= Solution
Write $T$ for transit time and $\ell(y)=t_0-\alpha y^2$ for lens thickness to avoid using $t$ for both. In the <thin lens> approximation, the two geometrical distances to the lens plane are $\sqrt{a^2+d^2}$ and $\sqrt{b^2+d^2}$. Their sum counts the lens segment as vacuum. Replacing that segment by material of <refractive index> $n$ adds $(n-1)\ell(d)$ to the <optical path length>. Thus, to the retained thin-element accuracy,
$$
cT=\sqrt{a^2+d^2}+\sqrt{b^2+d^2}+(n-1)(t_0-\alpha d^2).
$$
Using the <Taylor expansion> $\sqrt{1+u}=1+u/2+O(u^2)$ for $d\ll a,b$ gives
$$
\boxed{T=\frac1c\left[a\left(1+\frac{d^2}{2a^2}\right)+b\left(1+\frac{d^2}{2b^2}\right)+(n-1)(t_0-\alpha d^2)\right].}
$$
The omitted geometrical terms include $O(d^4/a^3+d^4/b^3)$ in <optical path length>, together with finite-thickness corrections discarded by the <thin lens> and <paraxial approximations>. The addition is $n-1$, because the underlying distance already includes propagation across the thickness once in vacuum.
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