340.] BODY WITH FIXED POINT. 



340. If the axes Ox\ Oy\ Oz 1 be the principal axes at O, the 

 equations (7) exhibit the relations between the system of the 

 principal axes and a fixed system with the same origin O, by 

 means of the cosines of the 9 angles between the axes of the 

 two systems. They can be used to derive Euler's equations by 

 .a purely analytical process from the equations (7), Art. 224. 



To accomplish this we must form the quantities ^m(yzzy}, 

 ^m(zxxz}, ^m(xyyx\ We need therefore x, j/, z. Now, 

 differentiating the expressions (7) with respect to the time and 

 remembering that x' t /, z' are independent of the time, we 

 find: 



Czz'. (14) 



To introduce the angular velocities to lt co 2 , o> 3 about the prin- 

 cipal axes at O, we observe that the direction-cosines a lt a 2 > a s 

 of the axis Ox can be regarded as the co-ordinates of the point 

 situated on Ox at unit distance from O. The components of 

 the linear velocity of this point, arising from w^ o> 2 , 3 are 



a l = <2 2 a> 3 # 3 o> 2 , tf 2 = a &>\ ~~ ^1^3* ^3 = ^1^2 7 a <iF>\ J 



and similarly we have for points at unit distance from O on Oy 

 .and Oz\ 



It should be noticed that the motion of a body with a fixed 

 point is fully determined by the motion of two of its points, not 

 in the same line with the fixed point ; the third point is here 

 only introduced to preserve the symmetry. 



Substituting these values in (14), we find 



X = 



y = ( 2 a> 3 - 3 o> 2 X + (^ft?! - ^wj})/ + (bi&i b#>^z\ (15) 



