## 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.

Time-stamp: "11.08.2016 23:11"