CONTENTS. is integrable, or represents a system of surfaces, without the expression dv=pdo, d. nv = ddp, (e). In this manner it is found (p. 702), that the Condition* of In- Syv=0; (Y2") in which y is a vector function of p, not generally linear, and deduced being the conjugate of p, but not here equal to it. (Y2"') (d). Connexions (pp. 702, 703) of the Mixed Transformations in (). The equation (p. 704), T-V.ẞVya)=T(a - V. & Vẞp), (Z2) in which a, B, y are any three vector constants, represents a central work. The vector of the centre of the quadric, represented by the equation fp - 28ep = const., with fp = Sppp, is generally x=-1ε = (p. 704); case of paraboloids, and of cylinders. (9). The equation (p. 705), Sqpq'pq"p+ Spop + Syp+ C = 0, (Z2') represents the general surface of the third degree, or briefly the General (4). The General Cubic Cone, with its vertex at the origin, is thus xliii *It is shown, in a Note to p. 702, that this monomial equation (Y”2) be- In a Note to p. 649 (already mentioned in p. xxviii), the reader will find Pages. each a is a vector of application; ẞ the corresponding vector of applied force; y an arbitrary vector: and this one quaternion formula (A3) is equivalent to the system of the six usual scalar equations the applied forces have an unique resultant = 23, which acts along the line whereof (A3) is then the equation, with y for its variable (c). When the condition (C3) is satisfied, the forces compound themselves generally into one couple, of which the axis = £Vaß, what- (d). When Vaẞ=0, (D3), with or without (C3), the forces have no tendency to turn the body round that point o; and (e). In the general case, when neither (C3) nor (D3) is satisfied, if then Vq is the vector perpendicular from the origin, on the central axis of the system; and if e = Sq, then cẞ represents, both in quan- with T2ẞ>0, then SQ = c = central moment divided by total force; *It is easy to prove that the moment of the force ß, acting at the end of the CONTENTS. and VQ is the vector y of a point c upon the central axis which does not vary with the origin o, and which there are reasons for considering as the Central Point of the system, or as the general centre of applied forces in fact, for the case of parallelism, this point c coincides with what is usually called the centre of parallel forces. (9). Conceptions of the Total Moment Laß, regarded as being generally a quaternion; and of the Total Tension, – Σaß, considered as a scalar to which that quaternion with its sign changed reduces itself for the case of equilibrium (a), and of which the value is in that case independent of the origin of vectors. (h). Principle of Virtual Velocities, xlv Pages. (G3) 709-713 (H3) the vector representing the accelerating force, or me the moving force, acting on a particle m of which the vector at the time t is a; and da being any infinitesimal variation of this last vector, geometrically compatible with the connexions between the parts of the system, which need not here be a rigid one. (b). For the case of a free system, we may change each da to ε + Via, ε and being any two infinitesimal vectors, which do not change in passing from one particle m to another; and thus the general equation (H3) furnishes two general vector equations, namely, Em (Da-)=0, (13), and ΣmVa (Di3α-)=0; (J3) which contain respectively the law of the motion of the centre of grarity, and the law of description of areas. (e). If a body be supposed to be rigid, and to have a fixed point o, then only the equation (J3) need be retained; and we may write, Dia = Via, (K3) being here a finite vector, namely the Vector Axis of Instantaneous Rotation: its versor U denoting the direction of that axis, and its tensor Ti representing the angular velocity of the body about it, at the time t. (d). When the forces vanish, or balance each other, or compound themselves into a single force acting at the fixed point, as for the case of a heavy body turning freely about its centre of gravity, then 2mVag=0, (L3); and if we write, p=2maVai, (M3) so that again denotes a linear, vector, and self-conjugate function, we shall have the equations, the vector y being what we may call the Constant of Areas, and the scalar 2 being the Constant of Living Force. (e). One of Poinsot's representations of the motion of a body, under the circumstances last supposed, is thus reproduced under the form, that the Ellipsoid of Living Force (P3), with its centre at the fixed point o, rolls without gliding on the fixed plane (Q3), which is parallel to the Plane of Areas (Sty = 0); the variable semidiameter of contact, , being the vector-axis (c) of instantaneous rotation of the body. (f). The Moment of Inertia, with respect to any axis through o, is equal to the living force (h2) divided by the square (Ti2) of the semidiameter of the ellipsoid (P3), which has the direction of that axis; and hence may be derived, with the help of the first general construction of an ellipsoid, suggested by quaternions, a simple geometrical representation (p. 711) of the square-root of the moment of inertia of a body, with respect to any axis AD passing through a given point A, as a certain right line BD, if CD = CA, with the help of two other points B and C, which are likewise fixed in the body, but may be chosen in more ways than one. (g). A cone of the second degree, Siv = 0, (S3), with v=y2pt - h2p2, (T3) is fixed in the body, but rolls in space on that other cone, which is the locus of the instantaneous axis ; and thus a second representation, proposed by Poinsot, is found for the motion of the body, as the rolling of one cone on another. (h). Some of Mac Cullagh's results, respecting the motion here considered, are obtained with equal ease by the same quaternion analysis; for example, the line y, although fixed in space, describes in the body an easily assigned cone of the second degree (p. 712), which cuts the reciprocal ellipsoid, in a certain sphero-conic: and the cone of normals to the last mentioned cone (or the locus of the line + hey ́1) rolls on the plane of areas (StY = 0). (i). The Three (Principal) Axes of Inertia of the body, for the given point o, have the directions (p. 712) of the three rectangular and vector roots (comp. (P), p. xii., and the paragraph 415, (a), p. xlii.) of the equation Vipi=0, (V3), because, for each, Du=0; (V3') and if A, B, C denote the three Principal Moments of inertia corresponding, then the Symbolical Cubic in p (comp. the formula (N) in page xii.) may be thus written, (j). Passage (p. 713), from moments referred to axes passing through a given point o, to those which correspond to respectively parallel axes, through any other point 2 of the body. Pages. CONTENTS. ARTICLE 418.-On the motions of a System of Bodies, considered (a). Equation of motion of the system, EmSD ada + ¿P= 0, (X3), if P= Σmm'T (a− a')1; (Y3) a is the vector, at the time t, of the mass or particle m; P is the po- (b). Extension of the notation of derivatives, (c). The differential equations of motion of the separate masses mD2a + DaP= 0, . . ; (A1) and the laws of the centre of gravity, of areas, and of living force, xlvii 713-717 B, y being two vector constants, and H a scalar constant. (d). Writing, (CA) (P+T) dt, (Es), and V- =f [ 2 Tαt = F+ tH, (F.) F= ['" (P+T) α, F may be called the Principal Function, and V the Characteristic (e). We are led thus to equations of the forms, mDia + DaF=0,.. (Gs); mDoa + DaF=0,.. (H1); whereof the system (G) contains what may be called the Interme- (ƒ). In like manner we find equations of the forms, DaV=-mDia,.. (J1); DaV=mD。a,.. (K1); DμV=t; (Ls) * References are given to two Essays by the present writer, "On a General |