(7)
which is equivalent to Helmholtz’s result, since we may suppose the symbol Δ to refer to the reversed motion, provided we change the signs of the Δp. In the most general motion of a top ([Mechanics], § 22), suppose that a small impulsive couple about the vertical produces after a time τ a change δθ in the inclination of the axis, the theorem asserts that in the reversed motion an equal impulsive couple in the plane of θ will produce after a time τ a change δψ, in the azimuth of the axis, which is equal to δθ. It is understood, of course, that the couples have no components (in the generalized sense) except of the types indicated; for instance, they may consist in each case of a force applied to the top at a point of the axis, and of the accompanying reaction at the pivot. Again, in the corpuscular theory of light let O, O′ be any two points on the axis of a symmetrical optical combination, and let V, V′ be the corresponding velocities of light. At O let a small impulse be applied perpendicular to the axis so as to produce an angular deflection δθ, and let β′ be the corresponding lateral deviation at O′. In like manner in the reversed motion, let a small deflection δθ′ at O′ produce a lateral deviation β at O. The theorem (6) asserts that
| β | = | β′β′ | , |
| V′δθ′ | Vδθ |
(8)
or, in optical language, the “apparent distance” of O from O′ is to that of O′ from O in the ratio of the refractive indices at O′ and O respectively.
In the second reciprocal theorem of Helmholtz the configuration O is slightly varied by a change δqr in one of the co-ordinates, the momenta being all unaltered, and δq′s is Helmholtz’s second reciprocal theorem. the consequent variation in one of the momenta after time τ. Similarly in the reversed motion a change δp′s produces after time τ a change of momentum δpr. The theorem asserts that
δp′s : δqr = δpr : δq′s
(9)
This follows at once from (2) if we imagine all the δp to vanish, and likewise all the δq save δqr, and if (further) we imagine all the Δp′ to vanish, and all the Δq′ save Δq′s. Reverting to the optical illustration, if F, F′, be principal foci, we can infer that the convergence at F′ of a parallel beam from F is to the convergence at F of a parallel beam from F′ in the inverse ratio of the refractive indices at F′ and F. This is equivalent to Gauss’s relation between the two principal focal lengths of an optical instrument. It may be obtained otherwise as a particular case of (8).
We have by no means exhausted the inferences to be drawn from Lagrange’s formula. It may be noted that (6) includes as particular cases various important reciprocal relations in optics and acoustics formulated by R.J.E. Clausius, Helmholtz, Thomson (Lord Kelvin) and Tait, and Lord Rayleigh. In applying the theorem care must be taken that in the reversed motion the reversal is complete, and extends to every velocity in the system; in particular, in a cyclic system the cyclic motions must be imagined to be reversed with the rest. Conspicuous instances of the failure of the theorem through incomplete reversal are afforded by the propagation of sound in a wind and the propagation of light in a magnetic medium.