A Treatise on the Theory of Screws

4 INTRODUCTION. have three double points. The special feature of this homographic trans- formation is that every circle is transformed into a circle. Each circle passes through the two circular points at infinity in its plane. These two points in A are therefore two of the double points of the plane homographic trans- formation. There remains one real point X in A which is common to the two systems. The normal S to A drawn through X is therefore the one and only ray which the homographic transformation does not alter. This shows that in the most general change of a rigid system from one position to another there is one real ray unaltered Hence every point on »S' before the transformation is also on S afterwards. There must therefore be two double points distinct or coincident on S. But we have already proved that in the general case there is no finite double point. Hence S must have two coincident double points at T. Thus we learn that in the general transformation of a system which is equivalent to the displacement of a rigid body, there is one real point at infinity which is the result of two coinciding double points, and the polar of this point with respect to the imaginary circle on the plane at infinity cuts that circle in the two other double points. The displacement of the rigid body can thus be produced either by rotating the body around ,S’ or by translating the body parallel to S, or by a combination of such movements. We are therefore led to the funda- mental theorem discovered by Chasles. Any given displacement of a rigid body can be effected by a rotation about an axis combined with a translation parallel to that axis. Of much importance is the fact that this method of procedure is in general unique. It is easily seen that there is only one axis by rotation about which, and translation parallel to which, the body can be brought from one given position to another given position. Suppose there were two axes P and Q, which possessed this property, then by the movement about P, all the points of the body originally on the line P continue thereon; but it cannot be true for any other line that all the points of the body originally on that line continue thereon after the displacement. Yet this would have to be true for Q, if by rotation around Q and translation parallel thereto, the desired change could be effected. We thus see that the displacement of a rigid body can be made to assume an extremely simple form, in which no arbitrary element is involved. ON THE REDUCTION OF A SYSTEM OF FORCES APPLIED TO A RIGID BODY TO ITS SIMPLEST FORM. It has been discovered by Poinsot that any system of forces which act upon a rigid body can be replaced by a single force, and a couple in a plane