I would recommend to prime the entire ceiling space, so that you don't by chance get a new area and an old area.
Barbara, rolling actually requires much more knowledge than many people realize. First I would cut in the ceiling and wall intersection. Go all around the room and paint in a good 6 inch border with the brush.
Six inch is alot but when rolling it allows you to roll over this area a bit for elimination of border between brush and roller.
Now, rolling, you know that when you load a roller up the majority of the paint is off loaded in the first few rolls and then it decreases rather rapidly.
You can start at the top or bottom, makes no difference.
But paint an area about 5 foot high by 4 foot wide at a time. When you first load the roller up, do a "W" in that area. That pretty much evens up the paint on that area to be rolled out.
Then take that "W" and smooth it out, I like to roll in one direction working that paint out on the surface within your 5 x 4 area. Now, load up again and do a "W" next to that area moving to the right. Do the same feathering and blend in the previous area with the new area. Keep doing this to the right until you hit the wall, then like a type writer carriage return go to the area above square one.
Each time, you want to re paint the previous area border with the new.
Lines appear because of uneven loads of paint being applied and a dry line meeting a wet line. Keep the lines of demarcation wet and overlaped.
Barbara, finish that ceiling in one setting, do not break for lunch and come back and do not do one half and the other the next day.
Now, when you have worked your way to the top and finish area, go over the corners where the brush area meets the roller area and get rid of all those shaded areas and light and dark area of paint meet. Paint out any holidays or missed spots and when you do this, roll in a square feathering out each side.
Now, tools. Get a good roller with locking devices on the end so that the roller cover will not walk off the roller frame. Then get a good extension pole that will lock in place so that it will not de telescope on you and cause a mess. Get drop cloths all over the area and remove furniture. You will become covered in paint because when you roll the roller, spray will come off the roller and on your face. So cover your face with petroleum jelly so that the paint will come off easily.
Now, in conclusion, Barbara, seriously, have this paint job done by a professional.