Some companions asked the Prophet (peace be upon him) about the rule governing hunting with the help of a trained dog and falcon.
The answer was given in the following verse,
"They ask you as to what has been made lawful for them. Say, 'Made lawful for you are good things, and (hunting through) birds and beasts of prey that you train teaching them out of what Allah has taught you. So, eat of what they hold for you, and recite the name of Allah upon it.' And fear Allah. Surely, Allah is swift at reckoning. (Surah Al-Aa'idah: 5:4)
To make an animal lawful when hunted through a dog or falcon, four conditions have been mentioned in the question and its answer mentioned above.
1. The dog or falcon should be taught, trained and disciplined.
The dog should be groomed in a manner that it should, when released at the prey, catch it and bring it back to you - not that it starts eating it.
The falcon should immediately return when called by you, even if it was chasing the prey.
So when trained, it will prove that these beasts of prey hunt for you and not for themselves. Now the animals hunted by these beasts of prey will be considered your own.
And if they act against this training once in a while, for example, the dog itself starts eating the prey, or the falcon does not return at your call, then this game is not yours anymore, therefore, eating it is not permissible.
2. You should release the dog or falcon immediately at your choice and will.
3. The beasts of prey do not themselves start eating the game.
4. Say 'Bismillah' (With the name of Allah) when you release the dog or falcon after the game.
With Imam Abu Hanifah, there is a 5th condition as well: That this beast of prey should have also wounded the game.
Ruling: This injunction covers wild animals outside of one's possession and control. In case a wild animal has been captured, it will not become lawful without having been properly slaughtered.
Hunting through a beast of prey has been made lawful by Almighty Allah, but, it is not permissible to ignore Salah and other necessary obligations for the sake of having fun chasing game.