IG 
Proceedings of the Royal Society 
1. The centre of inertia has constant velocity. 
2. The vector moment of momentum about it is constant. 
3. So is that of the system relative to any uniformly moving 
point. 
4. '%fmvds is obviously a minimum. 
(10) The result of (7) points to an independence between two 
parts of the motion of a system, z.e., that relative to the centre of 
inertia, and that of the whole mass supposed concentrated at the 
centre of inertia. Maxwell’s reasoning is applicable directly to the 
latter, if the system be self-contained, ^.e., if it do not receive 
energy from, or part with it to, external bodies. Hence we may 
extend the axiom 3 to the centre of inertia of any such self- 
contained system, and, as will presently be shown, also to the 
motion of the system relative to its centre of inertia. This, though 
not formally identical with Hewton’s Lex III., leads, as we shall see, 
to exactly the same consequences. 
(11) If, for a moment, we confine our attention to a free system 
consisting of two particles only, we have 
^iPi + ^^2^2 = (% + ^2K 
or 
+ = 0 ( 1 ) 
This must be consistent with the conservation of energy, which 
gives 
+ m2P|)=/(T(/)i-p2)) ...... (2) 
since the potential energy must depend (so far as position goes) on 
the distance between the particles only. Comparing (l) and (2) we 
see that we may treat (2) by partial differentiation, so far as the 
coordinates of and are separately concerned. For we thus 
obtain 
m^pi = Vpi “ P2) 
^ 2/^2 ~ Vp2 • f~ ~f •^(Pi ~ ^ 2 )* 
Each of these, again, is separately consistent with the equation 
in § (5) for a lone particle. Hence, again, the integral f{m^fis^ + 
m^v^dsc^ has a stationary value. 
Hence also, whatever be the origin, provided its velocity be 
constant, 
'XmY pp = 0 . 
