(17)
where terms have been cancelled in virtue of § 5 (2). The last member of (17) represents a variation of the integral
∫t′t (T − V) dt
on the supposition that δX = 0, δX′ = 0, δX″ = 0, ... throughout, whilst δq1, δq2, δqm vanish at times t and t′; i.e. it is a variation in which the initial and final configurations are absolutely unaltered. It therefore vanishes as a consequence of the Hamiltonian principle in its original form.
Larmor has also given the corresponding form of the principle of least action. He shows that if we write
A = ∫ (2T − κχ˙ − κ′χ˙′ − κ″χ˙″ − ...) dt,
(18)
then
δA = 0,
(19)