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KINETICS OF A RIGID BODY. 



[336. 



336. The relation between two rectangular systems of axes 

 with the same origin can also be expressed by means of the 

 9 cosines of the angles between the axes. 



Let O be the common origin, x> y, z the co-ordinates of any 

 point with respect to the fixed system, x\ y\ 2' its co-ordinates 

 in the moving system ; then we have, evidently, 



(7) 



where the coefficients of x\ y\ z' are the cosines of the angles 

 between the axes, which can best be remembered in the form 



Thus, 9 angles are used in order to fix the position of the 

 moving axes with respect to the fixed axes, instead of Euler's 

 3 angles. But their 9 cosines (8) are connected by 6 in- 

 dependent relations, which can be written in either one of the 

 equivalent forms : 



or 



I, 



- 



The meaning of these equations is easily perceived from the 

 meaning of the angles involved. Thus, the first of the equations, 

 (9) expresses the fact that a lf b lt c^ are the direction-cosines 



