As Dr. Herman Tarnower's life ebbed away, "precious time was wasted because the police surgeon was not informed by detectives before he came to the scene that the victim had been shot in the chest."

This picture of the famous diet doctor's last minutes emerged today in testimony by Harrison, N.Y., chief police surgeon Howard C. Roth at the Westchester County murder trial of Jean Harris, 57.

Under cross-examination by defense lawyer Joel Aurnou, Roth, a prosecution witness, conceded that the 13 minutes between the time the surgeon first got the call on the night of the shooting and the time he arrived at the Tarnower residence "might very well" have made the difference between life and death.

"Any minutes could have made a difference," Roth said.

The victim should have been rushed in a squad car to the hospital for emergency cardiac treatment instead of waiting for an ambulance to arrive, Roth said, when coaxed by Aurnou. Roth arrived when the victim, on a stretcher, was being placed in the ambulance.

If the jury is convinced that police negligence contributed to Tarnower's death, this testimony looms important in any future moves by the defense to have the charge reduced to manslaughter. Harris is charged with second-degree murder in the March 10 shooting of Tarnower, author of the Scarsdale Diet.

Roth gave a vivid description of his vain attempts to save the life of his medical colleague in the ambulance on the way to the hospital. After Tarnower's pulse and heartbeat stopped, Roth began massaging the dying cardiologist's chest. But it was too late. Further efforts to revive him in the emergency room were also futile.

"It would be my opinion," Roth testified, "that he expired in the ambulance."

Roth also conceded that Tarnower's four gunshot wounds might have been inflicted during a struggle.

"Do you have a medical opinion," Aurnou asked, "as to whether the wounds could have been caused in a struggle over a gun between a man of five feet 10 inches and 175 pounds and a woman of five feet four inches and 105 pounds?"

"They could have occurred during a struggle," Roth replied.

The impact of this opinion was softened in later testimony when Roth told prosecutor George Bolen that "it is just as possible that none of these wounds was sustained during the struggle."

Tarnower had wounds in his right hand, right upper arm, chest just below the collar bone, and back near the spinal column. Roth said that the conclusion about a struggle also hinged on the sequence of the shots. Pain and impairment from the upper arm wound could have made further struggle impossible. The sequence of the shots has not yet been brought out in testimony.

The defense contends that Harris went to Tarnower's house to kill herself and that he was killed accidentally in a struggle over the gun. The prosecution has tried to show that Harris shot her lover of 14 years "deliberately, consciously" in a jealous rage.

Harris sat impassively throughout today's testimony. The demeanor of the former headmistress of the Madeira school in McLean, Va., was in marked contrast to her volatile behavior last week when she stormed out of the trial and complained to reporters about a ruling by Judge Russell R. Leggett barring pre-trial testimony by another witness, Detective Arthur Siciliano.

Harris remained very cool today even as the jury was shown gory evidence taken from the Tarnower mansion in Purchase, N.Y. -- a revolver, pajamas and bedclothes all stained with blood, presumably the victim's.