138 THEORY OF NEWTONIAN FORCES. [FT. I. CH. III. 



and introducing these instead of the velocities 



We have for the positional force 



ar ., 2 dT p 



P r = ^- = - mr6 2 = -~* = 



dr dr 



This being negative denotes that a force P r toward the axis 

 must be impressed on the mass m in order to maintain the cyclic 

 state. This may be accomplished by means of a geometrical 

 constraint, or by means of a spring. The force or reaction P r 

 which the mass m exerts in the direction from the axis in virtue 

 of the rotation is called the centrifugal force. We see that if 

 the motion is isocyclic, the positional force increases with r, while 

 if it is adiabatic, it decreases when r increases. The verification 

 of the theorems of 69 is obvious. The cyclic force 



dt dt 



vanishes when the rotation is uniform, and the radius constant. 

 If, the motion being isocyclic, that is, one of uniform angular 

 velocity, the body moves farther from the axis, P^, the cyclic 

 force is positive, that is, unless a positive force P$ is applied, 

 the angular velocity will diminish. In moving out from r t to r 2 

 work will be done against the positional force P r of amount 



/***2 C r 2 



-A = - P r dr = m^ rdr = 



Jfl J T! 



while the energy increases by the same amount. 



Thus the first theorem of 70 is verified. If the motion is 

 adiabatic, 



Pt = rar 2 </>' = c. 



If the body move from the axis, <' will accordingly decrease. 

 The change in P r due to a displacement Sr is 



which, being of the same sign as Sr, does a positive amount of 

 work in the displacement, illustrating the second theorem of 

 70. 



