MECHANICS passing through oc,y,z and (1). It is therefore perpendicular to (1). Also the whole displacement is /-= - ; - :, fit Vi + ; + ,- Hence the The last factor is the distance of x, y, z from (1). second is the angular velocity about (1). It appears at once from this result, and from the form of (1), that 12 Q n are the direction cosines of the instantaneous axis. If the figure be rotating simultaneously about a number of axes, say with angular velocity w lt about an axis whose direction cosines are Z a , m lt n lt &c. , we have evidently Rigid figure anyhow dis placed. Angular accelera tion about a moving axis. From these the single instantaneous axis is found immediately as above. 78. Any displacement whatever of a rigid figure may be effected by means of a screw-motion, i.e., translation parallel to some definite line, accompanied by a proportion ate rotation about that line. Let A and A be successive positions of any point in the figure, and suppose the body to be brought back by a mere translation so that A coin cides again with A. Then we have seen ( 75) that one line of the figure through A is necessarily restored to its original position. Let P be any plane section of the figure, perpendicular to this line, P its position after displacement. These fully determine the initial and final positions of the whole figure. Shift P into the plane of P by a translation perpendicular to either, and let P" be its position. P" can ( 71) be brought to coincide with P by a rotation in its own plane. Hence the proposition. There is an exceptional case when P" requires only translation to make it coincide with P . But then the whole figure is merely translated. 79. We have seen that the straight line representing an angular velocity is to be resolved by the same process as that representing a linear velocity, If we consider a figure to be rotating about axes fixed relatively to it, accelerations of angular velocity about these will be represented by changes in the lengths of the lines representing the angular velocities, and will therefore be subject to the same con ditions as the angular velocities themselves. Thus, as it is obvious that a figure is rotating at any instant with the same angular velocity about an axis fixed relatively to itself, and about another axis fixed in space, which at the given instant coincides with the former, it follows that the angular accelerations about these axes are equal at that instant. This is really the same proposition as that r is the velocity along a fixed line coinciding with the radius-vector r ( 47). But, just as r is not the complete acceleration parallel to r, if r be rotating, so the proposition above, though true for the first fluxion of the angular velocity about a moving line, is not generally true for fluxions of higher orders. As this subject is commonly regarded as somewhat obscure, we may give a more formal examination of it by an analytical process. Suppose a>], <a v co 3 to be the angular velocities about rectangular axes OA, OB, 00 fixed relatively to a figure, and o> the angular velocity of the figure relatively to a line OS fixed in space. Let I, m, n, be the direction cosines of the latter line with regard to the former three, then W = Ztoj + mu. 2 + n o> 3 , and co = ?w x + mw. 2 + n& 3 + la>i + mw. 2 + nu 3 . Hut ll + m m + nil = ; and if, at_a particular instant, we have Z=l, m=0, ?i = 0, this gives also l = Q , so that we have <a = toj + VI ca.} + 71 o).j . Now TO = cos BOS = cos 6 suppose. Hence m= sin 6.6. But, at the instant in question, 0--^^ir and 6 = u A , so that in = - cog . In the same way we see that and thus we have o> = Q>J , which is the proposition above given. 80. To complete the kinematics of a rigid figure of which one point is fixed, we require to have the means of calculat ing its position, after the lapse of any period during which it has been rotating with given angular velocities about given axes. If the axes about which the angular velocities are given be fixed in space, the formulae of 77 give at once, for a unit line fixed in the figure, the expressions t= n<oa vnwz t Position of rigid figure in terms o: rotation about axes fixed in space ; ure. Here I, m, n are the direction cosines of the unit line at time t ; and they satisfy, of course, the condition ll + mm + nn . But, except in some special cases, these equations are in tractable. This, however, is of little consequence, because in the applications to kinetics of a free rigid body the physical equations usually give the angular velocities about lines fixed in the body. Our problem, then, takes the form 81. Given the angular velocities of a figure about each of fixed in a system of three rectangular axes which are rigidly attached tlr to it, find at any time its position in space. It is clear that, if we know the positions of the revolving axes, referred to a fixed system, with which they at one instant coincided, the corresponding position of the figure is determined. The method usually employed is as follows. About the common origin of the two sets of axes suppose a sphere of unit radius to be described. Let X, Y, Z (fig. 29) be the traces on this sphere of the fixed axes, and A, B, C those of the revolving axes. Draw a great circle ZC so as to meet in A the quad rant BA produced. Then it is clear that the figure can be constructed (i.e., that the data x are sufficient for calculation) if we know (a) the angle XZC, this we call J/; (b) the arc ZC, called ; (c} the angle A CA, or the arc A A, called <. For X, Y, Z are given. Then (a) shows how to draw the great circle ZC, whose pole is N on the great circle XY. Hence (b) gives us the points C and A . We can next draw the great circle A N, and A and B are found on it by (c), for A A = NB = <f>. We have now only to determine these angular coordinates in terms of the angular velocities of the figure about OA, OB, OC, which we denote by w a , w 2 , w., respectively. The velocity of C along ZC is 0. But it is produced by the rotations about A and B. Thus we have 6 = o) 2 cos </> + ! sin . The velocity of C perpendicular to ZC is sin 6. j/. This also is part of the result of the rotations about A and B, so that sin 6.if/ = u. 2 sin <f> - f>>i cos <p . The velocity of A along AB is that of A together with the rate of increase of A A. Also it is entirely due to rotation about OC. Hence COS0. These three equations determine 6, j/, are given as functions of t.
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