4 GEOMETRY OF MOTION. [8. 



the former, or in the opposite sense. In either case the total, 

 or resultant, displacement is the algebraic sum of the two dis- 

 placements PoPv P\P%> which are called its components; i.e. 

 we have /> /> 2 = /y^ + /y> 2> or /y\ + /y> 2 + /y> == o, what- 

 ever may be the positions of the points P Q , P I} P% in the line. 



This reasoning is easily extended to any number of compo- 

 nent displacements ; that is, the resultant of any number of 

 consecutive displacements of a point in a line is a single displace- 

 ment equal to the algebraic sum of the components. 



Similar considerations apply to the motion of a point in a 

 curved line provided the displacements be always measured 

 along the curve. 



8. Let us next consider the motion of a rigid body. The 

 term rigid body, or simply body, is used in kinematics to denote 

 a figure of invariable size or shape, or an aggregate of points 

 whose distances from each other remain unchanged. Examples 

 are : a segment of a straight line, a triangle, a cube, an ellipsoid, 

 etc. 



Imagine such a body M brought in any manner from some 

 initial position M Q into any other position M v This displace- 

 ment M M l is determined by the displacements of the various 

 points of the body. We shall see that, even in the most general 

 case, the displacements of three points of the body determine 

 those of all other points, and consequently the displacement of 

 the whole body. 



There are, however, two special cases of motion, translation 

 and rotation, in which the displacement of the body is fully 

 determined by the displacement of a single point : such motions 

 can be called linear. There is also a class of motions deter- 

 mined by the displacements of only two points of the body : 

 this is called plane motion. 



9. The displacement of a rigid body is called a translation 

 when the displacements of all of its points are parallel and equal. 

 It is evident that in this case the displacement of any one 



