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 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 PQR 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 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 T, is similar and equal to BS, and is therefore a cycloid similar and 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 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. R H 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 Hypocycloid, 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 (§§ 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 another of equal size. The curve in this case is rolling externally on called the Cardioid. P P 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 ○ thus 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. 2 X 6 107. Let OA, OB be two axes about which a body revolves with angular velocities w, w, respectively. A a With radius unity describe the arc AB, and in it take any point I. Draw Ia, Iẞ perpendicular to OA, OB respectively. Let the rotations 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 w,Iẞ.T and wIa.T. The point I, and therefore every point in the line OI, will be at rest during the interval 7 if the sum of these displacements is zero—i. e. if B Ρ Hence the line OI is instantaneously at rest, or the two rotations about OA and OB may be compounded into one about OI. Draw Ip, Iq, parallel to OB, OA respectively. Then, expressing in two ways the area of the parallelogram IpOq, we have In words, if on the axes OA, OB, we measure off from O lines Op, Oq, 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 be the angular velocity about OI, the whole displacement of B may evidently be represented either by Hence w.Bb or 1.1.B :w:: Bb: IB 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 tending to produce rotation about a perpendicular axis, the result will be a change of direction of the axis about which the body revolves, but no change in the angular velocity. 109. If a pyramid or cone of any form roll on a similar pyramid (the image in a plane mirror of the first position of the first) all round, it clearly comes back to its primitive position. This (as all rolling of cones) is exhibited best by taking the intersection of each with a spherical surface. Thus we see that if a spherical polygon turns about its angular points in succession, always keeping on the spherical surface, and if the angle through which it turns about each point is twice the supplement of the angle of the polygon, or, which will come to the same thing, if it be in the other direction, but equal to twice the angle itself of the polygon, it will be brought to its original position. 110. The method of § 100 also applies to the case of § 106; and it is thus easy to show that the most general motion of a spherical figure on a fixed spherical surface is obtained by the rolling of a curve fixed in the figure on a curve fixed on the sphere. Hence as at each instant the line joining C and O contains a set of points of the body which are momentarily at rest, the most general motion of a rigid body of which one point is fixed consists in the rolling of a cone fixed in the body upon a cone fixed in space-the vertices of both being at the fixed point. 111. To complete our kinematical investigation of the motion of a body of which one point is fixed, we require a solution of the following problem :-From the given angular velocities of the body about three rectangular axes attached to it to determine the position of the body in space after a given time. But the general solution of this problem demands higher analysis than can be admitted into the present treatise. 112. We shall next consider the most general possible motion of a rigid body of which no point is fixed-and first we must prove the following theorem. There is one set of parallel planes in a rigid body which are parallel to each other in any two positions of the body. The parallel lines of the body perpendicular to these planes are of course parallel to each other in the two positions. Let C and C' be any point of the body in its first and second positions. Move the body without rotation from its second position to a third in which the point at C' in the second position shall occupy its original position C. The preceding demonstration shows that there is a line CO common to the body in its first and third positions. Hence a line C'O of the body in its second position is parallel to the same line CO in the first position. This of course clearly applies to every line of the body parallel to CO, and the planes perpendicular to these lines also remain parallel. 113. Let S denote a plane of the body, the two positions of which are parallel. Move the body from its first position, without rotation, in a direction perpendicular to S, till S comes into the plane of its second position. Then to get the body into its actual position, such a motion as is treated in § 91 is farther required. But by § 91 this may be effected by rotation about a certain axis perpendicular to the plane S, unless the motion required belongs to the exceptional case of pure translation. Hence (this case excepted), the body may be brought from the first position to the second by translation through a determinate distance perpendicular to a given plane, and rotation through a determinate angle about a determinate axis perpendicular to that plane. This is precisely the motion of a screw in its nut. A G D E F C 114. To understand the nature of this motion we may commence with the sliding of one straight-edged board on another. Thus let GDEF be a plane board whose edge, DE, slides on the edge, AB, of another board, ABC, of which for convenience we suppose the edge, AC, to be horizontal. By $30, if the upper board move horizontally to the right, the constraint will give it, in addition, a vertically upward motion, and the rates of these motions are in the constant ratio of AC to CB. Now, if both planes be bent so as to form portions. of the surface of a vertical right cylinder, the motion of DF parallel to AC will become a rotation about the axis of the cylinder, and the necessary accompaniment of vertical motion will remain unchanged. As it is evident that all portions of AB will be equally inclined to the axis of the cylinder, it is obvious that the thread of the screw, which corresponds to the edge, DE, of the upper board, must be traced on the cylinder so as always to make a constant angle with its generating lines (§ 128). A hollow mould taken from the screw itself forms what is called the nut-the representative of the board, ABC-and it is obvious that the screw cannot move without rotating about its axis, if the nut be fixed. If a be the radius of the cylinder, & the angular velocity, a the inclination of the screw thread to a generating line, u the linear velocity of the axis of the screw, we see at once from the above construction that aw: u:: AC: CB:: sin a: cos a, which gives the requisite relation between ∞ and u. 115. In the excepted case of § 113, the whole motion consists of two translations, which can of course be compounded into a single one: and thus, in this case, there is no rotation at all, or every plane of it fulfils the specified condition for S of § 113. 116. We may now briefly consider the case in which the guiding cones (§ 110) are both circular, as it has important applications to the motion of the earth, the evolutions of long or flattened projectiles, the spinning of tops and gyroscopes, etc. The motion in this case may be called Precessional Rotation. The plane through the |