‘You and I, my friend,’ Poirot leaned towards Race, ‘started our investigation with a preconceived idea. That idea was that the crime was committed on the spur of the moment, without any preliminary planning. Somebody wished to remove Linnet Doyle and had seized their opportunity to do so at a moment when the crime would almost certainly be attributed to Jacqueline de Bellefort. It therefore followed that the person in question had overheard the scene between Jacqueline and Simon Doyle and had obtained possession of the pistol after the others had left the saloon.

‘But, my friends, if that preconceived idea was wrong, the whole aspect of the case altered. And it was wrong! This was no spontaneous crime committed on the spur of the moment. It was, on the contrary, very carefully planned and accurately timed, with all the details meticulously worked out beforehand, even to the drugging of Hercule Poirot’s bottle of wine on the night in question!

‘But yes, that is so! I was put to sleep so that there should be no possibility of my participating in the events of the night. It did just occur to me as a possibility. I drink wine; my two companions at table drink whisky and mineral water respectively. Nothing easier than to slip a dose of harmless narcotic into my bottle of wine – the bottles stand on the tables all day. But I dismissed the thought. It had been a hot day, I had been unusually tired – it was not really extraordinary that I should for once have slept heavily instead of lightly as I usually do.

‘You see, I was still in the grip of the preconceived idea. If I had been drugged, that would have implied premeditation, it would mean that before 7.30, when dinner is served, the crime had already been decided upon; and that (always from the point of view of the preconceived idea) was absurd. ‘The first blow to the preconceived idea was when the pistol was recovered from the Nile. To begin with, if we were right in our assumptions, the pistol ought never to have been thrown overboard at all … And there was more to follow.’ Poirot turned to Dr Bessner. ‘You, Dr Bessner, examined Linnet Doyle’s body. You will remember that the wound showed signs of scorching – that is to say, that the pistol had been placed close against the head before being fired.’

Bessner nodded.

‘So. That is exact.’

‘But when the pistol was found it was wrapped in a velvet stole, and that velvet showed definite signs that a pistol had been fired through its folds, presumably under the impression that that would deaden the sound of the shot. But if the pistol had been fired through the velvet, there would have been no signs of burning on the victim’s skin. Therefore, the shot fired through the stole could not have been the shot that killed Linnet Doyle. Could it have been the other shot – the one fired by Jacqueline de Bellefort at Simon Doyle? Again no, for there had been two witnesses of that shooting, and we knew all about it. It appeared, therefore, as though a third shot had been fired – one we knew nothing about. But only two shots had been fired from the pistol, and there was no hint or suggestion of another shot.

‘Here we were face to face with a very curious unexplained circumstance. The next interesting point was the fact that in Linnet Doyle’s cabin I found two bottles of coloured nail polish. Now ladies very often vary the colour of their nails, but so far Linnet Doyle’s nails had always been the shade called Cardinal – a deep dark red. The other bottle was labelled Rose, which is a shade of pale pink, but the few drops remaining in the bottle were not pale pink but a bright red. I was sufficiently curious to take out the stopper and sniff. Instead of the usual emphasis odour of pear drops, the bottle smelt of vinegar! That is to say, it suggested that the drop or two of fluid in it was red ink. Now there is no reason why Madame Doyle should not have had a bottle of red ink, but it would have been more natural if she had had red ink in a red ink bottle and not in a nail polish bottle. It suggested a link with the faintly stained handkerchief which had been wrapped round the pistol. Red ink washes out quickly but always leaves a pale pink stain.

‘I should perhaps have arrived at the truth with these slender indications, but an event occurred which rendered all doubt superfluous. Louise Bourget was killed in circumstances which pointed unmistakably to the fact that she had been blackmailing the murderer. Not only was a fragment of a mille franc note still clasped in her hand, but I remembered some very significant words she had used this morning.

‘Listen carefully, for here is the crux of the whole matter. When I asked her if she had seen anything the previous night she gave this very curious answer: “Naturally, if I had been unable to sleep, if I had mounted the stairs, then perhaps I might have seen this assassin, this monster enter or leave Madame’s cabin…” Now what exactly did that tell us?’

Bessner, his nose wrinkling with intellectual interest, replied promptly:

‘It told you that she had mounted the stairs.’

‘No, no – you fail to see the point. Why should she have said that, to us?’

‘To convey a hint.’

‘But why hint to us? If she knows who the murderer is, there are two courses open to her – to tell us the truth, or to hold her tongue and demand money for her silence from the person concerned! But she does neither. She neither says promptly: “I saw nobody. I was asleep.” Nor does she say: “Yes, I saw someone, and it was so and so.” Why use that significant indeterminate rigmarole of words? Parbleu, there can be only one reason! She is hinting to the murderer; therefore the murderer must have been present at the time. But, besides myself and Colonel Race, only two people were present – Simon Doyle and Dr Bessner.’

