15 
of Edinburgh, Session 1882-83. 
some quasi-kinematical preliminaries. These are summed up in the 
following self-evident proposition : — If with each particle of a system 
we asssciate two vectors, e.g., ©j, with the mass &c., we have 
^m©$ = %{m) . 
where 
® = ©o + ^, 
d> = d>o + <^, 
and 
. ©0 
^m<J> = ^{m) . <I>Q, 
so that ©0 and are the values of © and $ for the whole mass 
collected at its centre of inertia ; and 0, those of the separate 
particles relative to that centre. 
(8) Thus, if © = P = Pq -P p he the vector of d> = © = P = P^ -I- p, 
its velocity, we have 
^mVV = S(m) . PqPq '%mpp 
the scalar of which is, in a differentiated form, a well-known pro- 
perty 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 
SmP2 = :S(m).P2 + Smp2, 
^.e., the kinetic energy, referred to any point, is equal to that 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 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 w^e 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 inde- 
pendent alike of the configuration of the system and of its position 
in space. 
