312 VII. DYNAMICS OF ROTATING BODIES. 



as two equations to determine p and r as functions of #. When 

 they are thus determined, the equation of energy 



208) M(v 2 + v 2 + v^+A(p 2 +q 2 ) + Cr 2 = 2{h-Mg(a8m&-ccos&)}, 

 or by 201), 



209) (A + Mc 2 }p 2 + (A + Ma 2 + Mc 2 }q 2 + (C+ Ma 2 )r 2 - ZMacpr 



suffices to determine q as a function of #. Thus we see that when- 

 ever & returns to a former value, the circumstances of the rolling 

 are repeated, so that the motion is periodic. 



Eliminating -~ from 207), we obtain 



210) MAa 2 p = - {AC + M (Aa 2 + Cc 2 }} | - MCacr, 



differentiating which, we may eliminate p, obtaining for r, 



.,-,\ d*r , _ dr . MCa 



211) 



a linear differential equation for r, with variable coefficients. In the 

 case of the disk, where c = 0, by introducing the new variable 



x = cos 2 #, 

 we reduce the equation to the form 



,. x d*r . /I 3 \ dr 



212) 



which is the differential equation of Gauss 



213) ^(l-^l^ + ^- 

 if we put 



MCa 



This differential equation is satisfied by the hypergeometric series 



214) F (K , ft , r , x ^ i + + 



tt( + i)( + a)p(/? + i)(/? + 8) 3 



and by the theory of linear differential equations we find that the 

 general integral is 



-f 



