For the following setup, what is the second holonomic constraint to limit the bead to one dimension?

Question

We have a bead sliding along a smooth wire in the shape of a cycloid with equations, $$ x=a(\theta-\sin\theta),\space y=a(1+\cos\theta), $$where $0\leq\theta\leq2\pi$. To find the number of coordinates required to describe the motion of the particle (call it $N$) we can use $3n-c=N$ where $n$ is the number of particles and $c$ is the number of constraints on the particles. I know that $N=1$ so we must have $c=2$. I also know that the first constraint is, $$\phi_1(\textbf{x},t) = \textbf{x}\cdot\textbf{n},$$where $\textbf{n}$ is perpendicular to the plane of the wire.
My question is what would be the function $\phi_2(\textbf{x},t)$ to constrain the bead to the wire?