154 
THE QUANTUM [ Xi . 3 
means of which measurements with reference to one particular 
frame of reference may be transformed so as to apply with 
respect to a second frame moving with velocity v with regard 
to the first. 
The fundamental electrodynamic equations are usually based 
on the experimental conclusions of Coulomb, Ampère, and Fara 
day. The Maxwell field equations, which are used in describing 
a beam of light, will always be the same, even though a second 
frame of reference is employed moving with high velocity with 
respect to the first, provided the transformation is that given 
by Lorentz : 
a;' = ${% — vt), y' = y, z' — z, V = (}(t — vx/c) n : i 
where = \/(i — v 2 /c 2 ). 
The equations are said to be covariant with respect to the 
transformation of co-ordinates. 
Einstein, in his general theory of relativity, has taken a 
broader view, and stated as a fundamental principle : “ The 
general laws of nature are expressed by mathematical equations 
which hold for all systems of co-ordinates, that is, they are 
co variant with respect to arbitrary substitutions.” 
Adopting this standpoint Leigh Page in An Introduction to 
Electrodynamics, instead of deducing the electrodynamic equa 
tions from the experimental results has pursued the opposite 
course and derived them from the principle of relativity. 
Einstein introduced a general non-Euclidean four-dimensional 
time-space, and enunciated his law of motion by saying : “ Par 
ticles which are not interfered with follow a geodesic line in the 
manifold.” A geodesic in curved space or on a surface corre 
sponds to a straight line in flat space. Larmor has laid stress 
on the application of the principle of least action in mechanical 
and electrodynamic problems. For example, an electron in mov 
ing through a magnetic field follows a geodesic line on a surface. 
It is this principle which led Einstein to his mode of stating 
the motion of bodies in space-time. The quantum theory may 
perhaps be thought of as providing the time-space world with 
certain partitions which impose restrictions on the motion of 
electrons or magnetons, which follow geodesic paths in accord 
ance with the principle of minimum action. As Murnaghan * has 
expressed it : “ Our guiding idea (the impossibility of action at 
a distance) will prompt us to say, following the example of 
Faraday in his electrical researches, that the geodesics of a gravi 
tational space have a physical existence as distinct from a mere 
mathematical one.” 
The chief advantage of the variation principle of Hamilton 
(page 20) is its independence of the system of co-ordinates. 
* Murnaghan in Bird’s Relativity and Gravitation, p. 286 (Methuen).