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Background

At some point after completing the quest Under Saarthal, when you approach J'zargo in the College of Winterhold, he'll ask you for some help (if he's practicing in the Hall of the Elements, then you might need to start and end a conversation with him so he notices that you're there). Agreeing to assist him will trigger the quest.


Walkthrough

J'zargo will tell you that he's been creating new spells, but that he's been so busy, he hasn't been able to test them. He'll then give you ten of J'zargo's Flame Cloak Scrolls, which are supposed to work wonders against the undead, and he'll ask you to test them out.

You'll need to use J'zargo's scrolls on at least three undead creatures (skeletons or draugr) to complete the quest. You can encounter undead creatures in just about every crypt in Skyrim (including the Midden under the college), and so you shouldn't have any trouble finding test subjects. Just be careful when you use the scrolls, though, because they'll explode, damaging you (and your companion if applicable) along with anything else nearby.

Note: If you run out of scrolls and ask J'zargo for more, then he'll get mad at you and you'll fail the quest.

After testing out the scrolls, when you return to J'zargo, he'll thank you for your help, and he'll become available as a follower.