When Air Force 2nd Lt. Christopher M. Cooke had car trouble two days before Christmas during a stopover in Washington, he called his father in Richmond to say he was going to be late getting home for the holidays.

That courtesy got him in trouble because he placed the call from the Soviet Embassy and the FBI was monitoring it. He now faces court-martial for allegedly passing nuclear secrets to the Soviets.

When former Army cryptographer Joseph George Helmich got a little short of cash last summer, he took a trip from his home in Niagra Falls, N.Y., to Ottawa. Helmich contacted the Soviet Embassy there, sources said, and Canadian security forces were watching and told the FBI.

Last month he was charged with espionage for allegedly selling the Soviets Vietnam war era secret codes in 1963.

When Marian W. Zacharski, a Polish state trade group official and suspected intelligence agent, got chummy with a neighbor in California, Hughes Aircraft Co. engineer William Holden Bell, the FBI noticed because they had been watching Zacharski since his entry into the United States in 1977.

In fact, after the men were indicted on spyu charges in June, Zacharski's attorney said his client was so friendly with his FBI shadows that they all went to a fast food restaurant for his daughter's birthday.

"Catching spies is hard work; phone taps and surveillances and about a whole lot of luck," one former intelligence officer said.

Although it is well-known that adversaries like the United States and the Soviet Union bug, wiretap, photograph and follow the other's secret agents, and try to intercept their coded messages, it is considered bad form diplomatically to talk publicly about the methods of spying.

Not everything is cloak-and-dagger seriousness, as the Zacharski-restaruant story shows. For instance, spy chasers reminisce about moments of humor that make the monotonous hours of surveillance bearable.

FBI agents have kept foreign agents they were tailing from getting mugged. One former counter-intelligence agent recalls a Soviet agent who once made weird turns in traffic in an apparent attempt to lose his followers. He finally stopped to tell his adversaries he was lost and needed directions.

An examination of the court papers the Air Force filed Thursday in the Cooke case reveals much more than had been known about how federal authorities went about discovering and apprehending the young lieutenant.

When first confronted in early May with the evidence against him, Cooke claimed he went to the Soviet Embassy as part of a research paper he wanted to write.

But when asked to take a polygraph test, he balked and said he wanted to be promised first that none of the material obtained during the investigation, including "tapes of phone calls to the embassy" or "films/photographs of embassy contacts," would be used against him.

Sources said the FBI had photographs of Cooke entering the Soviet Embassy from routine surveillance of the 16th Street building. But it wasn't until his car trouble and the call home that they could begin identifying him, sources said.

The Air Force said in court papers that the 25-year-old deputy commander of a Titan missile crew confessed May 9 to photographing, copying and giving the Soviets nuclear weapons secrets for nearly a year.

A court-martial has been recommended.

The court papers also show investigators were confused at first about whether the caller was Cooke or his brother Richard, 20, an airman assigned to the same Strategic Air Command base in Kansas. The lieutenant, who was on leave, became a prime suspect when a records search showed his brother was on duty at the time.

Cooke was due to go on leave again in early May. So in late April he and some other Air Force missilemen at his base were briefed about the requirement to report contacts with foreign governments. When he did not do so, the papers said, investigators recommended he not be allowed to go on leave.

But SAC Major Gen. Andrew Pringle Jr. decided Cooke should not be detained because of the crew scheduling problems that would result and because Cooke "did not have access to information which the Soviets didn't already know about."

Pentagon officials have said since then that he did give the Soviets information important enough to cause the Air Force to change Titan missile codes.

Air Force agents lost Cooke as soon as he got into a cab at National Airport May 2, but later learned he had gone to the embassy again.

Cooke said he could not find anyone to talk to him that day because the Russians were still celebrating May Day, the May 1 socialist workers holiday.