A liquid has been poured into a cylindrical vessel of mass M (the mass of the vessel bottom can be ignored) and height H. The linear density of the liquid, that is the ratio of the mass of the liquid column to its height is  $\lambda$. The height x of the column of liquid at which the common center of gravity of the liquid plus the vessel is in the lowest position,

The height of the center of mass of the vessel with the liquid is determined by the formula  $h_c=\frac{M(H/2)+m(x/2)}{M+m}$, where m is the mass of the liquid. We rewrite by replacing the mass of the liquid with $\lambda x$ :
 $h_c=\frac{1}{2}\frac{MH+\lambda x^2}{M+\lambda x}$
Nullifying the derivative of h with respect to x,  $\frac{d{h}_c}{dx}=\frac{1}{2}\frac{2\lambda x(M+\lambda x)-\lambda (MH+\lambda x^2)}{{(M+\lambda x)}^2}=0$
We get,  $x=\pm \sqrt{\frac{M^2}{{\lambda }^2}+\frac{MH}{\lambda }}-\frac{M}{\lambda }$
Of course, only the positive value of the root has physical meaning. Substituting these values, we will find the position of the centre of mass. After elementary transformations we get
 
$h_c=\sqrt{\frac{M^2}{{\lambda }^2}+\frac{MH}{\lambda }}-\frac{M}{\lambda }$