Wrench and the twist be inclined at an angle 0, and let h be the shortest distance between them.
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.
Again, that wrenches of arbitrary amounts about two given screws compound into a wrench the locus of whose axis is a cylindroid.
P is a rotor and coo- a vector), is called a motor, and has the geometrical significance of Ball's wrench upon, or twist about, a screw.
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.
Inserted between the scales or into the pome, but on opening the mouth still more widely, the lateral motion of the mandible is once more brought to bear with great force to wrench aside the portion of the fruit attacked, and then the action of the tongue completes the operation, which is so rapidly performed as to defy scrutiny, except on very close inspection.
This expression gives the work done by a given wrench when the body receives a given infinitely small twist; it must of course be an absolute invariant for all transformations of rectangular axe~.
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.
The factor (P+P) cos 0h sin 0 is called the vIrtual coefficient of the two screws which define the types of the wrench and twist, respectively.
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.
If a line is a null-line with respect to the wrench (X, Y, Z, L, M, N), the work done in an infinitely small rotation about it is zero, and its coordinates are accordingly subject to the further relation Lf+M~+Nl+XX+Yp+Zvo, (5~
Conversely it is seen that any wrench can be replaced in an infinite number of ways by two forces, and that the line of action of one of these may be chosen quite arbitrarily.
It appears also from (II) 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 righthanded, and vice versa.
Again, any plane w is the locus of a system of null-lines meeting in a point, called the null-point of c. If a plane revolve about a fixed straight line p in it, its ntill-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.
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.