It is the 1236th day in the investigation into the mysterious disappearance of Sheila and Katherine Lyon from the Wheaton Plaza shopping center, and Montgomery County police, detective Rodney T. Ingels is at his desk trying to follow up on the last three leads he has received.

Behind Ingels, three black, loose-leaf notebooks hold more than 800 pages of written reports and interviews telling the story of the 3 1/2-year-old case. In addition to the notebooks, three large cardboard boxes contain records of interviews with hundreds of people believed at one time to have had some knowledge of the circumstances surrounding the disappearance of the girls on March 25, 1975.

Ingels, who took over as chief investigator of the case in January, admits readily that after hundreds of leads, interviews and searches, the police are no closer to solving the bizarre disappearance of the two girls than they were when their parents, John and Mary Lyon, first reported that they were missing at 7 p.m. almost 3 1/2 years ago.

"We have never been able to establish any direct motive on the part of anyone, including any family members," Ingels added.

The unsolved case began when Shelia, then 13, and Katherine, then 11, strolled the half mile from their home in Kensington to the Wheaton Plaza shopping center to look at the Easter exhibits and have lunch. They were last reported seen by their brother, Jay, eating a pizza at the Orange Bowl restaurant and by several witnesses who said they saw the girls and several other children talking into a tape recorder carried by a man described as being well dressed and in his 50s.

Since then, there has been no trace of either the girls or the man.

But the leads keep coming in. Last January, for instance, the former manager of Washington radio station WAVA was in Memphis where he read a local article about a 15-year-old girl who could not be identified by authorities.

The former radio man, who was familiar with the Lyon case, through the picture of the girl that appeared in The Tennessenn resembled one of the Lyon sisters. When he heard that the girl, who was described in the article as being "severely traumatizedhad burns on her feet and ankles as well as older stars on her body, the man called Ingels.

The lost child could not speak and was at first thought by authorities to be a deaf-mute. Ingels asked the man to send him a copy of the article and got in touch with Tennessee authorities. A subsequent physical examination, however, proved that the girl was not one of the Lyon girls.

The girl was later identified by her father, as an escapee from a mental institution in Omaha, Neb., and was returned there.

The girls' father, John, a disc jockey with radio station WMAL, declined to talk about his daughters' disappearance, explaining through a secretary that there was no longer any point in further publicizing the case.

But Ingels, who last met with John and Mary Lyon in March and speaks to them on the phone several times a month, said they are well aware of the possibility that their daughters are dead. "I think that at this point they want to wash it out of their minds as much as possible," the detective said.

Ingels himself spends about half his working time on the Lyon case, mainly going through reports from across the country of sex offenses against minors and homicides and tracing the leads that still come from the public through the mail or on the telephone.

This appraoch, which Ingels describes as a "daily drudgery," is in sharp contrast to the search tactics used by police in the weeks following the girls' disappearance. On one occasion, 135 national guardsmen as well as a contingent of Montgomery County police combed 2 square miles of rock Creek parkland in the Granby woods area east of Gaithersburg and midway between Laytonville and Oleny in search of the girls.

Police were acting on a tip from a Dutch-born psychic who had told them that the girls could be found in the wooded area. Real or self-proclaimed [WORD ILLEGIBLE] still call Ingels. One call last week, for instance, came from a man who said "he felt" the two girls "had passed" over an undisclosed road near Fredericksburg.

Ingels say he listens carefully to the callers and then looks into their background to see if they are in any way connected with the case.

"The only person that's not a suspect in this case is anyone that can verify where he was between noon and 4 p.m. on March 25, 1975," Ingels said.

Two of the most notorious suspects that have been investigated by the police were Montie Eissell, convicted of murdering five Northern Virginia women last year, and Arthur F. Goode III, convicted of murdering an 11-year-old boy in Fairfax in 1976. "We spent a lot of time on Arthur Goode, a lot of time," recalled Ingels, but no connection was ever found between the disappearance of the two Lyon girls and either Goode or Rissell.

Even the Lyon family's background has been intensively investigated by police to determine if any family member could provide clues to the case, or in fact, if they were directly involved in the girls' disappearance.

No family member, however, has been linked in any way to the girls disappearance, and Ingels says that he has developed an excellent relationship with the family.

For the police, the Lyon case has been among their most difficult. As Ingels points out, there is no scene of a crime or physical evidence relating to any crime. "We have a house, we have the Wheaton Plaza, and we have the distance between those two points, that's all we have," explained Ingels.

The only thing he can do with a suspect accused or convicted of homicide or sex offenses, Ingels continued, is to establish where he was on March 25, 1975. Most of those questioned, Ingels said, were in jail on that day.

"When you have a suspect that's committed a crime like this," Ingels said, "What can you ask him?"

Nevertheless, Ingels said, he and the Montgomery County Police Department intend to keep at it.