Skyrim:J'zargo's Experiment

Quick Walkthrough
#Obtain the scrolls from J'zargo.
#Use the scrolls to kill at least three undead enemies.
#Return to J'zargo with your results.

Detailed Walkthrough
A Simple Request
If you attended Tolfdir's lecture when you joined the College of Winterhold you may have run into a conceited Khajiit named J'zargo. J'zargo is ambitious and after your adventures under Saarthal, he tries to ingratiate himself and ask for your assistance: ''"J'zargo has seen that Tolfdir likes you. You must be very skilled with magic, yes? J'zargo is working hard, but cannot do it alone. Can you help with this?"'' It seems he has developed a new Destruction scroll that is especially effective against the undead. J'zargo will give you ten scrolls and send you on your way with instructions to report back on their efficacy.

The Scrolls vs. The Undead
Candidates for J'zargo's scrolls are draugr, skeletons, ghosts and zombies, any of which can be encountered in Skyrim's numerous barrows, tombs or ruins. Undead may also be encountered in The Midden, allowing convenient completion of this quest without travel if the quest has been accepted before the first venture down there. Otherwise, the Halls of the Dead in Whiterun and Solitude are also home to a sufficient number of easily accessible, weak undead. Just be sure to order your followers to wait outside, as they may kill the undead before you, or be harmed by the scrolls.

To complete the quest, you will need to destroy three undead enemies while employing the scrolls. A quest marker will appear with each undead killed while under the effects of J'zargo's Flame Cloak. Undead do not need to be killed by the scrolls' damage; dying of other weapon or spell damage while under the effects of the scroll will satisfy the quest conditions. However, undead that are killed indirectly by the explosions of other undead do not count. If you use up all ten scrolls without killing three undead enemies, the quest will fail.

A Slight Oversight
The scrolls inflict ten points of fire damage per second for thirty seconds. Enemies on fire take extra damage. However, to accomplish this, J'zargo's cloak causes a massive explosion making the scrolls as harmful to you and your allies as to the undead, so ask your follower to stay behind when you test the scrolls, as the damage may kill them or turn them hostile. The scrolls can kill instantly if you are weakened with low reserves or health. Also, be wary of using a scroll when dealing with multiple undead enemies, as the multiplied effects of the ensuing explosions can be extremely damaging.

A Job "Well Done"
Return to J'zargo after you have killed three undead to report the unfortunate side-effect. Knowing J'zargo and his ambitious intentions, you can ask him about the dubious nature of this experiment and your involvement in it. He will assure you that he meant you no harm, and was quite unsure of the effects the scrolls would have on the wielder. Upon finishing the quest you may keep the remaining scrolls and he will be accessible as a follower.

Notes
The scrolls also work on undead that you have reanimated or conjured yourself. This does not work on reanimated animals or non-enemy humanoids. 
The scroll must actually damage the undead for the quest to register the kill, activating the scroll from a distance and killing the enemy by other means won't work.
Activating a scroll near a draugr sarcophagus will trigger the draugr to come out.
Activating a scroll near most followers will cause them to either retaliate or leave your service, sometimes resulting in a bounty even if you are within a dungeon. 
The active effect "J'zargo's Flame Cloak" is listed as a harmful effect, presumably because the explosion it causes damages you as well as your enemy. You can counteract this by using the Become Ethereal shout to take advantage of the explosion without receiving damage.
It is suggested not to give the scrolls to a follower to carry, as they will spam the fireballs at any enemy regardless of if they're undead or not. This is doubly important if you have not yet completed the quest as there is no way to obtain more copies of the scroll.

Bugs
 
 

Quest Stages