Hollymeade
Adlington Road
Wilmslow
Cheshire
Dear Philip
Many thanks for your congratulatory letter. It is very gratifying to be about to join the Olympians.
The 'waves on cows' is just an example in my mathematical theory of embryology which I am busy on now. 'Waves on leopards' are rather more elementary. A leopard skin, before the spots arise is supposed an infinite thin sheet containing two chemical substances with concentrations U,V which react and diffuse. This gives you equations of the form
\partial U/\partial t = f(U,V) + mu \nabla^2 U
\partial V/\partial t = g(U,V) + nu \nabla^2 V
If you take f and g to be linear (near U0, V0) and the whole system to be just unstable you find if the constants satisfy certain inequalities, that U=U0+\alpha Z, V=V_0+\beta Z where Z satisfies a 'wave equation'
\nabla^2 Z=4pi^2/\lambda^2 Z
the wavelength \lambda depending on the reaction rates f, g and diffusion rates \mu, \nu. What particular solution you get will depend on random disturbances just before instability started. Roughly speaking you get a 'random solution'. By assuming that this is black when Z>Z0, yellow for Z<Z0 you get very reasonable leopard skins. Certain slight variation of assumptions give you giraffes, zebras, cows. Cows are dappled.
I am delighted to hear Maurice Pryce is also in the list. I met him first when up for scholarship exam in 1929, but knew him best in Princeton. He was quite my chief flame at one time.
Yours ever
Alan
I hope I am not described as 'distinguished for work on unsolvable problems'
[No date. Filed with an envelope postmarked 17 Apr 1952 and another letter recording the trial and discussing a forthcoming visit to Cambridge on May 2nd. ]