Pokémon Gold and Silver/The Three Legendary Beasts

Background Info
The three Legendary Beasts are Suicune, Raikou and Entei, their moves depend on the versions, in Silver suicune has Leer, Water Gun, roar and gust, in crystal it has Leer, Bubblebeam, Rain Dance and Gust, this is probably because you can catch Suicune in Tin Tower in Crystal and the developers didn't want him using Roar. Entei knows Leer, Ember Roar and Fire Spin and Raikou knows Leer, Roar, Thundershock and Quick attack, now on to strategies.

Finding Them
To find them first you need to see them, then you can find which area they are in. Good places to be are Route 36 - beecause of it's links to other routes, if the Pokémon is to the left of Ecruteak, then go into the link place between Ecruteak and Route 38. It is a good place to check the PokéDex because if The Pokémon's in Route 39 you've got it.

Incapacitating Them
The easiest way is to make the Pokémon fall asleep, then swap to a Pokémon that can lower it's health, Mean Look and others are good, but they can still use Roar to escape.

Lowering HP
There are two ways to do this, either get a sycther and train it up to Level 16 where it learns False Swipe, this will bring it's health down to 1HP, it misses after that, so it is a good move for Lowering HP. The other way i if you Don't have a Sycther (found in National Park), get a weak Pokéon like level 5 and use their attack move, it will probably do 1 HP damage, do this then exit the battle and save until the beast faints (obiously don't save). Restart the game with the off switch then you should start flinging Ultra Balls when it's asleep, you should get it sooner or later that way but obviously the first way's quicker.