xxii INTRODUCTION. 



equal and opposite. We proceed to consider this case. 

 Let the angle of rotation be a, the axes A and B, and 

 their distance d. Let x be the distance of any point P 

 from A ; then the rotation about A elevates P above 

 the plane of A and B to a distance ax. The rotation 

 around B depresses P below the plane of A and B to a 

 distance a (x + d). The net result, therefore, is that P 

 is depressed below the plane of A and B to a distance a</. 

 Now it is remarkable that this result is independent 

 of the position of P in the plane of A and B ; con- 

 sequently all points in the plane are moved through 

 equal distances, and thus we have the important result 

 that a pair of equal parallel and opposite rotations are equi- 

 valent to a translation in the direction perpendicular to the 

 plane of the axes, and through an interval proportional to 

 the distance between them. 



The converse of this result is also of great import- 

 ance namely, that a translation can always be decom- 

 posed into a pair of equal parallel but opposite rotations, 

 in a plane perpendicular to the direction of the trans- 

 lation. 



Composition of a Rotation with a Translation Perpendicu- 

 lar to the Axis of Rotation. The translation may be 

 resolved into a pair of equal parallel and opposite rota- 

 tions in a plane which contains the given axis of rota- 

 tion. This couple of rotations may be compounded with 

 the given rotation in precisely the same way as a couple 

 is compounded with a force in the same plane. It 

 follows that the result of compounding a rotation with a 

 translation perpendicular thereto is merely to transfer the 

 rotation to a parallel position, without altering its mag- 

 nitude. 



Displacement of a Rigid Body about a Fixed Point. 

 A rigid body is supposed to be free to turn around a 

 fixed point in every way. If we fix our attention on 



