1. A particle with mass m and coordinates x1x2, x3 relative to a set of rectangular axes fixed in Newtonian space is moving in a field of conservative forces with a potential energy V(x1, x2, x3) and a kinetic energy
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100030474/resource/name/S0305004100030474_eqnU1.gif?pub-status=live)
The equations of motion, written
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100030474/resource/name/S0305004100030474_eqn1.gif?pub-status=live)
(representing the three equations i = l, i = 2, i = 3 in a way to be used in this paper), constitute, as they stand, a sufficient condition in order to ensure
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100030474/resource/name/S0305004100030474_eqnU2.gif?pub-status=live)
in the sense that the Hamiltonian integral has a stationary value if the actual motion is compared with neighbouring motions with the same terminal positions and the same terminal values of the time as in the actual motion.