# George Green's definition of Green's function

This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\delta(x−x′)$ function as their source on the RHS. But the Dirac-delta was defined about a century after Green's work.

I just want to know, how George Green have defined his function (in case he defined it)? In case he didn't define his Green's function, how did he give a hint about this method (without Dirac-delta function) ?

It is by no means necessary to introduce the Dirac delta. For instance, $G(x,x')$, singular for $x=x'$ is a Green's function of $\Delta$ if, defining, $$\phi(x):= \int_A G(x,x')f(x') dx'$$ we have $$\Delta \phi(x) = f(x)$$ where further details on the behaviour of $\phi$ and the regularity of $f$ and the nature of the integration domain $A$ are assumed and I omit them here...
The general idea is that $G$ is the integral kernel of the inverse of a given linear operator, nothing further. In some cases the inverse exists and is described by an integral kernel. Under some regularity hypotheses this integral kernel is a distribution in a proper sense and the equation defining it can be recast into the modern language of distributions.