312 MOTION OF EIGID BODIES 



horizontal axis. The former may be replaced by sin 2 6 about 



dd> 



the vertical, and sin 9 cos 6 -~ about^a horizontal axis. Thus the 



clt 



moment of momentum about the vertical contributed by part (6) 

 of the motion is j j 



***$ 



and since the moment of momentum about the vertical has a con- 

 stant value, say G, we have 



AD, cos + B sin 2 ( Q = G. (141) 



cut 



If we eliminate from this equation and equation (140), we 

 obtain 



B sin 2 \AW + B( Y+ 2 Mgh cos 6-E\ 



L\*/ J 



+ (G AQ, cos 0) 2 = 0, (142) 



a differential equation giving the variations in the value of 6, and 

 therefore allowing us to trace the changes in the inclination of the 

 axis of the top to the vertical. _ 



The maxima and minima of 6 are given by putting = 0, and 

 are therefore the roots of 



B(l cos 2 0) [AW + 2 Mgh cos 6 E] 



+ (GAQ, cos ) 2 = 0. (143) 



Let us call the left hand of this equation /(cos 6). Since/ is a 

 function of degree three, there will be three roots for cos 6. Let 

 us suppose that the top is started at an angle 6 = , and with the 



value of equal to ( ) Then, from equation (142), 

 dt \Vi 



B sin 2 AQ* + B l^- V + 2 Mgh cos - E\ 



\ 



so that /(cos Q ) =B sin 2 (9 [^n 2 + 2 Mgh cos e E] 



+ (G - Al cos 6 ) 2 



