You need to install ruby-Dev and mysql-Dev packages.
sudo apt-get install ruby-Dev mysql-dev
Could also be Devel instead of Dev, not sure about that at the moment.
Also it may be that you have to purge and reinstall ruby after that to get it working. This part is covered in any ruby with Ubuntu tutorial I am aware of.
Another one is that you really shouldn’t use 10.10 any more. 12.04 is the latest Ubuntu with Long Term Support. 10.10 not supported anymore as far as I remember.