text: # Transcription
  
  ## 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.
  
  Titre
  Thomson à Poincaré, 1892-12-23
  Incipit
  In writing you this forenoon before I left Glasgow ...
  Date
  1892-12-23
  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
  Langue
  en
  Licence
  CC BY-ND 4.0
  
  # Citer ce document
  
  « 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 juin 2024, https://henripoincare.fr/s/correspondance/item/8680
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