96, 97, 98] TOP ON" TABLE. FRICTION. 303 



The potential energy is as before Mglcosfr. Consequently the only 

 difference in the problem from that treated in 90 is in the extra 

 term in #', Ml* sin* & &' 2 in the kinetic energy. Carrying out the 

 various steps of 90, 91, we find instead of the first equation 76) 

 the equation 



176) # 



and putting s = cos #, 



d 



( d *\* 



\dtf- 



where we denote the roots of the denominator by # 4 , z 5 . It is to be 

 noticed that they lie outside the interval 1, 1, for evidently the 

 coefficient of #' 2 in 176) cannot vanish for real values of -9-. 



The square of -j-- being now the quotient of two polynomials 



in z, s is a hyperettiptic function of t. We may however, without a 

 knowledge of these functions, treat the problem just as we did the 

 former one, and we shall find that the top in general rises and falls 

 between two of the roots of the numerator, and that the motion 

 resembles the motion already discussed. The path of the peg has 

 loops, cusps, or inflexions, according to the initial conditions, as 

 before, while the regular precession and the small oscillations may 

 be investigated as before. Whereas accordingly the functional relations 

 involved are considerably different, physically this motion, which is 

 that of the common top, closely resembles that already studied. 



98. Effect of Friction. Rising of Top. We have now to 

 take account of the effect of friction. Here we have in addition to 

 the normal component of the reaction a tangential component called 

 the force of friction, and the ordinary law assumed is that the 

 tangential component is equal to the normal component multiplied 

 by a constant depending on the nature of the two surfaces in contact, 

 called the coefficient of friction. If the friction is less than a certain 

 amount, the two surfaces will slide one upon the other, and the 

 direction of the friction will be such as to oppose the sliding, being 

 in the direction of the relative motion of the points instantaneously 

 in contact. The bodies are then said to be "imperfectly rough". If 

 the friction is greater than a definite amount, it will prevent the 

 sliding, and there is then no relative motion of the points of contact, 

 so that there is a constraint due to the friction, which is expressed 

 by an equation stating that the velocities of the points of the two 

 surfaces in contact are equal. If one of the surfaces is at rest, as 

 is usually the case, the instantaneous axis then always passes through 



