## 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’’.^{1}^{1}endnote:
^{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 $a$ 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.^{2}^{2}endnote:
^{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. 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"