48] KINETIC POTENTIAL. 



Equations 45) or 46) 36 accordingly become 



1 A r-\ 



145) 



(or equal to P s if this represents an extraneous force not included 

 in the potential energy, or any dissipative force). The function 3> 

 is called by Routh the modified Lagrangian function, and its negative 

 _by Helmholtz the kinetic potential. It is to be understood that is 

 to be expressed in terms of the velocities , 5/4-1, . . . q' n by means of 

 equations 139) in which p lf . . . p r have been replaced by C 1; . . . c r . 

 The important thing to notice about <P is that it contains linear 

 terms in the velocities, as well as a homogeneous quadratic function 

 of the c's whose coefficients depend only on the coordinates q r +i, - q. m > 

 like the Q's from which they are derived. The terms of the latter 



sort in -rr cause precisely the same effect as if they were added 



to the potential energy. The effect of cyclic motions in a system is 

 accordingly partly represented by an apparent change of potential 

 energy, so that a system devoid of potential energy would seem to 

 possess it, if we were in ignorance of the existence of the cyclic 

 motions in it. The effect of the linear terms in is quite different 

 and will be discussed in 50. 



A system is said to contain concealed 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 case just treated, 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. 



The process of eliminating the cyclic coordinates of the concealed 

 motions as above described is termed by Thomson and Tait ignoration 

 of coordinates. 1 ) 



Examples of the process may be obtained in any desired number 

 from the theory of the motion of rigid bodies rotating freely about 



1) Thomson and Tait, Natural Philosophy, Part I, 319, example G. 



12* 



