ONE OF THE REASONS for Vincent Reed's success as superintendent of the District public schools has been his determination to establish administrative order throughout the system - from management downtown to all the classrooms of all 189 schools. That and a healthy reemphasis on basic language and math skills - including tests to measure relative rankings of students and their schools - have produced some encouraging changes that everyone hopes will lead to new levels of achievement citywide. Still, the classroom atmosphere that Mr. Reed and others clearly seek depends heavily on the top person in each school: the principal. Just as teachers need to be evaluated on their performances, principals must be held accountable for the atmosphere in which teachers try to work and students try to learn.

There appears to be good news on that front, too. According to a report this week by staff writer Juan Williams, school-board members have begun to pay more attention to principals in their wards - and have been asking Mr. Reed to do whatever is necessary to get rid of those who aren't measuring up. The board members have asked the superintendent for access to the records and performance evaluations of all principals; Mr Reed so far has been supplying information, on a case-by-case basis, when it is requested. Though the board members do not have authority to hire and fire principals, they certainly can - and should - make recommendations to the superintendent, with the most complete information at their disposal. Also, they do have final approval authority over candidates for principalships and are on panels that screen principal-applicants.

The idea isn't to start a massive witch hunt, but to make sure that policies are being carried out, that discipline is enforced and that each school is being administered properly. Sensitive board members should have some feel for what is or isn't working in their wards' schools. So should parents, whose complaints and praise should carry some weight with the board as well as the superintendent. Already, various school communities around the city have seen the dramatic changes - for better or worse - that a new principal can make. If the changes are to be improvements, the board and the superintendent must be prepared to act strictly when a principal is found to be failing.