TEN YEARS is longer than the usual period required for Washington to break its word, but the Department of Health, Education and Welfare has now finally broken another promise to the poor.

Though few may remember it, in the tense time after Martin Luther King's assassination, when the government made a series of commitments to the Poor People's Campaign, one concerned a relatively small problem with the welfare system.

It was just a detail: Welfare payments began when a state decided a family was eligible -- which could take weeks or even months -- while the family was in need from the moment it asked for help. It seemed eminently reasonable for payments to be figured from the day families applied. Indeed, Social Security did more than that, paying for the entire month in which a person applied. (The same holds for the Supplemental Security Income program developed in 1974 for the needy aged and disabled.)

But when then-HEW Secretary Wilbur Cohen ordered new regulations written, the snags developed. The comptroller general had once ruled against such a a change, and it took time to get a reversal. Meanwhile, a Republican administration took office, and nobody expected it to make good on the promise.

When President Carter took office, some states were paying from the date of application -- but 31 continued to follow the cheaper path of paying from the date of decision, provided that wasn't more than 30 days after poor families had asked for help.

So in February 1977, when HEW Secretary Joseph A. Califano asked a number of welfare experts how to proceed with welfare reform, several replies mentioned the payment-date promise. Some letters stressed that this was a personal as well as a government responsibility, since Califano had been chief domestic aide to President Johnson when the promise was made.

In November, Califano wrote to the Center on Social Welfare Policy and Law, an advocacy organization, that he would require payment for the full month in which application was made. To make welfare consistent with Social Security and SSI, he was going to make good on more than the original promise.

But it was not to be. Studies floated from desk to desk. Technicians estimated that the cheaper change originally promised would cost $250 million a year, later scaled down to about $150 million a year. (From the perspective of poor families, of course, that means they have been done out of $1.5 billion. So far.) Delay had assured consideration in the midst of the president's draconian budget proposals.

Even so, virtually every high HEW official, including the general counsel, the secretary's own staff and the administrator of public assistance, recommended the twice-promised change. But Califano said no.

How does an administration supposedly committed to families refuse to do for poor mothers and children what is thought just for the aged and disabled? How many dollars weigh as much as a promise on the administration's moral scales? Though the welfare payment date is, relatively, a small issue, it carries a larger lesson about how much this administration can be relied upon.