Kelvin’s circulation theorem tells  that the time rate of change of circulation around a closed curve consisting of the same fluid elements is zero,that is $\frac{D\tau}{Dt}=0$.

Circulation is defined as $$\tau=\oint_{c}\overrightarrow{V}\cdot \overrightarrow{ds}$$

$$\frac{D\tau}{Dt}=\oint_{c}\frac{D\overrightarrow{V}}{Dt}\cdot \overrightarrow{ds}+\oint_{c}\overrightarrow{V}\cdot \overrightarrow{ds}$$

$$\frac{D\overrightarrow{ds}}{Dt}=\overrightarrow{dV}$$

$$\oint_{c}\overrightarrow{v}\cdot \overrightarrow{dV}=\oint_{c}d\left ( \frac{V^{2}}{2} \right )=0$$

$$\frac{D\overrightarrow{V}}{dt}=-\frac{1}{\rho}\nabla p$$

$$\oint_{c}\frac{D\overrightarrow{V}}{Dt}\cdot \overrightarrow{ds}=-\oint_{c}\frac{1}{\rho}\nabla p\cdot \overrightarrow{ds}=-\oint_{c}\frac{dp}{\rho}$$

when $\rho$=constant or$\rho$=$\rho(p)$,then

$$-\oint_{c}\frac{dp}{\rho }=0$$

Therefore $$\oint_{c}\frac{D\overrightarrow{V}}{Dt}\cdot \overrightarrow{ds}=0$$

or $$\frac{D\tau}{Dt}=0$$