December 1963. High in the Swiss Alps there exists a finishing school for young girls, presided over by the eccentric Miss Tremayne -- a Scots spinster with a religious fixation and a conviction that cold air purifies the soul. It's an unpleasant place to be stuck three days before Christmas, but Alison Speers' parents failed to arrive before winter storms closed down the roads, and her friend Peril Bellamy's family has decided -- based on past experience -- that Christmas will be much more peaceful if Peril isn't around. Thus Alison and Peril are trapped in a cold, ghostly academy, haunted by the gloomy strains of French mistress Mlle. Maupassant's piano playing... and Peril, for one, is having none of it. Thus it is that one night, Alison finds herself rappelling down the side of the academy and fleeing into the night with Peril, helping her friend not just to escape, but to elope. She suspects, however, that Peril is doing it just to annoy her mother... and as the blizzard closes in, she starts to believe that this was not a terribly wise move.

Nyssa is also trapped on the mountainside, wearing clothing far more appropriate for the warm interior of the TARDIS than the oncoming blizzard to which she has unexpectedly been teleported. She is rescued by Lieutenant Peter Sandoz of the mountain patrol, who is on his way to check on the school's inhabitants. She soon determines that she is unable to keep up with Sandoz, even on his spare pair of skis, and he thus gallantly offers to walk her to the school, apparently assuming that her talk of a teleportation experiment gone wrong and a ship which travels between worlds is the result of her having been out in the cold for too long. Upon arriving at the school, they find Miss Tremayne in a panic, and when Sandoz learns that Alison and Peril are out in the storm he sets off to find them before she can warn him that they face other, stranger problems. The spirits of the dead have risen in the academy...

Mlle. Maupassant takes Nyssa into the drawing room, where Miss Tremayne opens up all of the windows to ensure that a cold breeze blows away the spiritual staleness of the room. Mlle. Maupassant apologises to Nyssa for her employer's eccentricity, but claims that she has some grounds for her concern -- and she and Nyssa then hear Miss Tremayne screaming in the corridor, where all of the windows she's just opened have closed and latched themselves. Terrified for her very soul, Miss Tremyane goes to the chapel to pray, and as Nyssa tries to come up with a logical explanation, she smells a faint scent of roses -- and a heavy wooden wardrobe drags itself from one side of the doorway to the other without anybody touching it. Sandoz then arrives with Peril and Alison, and as Nyssa cares for the shivering Alison, Alison claims to be suffering from a recurring migraine brought on by the phantom scent of roses...

Miss Tremayne is praying in the chapel, in a state of shock, and she doesn't respond to either Nyssa or Mlle. Maupassant -- but then Sandoz arrives and announces that Peril ran away because she was going to elope with him, a man old enough to be her father. This shock acts like a slap in the face, and the furious Miss Tremayne storms out of the chapel, vowing to have Sandoz fired. Nyssa tries to warn the skeptical Sandoz that there may be a poltergeist in the school, and although he scoffs at first, she smells a distant scent of flowers -- and the giant mirror on the refectory wall falls to the ground and smashes to pieces. Sandoz tries to insist that the mirror simply wasn't fastened properly, but the shards of glass then rise into the air and pursue them like a swarm of wasps. The wardrobe slides across the corridor, blocking their escape, and they are forced to flee to the attic -- where the TARDIS suddenly materialises before their eyes, causing the terrified Miss Tremayne to faint dead away.

Nyssa recognises the gleam of curiosity in the Doctor's eye and despairs, realising that they'll be remaining at the academy for some while longer. The Doctor sets off to see whether Sandoz has had any luck with the radio, but finds the lieutenant and Peril locked in an embrace. Sandoz still refuses to accept that ghosts are at large within the school, and prefers to believe that the Doctor is a charlatan of some kind, although even he cannot explain what the Doctor hopes to accomplish. Irritated, the Doctor retunes the radio and clears up the interference which he believes was caused by the electromagnetic disturbance of the haunting. But when he leaves the radio room to speak with Nyssa and fetch Sandoz, somebody places a blanket over the radio and smashes the tubes to pieces. It's not a poltergeist manifestation this time; the blanket was used to muffle the sound of breaking glass. Someone has deliberately cut them off.

