89, 90] 



HEAVY SYMMETRICAL TOP. 



275 



(Fig. 91). We may call the XT- plane the equator of the top. 



Let ON (Fig. 91) be the line of nodes, or the line in which the 



equator intersects the fixed 



X' F'-plane. Let ij> be the 



longitude of the line of 



nodes, or the angle X'ON 



measured positively from X' 



to Y'. Let cp be the angle 



from the line of nodes to 



the X-axis, the positive 



direction of increase being 



from X to Y. By means 



of the three angles -O 1 , ^, cp, 



we may express the nine 



direction cosines, and the 



position of the body is 



completely determined. The 



meaning of the angles is 



easily seen on the gyroscope 



in gimbals (Fig. 92). It 



will not be necessary for us to express the cosines, as we need 



only the values of p, q, r in terms of Euler's angles and their velocities, 



Fig. 91. 



dt 



dt dt 



As these are the angular velocities about ON, OZ' and OZ, respectively, 

 we need only the cosines of the angles made by these lines with the 

 X, Y, ^-axes, which are evidently as given in the following table. 



Resolving now in the direction of the three axes, we obtain 



65) 



. - . . 



= ~It Sm m ^ "^ ~dt COS *?' 

 dib . 



18' 



