On the evening of 31 January 1910, two couples dined together at a house in Hilldrop Crescent, on the borders of Holloway, London. The hosts, Dr Crippen and his wife, Belle Elmore, had been entertaining their friends, Clara and Paul Martinetti, until the small hours. After some difficulty fetching a cab, the visitors headed home around 1.30am. It was the last time they, or anyone else, would see Elmore alive. When her colleagues at the Music Hall Ladies’ Guild made inquiries about their friend – she was treasurer of the organisation – Crippen told them she had gone off to America to deal with a family crisis. Some weeks later they were informed she had died of double pneumonia in Los Angeles.

Thus was sparked an international murder case, one of the most notorious in Britain, later called “the crime of the century”. But Hallie Rubenhold’s engrossing account begins a generation earlier when Hawley Harvey Crippen, a homeopathic doctor, met and married a nurse, Charlotte Bell, in New York. The couple moved west to San Diego, had a son, moved again. In the US of the 1880s, with its burgeoning railroads, you could always change towns, disappear, shed your mistakes along with your creditors, your given name, your dependents. This was the shifty Crippen way, and when Charlotte died of a stroke, aged 33, he was on the move and marrying again. His second wife, a Brooklynite born Kunigunde Mackamotzki, changed her name more than once, eventually settling on Belle Elmore, and after crisscrossing the US the couple emigrated to London, he to peddle his quack remedies for the Drouet Institute, she to pursue a career as an opera singer.

This is where the story catches fire, as Rubenhold adroitly conjures a seedy Edwardian era of gaslight, grift and gullibility. As blowsy Elmore turns to music hall and splashes out on jewellery and furs, Crippen continues flogging medicines for the “petty rat swindlers”, and falls for his new typist, 27-year-old Ethel Le Neve. The patina of middle-class respectability begins to tarnish as the doctor desperately tries to run two horses at once, playing along with his childless wife (he supervised an early operation to remove her ovaries) while promising marriage to his young protege. The atmosphere of these pages brings to mind Sickert’s great gloomy painting Ennui (circa 1914) with its estranged couple confined in a rectangle of domestic discontent: something has to give. Mere weeks after Elmore’s mysterious disappearance, Crippen makes an incredible error in taking Le Neve to a Benevolent Fund ball, where his wife’s friends notice that his new partner is wearing a diamond brooch that belonged to Elmore. By June 1910 she has supplanted the wife altogether, appropriating her home, her clothes, her money and her husband. Every trace of Belle Elmore had been removed – or so Crippen thought.

Rubenhold’s energy as a researcher is indefatigable, and admirable. It helped to burnish her award-winning previous book The Five, which ingeniously flipped the mythology of Jack the Ripper to focus upon the lives of his victims. She means to repeat the trick here by shifting the centre of gravity from the murderer to the women his shadow crossed, but it doesn’t quite come off. Unlike the Ripper, we know the killer’s identity, and Crippen, for all his mild-mannered blankness, remains the fiendish engine of the story. The mystery here is not who, but why, and the book cannot supply a convincing answer. Time and again, Rubenhold has to admit to the limitations of life-writing when her research meets a dead end: “is unknown”…“is a mystery”… “There is no indication of how” …“is uncertain”…“will never be known”. The truth has receded to some undiscoverable place, a darkness that can only be (partly) illuminated by recourse to another genre – which is fiction.

Not that Story of a Murder: The Wives, the Mistress and Doctor Crippen wants for excitement or intrigue. The discovery of a dismembered corpse beneath the cellar floor at Hilldrop Crescent was the first of two sensational turning points in the tale, the bodily remains bearing traces of hyoscine, a poison that Crippen had recently purchased, alongside a scrap of his own pyjamas. The ineptitude was almost as remarkable as the callousness. Crippen and Le Neve fled to Belgium, the latter disguised as a boy, and then boarded the steamer SS Montrose on a voyage to Canada.

In hot pursuit was DCI Walter Dew, who had begun his investigation of Crippen back in London and now worked out a way he could bring the fugitives to justice. Here lay the second startling development: how technology caught a killer. The captain of the Montrose, Henry George Kendall, had identified Crippen and Le Neve onboard, and by means of the new wireless telegraph he alerted the authorities. This allowed Dew to board a faster ship across the Atlantic – the papers called it the “Great Ocean Chase” – and to apprehend the couple as they reached Canada.

By now the world had been apprised of the “north London cellar murder”, and the suspects’ return to trial in England had inflamed the popular imagination. But while technology had made bold advances, Edwardian morality was still dragging its feet. Crippen, his wax effigy already on display at Madame Tussaud’s, was tried separately at the Old Bailey, where the prosecution made short work of his lies and evasions. Le Neve, on the other hand, was presented to the court as an innocent dupe, kept ignorant under Crippen’s tight supervision. How could this delicate young lady have connived in such an atrocity? The press bought her story, and so did the jury. Crippen was hanged at Pentonville, and Le Neve walked free. Rubenhold pertinently contrasts this with an incident in London only three weeks after her acquittal. During a suffragette march to Parliament, police and male bystanders brutally set upon the protesters in the streets, blackened eyes, broke limbs and in some cases sexually assaulted them. It became known as Black Friday and served as a reminder to those women who defied the “natural order” – know your place, or suffer the consequences.

An astonishing coda closes the book. Rubenhold quotes from different interviews Le Neve gave in the 1920s, designed to exonerate herself but only hardening suspicion of her awareness of the crime, if not her actual complicity. She later married, had a family and died in 1967. Her two children knew nothing of their mother’s past until a true-crime writer contacted them in their middle age. Both were stunned to learn that Ethel Smith was in fact Ethel Le Neve, mistress of Dr Crippen and one-time suspect in a murder case that had gripped the age. Quite an imposture to maintain for more than half a century, but then she had been schooled by a master in the art.

The Mouthless Dead by Anthony Quinn is published by Abacus