見にくいかもしれませんが
Zss=(d\(\frac{x}{d}\)t∂/∂x+d\(\frac{y}{d}\)t∂/∂y\()^{2}\)Z
=e^(2s)(-sint∂/∂x+cost∂/∂y\()^{2}\)Z

Ztt=(d\(\frac{x}{d}\)s∂/∂x+d\(\frac{y}{d}\)s∂/∂y\()^{2}\)Z
=\(e^{2}\)s(cost∂/∂x+sint∂/∂y\()^{2}\)Z
したがって
Zss+Ztt=\(e^{2}\)s(∂^2Z/∂\(x^{2}\)+∂^2Z/∂\(y^{2}\))
=\(e^{2}\)s(fss+fyy)

∴fss+fyy=(Zss+Ztt)/\(e^{2}\)s