The doctor sprang up with a roar.

‘Ach! what is that you say? You accuse me? Again? But it is ridiculous – beneath contempt.’

Poirot said sharply:

‘Be quiet. I am telling you what I thought at the time. Let us remain impersonal.’

‘He doesn’t mean he thinks it’s you now,’ said Cornelia soothingly.

Poirot went on quickly.

‘So it lay there – between Simon Doyle and Dr Bessner. But what reason has Bessner to kill Linnet Doyle? None, so far as I know. Simon Doyle, then? But that was impossible! There were plenty of witnesses who could swear that Doyle never left the saloon that evening until the quarrel broke out. After that he was wounded and it would then have been physically impossible for him to have done so. Had I good evidence on both those points? Yes, I had the evidence of Mademoiselle Robson, of Jim Fanthorp and of Jacqueline de Bellefort as to the first, and I had the skilled testimony of Dr Bessner and of Mademoiselle Bowers as to the other. No doubt was possible.

‘So Dr Bessner must be the guilty one. In favour of this theory there was the fact that the maid had been stabbed with a surgical knife. On the other hand Bessner had deliberately called attention to this fact.

‘And then, my friends, a second perfectly indisputable fact became apparent to me. Louise Bourget’s hint could not have been intended for Dr Bessner, because she could perfectly well have spoken to him in private at any time she liked. There was one person, and one person only, who corresponded to her necessity – Simon Doyle! Simon Doyle was wounded, was constantly attended by a doctor, was in that doctor’s cabin. It was to him therefore that she risked saying those ambiguous words in case she might not get another chance. And I remember how she had gone on, turning to him: “Monsieur, I implore you – you see how it is? What can I say?” And this answer: “My good girl, don’t be a fool. Nobody thinks you saw or heard anything. You’ll be quite all right. I’ll look after you. Nobody’s accusing you of anything.” That was the assurance she wanted, and she got it!’

Bessner uttered a colossal snort.

‘Ach! it is foolish, that! Do you think a man with a fractured bone and a splint on his leg could go walking about the boat and stabbing people? I tell you, it was impossible for Simon Doyle to leave his cabin.’

Poirot said gently:

‘I know. That is quite true. The thing was impossible. It was impossible – but it was also true! There could be only one logical meaning behind Louise Bourget’s words.

‘So I returned to the beginning and reviewed the crime in the light of this new knowledge. Was it possible that in the period preceding the quarrel Simon Doyle had left the saloon and the others had forgotten or not noticed it? I could not see that that was possible. Could the skilled testimony of Dr Bessner and Mademoiselle Bowers be disregarded? Again I felt sure it could not. But, I remembered, there was a gap between the two. Simon Doyle had been alone in the saloon for a period of five minutes, and the skilled testimony of Dr Bessner only applied to the time after that period. For that period we had only the evidence of visual appearance, and, though apparently that was perfectly sound, it was no longer certain. What had actually been seen – leaving assumption out of the question?

‘Mademoiselle Robson had seen Mademoiselle de Bellefort fire her pistol, had seen Simon Doyle collapse on to a chair, had seen him clasp a handkerchief to his leg and seen that handkerchief gradually soak through red. What had Monsieur Fanthorp heard and seen? He heard a shot, he found Doyle with a red-stained handkerchief clasped to his leg. What had happened then? Doyle had been very insistent that Mademoiselle de Bellefort should be got away, that she should not be left alone. After that, he suggested that Fanthorp should get hold of the doctor.

‘Accordingly Mademoiselle Robson and Monsieur Fanthorp got out with Mademoiselle de Bellefort and for the next five minutes they are busy, on the port side of the deck. Mademoiselle Bowers’, Dr Bessner’s and Mademoiselle de Bellefort’s cabins are all on the port side. Two minutes are all that Simon Doyle needs. He picks up the pistol from under the sofa, slips out of his shoes, runs like a hare silently along the starboard deck, enters his wife’s cabin, creeps up to her as she lies asleep, shoots her through the head, puts the bottle that has contained the red ink on her washstand (it mustn’t be found on him), runs back, gets hold of Mademoiselle Van Schuyler’s velvet stole, which he has quietly stuffed down the side of a chair in readiness, muffles it round the pistol and fires a bullet into his leg. His chair into which he falls (in genuine agony this time) is by a window. He lifts the window and throws the pistol (wrapped up with the tell-tale handkerchief in the velvet stole) into the Nile.’

