LettreThomson à Poincaré, 1892-12-23

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2-56-6. William Thomson to H. Poincaré

In train to London Dec. 23/92

Dear Mr. Poincaré,

In writing to you this forenoon before I left Glasgow, I inadvertently said “a quarter of …” instead of “four times the focal length”.¹¹See Thomson to Poincaré, 23.12.1892 (§ 2-56-5). Will you make the correction and kindly excuse my troubling you with it. Here is the whole affair of the periodic lense problem. Let $a$ be the distance from lense to lense, $f$ the focal length of each lense: $y_{i}$ the distance from the axis, and $\theta_{i}$ the inclination to the axis of the ray at mid-distance between two lenses, after it has crossed $i$ lenses. We have

	$\displaystyle y_{i}$	$\displaystyle=\bigl{(}1-\frac{a}{2f}\bigr{)}y_{i-1}+a\bigl{(}1-\frac{a}{4f}% \bigr{)}\theta_{i-1}$
	$\displaystyle\theta_{i}$	$\displaystyle=-\frac{1}{f}y_{i-1}+\bigl{(}1-\frac{a}{2f}\bigr{)}\theta_{i-1};$

Whence

y_{i+1}-2\bigl{(}1-\frac{a}{2f}\bigr{)}y_{i}+y_{i-1}=0;

which shows that when $a$ is between $0$ and $4f$ the inclined ray keeps always infinitely near to the axis.²²In the limiting case of $a=4f$ , we have $y_{i+1}=-y_{i}=y_{i-1},$ such that the light ray’s incidence angle is conserved in the lens system. Hence motion along the axis (in the corresponding kinetic problem) is stable. The inclination increases indefinitely if $f$ is negative, or if $f>\frac{1}{4}a$ .

For one lense we may of course substitute a group of lenses, according to well known principles.

Yours very truly,

Kelvin

ALS 3p. Private collection, Paris 75017.

Métadonnées

Titre: Thomson à Poincaré, 1892-12-23
Incipit: In writing you this forenoon before I left Glasgow ...
Date: 1892-12-23
Expéditeur: Thomson, William (1824-1907)
Destinataire: Poincaré, Henri (1854-1912)
Adresse: Paris
Chapitre: William Thomson
Lieu d’archivage: Private collection 75017
Type: fr Lettre autographe signée
Section (dans le livre): 6
Identifiant dans les archives locales: CD n° 167
Droits: Archives Henri Poincaré
Nombre de pages: 3
Est une partie de: La correspondance entre Henri Poincaré et les physiciens, chimistes et ingénieurs
Langue: en
Éditeur: Archives Henri Poincaré
Licence: CC BY-ND 4.0

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« Thomson à Poincaré, 1892-12-23 ». La Correspondance Entre Henri Poincaré Et Les Physiciens, Chimistes Et ingénieurs. Archives Henri Poincaré, s. d, Archives Henri Poincaré, s. d, La correspondance d'Henri Poincaré, consulté le 24 avril 2024, https://henripoincare.fr/s/correspondance/item/8680