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”. 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

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

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

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. 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.

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 $a=4f$, we have $$y_{i+1}=-y_{i}=y_{i-1},$$ such that the light ray’s incidence angle is conserved in the lens system.