IT WAS LATE in the afternoon that Mrs. William Johncox tried to call the veterinarian because Morris was looking so woebe-gone.

Morris, a rangy orange-and-white tomcat who bears a certain resemblance to the television commercial star of the same name, had straggled home after a night out and his malaise seemed to be more serious than a case of simple overexertion.

By then, it was too late for the vet's regular office hours, so Mrs. Johncox waited until her husband came home.

"He went to the basement to check on Morris and came up shaking his head and saying: 'He'll never last till morning,"' she recalls.

And the chances are that her husband was right in his prognosis except for the fact that the Johncoxes bundled an unhappy and doeful-pussed Morris off for emergency treatment at the after-hours vet clinic in Rockville. There his problem was diagnosed as cystitis with bladder blockage threatening uremic poisoning.

"And that is an emergency. It can mean death in as little as 8 to 10 hours if untreated," explains Dr. Michael Wise, the vet who helped treat Morris at the Metropolitan Emergency Animal Clinic.

That night, along with the orange tom, the clinic was treating Maggie, the beagle, who had dashed to freedom while her owners, Mr. and Mrs. David Johnson, were giving a cocktail party.

"When Maggie came home, she was gushing blood from a deep gash. We don't know yet if it was a raccoon, another dog, or perhaps a car," Johnson recalls of Maggie's plight and the trip for emergency care.

The after-hours animal clinic, open only at night and on weekends, was organized by 29 veterinarians in Montgomery County and Northwest Washington. As any pet owner knows, a dog is more likely to swallow a rubber band and a cat more likely to get injured when the vet's office is closed.

"We handle only emergencies. No worming, no shots, no scratching," Dr. Wise, chief of medical services at the animal clinic, explains.

Last week both Morris and Maggie were recovering after the emergency treatment and later visits to their regular veterinarians. Morris, his owners report, is now frisking in the leaves in Potomac, Md.

The Metropolitan Emergency Animal Clinic, housed in a shopping center at 12213 Nebel St., off the Rockville Pike near Randolph Road in Rockville, is open from 6 p.m. to 8 a.m. week-nights and from Saturday noon until 8 a.m. Monday.

Jill Bonart, who cheerily describes herself as an unemployed poet, certainly was glad the clinic was open for emergencies on a recent Sunday afternoon. She had been walking her chocolate toy poodle near the National Air and Space Museum.

"I was carrying him from the museum lobby, where we had stopped, and then put him down on the sidewalk. Rinx started staggering. His eyes were bulging. I thought he had been poisoned," Bonart said, all the while reassuring Rinx in the strange surroundings of the clinic treatment room with its stainless steel examining table.

The preliminary diagnosis was encouraging: Rinx may have eaten something to cause the trouble, but he should be watched for any other seizures although epilepsy didn't seem a probable cause.

At the after-hours clinic, the patients are not only the cat or dog hit by a car or poisoned or wounded in battle. There are other life-and-death situations that include illnesses where immediate care can mean the difference with only hours to spare: cystitis, epileptic seizures, hemorrhaging diarrhea, gastric torsion cases in which the stomach twists to prevent the relief of upchucking, the emergency Caesarean for the Yorkshire terrier having her first litter of three pups.

"A regular veterinary practice can get to be rather routine at times. In the first three months here, "I've seen cases that I never handled before," Dr. Wise points out.

On that Sunday afternoon, Dr. Wise and Dr. Harrison Foy, who works weekends at the emergency clinic while off-duty from the veterinary section at Walter Reed Army Hospital, had been busy.

Bismarck, a 10-year-old miniature dachshund, feeling as sprightly as a pup, had leaped off a sofa to come up limping (no fracture, probably a knee problem with a ruptured cruciate). A scarred veteran tomcat was dozing under sedation after an ugly abscessed wound had been lanced and cleaned. In a nearby cage, Patches, a black-and-white cat, was being treated after a brush with a car. Ace, a black-and-tan beagle, was taking intravenous fluids and antibiotics while waiting for major surgery on a tumor. And there were the unnamed dogs, one white and another a mixed-breed brown, brought in for treatment by the Montgomery Humane Society after being hit by cars.

The clinic charges a $20 emergency fee on top of the regular treatment charge. The pet will be kept only overnight. The owner is told to return to his own veterinarian during regular office hours for follow-up care.

"That was a call from an owner of a dog," Dr. Foy said hanging up the phone. "He said he noticed a lump the size of a grapefruit had popped out in just three or four hours. It's probably a salivary cyst and could wait until tomorrow but the owner was worried and wanted to bring the dog over now so he wouldn't worry overnight."