14 
Proceedings of the Boy al Society 
wise, it would be possible for a human mind to have knowledge of 
absolute position and of absolute velocity.) 
(5) But Maxwell’s reasoning is easily seen to apply equally to 
any component of the velocity. Hence, when we come to the case 
in which the potential energy depends on the position, the only 
change in the particle’s motion at any instant is a change of the 
speed in the normal to the equipotential surface on which the 
particle is at that instant situated. The conservation of energy 
assigns the amount of this change, and thus the motion is com- 
pletely determined. In fact, if a? be perpendicular to the equi- 
potential surface, the equation 
-f 2/^ + — const, 
gives 
mx= - 
dY 
dx ’ 
Generally, in the more expressive 
for y and z are independent of x. 
language of quaternions, 
mp— - vy. 
In fact, this problem is precisely the same as was that of the 
motion of a luminous corpuscle in a non-homogeneous medium, 
the speed of passing through any point of the medium being 
assigned. 
(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 vds 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 extended 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 con- 
strained. Hence the true statement of such cases requires further 
investigation. 
(7) To pass to the case of a system of free particles we require 
