All points of this sphere attached to the body will move on a sphere fixed in space. Hence the construction of §.91 may be made, only with great circles instead of straightnes: and the same reasoning will apply to prove that the point Gius obtained is common to the body in its two positions. Hence every point of the body in the line OC, joining O with the fixed point, must be common to it in the two positions. Hence the body may pass from any one position to any other by a definite amount of rotation about a definite axis. And hence, also, successive or simultaneous rotations about any number of axes through the fixed point may be compounded into one such rotation.

107. Let OA, OB be two axes about which a body revolves with angular velocities w, w, respectively.

With radius unity describe the arc AB, and in it take any point I. Draw Ia, Iẞ perpendicular to OA, OB respectively. Let the rota

a

T

tions about the two axes be such that that about OB tends to raise I above the plane of the paper, and that about OA to depress it. In an infinitely short interval of time 7, the amounts of these displacements will be wẞ. r and — wła. T. The point I, and therefore every point in the line OI, will be at rest during the interval r if the sum of these displacements is zero-i.e. if w1. Iẞ=w Fa

Hence the line Ol is instantaneously at rest, or the two rotations about OA and OB may be compounded into one about OI. Draw