> 
> In his paper "On the Electrodynamics of Moving Bodies", Einstein writes the equation
> 
> 
> $$\dfrac{\partial \tau}{\partial x'}+\dfrac{v}{c^2-v^2}\dfrac{\partial \tau}{\partial t}=0$$
> 
> 
> where
> 
> 
> * $\tau=\tau(x',y,z,t)$ is a linear function (i.e. $\tau=Ax'+By+Cz+Dt$)
> * $x'=x-vt$
> * $\dfrac{\partial \tau}{\partial y}=0$ (i.e. $B=0$)
> * $\dfrac{\partial \tau}{\partial z}=0$ (i.e. $C=0$)
> * $c$ is a constant
> * $x,x',y,z,t,v$ are variables
> 
> 
> and he derives that
> 
> 
> $$\tau=a\left(t-\dfrac{v}{c^2-v^2}x'\right)$$
> 
> 
> where $a=a(v)$
> 
> 
> 

Could someone please walk me through step-by-step how he derived this? I am not very familiar with integrals invovling partial derivatives, so I would be appreciative if any answers are quite explicit.

Additionally, if I modified the question to say that $\tau$ was an affine function (i.e. $\tau=Ax'+By+Cz+Dt+E$), would it make any difference to the result (I suspect it wouldn't)?