Sandoz still treats the Doctor with suspicion, and informs him that he had laid in contingency plans before departing the village; if he does not report in, then a helicopter will be sent to evacuate the school. They have a couple of hours yet before it arrives, and the Doctor intends to make use of the time remaining to him. He and Nyssa return to the attic to fetch equipment from the TARDIS, which was forced to appear in such an out-of-the-way location because the Doctor was tracking Nyssa's location and that's where she happened to be. Mlle. Maupassant is studying the Doctor's ship, fascinated, but the Doctor refuses to let her inside as he fetches the psychic tracking equipment he needs. Miss Tremayne has been coaxed to sleep with the aid of seconal, and there is thus nobody to disturb the Doctor and Nyssa as they scan the chapel and determine that it is indeed an epicentre of electromagnetic activity. The Doctor theorises that Miss Tremayne's religious convictions have imprinted a deep psychic signature on the chapel, and determines that one particular pew is strongly irradiated with psychic energy. But there is far more than can have come from any human being.

Alison pops in to see how the Doctor and Nyssa are progressing, and is amazed by the technology which the Doctor is using so casually. The Doctor notes that Alison shares his concerns about the age difference between Peril and Sandoz, and wonders why the girls of the academy seem so impressionable. Nyssa sees a plaque commemmorating the late Harding Wellman, a young mountaineer who died in an avalanche, but the Doctor finds no psychic residue when he scans the plaque. He, Nyssa and Alison return to the kitchen, where they find the Lieutenant's ski equipment drying out by the fire. Peril, given the run of the kitchen, has made a plate of ham sandwiches for lunch. As the Doctor sits down to eat he claims to have the first workings of a theory -- but it's one he isn't ready to share until it's fully formed.

The Doctor knocks Nyssa out of the way, and the ski poles embed themselves in the wall. He urges Alison to drink her tea immediately, and although confused, she does so -- and falls into a trance. The "tea" was in fact a mild sedative, and the poltergeist activity has now stopped. Much to the Doctor's surprise, however, when he scans Alison for traces of telekinetic activity, he finds nothing. Sandoz arrives to tell them that the helicopter is arriving, and becomes openly hostile when he finds that the Doctor has drugged Alison; these girls are in his care, and he will not permit the Doctor to conduct his experiments upon them. The Doctor insists that they need to understand what is happening in order to control it, and loses his temper when Sandoz ignores him and carries Alison out of the room. Nyssa is not terribly sympathetic; she still has not forgiven the Doctor for teleporting her here with little explanation. However, he still refuses to tell her what the Spillagers are; if she's lucky she'll never find out. As he and Nyssa watch the helicopter approaching, however, the Doctor notices that the wind is blowing the snow into patterns around the playing field -- and smells the scent of flowers. He and Nyssa rush to warn the others, but are too late; the poltergeist whips the wind into a cyclone, and the helicopter is dashed against the mountainside, exploding and killing the pilot.

Miss Tremayne is even more hysterical, convinced that she saw the face of Satan in the flames. The shaken Sandoz tries to comfort Peril and Alison, while the Doctor and Nyssa return to the chapel, hoping to find out what is happening before anyone else dies. This time, the Doctor realises why Harding Wellman's plaque is not the focus of the psychic disturbances; the pew is where dozens of girls have sat over the years, looking at the plaque and thinking about it. He and Nyssa return to the drawing room, where the Doctor places Alison in a hypnotic trance and pulls out another testing device -- which Nyssa recognises as being similar to the one which transported her here. Still refusing to explain why he's so interested in detecting the Spillagers, the Doctor scans Alison and finds that although she is not telekinetic, she is telepathic -- and possesses an active gift where Nyssa's own abilities are passive. Alison is not causing the poltergeist activity directly, but at one remove, by sending telepathic signals into someone else's mind...

To confirm his theory, the Doctor gives some of his special tea to Peril and run the same tests on her. This time the tests are more conclusive, and when Peril awakens the Doctor and Nyssa confront her with their findings. At first, Peril tries to laugh them off, but eventually she must give in and admit that she has been able to move things with her mind ever since she was a child. Her gran always told her that she was one of the Adept; the talent runs in her family, and in past generations they were often accused of witchcraft. The Doctor theorises that Perli possesses a recessive gene passed down through the females of her family, but assures Peril that she is not responsible for the helicopter crash and the death of the pilot. For some reason her powers were triggered by Alison and raged out of control, but the Doctor doesn't yet understand why, or how. When Sandoz arrives, the Doctor departs, still upset with him; if he had been allowed to complete his experiments earlier, perhaps the helicopter pilot would not have died. As the Doctor and Nyssa leave, Sandoz asks Peril what they were discussing, and warns her that Miss Tremayne was listening at the door...

