Why are Euler's angles picked exactly that way?

[illidan4426]

TL;DR

I'm wondering why exactly those angles are picked to describe the orientation of the rotating body.

So the Euler's angles are described like this:
xyz-x'y'z' (first rotation around z axis)
x'y'z'-x''y''z'' (second rotation around x')
x''y''z''-XYZ (third rotation around z'')
So I've been thought it goes like this, now I'm wondering why? Why exactly these angles and why this order? Why can't it go like this for example:
xyz-x'y'z' (rotate around x)
x'y'z'-x''y''z'' (rotate around y')
x''y''z''-XYZ (rotate around z'')
Can the motion be described this way? The equations of transformation of xyz-XYZ would be different for sure.

 

[vanhees71]

There are all kinds of conventions around to parametrize the rotations. The Euler angles are just the most often used ones.