The current through the potentiometer wire is
$I = {{{E_0}} \over {\left( {r + {r_1}} \right)}}$

and the potential difference across the wire is

$V = Ir = {{{E_0}r} \over {\left( {r + {r_1}} \right)}}$

The potential gradient along the potentiometer wire is

$k = {V \over L} = {{{E_0}r} \over {\left( {r + {r_1}} \right)L}}$

As the unknown e.m.f. E is balanced against length l of the potentiometer wire,

$\therefore$ $E = kl = {{{E_0}r} \over {\left( {r + {r_1}} \right)}}{l \over L}$