Heath scratched his head, and laughed mirthlessly.

“It don’t make sense, does it, sir?”

“What about the next apartment?” asked Vance, “the one with the door facing the rear passageway—No. 2, I think?”

Heath turned to him patronizingly. “I looked into that the first thing this morning. Apartment No. 2 is occupied by a single woman; and I woke her up at eight o’clock and searched the place. Nothing there. Anyway, you have to walk past the switchboard to reach her apartment the same as you do to reach this one; and nobody called on her or left her apartment last night. What’s more, Jessup, who’s a shrewd sound lad, told me this woman is a quiet, ladylike sort, and that she and Odell didn’t even know each other.”

“You’re so thorough, Sergeant!” murmured Vance.

“Of course,” put in Markham, “it would have been possible for some one from the other apartment to have slipped in here behind the operator’s back between seven and eleven, and then to have slipped back after the murder. But as Sergeant Heath’s search this morning failed to uncover any one, we can eliminate the possibility of our man having operated from that quarter.”

“I dare say you’re right,” Vance indifferently admitted. “But it strikes me, Markham old dear, that your own affectin’ recapitulation of the situation jolly well eliminates the possibility of your man’s having operated from any quarter. … And yet he came in, garroted the unfortunate damsel, and departed—eh, what? … It’s a charmin’ little problem. I wouldn’t have missed it for worlds.”

“It’s uncanny,” pronounced Markham gloomily.

“It’s positively spiritualistic,” amended Vance. “It has the caressin’ odor of a séance. Really, y’ know, I’m beginning to suspect that some medium was hovering in the vicinage last night doing some rather tip-top materializations. … I say, Markham, could you get an indictment against an ectoplasmic emanation?”

“It wasn’t no spook that made those finger-prints,” growled Heath, with surly truculence.

Markham halted his nervous pacing and regarded Vance irritably.

“Damn it! This is rank nonsense. The man got in some way, and he got out, too. There’s something wrong somewhere. Either the maid is mistaken about some one being here when she left, or else one of those phone operators went to sleep and won’t admit it.”

“Or else one of ’em’s lying,” supplemented Heath.

Vance shook his head. “The dusky fille de chambre[39], I’d say, is eminently trustworthy. And if there was any doubt about any one’s having come in the front door unnoticed, the lads on the switchboard would, in the present circumstances, be only too eager to admit it. … No, Markham, you’ll simply have to approach this affair from the astral plane, so to speak.”

Markham grunted his distaste of Vance’s jocularity.

“That line of investigation I leave to you with your metaphysical theories and esoteric hypotheses.”

“But, consider,” protested Vance banteringly. “You’ve proved conclusively—or, rather, you’ve demonstrated legally—that no one could have entered or departed from this apartment last night; and, as you’ve often told me, a court of law must decide all matters, not in accord with the known or suspected facts, but according to the evidence; and the evidence in this case would prove a sound alibi for every corporeal being extant. And yet, it’s not exactly tenable, d’ ye see, that the lady strangled herself. If only it had been poison, what an exquisite and satisfying suicide case you’d have! … Most inconsiderate of her homicidal visitor not to have used arsenic instead of his hands!”