338.] BODY WITH FIXED POINT. 185 



of a line, viz. the axis Ox 1 ; the last of the equations (10) 

 expresses the perpendicularity of the axes 'Ox and Oy ; and 

 similarly for the others. 



337. The relations between the 9 angles, whose cosines are 

 given in (8), and Euler's 3 angles 6, <, -\/r are readily found from 

 Fig. 42, by applying the formula cos c= cos a cos b + sin a sin b cosy 

 successively to the triangles 



XNX', XNY', XNZ', 



YNX', YNY', YNZ\ 



ZNX', ZNY 1 , ZNZ 1 . 



In this way the following relations are found : 

 ^ = cos i|r cos </> sin ty sin <f> cos 6, 

 &! = sin i/r cos $ + cos ^r sin < cos 0, 

 c l = sin <f) sin 6, 

 a%= cos 'v/r sin <^> sin -^ cos (/> cos ^, a 3 = sin A/r sin 6, 



. c^ = cos $ sin ^, ^ 3 = cos 6. 



338. It is evident, geometrically, from (8), that we must have 



(11) 



For just as the first of the equations (7) expresses that the sum 

 of the projections on Ox of the co-ordinates x',y' t z 1 is equal to 

 x, so the first of the equations (11) expresses the equality of x' 

 to the sum of the projections of x, y, z on the axis Ox'\ and 

 similarly for the other equations. 



Now the solution of the equations (7) for x f t y', z 1 should 

 give the values (11). Putting 



a \ a i <*s 



*i ^2 ^3 =A, 



