“That is certainly what the appearances suggest,” he agreed.

“It was undoubtedly Bendelow who was cremated,” said I.

“All the circumstances seem to point to that conclusion,” he admitted, “unless you can think of any that point in the opposite direction.”

“I cannot,” I replied. “Everything points in the same direction. The dead man was seen and identified as Bendelow by those two ladies, Miss Dewsnep and Miss Bonington, and they not only saw him here, but they actually saw him in his coffin just before it was passed through into the crematorium. And there is no doubt that they knew Bendelow by sight, for you remember that they recognized the photograph of him that the American detective showed them.”

“Yes,” he admitted, “that is so. But their identification is a point that requires further investigation. And it is a vitally important point. I have my own hypothesis as to what took place, but that hypothesis will have to be tested; and that test will be what the logicians would call the Experimentum Crucis. It will settle one way or the other whether my theory of this case is correct. If my hypothesis as to their identification is true, there will be nothing left to investigate. The case will be complete and ready to turn over to Miller.”

I listened to this statement in complete bewilderment. Thorndyke’s reference to “the case” conveyed nothing definite to me. It was all so involved that I had almost lost count of the subject of our investigation.

“When you speak of ‘the case,’ ” said I, “what case are you referring to?”

“My dear Gray!” he protested. “Do you not realize that we are trying to discover who murdered Julius D’Arblay?”

“I thought you were,” I answered; “but I can’t connect this new mystery with his death in any way.”

“Never mind,” said he. “When the case is completed we will have a general elucidation. Meanwhile there is something else that I have to show you before we go. It is through this side door.”