Most university presidents must answer to a board of 20, 30 or even 40 trustees. But the Rev. William J. Byron, newly installed president of The Catholic University of America, the only institution of higher education in the U.S that was chartered by the Vatican, has nearly 300 bosses: the Roman Catholic bishops of this country.

Relations between the university and the Catholic hierarchy have not always been cordial, especially during the last decade or so when many of the church's more conservative bishops took issue with some of the ideas coming out of the university's activist theology faculty, where, for example, some of the fierce theological resistance to the papal encyclical banning artificial birth control originated. Some bishops have even tried to ban CU theologians from speaking in their dioceses.

But as Byron took office this week there were indications that the era of discord may be passing. The National Conference of Catholic Bishops, in Washington for their annual meeting, rallied around the new president during events surrounding his formal installation on Thursday.

Byron, interviewed in his campus office on Harewood Road NE, a couple of blocks north of the Shrine of the Immaculate Conception, expressed a hope that he can overcome some of the bad feelings between bishops and the university, which largely grow out of theolgical conflicts.

"I got a significant amount of mail when I was appointed," Byron said, "including a number of notes from bishops I didn't know offering support."

"This university has a unique service relationship to us," Minneapolis Archbishop John R. Roach said at an installation dinner the university gave for the bishops and their new president Wednesday night.

"We do share with Byron and with this university a serious responsibility. With that responsibility goes our pledge of moral, spiritual and financial support," Roach reminded his brother bishops.

Byron, 57, comes to CU from the presidency of the University of Scranton and earlier associations with a string of other Jesuit institutions. He is the first priest to be president of CU in a decade and the first Jesuit ever to head the school.

He lives in a modest two-room suite in Curley Hall, a sort of ecclesiastical dormitory that houses clerical faculty members. Between official functions, Byron might be seen in the dining room, wearing a sports shirt instead of the usual clerical collar.

In addition to the administrative responsibilities of the presidency, he has made an effort to get acquainted with the students on an informal basis. Through question-and-answer sessions, sharing cokes in dormitories, public liturgies and open forums in the student center, he is getting to know his constituency.

"They know I am available" for the informal sessions, "and it really does work," he said. One reason it works is that Byron is an outgoing, friendly man who lets neither his priestly collar nor the dignity of his position inhibit his easy manner with people.

An economist, he serves on the boards of 27 education, health, business and social welfare institutions as diverse as the Northeastern Bank of Pennsylvania and the Lackawanna County United Way. He is a trustee of several other Catholic universities, including the competition across town--Georgetown--Loyola in Chicago and Notre Dame's program for constitutional studies.

Legally and technically, the school Byron now heads is known as The Catholic University of America. But for the average person, including most Catholics, mention of a Catholic university is far more likely to bring to mind Notre Dame, with its fighting Irish, or possibly Fordham, with its widely respected law school, or Georgetown, with its programs in government and diplomatic service.

"Part of the difficulty is our name," Byron said. " 'Catholic' is an adjective, not a noun. Then when we are recognized, the perception may be wrong. 'Catholic Univerity? Oh yes, that's a seminary, isn't it?' "

Catholic University, with its nearly 3,000 undergraduates and 4,000 graduate students in a variety of fields ranging from drama to chemistry, is not a seminary, but it does include among its graduate schools one of the finest theological faculties in the nation and the only school of canon church law in this country.

Such a school produces an alumni register long on priests and even including a few bishops, but shorter on successful lawyers and wealthy and generous businessmen that some Catholic schools can claim.

Like any college president, Byron must devote a sizable portion of his energies to fund-raising. His predecessor, Dr. Edmund Pelligrino, left the job because, he said, fund-raising took too much time away from his teaching and practice of medicine.

For CU, one of the keys to fund-raising is in fence-mending with the Catholic bishops, who are supposed to take an annual collection in each diocese for the support of the school. Currently, that comes to $3.7 million annually, Byron said.

"It would be helpful if we could increase that," the new president said, "but that's not going to happen overnight."

The Rev. Charles Curran, a CU theologian who has been at the eye of the storm of controversy over theological teachings, thinks Byron is "a very good choice" to turn things around with the bishops because "he appreciates theology" and will interpret what his theological faculty is trying to do.

Theologians, Curran insists, "should be on the cutting edge. If we are not controversial, something is wrong. That's what a university is all about. The university has always insisted that faith and reason can't contradict one another."

Difficulties arose in the past, Curran said, because "we had a president that was not able to explain that to our public . . . . We were apologizing" for stirring up theological controversy, when "we should have been praising it."

Auxiliary Bishop Edward T. Hughes of Philadelphia, chairman of the conference's education committee, which has special responsibilities for CU, describes it somewhat differently. "The role of the theologian is to speculate, yet the theologian must be responsive to the bishop," he said. "It's not an easy relationship."

Byron's view on CU's responsibility for the spiritual development of its student body, which is nearly 90 percent Catholic, recognizes the realities of modern Catholicism.

"Everybody knows in loco parentis [the school functioning in place of the parent] is dead," Byron said. "Nevertheless, parents have a right to expect that students at Catholic University have the opportunity to develop spiritually."

For students today, mandatory attendance at mass is out. Byron said he believes in a free-market spirituality: supply students with first-rate counselors and campus ministers who will provide good liturgies "and the demand will take care of itself," he said.

"If you can provide something really attractive, something that's hitting the heart of the matter, they'll be there.

"Do it well," he said, "and they'll respond."