President Reagan's display of power in the House last Thursday proved that the politics of personal attention worked as well as the nationally televised appeal his aides recommended to him on June 19 and then canceled one day later.

Reagan concentrated on cutting individual Democrats out of the herd, and the result was perhaps the most important single vote of his first term.

He zeroed in on freshman Democrat W. J. (Billy) Tauzin of Louisiana in midmorning, shortly before the critically important procedural vote on his radical budget cuts. If Tauzin and freshman Democrat Ralph Hall of Texas broke from the majority Democrats, it might start what budget director David Stockman calls "the loosening up process," winning over another half dozen Democrats. In the end, 29 deserted their party despite an unusual, written demand for loyalty by Speaker Thomas P. (Tip) O'Neill and his high command.

Reagan's deal with Tauzin and Hall: a separate vote on budget cuts from the deadlocked House Energy Committee to exempt natural gas states like Louisiana and Texas from switching their utility plants from gas to other fuels. Tauzin and Hall bought it fast, other Democrats swung into line and Reagan's radical budget cuts were suddenly alive and well (although in the clinching budget vote the next day Hall returned to the Democrats and voted against Reagan).

The importance of the House budget vote transcended food stamps, military retirement pay or impact-area school subsidies. If Reagan's radical effort to thin the governmental wedge of the economy had failed, his equally radical 33-month marginal tax-rate cuts would also have failed. That would have finished the Reagan revolution.

To maintain the working House coalition of Republicans and "boll weevil" Democrats -- conservatives centered in the Southern and border states -- for the approaching tax battle, Reagan must show harmony between savings gained by his budget cuts and revenue lost by his tax cuts.

Even with the draconian budge cuts whittled out by Stockman's artful knife, the mid-July federal spending estimates will show an upward creep of almost $18 billion over the March 10 estimate. Even with the Stockman cuts, the fiscal years' 1982-1984 reduction in entitlements, totaling $46 billion ($18 billion more than the House Democrats' plan) is still some $50 billion short of achieving the promised balanced budget in fiscal 1984. That work remains to be done.

The quintessential imperative of Reagan's victory in the battle of the budget thus becomes clear. Without it, conservative Democrats and even some Republicans squeamish about the president's massive tax cut program would have cause to desert the Reagan revolution. The pyramiding deficit would frighten them off, if it had not already undermined Reagan's tax-cut program.

On June 19, the White House was thoroughly alarmed by the prospect of a resurgent Democratic majority in the House trapping Reagan with a procedural rule tailored like an undertaker's suit to bury the Reagan budget cuts. Chief of Staff James Baker III, Stockman and chief congressional lobbyist Max Friedersdorf agreed that the three major television networks should be asked for free time for a presidential appeal to the nation the following Monday or Tuesday, just before Reagan would leave on his trip to California.

The networks were not enthusiastic: one said yes, one said no, one said maybe. "We needed a roadblock." one White House aide told us -- a guarantee of all three networks so that Reagan would dominate the tube with his address.

But on June 20, the mood inside the White House changed. Newspaper headlines that morning proclaimed that a mere $5 billion divided the Democrats' budget-cut measure from Reagan's (although, in fact, the nature and premanence of the cuts in the two measures left an immense fiscal and ideological gap). If a loss for Reagan could be made to appear so insignificant, Baker argued, his prestige should not be risked on a complex procedural vote that the head counters feared he might well lose anyway.

The decision was made not to protect Reagan's prestige, particularly with public opinion polls showing a sharp decline in his popularity. Instead, the president swung into the kind of politics he plays so well and began to pick off Democratic defectors. He bagged enough to keep his revolution going, and he saved the big TV gun for his tax bill -- but now he may not need it.