FOR THE FIRST time in the 10 years that he has ruled Chile, Gen. Augusto Pinochet's grip seems to be slipping. He has increasingly resorted to violence to contain the protests that began last May. Last week, police and troops went after the demonstrators with real ferocity, and some 20 people died.

In these darkening circumstances, the most hopeful sign is that the previous apparent unanimity of the armed forces is beginning to crack. Last June, the Navy reportedly prevented Gen. Pinochet from putting the country back under martial law. Now the chief of staff of the air force, a member of the junta, has publicly disassociated his service from the bloodshed and asserted that the time has come for an understanding with the political parties.

The first question is what follows Gen. Pinochet. Will it be the constitutional democracy that is Chile's tradition or something less satisfactory? The second is how many additional lives will be lost in the process of getting there. Both will be answered, at least in the near term, by Chilean military officers. But perhaps the generals and admirals will accept at least some discreet advice and encouragement from Chile's friends in North America.

The United States' influence over Latin affairs is a good deal less overwhelming than you might sometimes think, listening to the debate as it goes on up here. But neither is it negligible, and the time has arrived to do what can usefully be done. The State Department has been providing some careful nudges in the direction of democracy. That's splendid, but don't stop now.

It's useful to compare Chile with another unfortunate country, Poland. Both are under military governments, and standards of living are falling in both. Political protests in both have been met with harsh repression. The opposition in Poland is an illegal union and the Catholic Church; the opposition in Chile is the unions and the illegal political parties, among which the Christian Democrats have been most prominent.

The crucial difference between them is, of course, the heavy shadow of the Soviet army over Poland. Many Americans, President Reagan among them, have denounced the Soviets' influence in Poland. But since the cause of human rights is as greatly to be cherished in one country as in another, why not extend to Chileans the same warmth of public concern that this country has offered the Poles?