C' exception, viz. when the rotations about A and B are of equal amount and in opposite directions. In B this case A'B' is evidently parallel to AB, and therefore the compound result is a ation only. Th is, a body revolve in succession through equal angles, but in opposite directions, about two A B 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 A OA be the translation common to B с all points in the plane, and let BOC be the angle of rotation about 0, BO being drawn so that 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 seen, to an elementary rotation about some point in the plane. Let 0,, 02, 03, 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 02 0, (or 0, which coincides with it) till 0, coincides with then about the latter till 0, coincides with 03, and so on. Hence, if we join 01, 02, 03, etc., in the plane of the figure, and in the fixed plane, the motion will be the same as if the polygon 0,0,03, etc., rolled upon the fixed polygon 0,0,0g, etc. By supposing the successive displacements small enough, the sides 0$ 04 03 O 01 03 01 029 01,029 etc., 3) 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 ( at the circumference. Also O Q is a diameter of this circle, and is therefore constant. Hence, as ( 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, AOC, move along straight lines OC, angles. Hence the path of P may D be considered as that of a point in a line whose ends move on two F 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. A. 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. 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 B E P next section contains a simple investigation of those properties of the cycloid which are most useful in our subject. S 104. Let AB 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 POR be any subsequent position of the generating circle, Q and S the new positions of A and B, QPS is of course a right angle. If, therefore, QR be drawn parallel to B P C 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 Q the curve AT, which is the locus of T, is similar and equal to BS, A and is therefore a cycloid similar and equal to AC. But QR is perpendicular to Pl, and is therefore the instantaneous direction of motion of l, or is the tangent to the cycloid AQC. Similarly, PS is perpendicular to the cycloid BS at S, and therefore TQ is perpendicular to AT at T. Hence ($ 22) AQC is the evolute of AT, and arc AQ=QT=2QR. D Р 105. When a circle rolls upon another circle, the curve described by a point in its circumference is called an Epicycloid, or a Hypo P cycloid, as the rolling circle is without or within the fixed circle; and when the tracingpoint is not in the circumference, we have Epitrochoids and Hypotrochoids. Of the latter classes we have already met with examples (5$ 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. Р In the second figure a circle rolls externally 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. 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 1:2, $ 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 straight lines; and the same reasoning will apply to prove that the point O thus obtained is common to the body in its two positions. Hence every point of the body in the line OC, joining 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. In an a 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, IB perpendicular to OA, OB respectively. Let the rotations about the two axes be such that that about A OB tends to raise I above the plane of the paper, and that about OA to depress it. infinitely short interval of time t, the amounts of these displacements will be w, IB.t and - wla.t. The point I, and therefore every point in the р line 01, will be at rest during the interval , if the sum of these displacements is zero-i. e. if w..IB=w.la. 0 Hence the line OI is instantaneously at rest, or the two rotations about OA and OB may be compounded into one about 01. Draw Ip, Iq, parallel to OB, OA respectively. Then, expressing in two ways the area of the parallelogram 1p0q, we have Og.Iß= Op.la. Hence Oq:0p::w,:w. In words, if on the axes OA, OB, we measure off from O lines Op, 0q, proportional respectively to the angular velocities about these axes—the diagonal of the parallelogram of which these are contiguous sides is the resultant axis. Again, if Bb be drawn perpendicular to OA, and if s2 be the angular velocity about 01, the whole displacement of B may evidently be represented either by w.Bb or 12.IB. Hence 12:w::B6:18 ::01: Op. And thus on the scale on which Op, Oq represent the component angular velocities, the diagonal OI represents their resultant. 108. Hence rotations are to be compounded according to the same law as velocities, and therefore the single angular velocity, equivalent to three co-existent angular velocities about three mutually perpendicular axes, is determined in magnitude, and the direction of its axis is found, as follows:— The square of the resultant angular velocity is the sum of the squares of its components, and the ratios of the three components to the resultant are the direction-cosines of the axis. Hence also, an angular velocity about any line may be resolved into three about any set of rectangular lines, the resolution in each case being like that of simple velocities) effected by multiplying by the cosine of the angle between the directions. Hence, just as in § 38 a uniform acceleration, acting perpendicularly to the direction of motion of a point, produces a change in the direction of motion, but does not influence the velocity; so, if a body be rotating about an axis, and be subjected to an action |