I am trying to read Fritz John's article, here:

And for the proof of Thm 1.1 in page 303 he writes that one can easily verify the equation (7) is satisfied for $i=1,k=2$, is it really that easy, perhaps I am missing some symmetry here?

I got so far to:

$$[\frac{\partial^2}{\partial \xi_1 \partial \eta_2 } - \frac{\partial^2}{\partial \xi_2 \partial \eta_1} ] \frac{v(\xi , \eta)}{\sum_i (\xi_i -\eta_i)^2} = $$

$$= \frac {\(\frac{\partial^2 v}{\partial \xi_1 \partial \eta_2} -\frac{\partial^2 v }{\partial \xi_2 \partial \eta_1})|\xi-\eta| +2|\xi-\eta|^2[(\xi_2-\eta_2)v_{\xi_1}-(\xi_1-\eta_1)v_{\xi_2}]}{|\xi-\eta|^4}$$

Now somehow I need to calculate another several derivatives wrt to eqaution (6) in the paper, is there some easy way to calculate that I am missing, he writes that it should be easy... :-D

Thanks in advance, Alan.