244 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



Since the axes Y, Z are perpendicular, their direction cosines satisfy 

 the conditions, 



ill) 



and similarly, 



i j . 



Thus the nine cosines are not independent, but, satisfying six 

 conditions, may be expressed in terms of three parameters. These, 

 with the three |, ^, g, show the six degrees of freedom possessed 

 by a rigid system. 



By interchanging the roles of the axes, and considering the 

 direction cosines of X', T, Z' with respect to X, Y, Z we find the 

 equivalent conditions 



"I' + Ai' + yi'-l, 



112) 2 2 + /3 2 2 + r 2 2 = i, 



113) 



If we now differentiate the first of equations 109), supposing 

 y> % to be constant, we obtain 



for the components in the directions of the fixed axes of the velocity 

 of a point fixed to the moving axes. 



Let us now resolve the velocity in the direction which is at a 

 given instant that of one of the moving axes. To resolve in the 

 direction of the X-axis we have 



115) V x = tfX + C^Vy' + CC 3 V,' = ^ -^ + CC 2 ~ + ttj ^ 



The coefficient of x in this expression is the derivative of the left- 

 hand member of the first of equations 110), and is accordingly equal 



