65, 66] HAMILTON'S PRINCIPLE. 127 



The function <I> is called by Routh the modified Lagrangian 

 function, and on account of its importance has received from 

 Helmholtz the special name of the Kinetic Potential, by which 

 we shall designate it. (Helmholtz calls <E> the kinetic potential*.) 



It is to be noticed that the equations for the elimination of the 

 velocities, the equations (2) of 63 are now, instead of 64 (i) 



Qliqi + Quqi . + Qirqr = Pi- Qir+iq'r+i - Qmqn, 

 Qnqi + Qrsq* ' + Qrrqr' = Pr - Qrr+i q'r+l ~ Qm^n ', 



so that the q"s become linear functions of the right-hand sides of 

 these equations and hence of 



Pi,p 2 , ... p r , q'r+i q n '> 

 thus T becomes a homogeneous quadratic function of 



P!...p r and q'r+i...q n ', 



but is not homogeneous in either the p's or the <?"s alone, on account 

 of terms such as p^q/ which are linear in either the p's or <?"s. 



66. Concealed Motions. A system is said to contain con- 

 cealed masses, when the coordinates which become known to us by 

 observation do not suffice to define the positions of all the masses 

 of the system. The motions of such bodies are called concealed 

 motions. It is often possible to solve the problem of the motions 

 of the visible bodies of a system, even when there are concealed 

 motions going on. For it may be possible to form the kinetic 

 potential of the system for the visible motions, not containing the 

 concealed coordinates, and in this case we may use Lagrange's 

 equations, as in the preceding article, for all visible coordinates, 

 while the coordinates of the concealed masses may be ignored. 

 Such problems are incomplete, inasmuch as they tell us nothing of 

 the concealed motions, but very often we are concerned only with 

 the visible motions. Such concealed motions enable us to explain 

 the forces acting between visible systems by means of concealed 

 motions of systems connected with them. 



As an example of a concealed motion let us take the case of a 

 closed box containing a gyrostat, or fly-wheel, pivoted on an axis 



* Helmholtz's notation is quite different from that here employed. 



