for the line of the second force, and so on; we have, for the The equations of constraint being, as in § 553, (4), Ax+ By + C + G + Hp + Iσ = 0 ° .(9), suppose, for example, these equations to be four in number. ai + by + cż + gw + hp + io = w }... (10), where a, b, ... and a', b',... are any arbitrarily assumed quanti- _x = Aw + A'w', ÿ = Bw+B'w', ż= Cw + C'w', (11); The same analytically and in terms of rectangular co-ordi nates. ralized com velocities ing to two where A, B,... and A', B',... are known, being the determi- Two genenantal ratios found in solving (9) and (10). Thus the six rect- ponent angular component velocities are expressed in terms of two correspondgeneralized component velocities w, w', which, in virtue of the freedoms. four equations of constraint (9), suffice for the complete specification of whatever motion the constraints leave permissible. In terms of this notation we have, for the rate of working of the applied forces, Xx + Yỳ + Zż + L≈ + Mp + No =(AX+BY+CZ+GL+HM+EN) w + (A'X + B'Y + C'Z + G'L+Q'M + X'N) w') (12). This must be nil for every permitted motion in order that the forces may balance. Hence the equations of equilibrium are AX + BY + CZ + GL + WM + EN =0} and A'X+BY+ C'Z + G'L + W ́M + X'N = 0 ] (13). Two generalized component velocities corresponding to two freedoms. Equilibrant and result aut. Couples. Similarly with one, or two, or three, or five (instead of our example of four) constraining equations (9), we find five, or four, or three, or one equation of equilibrium (13). These equations express obviously the same conditions as those expressed by (8); the first of (13) is identical with the first of (8), the second of (13) with the second of (8), and so on, provided w, w',... correspond to the same components of freedom as the several screws of (8) respectively. The equations though identical in substance are very different in form. The purely analytical transformation from either form to the other is a simple enough piece of analytical geometry which may be worked as an exercise by the student, to be done separately for the first of (8) and the first of (13), just as if there were but one freedom. 558. Any system of forces which if applied to a rigid body would balance a given system of forces acting on it, is called an equilibrant of the given system. The system of forces equal and opposite to the equilibrant may be called a resultant of the given system. It is only, however, when the resultant system is less numerous, or in some respect simpler, than the given system that the term resultant is convenient or suitable. It is used with great advantage with respect to the resultant force and couple (§ 559 g, below) to which Poinsot's method leads, or to the two resultant forces which mathematicians before Poinsot had shown to be the simplest system to which any system of forces acting on a rigid body can in general be reduced. It is only when the system is reducible to a single force that the term "resultant" pure and simple is usually applied. 559. As a most useful commentary on and illustration of the general theory of the equilibrium of a rigid body, which we have completed in §§ 552-557, and particularly for the purpose of finding practically convenient resultants in a very simple and clear manner, we may now with advantage introduce the beautiful method of Couples, invented by Poinsot. In § 234 we have already defined a couple, and shown that the sum of the moments of its forces is the same about all axes perpendicular to its plane. It may therefore be shifted to any new position in its own plane, or in any parallel plane, without alteration of its effect on the rigid body to which Couples. it is applied. Its arm may be turned through any angle in the plane of the forces, and the length of the arm and the magnitudes of the forces may be altered at pleasure, without changing its effect-provided the moment remain unchanged. Hence a couple is conveniently specified by the line defined as its "axis" in § 234. According to the convention of § 234 the axis of a couple which tends to produce rotation in the direction contrary to the motion of the hands of a watch, must be drawn through the front of the watch and vice versa. This may easily be remembered by the help of a simple diagram such as we give, in which the arrow-heads indicate the directions of rotation, and of the axis, respectively. tion of 559 b. It follows from §§ 233, 234, that couples are to be composicompounded or resolved by treating their axes by the law of couples. the parallelogram, in a manner identical with that which we have seen must be employed for linear and angular velocities, and forces. Hence a couple G, the direction cosines of whose axis are λ, μ, v, is equivalent to the three couples GA, Gp, Gv about the axes of x, y, z respectively. solved into couple. 