Late at night, a crow is seen looking down on a pentagram/spell circle drawn on the floor of a spooky looking house, complete with mad scientist lab equipment, burning incense and wall sconces. A beautiful woman in robes is seen drawing symbols on the floor in chalk. She begins reciting an arcane spell, laughs, then stops and rubs her robe, saying she wishes she didn't have to wear the silly thing. Then she goes back to casting the spell to summon His Satanic Majesty for a little cha cha. At the lab equipment, she mixes up some "juice", and lights a fire in an urn. She announces that she's safe within the cast circle and pours the liquid in the urn. She demands that the spirits show her who her enemy is and there is a big puff of smoke. A hooded figure appears, holding a vial. The woman looks surprised; the figure throws liquid in the woman's face. She screams, staggers and falls, knocking the urn and fire over. Seen from the crows-eye-view, we see her body as the flames start to consume the house.
Burke is with his girlfriend, Sugar. She mixes him a suspicious foaming drink at the bar and Burke tells her she's a wretched bartender. She calls the drink a locust, like a grasshopper but better. It smokes dangerously. She tells Burke that it's important to base a relationship on hate, then you have nowhere to go but up. Just as Burke works up nerve to try the drink, Henry delivers a package that has just come special delivery. Sugar wonders if it's from one of her rivals.
A MAN WOULD HAVE TO BE OUT HIS MIND TO SEEK ELSEWHERE, WHEN YOU ARE OBVIOUSLY GODDESS OF THE NIGHT - BURKE'S LAW
Inside the package is a record which Burke plays. The voice is that of Purity Mather, a direct lineal descendent of Cotton Mather, who is supposedly a sorceress. Purity says that she is going to be murdered and has placed a list of her possible killers in the record jacket, so Burke won't have to waste his time. She claims to have a way of knowing these things, told to her by Baal-Shaman the crow, her familiar, but doesn't know when or who the murderer is. She only knows that by the time Burke gets the record, she'll be dead. Sugar thinks she's a nut, but Burke isn't so sure.
At the home of Purity Mather, fire and police crews have assembled and the house is mostly a charred ruin. McCleod says she didn't die from the fire but was asphyxiated by having had sulfuric acid thrown in her face. Between the acid and the fire, the only way the body has been identified is by a tattoo, a witch's symbol, on her shoulder. The neighbors identify it from having seen Purity sunbathing. Tim notes that there are no tan marks on the body. Purity's crow has escaped his chain. When caught, a medallion is found around Baal-Shaman's neck with the same symbol as the tattoo and a secret compartment. Inside the compartment is a note which reads: "Summon me from fire; the witching hour, full of woe". Les says any other clues were burned in the fire. Burke gives him the crow to care for and tells them about the record.
At the first address on the list, there is a sign on the door "Go Away I'm Sleeping Peace Be With You Love Is Truth". Tim knocks anyway and an angry Fakir George O'Shea answers the door. He calls the wrath of heaven on them, but lets them into his lab. He's been sleeping on a real bed of nails (Les doubts it until he touches them). An Irish Moslem "fakir", or holy man, he says he took up the religion when he was with the 10th Air Force flying over India during the war. He is now a consulting chemist for several cosmetic firms. Tim finds 3 crystal forms of sulfuric acid in O'Shea's lab. O'Shea says he uses them for etching, dyes, calico printing and varnishes. He angrily yells at them that he is not nervous, then lies on his bed of nails and wills them away. He calls Purity a "witch" but says he has learned to banish hate. Tim asks him why he doesn't work in the company lab, rather than in his rooms and he says he's busy with very "private" research. Les threatens him with the soft mattress and pillows at the station and he tells them he invented and finally perfected a "lipstick pill", the formula for which he says Purity stole. She tried to market it herself. Women would only have to take a pill and never have to put on lipstick. When Tim and Les tell O'Shea Purity was killed at midnight, he replies that he has an alibi - he was in a mosque with about 50 other people until about 12:30. Les warns him not to leave town. O'Shea replies that he can send his "astral body" anywhere, while his earthly shell stays in the room. Tim says they'll settle for that.
