The Senate is on course to finish its immigration bill this afternoon, with the penultimate pair of procedural test votes to be taken before lunch and the roll call on passage set to start at 4 p.m.

When that final vote begins, Majority Leader Harry Reid announced this morning, he expects all senators to be in the chamber and ready to vote from their desks.

Arranging that unusual and somber ceremony serves a couple of strategic purposes for proponents of the legislation.

Assembling all 100 senators in the room will put one final bit of unspoken peer pressure on the handful of waverers who could push the majority for passage close to or maybe above 70 — a benchmark of bipartisanship the bill’s advocates hope will create a groundswell of support in the country and, because of that, in the House.

As the alphabetical roll is called, and senators stand to say “aye” or “nay,” they will be able to see which way their uncommitted colleagues are trending. And each of the tossups will be compelled to make televised eye contact with, or endure stare-downs from, the leadership on both sides of the issue.

The proceeding is called for only on matters of particular import: authorizations of war, Supreme Court confirmations, major treaties, impeachments or matters of senatorial discipline. Passage of the debt limit and budget compromise two summers ago, and of the health care bill on Christmas Eve 2009, also got the treatment.

Using it now is an underscore of the historic nature of the moment — and also a sign of respect for the Hispanic community that has been waiting almost a decade for even this much legislative success.

The bill promises the most comprehensive change to the immigration system in more than a quarter century. It would accelerate the fundamental demographic reconfiguration of American citizenship by allowing almost all of the 11 million overwhelmingly Latino people residing in the country improperly to pursue the right to vote. And it would commit an unprecedented amount of federal resources to making the Mexican border less porous to another wave of illegal immigration.

The back-of-the envelope whip counts, and a series of votes Wednesday, suggest that a two-thirds majority is very likely but that getting more than 70 has become a decided longshot.

Partly that was because negotiations with two swing-vote Republicans have come to naught — Georgia’s Saxby Chambliss on restrictions for agriculture workers and Ohio’s Rob Portman on toughening the rules for making businesses use the E-Verify system to make sure new employees are legal.

But it’s also because there seems to be fading viability in the notion that a big majority in the Senate will compel the House to quickly pass a similarly comprehensive bill. That looks less likely by the day, no matter what the Senate does.

Many, if not most, Republicans in the House appear content with the more modest and piecemeal approach getting started in the Judiciary Committee. Many oppose the pathway to citizenship at the heart of the Senate bill.  And Speaker John A. Boehner has been explicit in saying he’ll only advance legislation that can muster a majority of his majority.

Whether the Senate vote for passage hits 69 or 70 probably won’t alter those unchanging House dynamics.