Miss Tremayne has always relied on the Good Book for guidance, and now she knows what she must do. She steals a butcher's knife from the kitchen and confronts Peril and Alison; she is an instrument of the Lord's wrath, and thou shalt not suffer a witch to live. The Doctor, Nyssa and Sandoz arrive to find the girls in danger, and although the Doctor tries to calm Miss Tremayne down, she lunges at the girls with her knife, and Sandoz shoots her dead. The Doctor storms out of the kitchen, furiously insisting that he could have talked her down without bloodshed, and Sandoz follows, insisting that he did what was necessary to protect Peril and Alison. As he protests, however, he and the Doctor smell flowers -- and the school's athletic trophies begin to fall out of their rack.

Sandoz refuses to participate in anything so absurd as a seance, and Mlle. Maupassant seems too afraid of what will happen to assist. The Doctor, Nyssa, Alison and Peril conduct the seance by themselves in the music room, and although Peril doesn't seem to be taking things seriously and Nyssa has trouble reconciling her rational beliefs with the thought of a seance, they eventually manage to concentrate -- and succeed in summoning the spirit of Harding Wellman. Or, if this is too much for Nyssa to accept, perhaps a being composed of pure energy who happens to have taken the memories and personality of a dead mountaineer. Harding remembers little about his death; only a rush of obliterating white and the scent of flowers at his funeral. But as he speaks, the Doctor and his friends smell the flowers he is describing, Alison feels another migraine coming on, Harding starts to feel odd -- and another poltergeist manifestation tears the room apart around them.

The Doctor and his friends flee to the kitchen, the disembodied Harding included, but although Harding claims that he was not responsible for the attack, he admits that he can remember little of what happened in the music room -- and that when he was alive, he suffered from epileptic seizures. The Doctor realises that Harding's presence causes the phantom smell of flowers, which triggers Alison's migraines, which cause her to trigger Peril's telekinesis, which triggers Harding's epilepsy and interacts with it to throw out dangerous surges of telekinetic power. The Doctor takes Nyssa back to the TARDIS, where he fetches a professionally designed Spillage detector, telling Nyssa that her teleportation may not have been an accident after all; the Spillagers are a particularly nasty form of life from another dimension, and the Doctor now believes that his prototype detector transported Nyssa to a centre of Spillager activity so that he would be forced to follow. He may be in time to prevent their advance scouts from paving the way for an invasion force... but when he takes the detector to the chapel to scan Miss Tremayne's body, he finds that she is entirely human. Which means that the scout must be someone else.

The Doctor confronts Mlle. Maupassant, who innocently confirms that she helped Miss Tremayne to select the students for the school. The Doctor isn't fooled by her apparent helpfulness, however; he now knows her to be a Spillager scout, who lured Adepts to this school in order to create a psychic gestalt and open a dimensional gateway through which her people can invade the Earth. This is what the Spillagers do; they spill out of gateways between dimensions, raiding them and moving on like a virus invading a cell. If they reach the Earth, humanity will be destroyed. The Spillager drops its human guise and pursues the Doctor back to the kitchen, where the Doctor urges Harding, Peril and Alison to manifest their powers again -- but deliberately, this time. With some effort, Harding creates a smell of cloves, Alison's migraine is triggered, and Peril manages to direct the energy flowing out of her, smashing the kitchen table through the door and crushing the attacking Spillager to death.

Sandoz and Nyssa arrive to find the crushed alien body lying outside the kitchen, and the shocked Sandoz finally appars to accept the Doctor's claims. He and the Doctor prepare to dispose of the body outside, and the Doctor tells the others to wait in the music room; they now have an opportunity to strike a telling blow against the Spillagers. Peril accompanies Sandoz outside, however, where the Doctor explains that they can now use the gestalt to crush the gateway down to a singularity, destroying the fleet. But Sandoz pulls a gun on the Doctor and reveals himself to be a second Spillager scout. He killed the real Sandoz some time ago, and pretended to be in love with Peril to keep her at the school until the gateway was open. Furious, Peril reveals that Harding has accompanied her, and they prepare to turn their powers against Sandoz -- but he shoots Peril and reveals that the gateway has been open for some time. The Spillagers were drawing energy from each poltergeist manifestation, and are even now preparing to come through. Before he can shoot the Doctor, however, they hear an ominous rumbling sound; the sound of Sandoz's first shot has triggered an avalanche. The Doctor grabs Peril and flees to safety as the avalanche buries Sandoz behind him.