A Treatise on the Theory of Screws

CHAPTER I. TWISTS AND WRENCHES. 1. Definition of the word Pitch. The direct problem offered by the Dynamics of a Rigid Body may be thus stated. To determine at any instant the position of a rigid body subjected to certain constraints and acted upon by certain forces. We may first inquire as to the manner in which the solution of this problem ought to be presented. Adopting one position of the body as a standard of reference, a complete solution of the problem ought to provide the means of deriving the position at any epoch from the standard position. We are thus led to inquire into the most convenient method of specifying one position of a body with respect to another. To make our course plain let us consider the case of a mathematical point. To define the position of the point P with reference to a standard point A, there can be no more simple method than to indicate the straight line along which it would be necessary for a particle to travel from A in order to arrive at P, as well as the length of the journey. There is a more general method of defining the position of a rigid body with reference to a certain standard position. We can have a movement prescribed by which the body can be brought from the standard position to the sought position. It was shown in the Introduction that there is one simple movement which will always answer. A certain axis can be found, such that if the body be rotated around this axis through a determinate angle, and translated parallel to the axis for a determinate distance, the desired movement will be effected. It will simplify the conception of the movement to suppose, that at each epoch of the time occupied in the operations producing the change of position, the angle of rotation bears to the final • angle of rotation, the same ratio which the corresponding translation bears to the final translation. Under these circumstances the motion of the body is precisely the same as if it were attached to the nut ot a uniform screw (in the ordinary sense of the word),