UNDER FLORIDA'S new gun law, local governments are no longer permitted to interpose their own "cooling-off" period between a customer's purchase of a weapon and his taking possession of it. The state has imposed its own 48-hour waiting period, which can begin the moment a customer merely signals his intent to purchase a particular gun.

Last Friday, Arthur Kane of Miami stopped by the Tamiami Gun Shop and indicated he was interested in buying a .357-magnum pistol, according to the gun shop; under Florida's strange new law, no paper work is required on such matters, so there's no evidence other than the shop's word that Mr. Kane made that visit.

There is evidence that on Sunday he paid for the weapon: a phone check was made on his credit card from the gun shop. That's about the only checking gun merchants need bother with in Florida; they certainly don't have to go looking into their customers' pasts. Mr. Kane, like many people who acquire handguns, had an interesting one. He was the former Arthur Katz, government witness in a couple of trials, prison inmate (for six months), recently pursuing a new life with a new name under the federal witness protection program. And he had just been a big loser in the stock market.

On Monday Mr. Kane went to a Merrill Lynch branch office near Miami and watched the stock ticker for a while. The market was going badly again. After a short time he left, went back to the gun shop, picked up his new .357 magnum and returned to Merrill Lynch, where he shot the branch manager to death, grievously wounded another official of the firm and then took his own life.

"Under the old gun law this tragic situation would not have taken place," said Cmdr. Bill Johnson, a spokesman for the Metro-Dade police department. "There are so many exemptions in the new gun law that effectively there is no cooling-off period for the purchase of handguns."

Earlier this month the director of the International Association of Chiefs of Police called the new law "a tremendous loss for the citizens of Florida who don't yet realize the impact." More and more they'll come to realize that impact as the news comes in on the ticker.