381.] Integrating once, ΣmVww = y+Sødt. (2) Again, as the motion considered is relative to the centre of inertia, it must be of the nature of rotation about some axis, in general variable. Let e denote at once the direction of, and the angular velocity about, this axis. Then, evidently, But, by operating directly by 2/Sedt upon the equation (1), we get Σm (Vew)2 = — h2 + 2ƒSeødt... .... (2) and (4) contain the usual four integrals of the first order. (4) 382.] When no forces act on the body, we have = 0, and therefore Σηπ Γεπ Sey=-h2. (5) (6) (7) and, from (5) and (6), One interpretation of (6) is, that the kinetic energy of rotation remains unchanged: another is, that the vector e terminates in an ellipsoid whose centre is the origin, and which therefore assigns the angular velocity when the direction of the axis is given; (7) shews that the extremity of the instantaneous axis is always in a plane fixed in space. Also, by (5), (7) is the equation of the tangent plane to (6) at the extremity of the vector e. Hence the ellipsoid (6) rolls on the plane (7). From (5) and (6), we have at once, as an equation which e must satisfy, y2Σ.m (Vew)2=—h2 (Σ.mwV€œ)2. This belongs to a cone of the second degree fixed in the body. Thus all the ordinary results regarding the motion of a rigid body under the action of no forces, the centre of inertia being fixed, are deduced almost intuitively and the only difficulties to be met with in more complex properties of such motion are those of integration, which are inherent to the subject, and appear whatever analytical method is employed. (Hamilton, Proc. R. I. A. 1848.) 383.] Let a be the initial position of w, q the quaternion by which the body can be at one step transferred from its initial position to its position at time t. Then and Hamilton's equation (5) of last section becomes or Let -1 Σ.mqaq-1V.eqaq-1 = y, Σ.mg {aS.ag-1eq-q ̄1€qa2 } q ̄1 = y. (1) where is a self-conjugate linear and vector function, whose constituent vectors are fixed in the body in its initial position. Then the previous equation may be written η (2) (3) 384.] It is easy to see what the new vectors ŋ and ¿ represent. For we may write (2) in the form η from which it is obvious that ŋ is that vector in the initial position of the body which, at time t, becomes the instantaneous axis in the moving body. When no forces act, y is constant, and is the initial position of the vector which, at time t, is perpendicular to the invariable plane. 385.] The complete solution of the problem is contained in equations (2), (3) above, and (4) of § 356*. Writing them again, we have We have only to eliminate and 7, and we get (4) (2) (3) (5) in which q is now the only unknown; y, if variable, being supposed known in terms of q and t. It is hardly conceivable that any simpler, or more easily interpretable, equation for q can be presented *To these it is unnecessary to add Tq=constant, as this constancy of Tq is proved by the form of (4). For, had Tq been variable, there must have been a quaternion in the place of the vector 7. In fact, until symbols are devised far more comprehensive in their meaning than any we yet have. 386.] Before entering into considerations as to the integration of this equation, we may investigate some other consequences of the group of equations in § 385. Thus, for instance, differentiating (2), we have which gives, in the case when no forces act, the forms 1 (6) (7) To each of these the term q1q, or q1yq, must be added on the right, if forces act. 387.] It is now desirable to examine the formation of the function 4. By its definition (1) we have Hence pp.m (aSap-a2p), =-Σ.ma Vap. -Spop = .m (TVap)2, so that -Spop is the moment of inertia of the body about the vector p, multiplied by the square of the tensor of p. Thus the equation Spopp-h2, evidently belongs to an ellipsoid, of which the radii-vectores are inversely as the square roots of the moments of inertia about them; so that, if i, j, k be taken as unit-vectors in the directions of its axes respectively, we have A, B, C, being the principal moments of inertia. φρ op={Ai Sip+BjSjp+ Ck Skp}. Thus the equation (7) for ʼn breaks up, if we put η n= iw1 + jwą + kwz, into the three following scalar equations Aw1+ (CB) w2wz = 0, which are the same as those of Euler. Only, it is to be understood that the equations just written are not primarily to be considered as equations of rotation. They rather express, with reference to fixed axes in the initial position of the body, the motion of the extremity, w1, W2, ws, of the vector corresponding to the instantaneous axis in the moving body. If, however, we consider w1, wg, wg as standing for their values in terms of w, x, y, z (§ 391 below), or any other coordinates employed to refer the body to fixed axes, they are the equations of motion. Similar remarks apply to the equation which determines (, for if we put (= iw1+ jw2+k@3, (6) may be reduced to three scalar equations of the form 388.] Euler's equations in their usual form are easily deduced from what precedes. For, let whatever be p; that is, let & represent with reference to the moving principal axes what represents with reference to the principal axes in the initial position of the body, and we have But perhaps the simplest mode of obtaining this equation is to start with Hamilton's unintegrated equation, which for the case of no forces is simply But from we deduce so that Σ.m Vaä = 0. a = Vew =Vew+Vεw = we2 - eSew + Vew, Σ.m (Vew Sew-éw2 + w Sew) = 0. If we look at equation (1), and remember that simply in having written differs from $ substituted for a, we see that this may be Vede+dé = 0, the equation before obtained. The first mode of arriving at it has been given because it leads to an interesting set of transformations, for which reason we append other two. But, by the beginning of this section, and by (5) of § 382, this is again the equation lately proved. Perhaps, however, the following is neater. It occurs in Hamilton's Elements. 389.] However they are obtained, such equations as those of § 387 were shewn long ago by Euler to be integrable as follows. with other two equations of the same form. Hence so that t is known in terms of s by an elliptic integral. Thus, finally, or may be expressed in terms of t; and in some of the succeeding investigations for q we shall suppose this to have been done. It is with this integration, or an equivalent one, that most writers on the farther development of the subject have commenced their investigations. 390.] By 381, y is evidently the vector moment of momentum of the rigid body; and the kinetic energy is so that we have, for the equations of the cones described in the |