The value of 'c' according to this formula is given by:

$c = \frac{1}{N} m^{1/6}$

where,

m = hydraulic mean depth

N = Manning's constant

Now,

$v = c \sqrt{m_i}$

$= \frac{1}{N} m^{1/6} \sqrt{m_i}$

$= \frac{1}{N} m^{(\frac{1}{6}+ \frac{1}{2})} i^{1/2}$

$= \frac{1}{N} m^{(\frac{1+3}{6})} i^{1/2}$

$\therefore$ $ [ v = \frac{1}{N} m^{2/3} i^{1/2} ]$