335-] BODY WITH FIXED POINT. 



334. The sense in which these angles are counted is best remembered 

 by imagining the two trihedral angles X YZ and X' Y'Z' originally coin- 

 cident. Now turn the system X'Y'Z' about the axis OZ in the posi- 

 tive sense (counter-clockwise) until the axis OX' coincides with the 

 assumed positive sense of the nodal line ON, i.e. the final intersection 

 of the planes XOY and X'OY' ; the amount of this rotation gives the 

 angle {j/. Next turn the trihedral X f Y'Z' about this line of nodes in 

 the positive sense until the plane X'OY' falls into its final position; 

 this gives the angle 6, as the angle between the planes XOY and 

 .X'OY' at N, or as the angle ZOZ' between their normals. Finally a 

 rotation of X'Y'Z 1 about the axis OZ', which has reached its final posi- 

 tion, in the positive sense until OX 1 comes into its final position, 

 determines the angle <f>. 



335. The angular velocity, represented by its rotor CD, whose 

 components along OX', OY', OZ' are co^ o> 2 , &> 3 , can be resolved 

 .along ON, OZ 1 , OZ into three components which are evidently 

 0, c/>, ^, respectively. The sum of the projections of these three 

 components on the line OX' should give o> l ; hence 



cw 1 = cos (j> -f <j) cos JTT 4- ^ cos ZX'. 

 .Similarly o> 2 = cos ($ + JTT) -j- </> cos |-TT + ^r cos Z Y', 



0) 3 = COS |- 7T H- (j) COS O + ^r COS 0. 



The spherical triangle ZNX' gives (by the fundamental formula 

 of spherical trigonometry, cos c = cos a cos b + sin a sin b cosy) 

 cos ZX ' = sin cos (J TT 0) = sin <f> sin ; and the triangle ZNY' 

 gives cos^F^sin^-hl-TrJcosd-Tr 0) = cos <p> sin 0. Hence, 

 iinally 



a) l = cos + ir sn sn , 



o> 2 = 6 sin < + ^ cos < sin 0, ($) 



G> 3 = + ^ COS0. 



.Solving these equations for 0, <^, njr, we find 

 = ft ) 1 cos < w 2 sin <^>, 



</>= ojj sin <^> cot o) 2 cos <^> cot 0+3 (6) 



^ = o) 1 sin (^ esc 0-fft> 2 cos </> esc 0. 



