## 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

${y}_{i}$ | $=\left(1-{\displaystyle \frac{a}{2f}}\right){y}_{i-1}+a\left(1-{\displaystyle \frac{a}{4f}}\right){\theta}_{i-1}$ | ||

${\theta}_{i}$ | $=-{\displaystyle \frac{1}{f}}{y}_{i-1}+\left(1-{\displaystyle \frac{a}{2f}}\right){\theta}_{i-1};$ |

Whence

$${y}_{i+1}-2\left(1-\frac{a}{2f}\right){y}_{i}+{y}_{i-1}=0;$$ |

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"