What is the name?

There is a classic English ghost story film done in black in white (40's) and later a remake of it in color of a nanny that comes to watch two children that have been orphaned by both parents, Father (very evil man) dies in an accident falling off his horse at the entrance to his castle and mother kills herself by the lake on the castle grounds. Nanny finds out that the parents posses the children to come together in carnal pleasure. The boy's name is Miles. She finally gets Miles to confess his dirty secret while the apparition of the father stands behind him, miles screams for the apparition to go away and the boy finally drops and dies. What is the name of the classic film? is it
The secret? or something to that effect?