MECHANICS 737 Thus we have Now if the fixed axes be taken so as to coincide at a particular instant with the principal axes of the body passing through the point which is regarded as fixed, the terms involving factors of the form ~S.(mxy) , &c., necessarily vanish ( 237). Also we have
- igi l
iody ~Sm(x- - y = 2ia so that, if the principal moments of inertia of the body be A, B, C respectively, the dynamical equations become Cw i + (B-A) I , = N, &c. But it was proved in 79 that when the angular velocities of a rigid body are referred to moving and to fixed axes, which coincide at a particular instant, not only are the angular velocities but also the angular accelerations equal at that instant in the two systems. Thus, if co 1; to.,, o> 3 be, the angular velocities of the body about its principal axes," the equations just obtained take the form (duo to Euler) Aw 1 + (C-B)w a a.3 = L, B 2 + (A ->,,!= M, When coj, to.,, co.j are found, in any particular case, from these equations, tin actual orientation of the body at any time can be calculated from them by the kinematical processes of 81. The position in space of the point of the body which has been hitherto treated as fixed is to be calculated separately by the processes already explained for kinetics of a particle, and thus the motion of the body is completely determined in the sense that the difficulties of the further steps are of a purely mathematical nature. "When the forces applied to the body have a single resultant, which either vanishes or passes through the origin, the right-hand .mler no terms disappear from Euler s equations. Multiply the equations >rce.s. by ca lt w.,, w^ respectively, and add. "NVe thus obtain A W i O) 1 + BtO-jCi^ + Ceo j&>3 = , whence (1), the statement that the kinetic energy is constant. Again, multiply by A 1? B^.,, Co> 3 respectively, and add. Then we have i 2 + C -C0 3 i> 3 = , A 2 w J + B 2 w| + C 2 aJ D a whence expressing the constancy of amount of moment of momentum. But if, in these equations, we now choose to regard wj, w. 2) u- A as the coordinates, parallel to the principal axes, of the extremity of a line which represents, in magnitude and direction, the instantaneous axis, we see that that axis is a central vector of each of the ellips oids (1) and (2). Hence the instantaneous axis describes, re latively to the body, a cone of the second order. Since T and D are the only arbitrary coefficients in the equations, all the ellipsoids (1) or (2) are similar and similarly situated. The curve of intersec tion of (1) and (2) projected on the plane of the axes A and B has the equation A(C - A)co= + B(C - B) j = 2TC - D 2 . This represents an ellipse if the terms on the left have the same sign, i.e., if C is either greater or less than each of A and B. Hence, if the body be originally rotating about an axis nearly coin ciding either with the axis of greatest or that of least moment of inertia, it will continue to do so. These two cases are exemplified respectively by a quoit and by an elongated rifle bullet, at least in so far as the resistance of the air does not interfere with their motion. But if the body be originally rotating about an axis nearly coinciding with the axis of intermediate moment of inertia, the curve indicated by the equation above is an hyperbola or a pair of straight lines through the origin ; and the instantaneous axis travels, in general, far away from its first position in the body. We will henceforth look on A, B, C as in descending order of magnitude. It is obvious from the mode in which they are formed that A=B + C only when the body is a plate. Hence, generally, any two of A, B, C are together greater than the third. Also by multiplying (1) by A, and comparing it term by term with (2), we see that 2AT >"D 2 ; similarly 2CT< D 2 . To complete the examination of the immediate results of Euler i equations in this case, let us find how the length of the instan taneous axis, considered as a common vector-radius of the ellipsoid; (1) and (2), depends on the time. Let n 2 = co^ + co^ + w=. .... ........ (3) C-B A-C B-A r I T> P - r , A 1> C vhere A is the determinant A- B 2 C 2 -(C-B)(B-A)(A-C). Tow (1), (2), (3) are linear equations in w*, *, wi;, and from them we have three equations like Aw; = BC(C - B)ft 2 + 2T(B 2 - C 2 ) + D-(C - B) = BC(C-B){Q 2 -}, where a .s, by the remark above, essentially positive. Hence AXX = A 2 B 2 C 2 (C - B)(B - A)( A - C)(0 2 - a)(0 2 - 6)(Q 2 - <) = A 2 B J C 2 A(a-i2 2 )(6 -ft 2 )(c-0 2 ) . Thus j - V(a - 2 )(6 - Q 2 )(c - 2 ) ; whence fl 2 is at once found by elliptic functions, fi known, we have !, &> 2 , o> 3 , and then by the method of 81 we have the complete analytical determination of the position of the body in terms of the time. 255. Such a solution, however, fails to give so clear a Poinsot s conception of the nature of the motion as is afforded by rolling the very elegant geometrical representation discovered by ellipsoid. Poinsot. We may arrive at it by considering the tangent plane to (1) at the extremity of the radius O. If x, y, z be the current coordinates of that plane, its equation is AW^W! -x) + Bo) 2 (co 2 -y) + Co) 3 (co 3 - ~) = , so that the perpendicular from the origin upon it is equal, by (2), to 2T/D, a constant. The direction cosines of this perpendicular are proportional to AW I} Bw 2 , Cw 3 , the com ponent moments of momentum. Hence it is the axis of resultant moment of momentum, and is therefore ( 166) fixed in direction in space. The tangent plane to (1) at the extremity of the instantaneous axis is therefore a fixed plane, and the ellipsoid (1) rolls upon it as if it were per fectly rough. From this we can of course find the equation of the curve of contact with the plane, and thence that of the cone, fixed in space, on which the cone of instantaneous axes in the body rolls, as in 75. The latter cone is of course given by the intersection of (1) and (2). To complete this beautiful representation of the motion, all that Sylves- is necessary is a method of measuring time, something to show the ter s rate of rolling of the ellipsoid on the fixed plane. Many construe- addition tions have been given for this purpose, such as Poinsot s " rolling to and sliding cone," &c., but none can compare in elegance with that Poinsot s invented by Sylvester. We can only sketch a particular case construc- sufficient, however, to completely solve the question. tion. Writing I, m, n for the direction cosines of the fixed line referred to the principal axes, we have Z = Ao> L /D, m = Bw.j/D, = Cw 3 /D .... (4). Our equations (1) and (2) may now be written 1-/A + m 2 /B + ?i 2 /C = 2T/D 2 (1). Z- + m 2 + ?i 2 = l (2). Introducing a factor p, of dimensions the same as I/A, we have from these a third equation Z-(l/A+^) + m 2 (l/B+^) + ?t 2 (l/C+p) = 2T/D 2 + ^ . (5). Now consider an ellipsoid of l/A+> TjB+p 1/C+p (6), which is similar, and similarly situated, to one of the ellipsoids confocal with (1) of 254. Draw to it a tangent plane perpen dicular to the line OL, whose direction cosines are (4). We obtain for the determination of the point of contact Q three equations of the form 7- x - y n tf ~S(1/A+|0 m 8(1/8+10 S(l/C+i;) where S 2 = r , , f -. + ,3 . ... + Here x, y, & are the coordinates of Q, and the distance of the tangent plane from the origin is E/S. Now (6) gives at once by means of (7) 93
XV.