RICHARD DUVALL rose, the following day, with a less troubled mind than at any time since his arrival in Paris.

His calculations of the night before had brought him to a definite conclusion.

After breakfasting in the cafÉ of the hotel he returned to his room, and rang up Monsieur Lefevre.

"I want the assistance of one of your men, Monsieur," he said.

"Ah!" laughed the Prefect. "You are—what you Americans call—up a tree, is it not?"

"Not at all. You have said that there existed between us a competition, to recover Mr. Stapleton's child. I think I am going to win. But since I am not in a position to make the necessary arrests, myself, I am going to share the glory with you, my dear friend, by allowing one of your men to do so for me."

"So you are confident?""Reasonably so. Can you spare Vernet for the day? He is a good man."

"One of my best. You shall have him. And if you succeed, I shall still regard myself the loser, and will buy the champagne, and the dinner at the CafÉ Royale, as I agreed."

"And I shall be most happy to do the same should I fail. Oblige me by requesting Vernet to come to my rooms at the hotel at once. Good by."

Duvall hung up the receiver, and sat down with the drawings he had made before him. He awaited the coming of Vernet with impatience.

The latter appeared in some twenty minutes.

"What can I do for you, Monsieur Duvall?" he asked.

"Good morning, Vernet. Sit down, and have a cigar. I have a little matter I wish to talk over with you."

"Concerning the missing child of Monsieur Stapleton, I understand," remarked Vernet, as he lit a cigar and drew his chair up to the table. He glanced at the drawings before him. "What are these, may I ask?"

Duvall took up his pencil. "This, Vernet, is a map of a small part of Paris. Here, as you see, is the Avenue Kleber, terminating at the Champs ÉlysÉes just in front of the Arc de Triomphe."

"I see. It is quite plain."

"Here—this black square—is Mr. Stapleton's house. From there to the arch is a matter of some six hundred yards."

"About that, I should say. What of it?"

"Wait. The black-bearded fellow—the kidnapper—who visited Mr. Stapleton last night, and escaped by the ruse of being arrested by one of his confederates, will arrive at Mr. Stapleton's house at eight o'clock tonight."

"Mon Dieu! If that is so, we have him!"

"Not so fast. We shall not interfere with him—then."

"But, Monsieur, would you let this fellow escape? It is my duty to arrest him, as soon as he puts in an appearance."

"You are mistaken, Vernet. Your duty is to do as I instruct you. Monsieur Lefevre has placed you under my orders for the day."

Vernet laughed. "That is so," he said. "What do you wish me to do?"

"The man will come to Mr. Stapleton's house at eight o'clock, and will be given a large sum of money. He has agreed, if he is not interfered with, to have the address where the boy may be found telephoned to Mr. Stapleton within half an hour."

"Ah! Then we shall follow, and get him after he has telephoned."

Duvall laughed. "We are dealing with a far shrewder man than that, Vernet. This fellow will do no telephoning."

"Then how will he let Monsieur Stapleton know?"

"That is just what I am trying to find out. Put yourself in his place. He is known—he dare not remain in Paris—he gets five hundred thousand francs to give up the child. Is it not natural to suppose that he will leave the city at once?"

"Yes. That is what I should do, in his place."

"Of course. Now I understand that the fellow will walk from Mr. Stapleton's house to the Arc de Triomphe, a distance of six hundred yards. He can do that easily in ten minutes."

"Yes."

"Once at the arch, he will stand awaiting a fast automobile, which will come along the Champs ÉlysÉes. This automobile will stop for an instant and pick him up, then proceed at high speed along the Avenue du Bois de Boulogne.""Why do you think that?"

"Because it will afford him the quickest and safest road out of Paris. From the arch to the Porte Dauphine is less than a mile. He can make it in five minutes. In fifteen minutes altogether then, he is outside the walls. In another fifteen minutes, he is beyond pursuit, in the country."

"But you forget, Monsieur Duvall, that he has not yet advised his confederates that all is well, and that the address of the place where the boy is hidden is to be telephoned to Mr. Stapleton."

"No, Vernet, I haven't forgotten that. In fact, I am coming to it now. Suppose you were in this fellow's place—how would you do it?"

Vernet scratched his head thoughtfully. "He might fire a pistol from the car."

"Too dangerous. The noise of the explosion would attract attention. He must work silently."

"A wave of the hand, perhaps, to someone along the street."

