"His ideas were somewhat fantastic, were they not?" asked Poirot carelessly.

"Some people have thought so. I do not agree."

"You considered them practicable?"

"Perfectly practicable. My own line of research has been somewhat similar, though not undertaken with the same end in view. I have been investigating the gamma rays emitted by the substance usually known as Radium C., a product of Radium emanation, and in doing so I have come across some very interesting magnetical phenomena. Indeed, I have a theory as to the actual nature of the force we call magnetism, but it is not yet time for my discoveries to be given to the world. Mr.

Halliday's experiments and views were exceedingly interesting to me."

Poirot nodded. Then he asked a question which surprised me.

"Madame, where did you converse on these topics. In here?"

"No, monsieur. In the laboratory."

"May I see it?"

"Certainly." 

She led the way to the door from which she had entered. It opened on a small passage. We passed through two doors and found ourselves in the big laboratory, with its array of beakers and crucibles and a hundred appliances of which I did not even know the names. There were two occupants, both busy with some experiment. Madame Olivier introduced them.

"Mademoiselle Claude, one of my assistants." A tall, serious-faced young girl bowed to us. "Monsieur Henri, an old and trusted friend."

The young man, short and dark, bowed jerkily.

Poirot looked round him. There were two other doors besides the one by which we had entered. One, madame explained, led into the garden, the other into a smaller chamber also devoted to research. Poirot took all this in, then declared himself ready to return to the salon.

"Madame, were you alone with M. Halliday during your interview?"

"Yes, monsieur. My two assistants were in the smaller room next door."

"Could your conversation be overheard-by them or any one else?"

Madame reflected, then shook her head.

"I do not think so. I am almost sure it could not. The doors were all shut."

"Could any one have been concealed in the room?"

"There is the big cupboard in the corner-but the idea is absurd."

"Pas tout d fait, madame. One thing more: did M.

Halliday make any mention of his plans for the evening?"

"He said nothing whatever, monsieur."

"I thank you, madame, and I apologise for disturbing you. Pray do not trouble-we can find our way out."

We stepped out into the hall. A lady was just entering the front door as we did so. She ran quickly up the stairs, and I was left with an impression of the heavy mourning that denotes a French widow.

"A most unusual type of woman, that," remarked Poirot, as we walked away.

"Madame Olivier? Yes, she-"

"Mais non, not Madame Olivier. Cela va sans dire! There are not many geniuses of her stamp in the world.

No, I referred to the other lady-the lady on the stairs."

"I didn't see her face," I said, staring. "And I hardly see how you could have done. She never looked at us."

"That is why I said she was an unusual type," said Poirot placidly. "A woman who enters her home-for I presume that it is her home since she enters with a key-and runs straight upstairs without even looking at two strange visitors in the hall to see who they are, is a very unusual type of woman-quite unnatural, in fact. Mille tonnerres! what is that?"

He dragged me back-just in time. A tree had crashed down on to the side walk, just missing us. Poirot stared at it, pale and upset.

"It was a near thing that! But clumsy, all the same-for I had no suspicion-at least hardly any suspicion.

Yes, but for my quick eyes, the eyes of a cat, Hercule Poirot might now be crushed out of existence-a terrible calamity for the world. And you, too, mon ami- though that would not be such a national catastrophe."

"Thank you," I said coldly. "And what are we going to do now?"

"Do?" cried Poirot. "We are going to think. Yes, here and now, we are going to exercise our little gray cells. This M. Halliday now, was he really in Paris? Yes, for Professor Bourgoneau, who knows him, saw and spoke to him." 

"What on earth are you driving at?" I cried.

"That was Friday morning. He was last seen at eleven Friday night-but was he seen then?"

"The porter-"

"A night porter-who had not previously seen Halliday.

A man comes in, sufficiently like Halliday-we may trust Number Four for that-asks for letters, goes upstairs, packs a small suit-case, and slips out the next morning. Nobody saw Halliday all that evening-no, because he was already in the hands of his enemies. Was it Halliday whom Madame Olivier received? Yes, for though she did not know him by sight, an impostor could hardly deceive her on her own special subject. He came here, he had his interview, he left. What happened next?"

Seizing me by the arm, Poirot was fairly dragging me back to the villa.

"Now, mon ami, imagine that it is the day after the disappearance, and that we are tracking footprints. You love footprints, do you not? See-here they go, a man's, Mr. Halliday's… He turns to the right as we did, he walks briskly-ah! other footsteps following behind-very quickly-small footsteps, a woman's.

See, she catches him up-a slim young woman, in a widow's veil. 'Pardon, monsieur, Madame Olivier desires that I recall you.' He stops, he turns. Now where would the young woman take him? She does not wish to be seen walking with him. Is it coincidence that she catches up with him just where a narrow alleyway opens, dividing two gardens. She leads him down it. 'It is shorter this way, monsieur.' On the right is the garden of Madame Olivier's villa, on the left the garden of another villa-and from that garden, mark you, the tree fell-so nearly on us. Garden doors from both open on the alley. The ambush is there. Men pour out, overpower him, and carry him into the strange villa."

"Good gracious, Poirot," I cried, "are you pretending to see all this?"

"I see it with the eyes of the mind, mon ami. So, and only so, could it have happened. Come, let us go back to the house."

"You want to see Madame Olivier again?"

Poirot gave a curious smile.

"No, Hastings, I want to see the face of the lady on the stairs."

"Who do you think she is, a relation of Madame Olivier's?"

"More probably a secretary-and a secretary engaged not very long ago."

The same gentle acolyte opened the door to us.

"Can you tell me," said Poirot, "the name of the lady, the widow lady, who came in just now?"

"Madame Veroneau? Madame's secretary?"

"That is the lady. Would you be so kind as to ask her to speak to us for a moment."

The youth disappeared. He soon reappeared.

"I am sorry. Madame Veroneau must have gone out again."

"I think not," said Poirot quietly. "Will you give her my name, M. Hercule Poirot, and say that it is important I should see her at once, as I am just going to the Prefecture."

Again our messenger departed. This time the lady descended. She walked into the salon. We followed her.

She turned and raised her veil. To my astonishment I recognised our old antagonist, the Countess Rossakoff, a Russian countess, who had engineered a particularly smart jewel robbery in London.

"As soon as I caught sight of you in the hall, I feared the worst," she observed plaintively.

"My dear Countess Rossakoff-"

She shook her head.

"Inez Veroneau now," she murmured. "A Spaniard, married to a Frenchman. What do you want of me, M.Poirot? You are a terrible man. You hunted me from London. Now, I suppose, you will tell our wonderful Madame Olivier about me, and hunt me from Paris?

We poor Russians, we must live, you know."

"It is more serious than that, madame," said Poirot watching her. "I propose to enter the villa next door, and release M. Halliday, if he is still alive. I know everything, you see."