Spoiler Alert! We have hidden this information so that you have the choice to try it on your own, or have the advantage of knowing how the sausage is made...
Their HP must be less than 30% before the Mystic Spell Stone will work, and it can be interrupted by them so you may need to cast it more than once.
Sometimes, your attempt to Calm the Angry Leopard Spirit will fail completely and he will become even more Frantic. When this happens, you sadly have no choice but to put it down.
Daily Quests are repeatable up to a total (for all Daily Quests) of 10 times per day. The count can be reset to zero by purchasing and using a Daily Quest Ticket from the Item Shop (for 39 ($1.16)† ) or from any Phirius Merchant (for 780 ).
In all other zones they do not reward XP, TP or Gold, only Public Event Points. In all other aspects they are the same as a Daily Quest (except for the infinitely repeatable part and the rewards). Public Event Points contribute towards the completion of Regional Quest objectives, which in turn rewards a number of Energy of Justice commensurate with your amount of participation in the Regional event.
Timesaver!
The following quests have steps, hunting areas, or mob targets in common:
The Affected Spirits triad wants you to drive away 10 Angry Leopard Spirits. Since the Mystic Spell Stone continues to work so long as you have not turned in Suffering Spirits the two quest triads work very well together.
Mystic Spell Stone to calm 3 Angry Leopard Spirits.
Daily Quest Ticket from the Item Shop (for 39
($1.16)† ) or from any Phirius Merchant (for 780
).
Dual-Function Daily Quest Potion.
Old Bag (hence the common sobriquet, Bag Quests). In Redhill Mountains, each repetition of [67][Public] Freedom Fighters rewards an 
Ancient Memento.