I have the following paper:


Fritz John, [*The ultrahyperbolic differential equation with four independent variables*](https://projecteuclid.org/euclid.dmj/1077490637), Duke Math. J. 4 (1938), no. 2, pages 300-322  

doi:[10.1215/S0012-7094-38-00423-5](https://doi.org/10.1215/S0012-7094-38-00423-5)


Now I want to check Fritz John's claim in the proof of Theorem 1.1, where he says that equation (7) can be easily verified for the case i=1, k=2.


I used maple to check the validity of this assertion, and I get false.
Can someone help me check if this assertion made in Fritz John's indeed valid?


Thanks in advance.
P.S
I am adding information crucial for this question of mine, since it maybe the case that there are some who cannot view the paper.


We have $\frac{\partial^2 u(x\_1,x\_2,x\_3,x\_4)}{\partial x\_1^2}+\frac{\partial^2 u(x\_1,x\_2,x\_3,x\_4)}{\partial x\_2^2}=\frac{\partial^2 u(x\_1,x\_2,x\_3,x\_4)}{\partial x\_3^2}+\frac{\partial^2 u(x\_1,x\_2,x\_3,x\_4)}{\partial x\_4^2}$.
And $v(\xi\_1,\xi\_2,\xi\_3,\eta\_1,\eta\_2,\eta\_3) = (\sum\_i (q\_i/q\_3)^2)^{1/2}u(\frac{p\_2+q\_2}{q\_3},\frac{-p\_1-q\_1}{q\_3},\frac{p\_2-q\_2}{q\_3},\frac{-p\_1+q\_1}{q\_3})$, where $q\_i=\xi\_i-\eta\_i$, and $p\_i=\xi\_j \eta\_k - \xi\_k \eta\_j$, where $ijk$ is a cyclic permutation of $\{ 1,2,3\}$.
Now, equation (7) is:
$$(\frac{\partial^2}{\partial \xi\_i \partial \eta\_k} - \frac{\partial^2}{\partial \xi\_k \partial \eta\_i})\frac{v}{|\xi - \eta|} = 0$$
where $|\xi - \eta| := \sum\_i (\xi\_i-\eta\_i)^2$.



A question on theorem 1.1 of Fritz John ultrahyperbolic pde
I don't have John's paper available at the moment, but are you sure


$$ | \xi - \eta| = \sum (\xi\_i - \eta\_i)^2 $$


and not


$$ | \xi - \eta|^2 = \sum (\xi\_i - \eta\_i)^2 ? $$


For one, I don't think John would define a "norm" with the wrong scaling, and for two, because if the latter holds you have that 


$$ \frac{v}{ |\xi - \eta|} = \frac{1}{|q\_3|} u( \dots) $$


and noting that $p\_1, p\_2, q\_1, q\_2$ are linear in $\xi\_1, \xi\_2, \eta\_1, \eta\_2$ you can write 


$$ \partial\_{\eta\_1} = \xi\_3 \partial\_1 + \partial\_2 + \xi\_3 \partial\_3 - \partial\_4 $$


with a change of variables and verify that 


$$ \partial^2\_{\xi\_1 \eta\_2} - \partial^2\_{\xi\_2 \eta\_1} = - q\_3 ( \partial^2\_{11} + \partial^2\_{22} - \partial^2\_{33} - \partial^2\_{44} ) $$


so that the claimed equation is true by virtue of $u$ solving the ultrahyperbolic PDE. 


