Recording his verdicts, the coroner said the deaths represented a "great darkness" and were the "sort of thing nightmares are made of".
After offering his condolences to members of the couple's family, the coroner told them: "It's clear to me that the relationship between Ruth and her husband had lasted for about 15 years and was generally quite good.
It appears there were ups and downs and there were separations but nothing so remarkable as to give a clue as to what followed.
A discussion about a separation had taken place on July 11, the coroner said, at a time when Mrs Fuller was clearly unwell.
Ruling that it was highly likely that the children were killed on July 12, the coroner added: "What exactly happened on that fateful day may never fully be known.
From the scene, the police, despite a painstaking search, could find no evidence to suggest any third-party involvement in the deaths.
The injuries sustained by the children were caused by a knife. The evidence that the injuries were caused by their father is overwhelming and that conclusion is inescapable.
I am satisfied and I am sure that he, and he alone, inflicted fatal injuries on his children.
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Sean Cawthray, who recently changed his name to Perry, failed to return to his probation-approved address on 19 June.