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’’.11endnote: 1 See 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 aa be the distance from lense to lense, ff the focal length of each lense: yiy_{i} the distance from the axis, and θi\theta_{i} the inclination to the axis of the ray at mid-distance between two lenses, after it has crossed ii lenses. We have

yi\displaystyle y_{i} =(1-a2f)yi-1+a(1-a4f)θi-1\displaystyle=\bigl{(}1-\frac{a}{2f}\bigr{)}y_{i-1}+a\bigl{(}1-\frac{a}{4f}% \bigr{)}\theta_{i-1}
θi\displaystyle\theta_{i} =-1fyi-1+(1-a2f)θi-1;\displaystyle=-\frac{1}{f}y_{i-1}+\bigl{(}1-\frac{a}{2f}\bigr{)}\theta_{i-1};

Whence

yi+1-2(1-a2f)yi+yi-1=0;y_{i+1}-2\bigl{(}1-\frac{a}{2f}\bigr{)}y_{i}+y_{i-1}=0;

which shows that when aa is between 0 and 4f4f the inclined ray keeps always infinitely near to the axis.22endnote: 2 In the limiting case of a=4fa=4f, we have yi+1=-yi=yi-1,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 ff is negative, or if f>14af>\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.

Time-stamp: " 3.05.2019 01:30"

Notes

  • 1 See Thomson to Poincaré, 23.12.1892 (§ 2-56-5).
  • 2 In the limiting case of a=4fa=4f, we have yi+1=-yi=yi-1,y_{i+1}=-y_{i}=y_{i-1}, such that the light ray’s incidence angle is conserved in the lens system.