Martin Luther King led extensive civil rights sit-ins  throughout the South in the 1950s and 1960s. He was pastor, for a time, at the Dexter Avenue Baptist Church in Montgomery, Alabama, where he helped coordinate sit-ins at lunch counters and boycotts of businesses refusing to serve all people.  

As a clergyman preparing sermons, he drew heavily from the Bible for words that would be meaningful to his congregation, including this quote from the book of Amos, chapter five, verse 24. 

A reference to the "mighty stream" of justice is found in a number of his sermons, as well as in his famous I Have a Dream speech, given before a crowd of more than 100,000 people gathered on the National Mall on August 28, 1963, for the March on Washington for Jobs and Freedom. 

Here is how he uses the stream image in his I Have a Dream speech:

But there is something that I must say to my people who stand on the warm threshold which leads into the palace of justice. In the process of gaining our rightful place we must not be guilty of wrongful deeds. Let us not seek to satisfy our thirst for freedom by drinking from the cup of bitterness and hatred.

We must forever conduct our struggle on the high plane of dignity and discipline. We must not allow our creative protest to degenerate into physical violence. Again and again we must rise to the majestic heights of meeting physical force with soul force. The marvelous new militancy which has engulfed the Negro community must not lead us to distrust of all white people, for many of our white brothers, as evidenced by their presence here today, have come to realize that their destiny is tied up with our destiny and their freedom is inextricably bound to our freedom. We cannot walk alone.

And as we walk, we must make the pledge that we shall march ahead. We cannot turn back. There are those who are asking the devotees of civil rights, "When will you be satisfied?" We can never be satisfied as long as our bodies, heavy with the fatigue of travel, cannot gain lodging in the motels of the highways and the hotels of the cities. We cannot be satisfied as long as the Negro's basic mobility is from a smaller ghetto to a larger one. We can never be satisfied as long as a Negro in Mississippi cannot vote and a Negro in New York believes he has nothing for which to vote. No, no, we are not satisfied, and we will not be satisfied until justice rolls down like waters and righteousness like a mighty stream.

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Go here for the full text and audio version of the I Have a Dream speech. This is the only place where I was able to find a video version.