KINETICS OF A RIGID BODY. [339. 



solving the equations (7) for x\ and comparing the coefficients 

 of x, y, z to those in (i i), we find the following relations : 



(12) 



339. Squaring and adding these equations and applying the 

 relations (9), we find after reduction 



A 2 =i. 



The two values of A, + 1 and i, correspond to the two 

 different relations between the two rectangular systems, which 

 might perhaps be called like and unlike. Two systems are alike 

 if their positive axes can be brought to coincidence ; tjjey are 

 unlike if this cannot be done. It is, of course, always possible 

 to bring the axes Ox' and Oy' to coincidence witli Ox and Oy, 

 respectively. But after having accomplished this, the axis Oz f 

 may fall along Oz, in which case the systems are alike, or it 

 may fall into the opposite direction, when the systems are 

 unlike. 



Now if Ox 1 coincides with Ox, Oy 1 with Oy, we have a^ = i , 

 ^ 2 = i ; and c%= + i for like systems, c z = i for unlike systems , 

 as the other 6 cosines are zero, we find that A= -f i corresponds 

 to like systems, and A= i to unlike systems. For it is evi- 

 dent that the motion of one system with respect to the other 

 cannot affect A so as to change from one of these values to the 

 other. 



In mechanics the systems should generally be such that they 

 can be brought to coincidence. We assume therefore A= i. 



With this value of A, the equations (12) and the similar rela- 

 tions obtained by cyclical permutation of the subscripts give 

 the identities : 