"Also dangerous. This fellow realizes that every possible step will be taken to capture not only himself, but his confederates. He anticipates, no doubt, that the road will be carefully watched. Why take chances, and run the risk of his confederates, at least, being arrested, when there are simpler, easier ways?"

"Such as what?"

"Do you not remember the signal, used on the Versailles road, the blue light?"

"Ah! Exactly. He will signal to some one in a house along the way."

"That would be easier and safer; but you will remember that there are no houses along the way—none, at least, in which a man of this sort could have a confederate hidden. But I should not say none. There is one, perhaps."

"Indeed, Monsieur. And what house is that?"

"Mr. Stapleton's. Look!" He drew toward him the sheet of paper. "Here," he placed the point of his pencil upon the black square which indicated the location of the banker's residence, "is the house. The north window of a room on the top floor commands a view of the Avenue du Bois de Boulogne, from a point some 500 feet west of the Arc de Triomphe, to where it intersects the Avenue Malakoff. Beyond there, the view is interrupted. In fact, the trees along the Avenue du Bois de Boulogne are to some extent an obstruction; but at the crossing with the Avenue Malakoff there is a wide and uninterrupted view."

"But a confederate in Monsieur Stapleton's own house?"

"Yes. The chauffeur, FranÇois."

"You astonish me, Monsieur. We have suspected the fellow, it is true. The very room of which you speak has been searched. We found nothing. How do you know that what you say is true?"

"Never mind how I know it—now. The point is this—FranÇois, I fully believe, will be in that room, tonight, at eight o'clock, watching carefully the automobiles which pass the intersection of the Avenue Malakoff—"

"Not necessarily, Monsieur. We can easily prevent it, by placing him under arrest."

"That is exactly what we must not do. Don't you see, it is absolutely necessary, for the recovery of Mr. Stapleton's child, that the signals go through uninterrupted?"

"Of course, I had forgotten that. And these signals?"

"Naturally I cannot tell—yet. I think, however, that the automobile for which FranÇois will be looking will show a brilliant blue light, while crossing the Avenue Malakoff. That is, of course, if our friend the kidnapper gets safely away, without being pursued."

"And otherwise?"

"I think the light would be red. He can make either, very simply, by means of a powerful electric searchlight—one of these pocket affairs, you know, fitted with colored glasses."

"You interest me wonderfully, Monsieur Duvall. What next?"

"It is, of course, most important that the signal given shall be the correct one. There must be no interference whatever with this fellow's escape—up to that point."

"Ah—I begin to see. And what after that?"

"First, let us continue with FranÇois. He will, I think, return a blue signal to the man in the automobile, to show that he has seen, and understood. He has the means to do so all ready, in his room."

"And then?"

"He will make, I think, a similar signal from his south window to some one who is on watch, in the direction of Passy. This second person, who no doubt has the child in his care, will then go to a telephone, transmit the address of the house where the child is hidden, to Mr. Stapleton, and quietly depart, to join his confederate in—say—Brussels. He will run not the slightest risk of capture. If, on the other hand, that message fails to go through, the address will not be telephoned, and the child will probably be killed."

Vernet frowned grimly. "It is a remarkable plan, Monsieur. These fellows are no bunglers. I think, however, that we shall be able to stop them."

"How?"

"I will station myself at the Porte Dauphine with a fast automobile, a racer. When these fellows pass, I will follow them, and overtake them."

"An excellent idea, Vernet; but how, may I ask, will you know the car, when it passes you? There are hundreds of cars on the Avenue du Bois de Boulogne, at eight o'clock in the evening."

Vernet laughed. "I confess, Monsieur, you have me there."

"Of course you might station a man at the intersection of the Avenue Malakoff and the Avenue du Bois de Boulogne; but I do not think he would be able to see the signal. By placing on the end of the searchlight a paper tube, the light would be invisible except in the direction in which it is pointed—and that, you will remember, is diagonally upward. A man on the sidewalk would not see it at all."

"Then, Monsieur, I fail to see that there is anything we can do."

"There is one thing, Vernet. You forget the answering signal, from the window."

The Frenchman looked at his companion with undisguised admiration. "SacrÉ!" he exclaimed. "You have a mind, Monsieur Duvall, in a thousand."

