From the Unofficial Elder Scrolls Pages wiki, "The Eye of Magnus" article (important parts emphasized by me):
Ancano and Tolfdir will talk for a short time before Tolfdir launches a spell at him, but with no effect. Tolfdir is knocked down and remains incapacitated while Ancano attacks. He will still have his protective barrier, which can be removed by using the Staff of Magnus on the Eye of Magnus.
It is important to use the Staff for a sufficient amount of time on the Eye of Magnus- until it returns to a solid, non-glowing sphere - because the protective barrier gives Ancano near immunity to magic and physical attacks when the Eye of Magnusis only partially weakened. When the glow surrounding Ancano dissipates he will be vulnerable to attacks.
During the fight he may approach the Eye and regain his protective barrier, which must be removed again. When the Eye separates into several hovering and rotating pieces, it is gaining strength, so Magic Anomalies may appear; be sure to use the staff on the Eye, so as to get it to lose more power and return to its original form for some time. The Magic Anomalies will yield soul gems when killed, allowing the Staff of Magnus to be recharged if it depletes during the battle. It is possible to leave Ancano to fight the magic anomalies and get himself killed that way, as they will attack him as well.
When Ancano is defeated several members of the Psijic Order will appear once more. The player is congratulated on their efforts, and declared worthy of being the new Arch-Mage. They will also state that the Eye of Magnus is unstable, and too dangerous for this world. Shortly after they will vanish in a glow of light, along with the Eye.