132 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. 



rigidly fixed in the axis of the pendulum. An ordinary pendulum 

 set vibrating in a plane continues to vibrate in a plane, with a 

 periodic reversal of its motion. The gyroscopic pendulum on the 

 other hand describes a curious looped surface, never remaining in 

 a plane nor returning on its course. This example is worked out 

 in Thomson and Tait's Natural Philosophy, 319, Example (D). 



68. Cyclic Systems. A system in which the kinetic energy 

 is represented with sufficient approximation by a homogeneous 

 quadratic function of its cyclic velocities is called a Cyclic System. 

 Of course the rigid expression of the kinetic energy contains the 

 velocities of every coordinate of the system, cyclic or not, for no 

 mass can be moved without adding a certain amount of kinetic 

 energy. Still if certain of the coordinates change so slowly that 

 their velocities may be neglected in comparison with the velocities 

 of the cyclic coordinates, the approximate condition will be ful- 

 filled. These coordinates define the position of the cyclic systems, 

 and may be called the positional coordinates or parameters of the 

 system. In the case of the gyrostat the two coordinates of the 

 gimbal rings may be taken for the positional coordinates, while 

 the cyclic coordinate determines the rotation of the gyrostat. In 

 the case of a liquid circulating through an endless rubber tube, 

 the positional co-ordinates would specify the shape and position of 

 the tube. The positional coordinates will be distinguished from 

 the cyclic coordinates by not being marked with a bar. The 

 analytical conditions for a cyclic system will accordingly be, for all 

 coordinates, either 



)/TF )'7 T 



<'> afT or Wr pi=Q ' 



or if we use the Hamiltonian form of T obtained by replacing the 

 velocities by the momenta, which we shall denote by T p , since the 

 non-cyclic momenta vanish 



(2} ^ p pV = _p =0 . 



dp s dq* dq s 



We accordingly have for the external forces tending to in- 

 crease the positional coordinates [see 62, (17)], 



d(T-W)_d_(T,+ 

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