33 Rotation of a Solid Body round a Fixed Point. 



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

 01. u/ sin K'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 zto x, x to y, y to s, 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 (w, 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 (a?, y, 2) ; and its 

 magnitude is represented by the area of the triangle whose 

 angles are situated at the origin, the point (a?, 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. 