559 c. If a force, F, act at any point, A, of a body, it may Force rebe transferred to any other point, B. Thus: by the principle of force and superposition of forces, introduce at B, in the line through it parallel to the given force F, a pair of equal and opposite forces Fand F. Then F at A, and F at B, form a couple, and there remains Fat B. Fat to equilirigid body. From this we have, at once, the conditions of equilibrium Application of a rigid body already investigated in § 552. For, each force brium of may be transferred to any assumed point as origin, if we introduce the corresponding couple. And the forces, which now act at one point, must equilibrate according to the principles of Chap. VI.; while the resultant couple, and therefore its components about any three lines at right angles to each other, must vanish. Forces re by 559 d. Hence forces represented, not merely in magnitude presentedes and direction, but in lines of action, by the sides of any closed of a polygon. polygon whether plane or not plane, are equivalent to a single couple. For when transferred to any origin, they equilibrate, by the Polygon of Forces (§§ 27, 256). When the polygon is plane, twice its area is the moment of the couple; when not plane, the component of the couple about any axis is twice the area of the projection on a plane perpendicular to that axis. The resultant couple has its axis perpendicular to the plane (§ 236) on which the projected area is a maximum. Forces proportional dicular to 559 e. Lines, perpendicular to the sides of a triangle, and and perpen- passing through their middle points, meet; and their mutual the sides of inclinations are equal to the changes of direction at the corners, a triangle. in travelling round the triangle. Hence, if at the middle points of the sides of a triangle, and in its plane, forces be applied all inwards or all outwards; and if their magnitudes be proportional to the sides of the triangle, they are in equilibrium. The same is true of any plane polygon, as we readily see by dividing it into triangles. And if forces equal to the areas of the faces be applied perpendicularly to the faces of any closed polyhedron, at their centres of inertia, all inwards or all outwards, these also will form an equilibrating system; as we see by considering the evanescence of (i) the algebraic sum of the projections of the areas of the faces on any plane, and of (ii) the algebraic sum of the volumes of the rings described by the faces when the solid figure is made to rotate round any axis, these volumes being reckoned by aid of Pappus' theorem (§ 569, below). Composition of force and couple. 559 f. A couple and a force in a given line inclined to its plane may be reduced to a smaller couple in a plane perpendicular to the force, and a force equal and parallel to the given force. For the couple may be resolved into two, one in a plane containing the direction of the force, and the other in a plane perpendicular to the force. The force and the component couple in the same plane with it are equivalent to an equal force acting in a parallel line, according to the converse of § 559 c. tion of forces act rigid body. 559 g. We have seen that any set of forces acting on Composia rigid body may be reduced to a force at any point and a any set of couple. Now (§ 559 f) these may be reduced to an equal force ing on a acting in a definite line in the body, and a couple whose plane is perpendicular to the force, and which is the least couple which, with a single force, can constitute a resultant of the given set of forces. The definite line thus found for the force is called the Central Axis. It is the line about which the sum of the moments Central of the given forces is least. With the notation of §§ 552, 553, let us suppose the origin to Σ[Z(y—y')—Y(z−2')], Σ[X(≈-≈')-Z(x-x')], Σ[Y(x−x')−X(y—y')]; or Gλ-R (ny' - mz'), Gμ-R (lz-nx), Gv-R (mx' - ly'). GA – R (mj – mấ) Gu - R ( nữ ) = m Gv-R (mx' - ly') = n These two equations among x', y', z' are the equations of the We find the same two equations by investigating the con- √[Gλ-R(ny'-mz')]2 + [Gμ−R (lz'— nx')]2 + [Gv − R (mx' — ly')]" may be a minimum subject to independent variations of x', axis. to two 560. By combining the resultant force with one of the Reduction forces of the resultant couple, we have obviously an infinite forces. number of ways of reducing any set of forces acting on a rigid body to two forces whose directions do not meet. But there is one case in which the result is symmetrical, and which is therefore worthy of special notice. cal case. Supposing the central axis of the system has been found, Symmetri draw a line, AA', at right angles to it through any point C of |