444 Prof. Tait on the Laws of Motion. 
(6) It is next shown that the above inertia-condition (that 
the velocity parallel to the equipotential surface is the same 
for two successive elements of the path) at once leads to a 
" stationary " value of the sum of the quantities vols for each 
two successive elements, and therefore for any finite arc, of 
the path. This is, for a single particle, the Principle of Least 
Action, which is thus seen to be a direct consequence of inertia. 
(It is then shown that the results above can be easily ex- 
tended to a particle which has two degrees of freedom only.) 
But it is necessary to remember that, in these cases, we take 
a partial view of the circumstances; for a lone particle cannot 
strictly be said to have potential energy, nor can we conceive 
of a constraint which does not depend upon matter other than 
that which is constrained. Hence the true statement of such 
cases requires further investigation. 
(7) To pass to the case of a system of free particles we require 
some quasi kinematical preliminaries. These are summed up 
in the following self-evident proposition : — If with each par- 
ticle of a system we associate two vectors, e. g. © 1? <E> 1? with 
the mass m 1; &c, we have 
tm®<$>=%(m) . ® <$> + $m6(f>, 
where ■ © = ® + <9, 
and 2m© = 2(m).© , 
2m<l> = 2(w) .<I> ; 
so that © and <I> are the values of © and <£> for the whole 
mass collected at its centre of inertia, and 6, cj> those of the 
separate particles relative to that centre. 
(8) Thus, if © = P = P + p be the vector of m, 
c|> = © = P = P + p 
its velocity, we have 
2?nPP = %(m) . P P + Xmpp, 
the scalar of which is, in a differentiated form, a well-known 
property of the centre of inertia. The vector part shows that 
the sum of the moments of momentum about any axis is equal 
to that of the whole mass collected at its centre of inertia, 
together with those of the several particles about a parallel 
axis through the centre of inertia. 
If © = <£ = P, 
we have 2mP* =2(m) . P* + 2™ p 2 ; 
i. e. the kinetic energy, referred to any point, is equal to that 
