In his "letter from birmingham city jail," how does martin luther king, jr. connect the public and private actions of the police?

To answer your question,

From his Letter from Birmingham City Jail, Martin Luther King, Jr. revealed to readers that the police have a different public and private persona. In the public, police are trying to exercise the utmost degree of discipline in handling demonstrators. They are conducting themselves in a non-violent way in public. But what is true behind bars is that not all lives matter the same. Injustice and oppression are still seen and are very evident.

The public actions of the police are supposed to be about protecting citizens and keeping order. However, the private actions of the police are the complete opposite when it comes to colored people. The policemen don't hesitate to shoot and hurt black people if they are so inclined.