A Treatise on the Theory of Screws

CHAPTER IV . SCREW CO-ORDINATES. 28. Introduction. We are accustomed, in ordinary statics, to resolve the forces acting on a rigid body into three forces acting along given directions at a point and three couples in three given planes. In the present theory we are, however, led to regard a force as a wrench on a screw of which the pitch is zero, and a couple as a wrench on a screw of which the pitch is infinite. The ordinary process just referred to is, therefore, only a special case of the more general method of resolution by which the intensities of the six wrenches on six given screws can be determined, so that, when these wrenches are com- pounded together, they shall constitute a wrench of given intensity on a given screw*. The problem which has to be solved may be stated in a more symmetrical manner as follows:— To determine the intensities of the seven wrenches on seven given screws, such that, when these wrenches are applied to a rigid body, which is entirely free to move in every way, they shall equilibrate. The solution of this problem is identical (12) with that of the problem which may be enunciated as follows:— To determine the amplitudes of seven small twists about seven given screws, such that, if these twists be applied to a rigid body in succession, the body after the last twist shall have resumed the same position which it occupied before the first. The problem we have last stated has been limited as usual to the case where the amplitudes of the twists are small quantities, so that the motion of a point by each twist may be regarded as rectilinear. Were it * If all the pitches be zero, the problem stated above reduces to the determination of the six forces along six given lines which shall be equivalent to a given force. If further, the six lines of reference form the edges of a tetrahedron, we have a problem which has been solved by Möbius, Crelle’s Journal, t. xvm. p. 207 (1838).