DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
311 
dT A 
d X\ ! 
dTl }>.(3) 
— —=c 2 l 
d Xi 
... J 
where c 13 c 2 . . . , c n are constants. 
We can find the values of Xi> Xs • • • from these equations in terms of c x , c 2 ... , 
and substitute them in the expression for T, which will then be of the form 
£+K 
where X is a homogeneous quadratic function of the differential coefficients of the 
x, y, z and w coordinates; in fact, the kinetic energy of the system (A) and K is a 
quadratic function of the c’s involving it may be the coordinates x , y, z and w, but 
not their differential coefficients; K is evidently equal in magnitude to the kinetic 
energy of the system (B). Equations (19) on page 323 of Thomson and Tait’s 
‘Natural Philosophy,’ vol. i., 2nd ed., show that if we consider only the kinetic 
energy, Lagrange’s equation takes the form 
dt dx dx dx 
( 4 ) 
but if the system A had possessed potential energy equal to V the equation 
(considering A alone) would have been 
d_ dX 
dt dx 
dX , dV n 
— H-= 0 
dx dx 
( 5 ) 
Thus the effect of the system (B) on (A) is the same as if (A) possessed potential 
energy equal to K, which is, as we saw, the kinetic energy possessed by the system B, 
which is fixed by the kinosthenic coordinates. Thus we may look on the potential 
energy of any system (A) as being the kinetic energy of a kinosthenic system (B) 
connected with A ; and so we may regard all energy as kinetic. If we do this it will 
simplify some of the dynamical principles very much. We may take as our funda¬ 
mental principle Hamilton’s Principle of Varying Action, as we can very readily 
deduce from this principle the ordinary dynamical methods, such as Lagrange’s and 
Hamilton’s equations. But if all the energy is kinetic, then by the principle of the 
conservation of energy the magnitude of the kinetic energy remains constant, and the 
principle of least action takes the very simple form that, with a given quantity of 
energy, any material system will by its unguided motion pass from one configuration 
to another in the least possible time, where, of course, in the phrase material system 
we include the kinosthenic systems whose motion produce the same effects as the 
2 s 2 
