66, 67] CYCLIC SYSTEMS. 129 



Let the solutions of these for the q"s be 



The R's being the quotients of the various subdeterminants of the 

 determinant 



Qn, Qi*, ... Ci 



ri > V>'2 > nJr 



by the determinant itself, are functions of the coordinates only, 

 and since by hypothesis the function T did not contain the cyclic 

 coordinates, the R's are functions of only the non-cyclic coordinates. 

 The kinetic potential consequently is a function only of the non- 

 cyclic coordinates and velocities, but on account of the presence of 

 the constants c s , it is not a homogeneous function of the velocities, 

 but contains a linear function of them, as was remarked in 65. 

 Cases in which the kinetic potential contains a linear function 

 of the velocities may thus be considered as cases with concealed 

 motions. A case of this nature will be found in considering 

 the mutual actions of magnets and electric currents. Physically 

 the difference between the two cases is that while if <& contains 

 only terms of the second degree in the velocities, if every velocity 

 is reversed the kinetic potential is unchanged, and hence the 

 motion may be reversed without change of circumstances, but if 

 on the other hand there are terms of the first degree in the 

 velocities, the motion cannot be reversed unless the concealed 

 motions are reversed as well. 



As an example we will take the case of a gyrostat hung in 

 gimbals. Let the outer ring of the gimbals A, Fig. 27, be 

 pivoted on a vertical axis, and let the angle made by the plane of 

 the ring with a fixed vertical plane be i/r. Let the inner ring B 

 be pivoted on a horizontal axis, and let its plane make an angle 6 

 with the plane of the outer ring. The gyrostat is pivoted on an 

 axis at right angles with the last, and let a fixed radius of the 

 gyrostat make an angle < with the plane of the inner ring. It is 

 shown in the theory of the dynamics of a rigid body that the 

 energy of a body revolving about an axis is one-half the product 

 w. E. 9 