In the highrise suite of millionaire I.A. Bugg, Burke enters through the terrace after getting no answer at the front door. He finds Bugg in lederhosen wielding a large feather as he chases an attractive young woman, who cowers at the fireplace, making terrified shrieks and squeals. She keeps squeaking at every slightest move, but says nothing. Bugg wonders if Burke has climbed 17 floors, but Burke reassures him that he only came around the terrace when they didn't hear him knock. Bugg tells Burke that the young woman likes to make noise and, no, she's not his wife, only a pleasant evening's entertainment. When Burke asks Bugg about sulfuric acid, the girl squeals. Burke asks him to get her to stop that. Bugg tells him she's easily frightened and may have thought Burke was a ghost. After all, he holds seances there often (assisted by Purity) and the girl was once frightened by a large glop of ectoplasm. When told of Purity's death, Bugg dances for joy and tells Burke he knew it had to happen - "Viola" has gotten even with Purity. He keeps dancing around, chanting "oh joy oh rapture". Bugg goes to change his clothes and tells the girl to talks to Burke while he's gone. As soon as he leaves, the girl climbs down from the ledge and says thank goodness now I can relax, and acts perfectly normal. She tells Burke she comes to Bugg's suite three nights a week and lets him chase her - he never catches her (probably doesn't want to). He's a millionaire with nothing else to do - plus he pays her well. "Viola" was Bugg's childhood violin teacher who died decades ago, and he is always trying to call up her ghost. She says Bugg told her some woman was trying to gyp him out of a lot of money with fake seances and he was going to get even with her. The girl assumes it was Purity. Just as she is getting cozy, Burke realizes that Bugg is taking awfully long to change. Burke goes to check and finds his room empty - he has left by a hidden elevator behind a secret panel which goes to the garage. Girl Girl tells Burke that explains what happened that night before. Bugg left her and went to his room about midnight for about a half hour, complaining of a stomach ache. When she tried the bedroom door it was locked. She complains to Burke that she hasn't been paid yet and tells him her name is "Girl". When he asks if that's her first or last name - she says "both - Girl Girl".
Sgt. Ames gives Burke information on Count Carlo Szipesti, who claims to be descended from a long line of Transylvanian royalty - with very long sharp teeth. Everyone knows his claim of being a vampire is a phony, but he carries it off well. He lives out of his car - which is a glass-sided hearse. He is currently making a "special personal appearance" in front of the Studio City Theater for an all-night horror festival.
Burke pulls up to the theater where there's a poster announcing the appearance of "the world's only living vampire". The marquee advertises a double bill of "The Creature That Ate NY" and "She". Burke talks to Gorgo the chauffeur who directs him to the "front door" - in the back. He knocks then enters the rear door of the hearse, bumping his head on the roof. On the wall is a picture of Bela Lugosi. The Count is engrossed watching cartoons on TV, drinking a Bloody Mary. He furrows his painted on eyebrows as he intensely, intensely, intensely squints at the cartoons through his spectacles, ignoring Burke. Burke finally gets his attention and the Count tells him how much violence on tv bothers him. He offers Burke a Bloody Mary, saying he hates them but they help keep up his image. Throughout the interview, passersby peer in the side windows, only to have the Count snarl or make "threatning" gestures at them. "So I'm a vampire - nobody's perfect". He says being a vampire is better than being unemployed. At midnight he was at a banquet with friends. He calls Purity a trollop and a strumpet and tells Burke he hated her because Purity called him a fraud in front of a group of the wealthiest, most influential society people in town. He could have killed her. When Burke tells him Purity was murdered with sulfuric acid and fire, he says that's not his style. It's too messy - the best way is with two little holes in the neck. (Another snarl at passersby). Burke says he thinks the Count is a fraud and the Count tells him not to make fun of him just because his teeth aren't sharp - he has dental troubles. An alarm goes off and the Count says he must leave soon as dawn is coming. The message in the crow's medallion means nothing to him, but the Count says it is Purity's handwriting. He tells Burke she was a strange girl - no one believed she was a witch. He assures Burke that he himself is authentic. He also says hat Rina Jacobs is a "doll". She runs the Health-Glo Nudist Colony. Rina and Purity used to be close until Purity stole Rina's fiancee; he later died in Europe in a plane crash. Rina never forgave Purity. The Count says he met Venus Hekate Walsh at a seance - she thinks she's the reincarnation of Venus - but she's nuts. Burke warns the Count not to turn into a bat, moose or aardvark or leave town. The Count reminds Burke that he's tied there by his native soil. He calls out to Gorgo to hurry home and beat the dawn. Burke bumps his head on the hearse roof again, and arranges to have the Count's alibi checked.
At the Health-Glo Sunbathing Colony, Burke and Tim come across a NO TRESPASSING sign. A nude woman appears from behind the foliage, then stands behind a discretely placed fence. She refuses to let them into the colony and says all her licenses are in order. When Burke identifies himself, she tells him she's Rina Jacobs. She doesn't recognize anything about the note when he shows it to her. She doesn't want to talk about Purity and tells Burke she never leaves the colony except for "special" reasons. When she thinks of Purity, she says she thinks of a goodlooking gentle man who turned to Purity and got destruction; she's still upset by his death. She thinks O'Shea is the most likely suspect because of the theft of something, she doesn't know what. She tells Burke that whoever killed Purity has her deepest gratitude.
The lab reports come in showing that the tattoo on the body was phony. Bugg is still missing.
