But seeing her lifeless on the table made his soul wrench in a way that reminded him of how he'd felt when he found his brother's lifeless body thousands of years ago.
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.
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.
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.
The right-hand expression is six times the volume of the tetrahedron of which the lines AA, BB representihg the forces are opposite edges; and we infer that, in whatever way the wrench be resolved into two forces, the volume of this tetrahedron is invariable.
By means of this theorem we can show that the previous reduction of any system to a wrench is unique.
The pitch of the equivalent wrench is (LX + MY + NZ)/(Xf + Y1 + Z2).
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.
MObius (1790-1868), a line such that the moment of a given wrench about it is zero is called a null-line.
The moment of the resultant force R of the wrench about this line is Rr sine, and that of the couple G is G cos 9.
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.
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.
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.
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.
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.
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.
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 work done by a wrench about a given screw, when the body twists about a second given screw, may be calculated directly as follows.
48 let R, G be the force and couple of the wrench,, r the rotation and translation in the twist.
Wrench and the twist be inclined at an angle 0, and let h be the shortest distance between them.
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.
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~