Answer
$$
\begin{aligned}
\frac{\partial^{2} z}{\partial t^{2}} &= \\
&=\frac{\partial^{2} z}{\partial x^{2}}\left(\frac{\partial x}{\partial t}\right)^{2}+2 \frac{\partial^{2} z}{\partial x \partial y} \frac{\partial x}{\partial t} \frac{\partial y}{\partial y^{2}}\left(\frac{\partial y}{\partial t}\right)^{2}+\frac{\partial^{2} x}{\partial t^{2}} \frac{\partial z}{\partial x}+\frac{\partial^{2} y}{\partial t^{2}} \frac{\partial z}{\partial y},
\end{aligned}
$$
and
$$
\begin{aligned}
\frac{\partial^{2} z}{\partial s \partial t} &=\frac{\partial^{2} z}{\partial x^{2}} \frac{\partial x}{\partial s} \frac{\partial x}{\partial t}+\frac{\partial^{2} z}{\partial x \partial y}\left(\frac{\partial y}{\partial s} \frac{\partial x}{\partial t}+\frac{\partial y}{\partial t} \frac{\partial x}{\partial s}\right)+\frac{\partial z}{\partial x} \frac{\partial^{2} x}{\partial s \partial t}+\frac{\partial z}{\partial y} \frac{\partial^{2} y}{\partial s \partial t}+\frac{\partial^{2} z}{\partial y^{2}} \frac{\partial y}{\partial s} \frac{\partial y}{\partial t}.
\end{aligned}
$$
Work Step by Step
Applying Case 2 of the Chain Rule, we get
(a)
$$
\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x} \frac{\partial x}{\partial t}+\frac{\partial z}{\partial y} \frac{\partial y}{\partial t}
$$
Then , we have:
$$
\begin{aligned}
\frac{\partial^{2} z}{\partial t^{2}} &=\frac{\partial}{\partial t^{2}}\left(\frac{\partial z}{\partial x} \frac{\partial x}{\partial t}\right)+\frac{\partial}{\partial t}\left(\frac{\partial z}{\partial y} \frac{\partial y}{\partial t}\right) \\
&=\frac{\partial}{\partial t}\left(\frac{\partial z}{\partial x}\right) \frac{\partial x}{\partial t}+\frac{\partial^{2} x}{\partial t^{2}} \frac{\partial z}{\partial x}+\frac{\partial}{\partial t}\left(\frac{\partial z}{\partial y}\right) \frac{\partial y}{\partial t}+\frac{\partial^{2} y}{\partial t^{2}} \frac{\partial z}{\partial y} \\
&=\frac{\partial^{2} z}{\partial x^{2}}\left(\frac{\partial x}{\partial t}\right)^{2}+\frac{\partial^{2} z}{\partial y \partial x} \frac{\partial x}{\partial t}+\frac{\partial^{2} x}{\partial t^{2}} \frac{\partial z}{\partial x}+\frac{\partial^{2} z}{\partial y^{2}}\left(\frac{\partial y}{\partial t}\right)^{2}+\frac{\partial^{2} z}{\partial x \partial y} \frac{\partial y}{\partial t} \frac{\partial^{2} y}{\partial t^{2}} \frac{\partial z}{\partial y} \\
&=\frac{\partial^{2} z}{\partial x^{2}}\left(\frac{\partial x}{\partial t}\right)^{2}+2 \frac{\partial^{2} z}{\partial x \partial y} \frac{\partial x}{\partial t} \frac{\partial y}{\partial y^{2}}\left(\frac{\partial y}{\partial t}\right)^{2}+\frac{\partial^{2} x}{\partial t^{2}} \frac{\partial z}{\partial x}+\frac{\partial^{2} y}{\partial t^{2}} \frac{\partial z}{\partial y},
\end{aligned}
$$
and
(b)
$$
\begin{aligned}
\frac{\partial^{2} z}{\partial s \partial t} &=\frac{\partial}{\partial s}\left(\frac{\partial z}{\partial x} \frac{\partial x}{\partial t}+\frac{\partial z}{\partial y} \frac{\partial y}{\partial t}\right) \\
&=\left(\frac{\partial^{2} z}{\partial x^{2}} \frac{\partial x}{\partial s}+\frac{\partial^{2} z}{\partial y \partial x} \frac{\partial y}{\partial s}\right) \frac{\partial x}{\partial t}+\frac{\partial z}{\partial x} \frac{\partial^{2} x}{\partial s \partial t}+\left(\frac{\partial^{2} z}{\partial y^{2}} \frac{\partial y}{\partial s}+\frac{\partial^{2} z}{\partial x \partial y} \frac{\partial x}{\partial s}\right) \frac{\partial y}{\partial t}+\frac{\partial z}{\partial y} \frac{\partial^{2} y}{\partial s \partial t} \\
&=\frac{\partial^{2} z}{\partial x^{2}} \frac{\partial x}{\partial s} \frac{\partial x}{\partial t}+\frac{\partial^{2} z}{\partial x \partial y}\left(\frac{\partial y}{\partial s} \frac{\partial x}{\partial t}+\frac{\partial y}{\partial t} \frac{\partial x}{\partial s}\right)+\frac{\partial z}{\partial x} \frac{\partial^{2} x}{\partial s \partial t}+\frac{\partial z}{\partial y} \frac{\partial^{2} y}{\partial s \partial t}+\frac{\partial^{2} z}{\partial y^{2}} \frac{\partial y}{\partial s} \frac{\partial y}{\partial t}.
\end{aligned}
$$
