Why does the search for Montag end in Part Three?

In Part Three of Fahrenheit 451by Ray Bradbury, the protagonist, Montag, finds himself on the run after it is discovered that he possesses books. To compound matters further, Montag resists his arrest, killing his boss and destroying a Mechanical Hound in the process. Montag runs to Faber's house first, asking him to leave town and seeking advice on his next step. Faber shows him the television and relays the information that a new hunt...

In Part Three of Fahrenheit 451 by Ray Bradbury, the protagonist, Montag, finds himself on the run after it is discovered that he possesses books. To compound matters further, Montag resists his arrest, killing his boss and destroying a Mechanical Hound in the process. Montag runs to Faber's house first, asking him to leave town and seeking advice on his next step. Faber shows him the television and relays the information that a new hunt for Montag is on, assisted by a Mechanical Hound imported from another district. After a brief exchange, Faber advises Montag to head for the river and the railroad tracks, where hobo camps are said to exist. 

Montag leaves Faber's house and makes a run for the river. The new Mechanical Hound begins the hunt for Montag. Montag finally makes it to the river: 

He was three hundred yards downstream when the Hound reached the river. Overhead the great racketing fans of the helicopters hovered. A storm of light fell upon the river and Montag dived under the great illumination as if the sun had broken the clouds. He felt the river pull him further on its way, into darkness. Then the lights switched back to the land, the helicopters swerved over the city again, as if they had picked up another trail. They were gone. The Hound was gone. Now there was only the cold river and Montag floating in a sudden peacefulness, away from the city and the lights and the chase, away from everything. (52)

The Hound is unable to track Montag once his scent is lost in the river. Montag continues on to join the Book People. In an attempt to save face, the government uses the Mechanical Hound to kill a "fake" Montag on camera. This allows the government to continue its farce of being completely in control of all of its people.