"Thanks," answered Duvall, dryly. "Now, my idea is, to have you select some point near the intersection of the two avenues, from which the window in the rear of Mr. Stapleton's house can be clearly seen. Station yourself there, tonight, with the fastest automobile you can secure. Let one man watch the window, another the vehicles passing in the Avenue du Bois de Boulogne. The moment you see the blue light, start after your man. He should be just across the intersection, on his way down the Avenue du Bois de Boulogne."Vernet rubbed his hands together with satisfaction. "We shall get him—never fear."

"Of course," said Duvall, slowly, "all this is pure assumption on my part, based upon what I have discovered in the chauffeur's room. It may not turn out as I say, but the chances are fifty to one that it will."

"And you, Monsieur? Where will you be?"

"I shall be in the room, with FranÇois. I do not propose that he shall escape. And further—I do not know that I am correct, in my assumption regarding his signals to Passy. He may go out, and send the telephone message himself. In that case, I shall follow. Or he may, through some unforeseen accident, get the wrong signal, in which case I propose to overpower him, and give the right one. Suppose we go, now, and take a look at the intersection of the Avenue Malakoff and the Avenue du Bois de Boulogne, and see what arrangements can best be made. Also, if Mr. Stapleton is out in his car, we may be able to take a few observations from his chauffeur's window." He took up his hat, lighted a cigar, and led the way to the door.

They drove to the Arc de Triomphe in a cab, and, after dismissing it, walked slowly down the Avenue du Bois de Boulogne. At the intersection with the Avenue Malakoff they stopped and gazed about carefully, although in such a way as not to attract attention. A brief inspection served to confirm all that Duvall had said. It took them some little time to locate the window in the rear of Mr. Stapleton's house; but after a time they managed to do so, and saw that it commanded an uninterrupted view of the point where they stood.

Vernet was highly satisfied, as they parted. It was deemed unnecessary for him to visit the chauffeur's room, and thereby run the risk of their being seen entering the banker's house together. Vernet departed to make his arrangements for the evening, strictly cautioned by his companion not to let Monsieur Lefevre into his secret. "It is a bet," he told Vernet. "I hope we shall succeed in winning it."

After his companion had departed, Duvall dropped in to see Mr. Stapleton. He learned that the banker was out, driving in the Bois with Mrs. Stapleton, who, overcome by anxiety and grief, had great need of the fresh air to retain her health. She was fast breaking down under the strain.Duvall went up to have another look at the chauffeur's room. He had been unable to get a thoroughly clear idea of the view from the window, the night before, owing to the darkness.

He found everything as he had left it,—the searchlight on the dresser, the colored glass ornaments hanging from their gay ribbons. The north window overlooked with perfect clearness the intersection of the two avenues, as he and Vernet had seen them from below. The other window presented a more distant view. Nearby roofs and chimneys obstructed it in part; but between them could be seen the villas and buildings in Passy, smiling in the sunlight. The sight impressed Duvall the more strongly with the cleverness of the men he sought to arrest. Somewhere in all that maze of buildings, that wide vista of houses and trees and distant fields, Mr. Stapleton's child lay concealed, and it needed but a flash of light from this window to set him free. Passing his fingers idly along the window sill, Duvall suddenly observed two parallel scratches in the white paint, which had apparently been made with the point of a knife. He knelt down, and sighted between them. His line of vision swept clear of the nearby roofs and chimneys, toward Passy.

The detective turned from the window, a smile of satisfaction on his face, and proceeded to make a careful examination of the chauffeur's closet. It was here that he intended to lie hidden. He felt certain that, in order the better to perceive and send his signals, as well as to escape detection from below, the chauffeur would allow his room to remain unlighted.

This, Duvall reasoned, would render it easy for him to lie concealed until the signal which would insure the safe return of the lost child had been given, after which he would call upon FranÇois with precision and despatch. Should anything occur to prevent the chauffeur from giving the favorable signal, he proposed to give it himself.

The closet was close to the north window, and its door opened in such a way that Duvall saw at once that in the darkened room he could readily open it sufficiently to see all that FranÇois did, without running any serious risk of detection.

He left the house at a little after noon and stopped in at a well known restaurant on the Boulevard des Italiens for lunch. He felt very well satisfied with the course that events were taking. If only he could get through with this thing, and get back to Grace, and the farm, he would be supremely happy. He became so absorbed in his thoughts that he failed to notice a gentleman who slipped quietly into the chair opposite him, until the latter leaned over and touched his arm.

He looked up suddenly. It was Monsieur Lefevre!