An Arlington cable television company is seeking permission from the Metro board to string its cable through the subway tunnels and thus save money on the cost of feeding the American public debates from the House of Representatives.

ARTEC, which sells cable television service in Arlington County for profit, would string the cable on behalf of a nonprofit company called Cable Satellite Public Affairs Network (CSPAN). CSPAN will provide te signal to cable TV outlets, schools and others when the House opens its debates to live television coverage for the first time next year.

Transportation Secretary Brock Adams and Rep. Charles Rose (D-N.C.), chairman of the Speaker's Committee on Broadcast Television in the House of Representatives, have both written letters supporting favorable action by the Metro board.

John D. Evans, vice president and chief operating officer of ARTEC. said his firm offered to establish and maintain the cable link as part of its contribution to CSPAN, the nonprofit firm. Nine of the 10 largest for-profit cable TV operators in the country are contributing cash to the CSPAN, according to CSPAN president Brian Lamb.

The problem is how to get the signal from House-provided gavel-to-gavel television coverage to the satellite in the sky. ARTEC's idea is to run a cable from the Capitol Hill area through the Blue Line tunnel to ARTEC's equipment in Arlington County. From there the signal would be transmitted to the satellite and made available to potential users.

ARTEC could use a microwave relay from Capitol Hill, but a two-way link would cost about $50,000 and care would have to be taken to avoid interfering with microwave frequencies on navigational aids at National Airport.

It would ARTEC cost between $75,000 and $125,000 to find its own Potomac River crossing and bury its own cable somewhere else. Evans said. But the cable could be run in the Metro tunnel for about $20,000 not counting whatever fee Metro might elect to charge.

CSPAN's Lamb said that cable systems taking the congressional feed would be charged 1 cent per month per subscriber, and that the money would be used to defray operating costs. If any is left over, he said, the money "would go into the pot to serve the nonprofit organization, add programming, etc."

The networks could take the satellite feed, but they could also plug in at the House of Representatives, Lamb said.

Metro board chairman Joseph S. Wholey brought the matter into public view yesterday when he asked General Manager Theodore C. Lutz what was happening on the ARTEC request.

After discussion it was agreed that the board would debate the matter in two weeks.

Lutz said later that there have been some other requests to use Metro's tunnels, but none has been granted. His staff has assured him that the cable would not interfere with Metro communications, he said.

"We really view this as a fantastic opportunity to take democracy back to the country" said ARTEC's Evans.