91, 92] GYROSCOPIC FORCES OF TOP. 279 



T> TT dip 



P = H *' 



88) 



and the same force is required for the rotation about the vertical as 

 if there were no spinning, whereas a force is developed tending to 

 turn about the horizontal axis, which must be balanced by the 



constraint, P$, proportional to - Thus the effect of the concealed 



motion would be made evident, even if the disposition of the concealed 

 rotating parts were unknown. The effect of the gyroscopic term may 

 be described by saying that if the apex of the top be moved in any 

 direction, the spinning tends to move it at right angles to that 

 direction, as shown in 50. 



In our present problem, we have Py = 0, 



^sin 2 # -^'4- C(<p' + cos# t/>')cos# = const. = H}, 

 or by 83), 



89) A sin 2 # ^ + H, cos # = HJ, 



which is the integral of 70). 



The , differential equation for # is 



90) A Aty ' 2 sin # cos # + H z sin # - ty ' = Mgl sin #, 



which, on replacing ty' by the value from 89), and using the constants 

 of 73), 75) becomes 



~^ - -_ a . 



~*~ 8 ~ =S: 



fj Qi 



If we now multiply this by 2sin 2 # ^r? it becomes an exact derivative, 



and integrates into 77). Thus our three integrals are immediate 

 integrals of Lagrange's equations. 



92. Nature of the Motion. Equation 78) which states that 

 the time -derivative of 2, the cosine of the inclination of the axis to 

 the vertical, is a polynomial of the third degree in 2, shows that 

 is an elliptic function of the time. As we do not here presuppose 

 a knowledge of the elliptic functions, we will discuss the motion 

 without explicitly finding the solution in terms of elliptic functions. 



We see at once that the solution depends on the four arbitrary 

 constants cc, a, of the dimensions [T~ 2 ], which enter equation 78) 

 linearly, and /3, &, of dimensions [l 7 " 1 ], which enter homogeneously 

 in the second degree, so that if we divide &, /3 by the same number 

 and a, a by its square, while we multiply t by the same number, 

 the two equations 78), 79) are unchanged, that is to any value of -0 



