"EVERYBODY agrees that we are going to complete the [Tenn-Tom]," Rep. Jamie L. Whitten told his colleagues in the House the other day. A few minutes later, the votes revealed that everybody does not agree. The Tennessee-Tombigbee Waterway -- the nation's most expensive ditch -- is in trouble. Support for its completion is wobbling in the House and a full-scale assault is under way in the Senate.

The Tenn-Tom was a bad idea when it was first proposed a century ago. It has not improved with age. Designed to create a 449-mile waterway from Tennessee to the Gulf of Mexico about 150 miles east of the Mississippi River, this project has never made economic sense. That fact is slowly settling in on members of Congress who have finally become concerned about the way federal money is spent on the pet projects of powerful members.

Construction of the Tenn-Tom was authorized by Congress in 1946. During the 1950s and 1960s, the House Appropriations Committee refused to provide funds for it because the projected return on the government's investment was marginal, at best. On its fourth try, however, the Corps of Engineers came up with a cost/benefit study that the committee (and Congress) found sufficiently persuasive to appropriate the first construction money in 1971.

Sine then, almost everything about the Tenn-Tom has been seriously questioned. The cost estimates have gone from $323 million in 1970 to $2 billion last year to $3 billion this year. The validity of the Corps' projection on how many barges the new waterway would attract (and, therefore, the validity of its cost/benefit analysis) is in hot dispute.

These complaints, and even a request for a GAO study, have been routinely turned aside. Federal spending on the Tenn-Tom has passed the three-quarter-billion mark, and the House just voted to give it another quarter million -- $58 million of that as supplemental funding to be spent in the next three months.

It was in the debate over that supplement appropriation that Rep. Whitten made his confident statement about the Tenn-Tom's future. Two relatively junior members of the House, Reps. Joel Pritchard and Robert Edgar, had challenged the appropriation on the House floor. They lost 210 to 185, that day and lost again, 216 to 196, last Wednesday when they challenged the 1981 appropriation.

But Mr. Whitten had to pull out all the stops to keep Tenn-Tom alive. There were veiled threats, and some not so veiled, about the fate of other water projects.For some reason, comments about Mount St. Helens and the money's being appropriated to help victims of its eruption kept cropping up in the debate; Rep. Pritchard is from Seattle. Yet, a majority of the members of the House elected since 1970 voted to defy Mr. Whitten's wishes and only the 2-to-1 support he got from the body's more senior members kept Tenn-Tom in the budget.

The effort to kill the Tenn-Tom and start reclaiming about half the federal investment in it by selling the land the Corps has acquired now shifts to the Senate. We wish it well. The idea of building an alternative to the Mississippi River makes even less sense now than it did in 1874.