Ideal geometry for a shaft terminating thrust bearing Following this PyDy example, I'm trying to understand the problem set 3.10 of the book:

Kane, Thomas R., and David A. Levinson. Dynamics, theory and applications. McGraw Hill, 1985.

page 272 (292 of the PDF from here):

                          
                              Fig.1 - image courtesy of Kane 1985

where given the rigid bodies $C$, $R$ and S are in perfect rolling, the ideal value of $b$ in regard to $r$ and $\theta$ is calculated as 
$$b = r \frac{1+ \sin\theta}{\cos \theta - \sin \theta}  \, , \tag{Eq. 1}$$
which I don't how it is calculated. To my best understanding as far as the condition $0 < \theta < \pi / 2$, which is already assured in the question, is met the perfect rolling is possible and $b$ can be any value bigger than $r$. I would appreciate if you could help me understand how the equation above is calculated. Thanks for your support in advance. 
P.S. It worth noting that the contact point between $S$ and $C$ is independent of the value of $b$. 
 A: You can solve this in a purely geometrical way. I assume the shaft rotates about the X  axis, and the housing is fixed. There are two contact points of each rolling ball which must have zero velocity and hence the rotation axis of each ball must pass through them, defining the Y axis.

Now comes the fun part. The relative rotation axis between two bodies must lie on an axis that is a linear combination of the two instant axes of rotation. But what is the linear combination of two lines? It is similar to considering the contribution of two forces that act on their own lines of action. You add up the components to get the direction, and the line must pass through the intersection point of the two axes.

This means that the cone of the shaft must have the apex where the two red lines intersect above. This defines the relative rotation axis Z (yellow line) and taking the moment arms from the contact point $x$ and $y$ we state that $$ \Omega x = \omega y$$ where $\Omega$ is the shaft rotation and $\omega$ is the ball rotation.
With some basic trigonometry, you find the angle $\theta$ that causes the apex to be at the correct point and the values for $x$ and $y$. In turn, this will give you the ball rotation speed.
