I'm a little bit stuck - where did I go wrong?

I was asked by a colleague why the amplitude of tangential modes of a ring or cylinder scale as \(cos(m\theta)/m\).

The answer was given by Rayleigh in his Theory of Sound, which you can read on The Internet Archive. Here's the relevant part:

The relevant solution to the differential equation (1) in Rayleigh for the tangential component has the  \(cos(m\theta)/m\) behavior.  

But, I wasn't able to start with Rayleigh's coordinates and end up with his equation (1).

Here's my work; where did I go wrong?