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) ?