President Carter yesterday signed into law the bill that authorizes $1.7 billion in federal dollars to complete the 101-mile Metro subway system. The signing placed the responsibility for Metro's future firmly in the hands of state and local governments.

In a ceremony in the Cabinet Room at the White House, Carter told about 50 Washington area politicians and officials that it "would have been tragic" to interrupt Metro's construction program because of a lack of federal, state or local funds.

About 33 miles of Metro are in operation; 27 are under construction and paid for. The federal pledge provides the money for the remaining 41 miles, the last of which are scheduled to be completed in 1989.

The bill Carter signed is a double-edged sword. While it authorizes the federal money that estimates say will be needed to complete the $7 billion system, the legislation requires the District of Columbia, Maryland and Virginia to provide a "stable and reliable" source of income to guarantee local capital and operating expenses.

Local and state officials historically have been better at complaining about what they described as federally imposed stumbling blocks than they have been about committing their own dollars to Metro's future. During the last three years, as Metro has undergone a federally directed reevaluation, local and state officials have said it was impossible for them to go to their constituents and seek the money needed for Metro until they knew what the federal government was going to do. Now they know.

The sometimes intense bickering was pushed aside yesterday for the signing ceremony. A veritable Who's Who of local and state politicians gathered around the president and congratulated each other as he signed the bill. The first pen went to Rep. Herbert E. Harris II (D-Va.), a former Fairfax County board member and Metro Board chairman who introduced the bill that, somewhat amended, became law yesterday.

Harris have Carter a Metro farecard just before the president signed the bill. Carter cited energy conservation and the need to halt urban decay as among the reasons for his administration's growing support of Metro specifically and mass transit programs generally.

Rep. Gladys N. Spellman (D-Md.), another former Metro Board member, called the president's signature "a dream come true," and praised him for having reached an understanding of the difficult 15-year political history of Metro. Carter has not always been perceived as a friend.

"Knowing everything that had gone on before was not one of my most delightful experiences," Carter quipped.

All the senior elected officials from each of the area's major local governments attended the ceremony except Prince George's "County Executive Lawrence Hogan. Hogan spokesman Jim Threat said Hogan had been invited on Wednesday and was not able to change his schedule for yesterday.

The ceremony was delayed about 10 minutes until Washington Mayor Marion Barry arrived. Carter is actively seeking Barry's support, plus a little campaign help, and the mayor is expected to publicly endorse him early next week.

Both the Maryland and Virginia legislatures are going to consider methods of meeting the "stable and reliable" requirements in the Metro funding bill. That requirement must be met by August 1982, or the authorization will lapse. Barry has promised that the District of Columbia will designate a "stable and reliable source."

In a related action last night the Northern Virginia Transportation Commission adopted a resolution seeking the state legislature's permission for Northern Virginia jurisdictions to impose a one percent sales tax and dedicate it to transportation purposes.

Such a tax, twice defeated in earlier legislative sessions, would meet the "stable and reliable" requirement. However, the commission's resolution was vague in that it suggested that a sales tax be considered "along with any recommendations the governor may make" to meet the "stable and reliable" requirement.

The commission also endorsed a proposal of the Virginia Department of Highways and Transportation that would have the state assume 95 percent of the Northern Virginia costs of completing Metro, plus 95 percent of the cost of debt service on existing bond issues sold for Metro construction by both the localities and by Metro.