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: yi the distance from the axis, and θi the inclination to the axis of the ray at mid-distance between two lenses, after it has crossed i lenses. We have

yi =(1-a2f)yi-1+a(1-a4f)θi-1
θi =-1fyi-1+(1-a2f)θi-1;

Whence

yi+1-2(1-a2f)yi+yi-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 yi+1=-yi=yi-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>14a.

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"