Since a wrench is defined by six independent quantities, it can in general be replaced by any system of forces which involves six adjustable elements. For instance, it can in general be replaced by six forces acting in six given lines, e.g. in the six edges of a given tetrahedron. An exception to the general statement occurs when the six lines are such that they are possible lines of action of a system of six forces in equilibrium; they are then said to be _in involution_. The theory of forces in involution has been studied by A. Cayley, J. J. Sylvester and others. We have seen that a rigid structure may in general be rigidly connected with the earth by six links, and it now appears that any system of forces acting on the structure can in general be balanced by six determinate forces exerted by the links. If, however, the links are in involution, these forces become infinite or indeterminate. There is a corresponding kinematic peculiarity, in that the connexion is now not strictly rigid, an infinitely small relative displacement being possible. See § 9. When parallel forces of given magnitudes act at given points, the resultant acts through a definite point, or _centre of parallel forces_, which is independent of the special direction of the forces. If P_r be the force at (x_r, y_r, z_r), acting in the direction (l, m, n), the formulae (6) and (7) reduce to X = [Sigma](P).l, Y = [Sigma](P).m, Z = [Sigma](P).n, (12) and L = [Sigma](P)·(n[|y] - m[|z]), M = [Sigma](P)·(l[|z] - n[|x]), N = [Sigma](P)·(m[|x] - l[|y]), (13) provided [Sigma](Px) [Sigma](Py) [Sigma](Pz) [|x] = -----------, [|y] = -----------, [|z] = -----------. (14) [Sigma](P) [Sigma](P) [Sigma](P) These are the same as if we had a single force [Sigma](P) acting at the point ([|x], [|y], [|z]), which is the same for all directions (l, m, n). We can hence derive the theory of the centre of gravity, as in § 4. An exceptional case occurs when [Sigma](P) = 0. If we imagine a rigid body to be acted on at given points by forces of given magnitudes in directions (not all parallel) which are fixed in space, then as the body is turned about the resultant wrench will assume different configurations in the body, and will in certain positions reduce to a single force. The investigation of such questions forms the subject of "Astatics," which has been cultivated by Möbius, Minding, G. Darboux and others. As it has no physical bearing it is passed over here. [Illustration: FIG. 45.]