33O Rotation of a Solid Body round a Fixed Point. 



point I by the rotations w and a/ are 01. to sin ROI and 

 01. w' sin R'OI ; and the directions of these velocities are per- 

 pendicular to the sides of the projected parallelogram. Hence, 

 if this parallelogram be turned in its plane through 90, its 

 sides will represent in magnitude and direction the actual 

 velocities : the resultant of these velocities is perpendicular to 

 the projection of the diagonal of the parallelogram (o>, a/) : 

 this projection, turned round through 90, will represent the 

 actual velocity, which is therefore the same in magnitude and 

 direction as would be produced by a single rotation represented 

 by the diagonal of (w, a/). Hence rotations may be resolved 

 along three rectangular axes by the same laws as couples, and 

 they must be counted positive when the motion produced is 

 from z to x, x to y, y to z, and vice versa. 



II. LINEAR VELOCITIES PRODUCED BY A GIVEN EOTATION. 



Let the origin of co-ordinates be assumed on the axis of 

 rotation, and let the magnitude of the rotation and of its com- 

 ponents be represented by (&>, p, q, r) : the velocity of any 

 point (x, y, z) is in a direction perpendicular to the plane con- 

 taining the axis of rotation and the point (x, y, z) ; and its 

 magnitude is represented by the area of the triangle whose 

 angles are situated at the origin, the point (#, y, s), and the 

 point (p, q, r). Hence, the components of the linear velocity 

 are represented by the projections of this triangle on the co- 

 ordinate planes. These projections are 



u = qz - ry ; 



v = rx - pz ; (1) 



w = py qx. 