Venus Hekate Walsh is in rooms are filled with cages with two of every sort of animal. When Burke finds her, she is garishly made up, wearing diaphonous robes covered in hearts and a fright wig, churning out leaflets on a mimeo machine to inform her "disciples" that she has returned to Earth. She thinks Burke and Tim are the new delivery boys (says the others were only 12 years old and not nearly so well dressed). She piles Burke's arms full of fliers to distribute as Tim discusses Hekate and Egyptian religion with her. Venus says she has lost most of her believers but still has her animal friends. She shows him the cages and reminds him you have to have two of everything. She tells Burke of her belief in Free Love - which is one of the advantages of worshipping Venus. Venus keeps getting confused, mixing up Tim's name and thinking Burke is the messenger boy. She's afraid they don't find her attractive and says she'll have to punish Purity for that. If she wasn't a goddess, she'd despise Purity. Venus tells Burke that Purity was like a giant magnet pulling all the loveliness out of her like a vacuum - she fears she is becoming "almost" plain. No wonder no one believes in her. Burke shows her the note, but she has no idea what it is, other than it's lovely handwriting. She tells them to call her "your celestial goddess". She gets Tim's name mixed up so many times, he gets it wrong when he corrects her, and she gets it right. Tim wants to write her off as just a kook, but Burke wonders if it's just an act.
Bugg is finally located at the Foley Cemetary, in Viola's mausoleum, hhaving a picnic while sitting on her crypt, with the inscription "Viola Duvoux". He asks Burke to sit and join him in peanut butter and jelly sandwiches. He had Viola moved from Kankakee, Illinois in 1937. He's still in love with her. She was beautiful and he keeps trying to let her know that one of her students still remembers and loves her. Purity bilked him out of a lot of money and he's glad someone got even, but it wasn't him. He just panicked when Burke came to see him. He keeps talking to Viola throughout the interview. Bugg reads the note Burke found and says he has no idea what it means. Burke and Tim leave him picnicking with "Viola". He asks Viola if it was right for Purity to cheat him and answers "no" for her.
When Burke returns home, a hooded figure comes out of his second floor bedroom window and runs away, dropping a bottle. There's no odor, but Tim takes it to the lab for tests. Les is at the station with the crow. They discuss how there are no rational motives in the case, or motives that only seem to be thrown in to confuse the situation. Later, Burke, getting changed in his room, is about to brush his hair when Tim receives a call from Les at the station. Tim rushes upstairs and violently knocks the hair brushes out of Burke's hand. It seems they had been treated with curare-spear poison. Burke wonders what the killer thinks Burke knows and hasn't figured out yet. Tim says that it had to O'Shea, but why curare? Burke replies:WHEN THE POSSIBLE SOLUTION IS NO LONGER POSSIBLE, ACCEPT THE IMPOSSIBLE. Tim asks: BURKE'S LAW? Burke replies, no, Sherlock Holmes.
He says it's all to pat to be O'Shea and that the list in the record jacket is hurting them more than helping. He delivers a real BURKE'S LAW
I DO NOT LIKE KILLERS DELIVERED TO ME IN AN ENVELOPE
What does Burke have that he doesn't know he has? Burke thinks of the crow again. Behind the first secret compartment in the mediallion, he finds a second, containing the formula for the lipstick pill. Tim figures out the message from the childhood rhyme "Monday's child is fair of face..." Wednesday's child is full of woe, so the message means to go to the burned ruins of Purity's home at midnight on Wednesday. Burke knows who the killer is. Among the ruined timbers of the house, a hooded figure enters, sees them and runs, but trips. They catch and unmask him - it's The Count. He swears he didn't kill Purity but was in on the lipstick deal. Purity was planning to get rid of O'Shea, and told the Count to ignore what the papers said. He was to get papers drawn up for their partnership. She would be away for a few days and would leave a message with the crow on where they were to meet to sign the paperwork. The Count figured it out when Burke showed him the note. This of course means that Purity isn't dead.
Just then they all hear a noise, Burke turns out the lights and the hide, as another hooded figure enters the scene. When the lights are turned on and the figure caught and un"hooded", Burke finds Rina Jacobs, dressed in a lame evening gown. Actually, who the figure is is the real Purity Mather - the real Rina Jacobs was the woman who was killed. The real Rina of course would have had no tan line and no real tattoo. Purity had covered the real tattoo on her own body with body paint - in fact Tim had noticed how pale she was for a nudist. Purity lured Rina to her home, "apologizing" for having stolen her fiance and promising to help relieve Rina of her "curse". She convinced Rina to don the robes (Rina, of course, was uncomfortable in clothes) and put on the fake tattoo as part of the ritual. She killed Rina to take her place and continue the theft of the lipstick pill. She had planned to kill The Count as well - the Count tells her she's not nice. She tried to poison Burke with curare because she was afraid he knew too much. As she finishes her confession, Purity lets loose with a spell and tosses a smoke bomb in order to escape. Instead, the smoke bomb chokes Purity - "too much unicorn horn" - and Burke nabs her again.
YOU WIN SOME AND YOU LOSE SOME - BURKE'S LAW
Back home, Burke and Sugar dance. Sugar breaks the news to Burke that they have to break their date for Thursday because she's getting married on Wednesday. Sugar tells Burke that she can meet him on Saturday, though.