Arthur Burns, chairman of the Federal Reserve Banking, looked around the room and felt humbled.

Seated around him at an oval table in the Fed's richly decorated Terrace Room were a couple dozen of the world's leading experts on electronic banking, gathered there recently to compare notes and exhange anecdotes.

'I know nothing about electronics," said Burns, to a wave of appreciative laughter. "I never have mastered the problems of funds transfer and bank clearing and shall leave that to you who are the experts."

Burns' deference to the computer pros is shared by many. In banking, as in other industries, they have been the brains behind major innovations in the last decade.

To a large degree, the banking revolution is still on. The Senate Banking Committee heard testimony last week on a Carter administration bill that would extend what are in effect interest-paying checking accounts nationwide. Called NOW accounts (for "negotiable order of withdrawal"), they would hardly have been possible without automated processes to transfer funds from savings to checking accounts with ease.

In another major financial development last week, the country's largest securities firm - Merrill Lynch, Pierce, Fenner & Smith - announced a new kind of brokerage account which will let customers earn interest on cash balances held by the firm. Customers will also be able to use VISA cards (formerly Bank Americards) to get immediate access to these funds.

But there are signs, particularly in the Washington area, that the revolution is slowing. The gadgetry which gave rise to automated tellers and BankQuik stations is the farthest Washington's largest banks say they want to go for now. And other automated systems already in service handling the direct deposit of company paychecks and the pre-authorized payment of bills, still have bugs to be worked out.

Perhaps in the banking world of tomorrow, people won't have checks to write, or bill notices to be left around the house until the last minute, or payroll checks to deposit. They may not even have to carry cash - just a pocket-full of credit cards, or maybe just one card with a long, long number.

But that day is sometime away.

"I wouldn't say it will never come about," said Alan Rothmayer, director of operations for the American Security Bank. "But it's certainly not in the immediate future."

The evidence for this conservative prediction is contained in a year-long study recently completed by the District's three largest savings and loan associations. Basically, the study concluded that both cost and fuzzy legal issues preclude a full-scale expansion of electronic funds transfer systems.

Besides which, no one is sure just how automated consumers want their money transactions to be.

The next logical step in banking's computer revolution would be the introduction of electronic terminals in retail stores, so that shoppers, by inserting a plastic card into the machine could switch funds immediately from their accounts to the store's without bothering with paper checks or credit card slips. In banking circles this system is referred to as "POS," for "point-of-sales."

"There's an obvious advantage in this for the merchant, who gets his money on the spot, and for the bank, which avoids handling checks," said Robert O'Donoghue, an operations expert at Riggs National Bank. "But what is the advantage to the consumer?The consumer is never going to go for it unless we can find a benefit in it for him."

Two years ago, Congress appointed a commission to study whether electronic banking is in the public interest. According to the commission's interim report, consumers think they are well-served by the current payment system and do not view it as inconvenient. In a poll taken in Virginia and quoted in the report, 67 per cent of the respondents agreed with the statement, "Bill payment is not too time consuming."

A highly automated system may in fact work to the consumer's disadvantage by erasing the luxury of the float, the time between a check is written and submitted to the bank for payment.

Also, electronic money transfers raise all sort of yet unresolved legal and security concerns, such as who should be liable for erroneous, unauthorized, or fraudulent use of automated accounts.

And there is concern that banks may get lazy the more computers take over, and will cut corners on customer services. For instance, some area banks which receive and deposit checks automatically and directly from the Social Security, Civil Service and Veterans Administrations for their customers, have stopped sending notices to depositors that the money has been received and credited. They expect depositors to assume the money is in the account when it should be.

Automation may yet contribute to the streamlining of bank activities but in internal rather than external operations. Check-processing between banks is nearly fully automated today. Checks are coded for machine reading with numbers indicating the bank and individual account number, and automated clearing houses are being developed which will substitute transfer of magnetic tapes in place of paper checks.