Given $$ g(\tau_e, t, \vec x, \vec \zeta) \equiv \tau_e - t + \frac{1}{c_0} |\vec x - \vec y(\vec \zeta, \tau_e)| = 0 $$
The textbook "Aeroacoustics" said applying for chain rule, will have $$ \left(\frac{\partial g}{\partial x_j}\right)_{\tau_e=const} + \left(\frac{\partial g}{\partial \tau_e}\right)_{\vec x=const}\frac{\partial \tau_e}{\partial x_j} =0$$
How did it come?
I guess it was because
assuming $$\vec y = \text{const}$$ $$ \vec \zeta = \text{const}$$
so consider $\tau_e$ as a function of $x_j$, the result is the got.