The formula therefore reduces to
δA = [Σm (ẋδx + ẏδy + z˙δz)]t′t + 2Τ′δt′ − 2Τδt.
(7)
Since the terminal configurations are unaltered, we must have at the lower limit
δx + ẋδt = 0, δy + ẏδt = 0, δz + z˙δt = 0,
(8)
with similar relations at the upper limit. These reduce (7) to the form (2).
The equation (2), it is to be noticed, merely expresses that the variation of A vanishes to the first order; the phrase stationary action has therefore been suggested as indicating more accurately what has been proved. The action in the free path between two given configurations is in fact not invariably a minimum, and even when a minimum it need not be the least possible subject to the given conditions. Simple illustrations are furnished by the case of a single particle. A particle moving on a smooth surface, and free from extraneous force, will have its velocity constant; hence the theorem in this case resolves itself into
δ ∫ ds = 0,
(9)