2-56-5. William Thomson to H. Poincaré
December 23 1892^{1}^{1}endnote: ^{1} Only the final lines are in Thomson’s hand.
The University, Glasgow
Dear Mr. Poincaré,
I don’t know whether you have ever chanced to see a statement in Thomson and Tait’s Natural Philosophy § 343 (m), (Vol. 1, Pt. 1, pp. 381–2) in which I pointed out a remarkable error into which both Lagrange and Laplace had fallen, regarding equal roots of the determinantal equation for the problem of infinitely small oscillations about a position of equilibrium.^{2}^{2}endnote: ^{2} Thomson and Tait study a system in motion where the “positional forces” are proportional to displacements and the “motional forces” are proportional to velocities (Thomson & Tait 1879, I, 370). They write the equation of motion, the general solution of which is the sum of special solutions of the form (updating the notation): $${\mathrm{\Psi}}_{1}={a}_{1}{e}^{\lambda t},{\mathrm{\Psi}}_{2}={a}_{2}{e}^{\lambda t}.$$ The roots $\lambda $ are found by introducing the latter sum in the equation of motion. For the case of small oscillations, Thomson and Tait characterize the error of Lagrange et Laplace, both of whom “fell into the error of supposing that equality among roots necessarily implies terms in the solution of the form $t{e}^{\lambda t}$ (or $t\mathrm{cos}\rho t$), and therefore that for stability the roots must be all unequal.” The authors were apparently unaware that Weierstrass had pointed out this error in 1858. I find quite a corresponding idea in § 10 p. 101 of your great paper “Sur le problème des trois corps et les équations de la dynamique”.^{3}^{3}endnote: ^{3} Poincaré 1890. Poincaré defines “exposants caractéristiques” $\alpha $, where the stability coefficients ${\alpha}^{2}$ are analogous to Thomson and Tait’s roots $\lambda $. He concludes as follows: “Si donc tous les coefficients ne sont pas réels, distincts, il n’y a pas en général de stabilité temporaire.” See also Poincaré to W. Thomson, 26.12.1892 (§ 2-56-7). That it is an error I think you will see by considering the simple case of a ray of light passing along the axis of a periodic arrangement of lenses. For the very simplest case consider an infinite number of convex lenses placed perpendicularly to a straight line with their centres at equal distances along it. It is easy (and very interesting, though very simple) to find the conditions of stability for a ray diverting infinitesimally from this straight line. But now what I want to ask you to think of is the corresponding problem when each lens as different vergenties for rays incident upon it in different planes through its axis as, for example, if each lens is bounded by a spherical surface on one side and a cylindrical surface on the other. Your general equations would give the full condition of stability for a ray infinitely near the axis in this case. But according to your statement in § 10 the equilibrium would be essentially unstable when the two principal vergenties are equal, that is to say, in the well known case of ordinary lenses, whereas you will see in a moment that for example for the case of convex lenses there is stability for a ray infinitely near the axis if the distance from lens to lens be less than $1/4$ the focal length.^{4}^{4}endnote: ^{4} As Thomson observed in his follow-up letter (§ 2-56-6), this should read: “the distance from lens to lens be less than four times the focal length.”
I hope you will kindly excuse my writing to you to point out what seems to be a mistake in a work for which we have admiration quite comparable with the admiration we have for the greatest works of Lagrange and Laplace.
Wishing you all the best wishes of this season. I remain, dear Mr. Poincaré, yours very truly,
Kelvin
ALS 1p. Private collection, Paris 75017.
Time-stamp: " 3.05.2019 01:30"
Notes
- ^{1} Only the final lines are in Thomson’s hand.
- ^{2} Thomson and Tait study a system in motion where the “positional forces” are proportional to displacements and the “motional forces” are proportional to velocities (Thomson & Tait 1879, I, 370). They write the equation of motion, the general solution of which is the sum of special solutions of the form (updating the notation): $${\mathrm{\Psi}}_{1}={a}_{1}{e}^{\lambda t},{\mathrm{\Psi}}_{2}={a}_{2}{e}^{\lambda t}.$$ The roots $\lambda $ are found by introducing the latter sum in the equation of motion. For the case of small oscillations, Thomson and Tait characterize the error of Lagrange et Laplace, both of whom “fell into the error of supposing that equality among roots necessarily implies terms in the solution of the form $t{e}^{\lambda t}$ (or $t\mathrm{cos}\rho t$), and therefore that for stability the roots must be all unequal.” The authors were apparently unaware that Weierstrass had pointed out this error in 1858.
- ^{3} Poincaré 1890. Poincaré defines “exposants caractéristiques” $\alpha $, where the stability coefficients ${\alpha}^{2}$ are analogous to Thomson and Tait’s roots $\lambda $. He concludes as follows: “Si donc tous les coefficients ne sont pas réels, distincts, il n’y a pas en général de stabilité temporaire.” See also Poincaré to W. Thomson, 26.12.1892 (§ 2-56-7).
- ^{4} As Thomson observed in his follow-up letter (§ 2-56-6), this should read: “the distance from lens to lens be less than four times the focal length.”