## 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 and Tait, 1879, 370). They write the equation of
motion, the general solution of which is the sum of special solutions of the
form (updating the notation):
$\Psi_{1}=a_{1}e^{\lambda t},\Psi_{2}=a_{2}e^{\lambda t}.$
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
$te^{\lambda t}$ (or $t\cos\rho t$), and therefore that for stability
the roots must be all unequal” (Thomson and Tait (1879, 381), replacing
$\epsilon$ with $e$). The authors were apparently unaware that
Weierstrass had pointed out – and corrected – 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
$\nicefrac{{1}}{{4}}$
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: "30.06.2023 11:47"

### 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 and Tait, 1879, 370). They write the equation of motion, the general solution of which is the sum of special solutions of the form (updating the notation): $\Psi_{1}=a_{1}e^{\lambda t},\Psi_{2}=a_{2}e^{\lambda t}.$ 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 $te^{\lambda t}$ (or $t\cos\rho t$), and therefore that for stability the roots must be all unequal” (Thomson and Tait (1879, 381), replacing $\epsilon$ with $e$). The authors were apparently unaware that Weierstrass had pointed out – and corrected – 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.”