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 8, 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 



