Prof. Tait on the Laws of Motion. 445 
of the mass collected at its centre of inertia, together with that 
of the separate particles relative to the centre of inertia. 
If we integrate this expression, multiplied by dt, between 
any limits, we easily obtain a similar theorem with regard to 
the Action of the system. 
Such theorems may be multiplied indefinitely. 
(9) From those just given, however, if we take them along 
with 3 above, we see at once that, provided the particles of the 
system be all free, while the energy of each is purely kinetic 
and independent alike of the configuration of the system and 
of its position in space: — 
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. X \mvds is obviously a minimum. 
(10) The result of (7) points to an independence between 
two parts of the motion of a system, i. e. that relative to the 
centre of inertia and that of the whole mass supposed concen- 
trated at the centre of inertia. Maxwell's reasoning is appli- 
cable directly to the latter if the system be self-contained, i. 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 Newton's 
Lex III., leads, as we shall see, to exactly the same con- 
sequences. 
(11) If, for a moment, we confine our attention to a free 
system consisting of two particles only, we have 
m iPx + m zP2 = ( m i + m 2 )u, 
or 
wh/>i + »n 2 p 2 = («) 
This must be consistent with the conservation of energy, which 
gives 
h{m lP \ + m 2 p^)=f\ll(p l -p 2 )\ . . . (£) 
since the potential energy must depend (so far as position 
goes) on the distance between the particles only. Comparing 
(«) and (/3), we see that we may treat (/3) by partial differ- 
entiation, so far as the coordinates of ?% and m 2 are separately 
concerned. For we thus obtain 
^iPi=V Pl •/=/'• U(pi-p 8 ), 
m 2 p 2 = V P2 •/= -/' • U(px-p 2 ). 
