These are the equations of the central axis. Since the moment
of the resultant couple is now
| G′ = | X | L′ + | Y | M′ + | Z | N′ = | LX + MY + NZ | , |
| R | R | R | R |
the pitch of the equivalent wrench is
It appears that X2 + Y2 + Z2 and LX + MY + NZ are absolute invariants (cf. § 7). When the latter invariant, but not the former, vanishes, the system reduces to a single force.
The analogy between the mathematical relations of infinitely small displacements on the one hand and those of force-systems on the other enables us immediately to convert any theorem in the one subject into a theorem in the other. For example, we can assert without further proof that any infinitely small displacement may be resolved into two rotations, and that the axis of one of these can be chosen arbitrarily. Again, that wrenches of arbitrary amounts about two given screws compound into a wrench the locus of whose axis is a cylindroid.
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
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
and
provided
| x̄ = | Σ(Px) | , ȳ = | Σ(Py) | , z̄ = | Σ(Pz) | . |
| Σ(P) | Σ(P) | Σ(P) |
These are the same as if we had a single force Σ(P) acting at the point (x̄, ȳ, z̄), 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 Σ(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.
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| Fig. 45. |
§ 9. Work.—The work done by a force acting on a particle, in any infinitely small displacement, is defined as the product of the force into the orthogonal projection of the displacement on the direction of the force; i.e. it is equal to F·δs cos θ, where F is the force, δs the displacement, and θ is the angle between the directions of F and δs. In the language of vector analysis (q.v.) it is the “scalar product” of the vector representing the force and the displacement. In the same way, the work done by a force acting on a rigid body in any infinitely small displacement of the body is the scalar product of the force into the displacement of any point on the line of action. This product is the same whatever point on the line of action be taken, since the lengthwise components of the displacements of any two points A, B on a line AB are equal, to the first order of small quantities. To see this, let A′, B′ be the displaced positions of A, B, and let φ be the infinitely small angle between AB and A′B′. Then if α, β be the orthogonal projections of A′, B′ on AB, we have
ultimately. Since this is of the second order, the products F·Aα and F·Bβ are ultimately equal.
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| Fig. 46. | Fig. 47. |
The total work done by two concurrent forces acting on a particle, or on a rigid body, in any infinitely small displacement, is equal to the work of their resultant. Let AB, AC (fig. 46) represent the forces, AD their resultant, and let AH be the direction of the displacement δs of the point A. The proposition follows at once from the fact that the sum of orthogonal projections of AB→, AC→ on AH is equal to the projection of AD→. It is to be noticed that AH need not be in the same plane with AB, AC.
It follows from the preceding statements that any two systems
