## 2-56-8. William Thomson to H. Poincaré

January 9, 1893^{1}^{1}endnote:
^{1}
This letter was penned by a copyist.

The University, Glasgow

Dear Mr. Poincaré,

Your letter of the 26^{th} followed me to England where I have been
passing the holidays. I am sorry thus to have been prevented from
writing to you sooner in answer.

I agree with you substantially in respect to the three cases which you describe, and which in reality comprehend the whole subject in question; except in a not unimportant addition to your case N° 1. The cases of instability are not found only for $\alpha^{2}=0$. They occur also when $\alpha^{2}=-\frac{1}{4}$; and all the cases of stability are included between $\alpha^{2}=0$ and $\alpha^{2}=-\frac{1}{4}$. No extension in respect to generality is obtained by considering negative values of $\alpha^{2}$ beyond $-\frac{1}{4}$.

The case $\alpha^{2}=-\frac{1}{4}$ makes $\lambda=-1$, (according to your notation § 10 on p. 100). All the cases of stability correspond to values of $\alpha^{2}$ between $0$ and $-\frac{1}{4}$, or to values of $\lambda$ corresponding to real values of $k$ from $0$ to $\frac{1}{2}$ in the formula,

$\lambda=\cos(2\pi k)\pm i\sin(2\pi k).$ |

All the cases of instability belonging to your cases $1$ and $2$ correspond to real values of $\lambda$ or $\frac{1}{\lambda}$ from $-\infty$ to $-1$ and from $+1$ to $+\infty$.

There are also the cases of instability belonging to case 3 of your letter. These correspond to pairs of cases in which $\lambda$ or $\frac{1}{\lambda}$ is equal to

$p\left\{\cos(2\pi k)\pm i\sin(2\pi k)\right\},$ |

where $p$ denotes any real positive numeric and $k$ may have any value
from $0$ to $\frac{1}{2}$.^{2}^{2}endnote:
^{2}
The term *numeric* is a neologism signifying any numerical
expression. It was introduced by James Thomson, as noted by his brother William
(Thomson and Tait, 1879, 389).

The consideration of all these cases is facilitated by putting
$\lambda+1/\lambda=2e$, as I have done in a short paper,
“Instability of Periodic Motion”, of which I send you a copy by
book-post.^{3}^{3}endnote:
^{3}
W. Thomson 1891. Thus we have an
algebraic equation for $e$ of degree $n$, instead of $\lambda$ of
degree $2n$.

All cases of stability correspond to real values of $e$ between $-1$ and $+1$. The case of instability belonging to your cases 1 and 2 corresponds to values of $e$ from $+1$ to $\infty$; and with the extension I have indicated, to values of $e$ from $-\infty$ to $-1$. All the cases of instability belonging to your case 3 correspond to complex values of $e$.

Particular cases of equalities among roots depending on annulment of $Q$ in the expression $e=P+iQ$* belong to the special limiting case of instability described in your case 3 (when $P$ is between $-1$ and $+1$), but the algebraic equation for $e$ may be, and generally is, such that equal roots not bordering on imaginary roots can occur. This class of equal roots does not involve any tendency to instability or any seeming indeterminateness in the assignment of all the constants required for a complete solution.

Yours very truly,

Kelvin

* $P$ and $Q$ being any real numerics.

ALS 4p. Collection particulière, Paris 75017.

Time-stamp: "16.04.2023 15:42"