Under what conditions current passing through the resistance $R$ can be increased by short circuiting the battery of emf $E_2$. The internal resistances of the two batteries are $r_1$ and $r_2$, respectively :

$E_1 r_2>E_2\left(R+r_1\right)$

Current, $i_1=\frac{E_1+E_2}{r_1+r_2+R}$   and   $i_2=\frac{E_1}{R+r_1}$

∴  $\frac{E_1}{R+r_1}>\frac{E_1+E_2}{r_1+r_2+R}$

or $E_1 r_2>E_2\left(R+r_1\right)$