The mathematical properties of a twist or of a wrench have been the subject of many remarkable investigations, which are, however, of secondary importance from a physical point of view. In the “Null-System” of A. F. Möbius (1790-1868), a line such that the moment of a given wrench about it is zero is called a null-line. The triply infinite system of null-lines form what is called in line-geometry a “complex.” As regards the configuration of this complex, consider a line whose shortest distance from the central axis is r, and whose inclination to the central axis is θ. The moment of the resultant force R of the wrench about this line is − Rr sin θ, and that of the couple G is G cos θ. Hence the line will be a null-line provided
tan θ = k/r,
(11)
where k is the pitch of the wrench. The null-lines which are at a given distance r from a point O of the central axis will therefore form one system of generators of a hyperboloid of revolution; and by varying r we get a series of such hyperboloids with a common centre and axis. By moving O along the central axis we obtain the whole complex of null-lines. It appears also from (11) that the null-lines whose distance from the central axis is r are tangent lines to a system of helices of slope tan−1 (r/k); and it is to be noticed that these helices are left-handed if the given wrench is right-handed, and vice versa.
Since the given wrench can be replaced by a force acting through any assigned point P, and a couple, the locus of the null-lines through P is a plane, viz. a plane perpendicular to the vector which represents the couple. The complex is therefore of the type called “linear” (in relation to the degree of this locus). The plane in question is called the null-plane of P. If the null-plane of P pass through Q, the null-plane of Q will pass through P, since PQ is a null-line. Again, any plane ω is the locus of a system of null-lines meeting in a point, called the null-point of ω. If a plane revolve about a fixed straight line p in it, its null-point describes another straight line p′, which is called the conjugate line of p. We have seen that the wrench may be replaced by two forces, one of which may act in any arbitrary line p. It is now evident that the second force must act in the conjugate line p′, since every line meeting p, p′ is a null-line. Again, since the shortest distance between any two conjugate lines cuts the central axis at right angles, the orthogonal projections of two conjugate lines on a plane perpendicular to the central axis will be parallel (fig. 42). This property was employed by L. Cremona to prove the existence under certain conditions of “reciprocal figures” in a plane (§ 5). If we take any polyhedron with plane faces, the null-planes of its vertices with respect to a given wrench will form another polyhedron, and the edges of the latter will be conjugate (in the above sense) to those of the former. Projecting orthogonally on a plane perpendicular to the central axis we obtain two reciprocal figures.
In the analogous theory of infinitely small displacements of a solid, a “null-line” is a line such that the lengthwise displacement of any point on it is zero.
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 Pr be the force at (xr, yr, zr), acting in the direction (l, m, n), the formulae (6) and (7) reduce to
X = Σ(P)·l, Y = Σ(P)·m, Z = Σ(P)·n,
(12)