The Arch Mage will mention that Mirabelle was talking about the Staff of Magnus not too long ago. Go and ask her what she knows about the artefact and its whereabouts.
She will direct you to the ancient Dwarven ruin known as Mzulft, as representatives of a group called the Synod, a faction of mages based in Cyrodiil, recently approached her in a secretive manner and enquired about the place.
When you arrive at Mzulft, you will immediately encounter one of the members of the Synod lying badly injured on the floor. Listen to the individual explain how the Falmer attacked and took an object called the “Focusing Crystal”. He then succumbs to his injuries and dies. Take a key from his body.
You will have to go through a fairly large Dwemer Dungeon. It contains a lot of loot and lots of Dwemer items. You will also encounter the Falmer, Chaurus and some dwemer automatons. You will eventually reach the Aedrome, where you’ll meet a levelled boss who has the Focusing Crystal in his possession. Defeat him and loot the Crystal. You’ll then find a locked door that requires a key.
Turn back and follow a little further into the ruins, and you’ll soon come to a room with the bodies of dead Falmer scattered across the floor and a levelled Dwemer Centurion who stands on guard ready to take care of the next intruder. Destroy the automaton and loot the Key to Oculory from the chest in the room.
With the key in hand, head back and open the closed door. In the space beyond will be another closed door. Move toward it and you’ll hear a voice coming from the other side. It belongs to Paratus Decimius, another Synod mage, who will assume that you’re Gavros (the dying man you spoke to earlier) and open the door for you. Talk with him and explain what you’re doing there. He’ll then give you some information about his mission in Mzulft. Finally, tell him that you’ve recovered the Focusing Crystal. Upon hearing this he’ll lead you deeper into the Oculory.
You will reach a room with huge spherical device in it; follow Paratus to the top, and he will tell you to place the Crystal in the device and after you do that, to attune the crystal using fire and frost spells. You are ready when the beams coming from the crystal are pointing into the middle of each of three rings on the celling. You have to use the spells Frostbite and Flames to do what you need; any other fire or frost spells will not produce the required results. The room with the Oculory contains copies of each of those two spells if you don’t already know them.
You now have to use the buttons on a pedestal above the sphere to fit the rings, so the light beams will point directly to the mirrors on the rings. When you’re done, Parathus will comment that this is not what he was expecting; go down and see what he’s talking about.
On a wall under the pedestal, a map of Tamriel (mostly just the northern provinces and Cyrodiil) is now being displayed. Parathus will tell you that the Synod were searching for a way to scan the continent in order to find powerful sources of magic, but that the map is currently being disturbed by two major magical sources, one in the College of Winterhold (it’s the Eye of Magnus, but you don’t have to tell him that if you don’t want to) and another in the Labyrinthian. Parathus will presume that those ancient ruins is where the Staff of Magnus is. You can go back to Winterhold now.
As you enter the Hall of the Elements, you’ll see Savos and Mirabelle looking at a strange magical barrier that is blocking your way to the chamber with the Eye of Magnus in it. It seems that this barrier has been erected by Ancano, who’s doing something with the Eye of Magnus. Mirabelle will start casting a Destruction spell on it, and Savos will ask you to do the same along with himself.
When the ward is dispelled, go with them inside the Hall. You’ll see Ancano casting a spell on the Eye. As Savos approaches him, he’ll recast the barrier around him without stopping to speak to any of you, and you and the others will be blown away by the force of it.
Soon you’ll regain clarity of your vision, and you’ll see Mirabelle sitting nearby, breathing heavily. After asking if you’re okay, she’ll ask you to search for the Archmage. This quest ends here.