Density and circumference relationship? From Example in book

[unicornflyers]

Homework Statement

Example 5.3 from The Marion Thornton book (fifth edition) of Classical Dynamics states the following problem:

Consider a thin uniform circular ring of radius a and mass M. A mass m is placed in the plane of the ring. Find a position of equilibrium and determine whether it is stable. I'm following the example in the book, and there's two things I don't understand. First is why /rho = \frac{M}{2*/pi * a}. I always thought that the density was the mass over the volume, so I don't see why this is mass over circumference.

Second, in the next piece, it says that d\Phi = -G \frac{dM}{b} = \frac{-Ga\rho}{b}d\phi

I'm failing to see two things. 1) why the dM element ends up with a small phi, and 2) why rho is as it is. From here, I believe I can find the rest of the example, but why are these two things true?

 

[TSny]

The textbook is apparently using the symbol $\rho$ to denote the linear mass density.

An infinitesimal arc length can be written in terms of $d \phi$.