446 Prof. Tait on the Laws of Motion. 
Each of these, again, is separately consistent with the equa- 
tion in § 5 for a lone particle. Hence, again, the integral 
J (mjt^dJsi + m 2 v 2 ds 2 ) 
has a stationary value. 
Hence also, whatever be the origin, provided its velocity be 
constant, 
Thus, even when there is a transformation of the energy of 
the system, the results of § 9 still hold good. 
And it is to be observed that if one of the masses, say m 8 , 
is enormously greater than the other, the equation 
m 1 p l + m 2 p 2 =0 
shows that "p 2 is excessively small, and the visible change of 
motion is confined to the smaller mass. Carrying this to the 
limit, we have the case of motion about a (so-called) " fixed 
centre.'' In such a case it is clear thaj, though the momenta 
of the two masses relative to their centre of inertia are equal 
and opposite, the kinetic energy of the greater mass vanishes 
in comparison with that of the smaller. 
These results are then extended to any self-contained system 
of free particles, and the principle of Varying Action follows 
at once. It is thus seen to be a general expression of the 
three propositions of § 2 above. 
(12) So far as we have gone, nothing has been said as 
to koto the mutual potential energy of two particles depends 
on their distance apart. If we suppose it to be enormously 
increased by a very small increase of distance, we have prac- 
tically the case of two particles connected by an inextensible 
string — as a chain-shot. But from this point of view such 
cases, like those of connexion by an extensible string, fall under 
the previous categories. 
The case of impact of two particles falls under the same 
rules, so far as motion of the centre of inertia and moment of 
momentum about that centre are concerned. The conserva- 
tion of energy requires in such cases the consideration of the 
energy spent in permanently disfiguring the impinging bodies, 
setting them into internal vibration, or heating them. But 
the first and third of these, at least, are beyond the scope of 
abstract dynamics . 
(13) The same may be said of constraint by a curve or 
surface, and of loss of energy by friction or resistance of a 
medium. Thus a constraining curve or surface must be looked 
upon (like all physical bodies) as deformable; but, if neces- 
