DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
479 
[* In considering this case it will be convenient to divide the kinetic energy into 
two parts, one part, which we shall denote by T 1? depending on the velocities of 
coordinates fixing the position of the molecule; the other, which we shall denote by 
T 3 , depending on the velocities of coordinates of the type p, which we have completely 
under our control. The coordinates fixing the configuration of the system with 
respect to strain, electrification, magnetisation, &c., are coordinates of this class. 
T 3 will be a quadratic function of the velocities of the p coordinates, since the total 
kinetic energy of the system cannot involve the product of the velocity of a p 
coordinate with that of one fixing the position of a molecule; otherwise the kinetic 
energy of the system would be altered by reversing the motion of all the molecules. 
We have, by Lagrange’s equation, 
now 
and by definition 
d dT 
dt dp L 
dT dY 
dp dp 
T = T x + T„ 
dTi 
dp Y 
= 0 
we have also, by the Conservation of Energy, 
8Q = 8T 1 + 8T 3 + SV—SP8 p; .(6*) 
now 
«T. = s(f8p + f*).(0 
since T a is a homogeneous quadratic function of the velocities of the “ p” coordinates, 
• dT„ 
and therefore 
2 8T, = S(fcf + *8^); 
( 8 ) 
subtracting (7) from (8), we have 
8T s =S(jp8-^-8p- 
If the change in the configuration is that which actually takes place, then we have 
Pi St = Sp lt 
ST — X Sp ^ d 
so that 
dt dp dp J’ 
* This portion 'within brackets re-written October 17, 1887. 
