75, 76] 



We have 



MOVING AXES. 

 if 



243 



Then the cylinder is dynamically equivalent to a sphere, as is 

 also the case for a cube. 



These examples furnish the means of treating the cases that 

 usually appear in practice. 



76. Analytical Treatment of Kinematics of a Rigid 

 System. Moving Axes. In 55 57 we have treated the 

 general motion of a rigid system, from the purely geometrical point 

 of view, without analysis. We shall now give the analytical treatment 

 of the same subject. Let us refer the position of a point in the 

 system to two different sets of coordinates. Let %', y\ #' be its co- 

 ordinates with respect to a set of axes fixed in space, and let x, y, z 

 be its coordinates with respect to a set of axes moving in any 

 manner. The position of the moving axes is defined by the position 

 of their origin, whose coordinates referred to the fixed axes are 

 |, iq, g, and by the nine direction cosines of one set of axes with 

 respect to the other. Let these be given by the following table 



X Y Z 



The equations for- the transformation of coordinates are then, 



109) 



Since c^, a?, cc 3 are the direction cosines of the X-axis with 

 respect to X', Y', Z f , we have 



110) 

 and similarly 



110) 



a* + a, 2 + ^ = 1, 



