2-56-8. William Thomson to H. Poincaré
January 9, 189311 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 . They occur also when ; and all the cases of stability are included between and . No extension in respect to generality is obtained by considering negative values of beyond .
The case makes , (according to your notation § 10 on p. 100). All the cases of stability correspond to values of between and , or to values of corresponding to real values of from to in the formula,
All the cases of instability belonging to your cases and correspond to real values of or from to and from to .
There are also the cases of instability belonging to case 3 of your letter. These correspond to pairs of cases in which or is equal to
where denotes any real positive numeric and may have any value from to .22 2 The term numeric is a neologism signifying any numerical expression. It was introduced by James Thomson, as noted by his brother William (Thomson & Tait 1879, I, 389).
The consideration of all these cases is facilitated by putting , as I have done in a short paper, “Instability of Periodic Motion”, of which I send you a copy by book-post.33 3 W. Thomson 1891. Thus we have an algebraic equation for of degree , instead of of degree .
All cases of stability correspond to real values of between and . The case of instability belonging to your cases 1 and 2 corresponds to values of from to ; and with the extension I have indicated, to values of from to . All the cases of instability belonging to your case 3 correspond to complex values of .
Particular cases of equalities among roots depending on annulment of in the expression * belong to the special limiting case of instability described in your case 3 (when is between and ), but the algebraic equation for 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,
* and being any real numerics.
ALS 4p. Collection particulière, Paris 75017.
Time-stamp: "22.09.2014 10:11"