with a similar expression for Δpr. Hence the right-hand side of (2) becomes
− Σr {Σs(r, s) δqs + Σs(r, s′) δq′s} Δqr + Σr {Σs(r, s)Δqs + Σs(r, s′) Δq′s} δqr
= ΣrΣs(r, s′) {δqr·Δq′s − Δqr·δq′s}.
(5)
The same value is obtained in like manner for the expression on the left hand of (2); hence the theorem, which, in the form (1), is due to Lagrange, and was employed by him as the basis of his method of treating the dynamical theory of Variation of Arbitrary Constants.
The formula (2) leads at once to some remarkable reciprocal relations which were first expressed, in their complete form, by Helmholtz. Consider any natural motion of a conservative system between two configurations O and O′ Helmholtz’s reciprocal theorems. through which it passes at times t and t′ respectively, and let t′ − t = τ. As the system is passing through O let a small impulse δpr be given to it, and let the consequent alteration in the co-ordinate qs after the time τ be δq′s. Next consider the reversed motion of the system, in which it would, if undisturbed, pass from O′ to O in the same time τ. Let a small impulse δp′s be applied as the system is passing through O′, and let the consequent change in the co-ordinate qr after a time τ be δqr. Helmholtz’s first theorem is to the effect that
δqr : δp′s = δq′s : δpr.
(6)
To prove this, suppose, in (2), that all the δq vanish, and likewise all the δp with the exception of δpr. Further, suppose all the Δq′ to vanish, and likewise all the Δp′ except Δp′s, the formula then gives
δpr·Δqr = −Δp′s·δq′s,