# Jaynes' Derivation of Herschel-Maxwell for Normal Distribution

I am reading the following paper:

http://www-biba.inrialpes.fr/Jaynes/cc07s.pdf

and cannot seem to figure out how Jaynes is deriving (P2) and below (specifically the log arithmetic log[f(x)/f(0)]....)

Can someone help me out?

• Can you put some details in your question please, firstly so readers of your question are not required to go read a paper to understand the question, and secondly, to protect your question from the possibility that the link will disappear (at which point it becomes not so much a useful resource for others as junk cluttering up the site) Commented Sep 1, 2014 at 23:01

Equation $$f(x)f(y)=g(\sqrt{x^2+y^2}) \qquad (*)$$ holds for every $x,y$. If $y=0$, then $(*)$ gives $g(|x|)=f(x)f(0)$ for every $x$. Using this in $(*)$ to "eliminate" $g$, we have $f(x)f(y)=f(\sqrt{x^2+y^2})f(0)$. Dividing by $(f(0))^2$ and taking the log on both sides we find $$\frac{f(x)f(y)}{f(0)f(0)}=\frac{f\left(\sqrt{x^2+y^2}\right)}{f(0)} \, ,$$ $$\log\left(\frac{f(x)}{f(0)}\right)+\log\left(\frac{f(y)}{f(0)}\right)= \log\left( \frac{f\left(\sqrt{x^2+y^2}\right)}{f(0)}\right) \, .$$