I thought that when you make it to the second school. After the admission there would be more. There is no additional spell training. What is the point? This place should be where you can get higher levels of magic.
Could you be a bit more specific of what you mean by second school? Second and Third school generally refer to, in my experience, learning spells from other schools via training points. If you mean from DS schools, keep in mind that the world is essentially dead, and new schools will be opened in Celestia.