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

January 9, 189311endnote: 1 This letter was penned by a copyist.

The University, Glasgow

Dear Mr. Poincaré,

Your letter of the 26th 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 α2=0\alpha^{2}=0. They occur also when α2=14\alpha^{2}=-\frac{1}{4}; and all the cases of stability are included between α2=0\alpha^{2}=0 and α2=14\alpha^{2}=-\frac{1}{4}. No extension in respect to generality is obtained by considering negative values of α2\alpha^{2} beyond 14-\frac{1}{4}.

The case α2=14\alpha^{2}=-\frac{1}{4} makes λ=1\lambda=-1, (according to your notation § 10 on p. 100). All the cases of stability correspond to values of α2\alpha^{2} between 0 and 14-\frac{1}{4}, or to values of λ\lambda corresponding to real values of kk from 0 to 12\frac{1}{2} in the formula,

λ=cos(2πk)±isin(2πk).\lambda=\cos(2\pi k)\pm i\sin(2\pi k).

All the cases of instability belonging to your cases 11 and 22 correspond to real values of λ\lambda or 1λ\frac{1}{\lambda} from -\infty to 1-1 and from +1+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 λ\lambda or 1λ\frac{1}{\lambda} is equal to

p{cos(2πk)±isin(2πk)},p\left\{\cos(2\pi k)\pm i\sin(2\pi k)\right\},

where pp denotes any real positive numeric and kk may have any value from 0 to 12\frac{1}{2}.22endnote: 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 λ+1/λ=2e\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.33endnote: 3 W. Thomson 1891. Thus we have an algebraic equation for ee of degree nn, instead of λ\lambda of degree 2n2n.

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

Particular cases of equalities among roots depending on annulment of QQ in the expression e=P+iQe=P+iQ* belong to the special limiting case of instability described in your case 3 (when PP is between 1-1 and +1+1), but the algebraic equation for ee 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

PP and QQ being any real numerics.

ALS 4p. Collection particulière, Paris 75017.

Time-stamp: "16.04.2023 15:42"

Notes

  • 1 This letter was penned by a copyist.
  • 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).
  • 3 W. Thomson 1891.

References

  • W. Thomson and P. G. Tait (1879) Treatise on Natural Philosophy, Volume 1, Part 1. Cambridge University Press, Cambridge. link1 Cited by: endnote 2.
  • W. Thomson (1891) On instability of periodic motion. Proceedings of the Royal Society of London 50, pp. 194–200. link1, link2 Cited by: endnote 3.