“No; though there are still some things about that lady I don’t understand. I don’t expect any man could understand her altogether, for the matter of that.”

“Mrs. Ballantine, then?”

“No, no. This isn’t a woman’s crime. Besides, she had long got past the stage of hating Ballantine enough to murder him, if I’m any judge of character. I think she was the kind of woman to go on making the poor sinner’s life a burden to him with her rectitude and patience, and to take a pride in putting up with all the unhappiness he caused her. The suspect we’ve got to put on the list is a more dangerous character than that.”

“Who do you mean?”

“X,” answered the inspector. “The unknown quantity, who may upset all our calculations. It’s fatal to forget him. Whenever you make a list of possible criminals, you are apt to put yourself in blinkers and forget that anyone exists outside your list. Always put in X, and keep a sharp lookout for him.”

“And how do we set about finding X?” asked Frant, with heavy irony.

“I propose to do a little quiet thinking,” said Mallett. “Unless we’re unexpectedly lucky, I don’t think anything more is going to turn up to help us. The facts are here,” he pointed to the desk, “and I’ve got to spell them out.”

“But you said just now,” Frant objected, “that there’s nothing there to justify a conviction.”

“Perhaps not. But when I know where to look, it won’t be too hard to find the material I want. At least, I hope not. There is such a thing, of course, as knowing who the murderer is and not being able to bring it home to him.”

Frant rose to go.

“While you are doing that,” he said, “do you mind if I do a little work on my own?”

“Of course not. What had you in mind?”

“I still think Eales is the likeliest man on our little list. Certainly he’s the one we know least about. I should like to apply for a warrant for his arrest.”

“A warrant?”

“Yes—for bigamy.”

Mallett stared at him. “But you can’t,” he said. “Not on Mrs. Eales’s story to me alone.”

“I don’t mean to. If Ballantine found out about his first marriage, there ought to be some record of it in his papers. Renshaw has them all. I shall ask him to let me go through them.”

Mallett was never slow to show appreciation of a subordinate’s ideas. “That’s a really good suggestion,” he said. “Do that. Find out if we can prove Mrs. Eales’s story to be true, and let me know.”

“And then, may I apply for a warrant?”

The inspector smiled, somewhat in the manner of a parent humouring a child anxious for a treat.