‘Impossible!’ said Race.

‘No, my friend, not impossible. Remember the evidence of Tim Allerton. He heard a pop – followed by a splash. And he heard something else – the footsteps of a man running – a man running past his door. But nobody could have been running along the starboard side of the deck. What he heard was the stockinged feet of Simon Doyle running past his cabin.’

Race said: ‘I still say it’s impossible. No man could work out the whole caboodle like that in a flash – especially a chap like Doyle who is slow in his mental processes.’

‘But very quick and deft in his physical actions!’

‘That, yes. But he wouldn’t be capable of thinking the whole thing out.’

‘But he did not think it out himself, my friend. That is where we were all wrong. It looked like a crime committed on the spur of the moment, but it was not a crime committed on the spur of the moment. As I say, it was a very cleverly planned and well thought out piece of work. It could not be chance that Simon Doyle had a bottle of red ink in his pocket. No, it must be design. It was not chance that he had a plain unmarked handkerchief with him. It was not chance that Jacqueline de Bellefort’s foot kicked the pistol under the settee, where it would be out of sight and unremembered until later.’

‘Jacqueline?’

‘Certainly. The two halves of the murderer. What gave Simon his alibi? The shot fired by Jacqueline. What gave Jacqueline her alibi? The insistence of Simon which resulted in a hospital nurse remaining with her all night. There, between the two of them, you get all the qualities you require – the cool, resourceful, planning brain, Jacqueline de Bellefort’s brain, and the man of action to carry it out with incredible swiftness and timing.’

‘Look at it the right way, and it answers every question. Simon Doyle and Jacqueline had been lovers. Realize that they are still lovers, and it is all clear. Simon does away with his rich wife, inherits her money, and in due course will marry his old love. It was all very ingenious. The persecution of Madame Doyle by Jacqueline, all part of the plan. Simon’s pretended rage. And yet – there were lapses. He held forth to me once about possessive women – held forth with real bitterness. It ought to have been clear to me that it was his wife he was thinking about – not Jacqueline. Then his manner to his wife in public. An ordinary, inarticulate Englishman, such as Simon Doyle, is very embarrassed at showing any affection. Simon was not a really good actor. He overdid the devoted manner. That conversation I had with Mademoiselle Jacqueline, too, when she pretended that somebody had overheard, I saw no one. And there was no one! But it was to be a useful red herring later. Then one night on this boat I thought I heard Simon and Linnet outside my cabin. He was saying, “We’ve got to go through with it now.” It was Doyle all right, but it was to Jacqueline he was speaking.

‘The final drama was perfectly planned and timed. There was a sleeping draught for me, in case I might put an inconvenient finger in the pie. There was the selection of Mademoiselle Robson as a witness – the working up of the scene, Mademoiselle de Bellefort’s exaggerated remorse and hysterics. She made a good deal of noise, in case the shot should be heard. En vérité, it was an extraordinarily clever idea. Jacqueline says she has shot Doyle, Mademoiselle Robson says so, Fanthorp says so – and when Simon’s leg is examined he has been shot. It looks unanswerable! For both of them there is a perfect alibi – at the cost, it is true, of a certain amount of pain and risk to Simon Doyle, but it is necessary that his wound should definitely disable him.

‘And then the plan goes wrong. Louise Bourget has been wakeful. She has come up the stairway and she has seen Simon Doyle run along to his wife’s cabin and come back. Easy enough to piece together what has happened the following day. And so she makes her greedy bid for hush money, and in so doing signs her death warrant.’

‘But Mr Doyle couldn’t have killed her?’ Cornelia objected.

‘No, the other partner did that murder. As soon as he can, Simon Doyle asks to see Jacqueline. He even asks me to leave them alone together. He tells her then of the new danger. They must act at once. He knows where Bessner’s scalpels are kept. After the crime the scalpel is wiped and returned, and then, very late and rather out of breath, Jacqueline de Bellefort hurries in to lunch.

‘And still all is not well. For Madame Otterbourne has seen Jacqueline go into Louise Bourget’s cabin. And she comes hot-foot to tell Simon about it. Jacqueline is the murderess. Do you remember how Simon shouted at the poor woman? Nerves, we thought. But the door was open and he was trying to convey the danger to his accomplice. She heard and she acted – acted like lightning. She remembered Pennington had talked about a revolver. She got hold of it, crept up outside the door, listened and, at the critical moment, fired. She boasted once that she was a good shot, and her boast was not an idle one.

‘I remarked after that third crime that there were three ways the murderer could have gone. I meant that he could have gone aft (in which case Tim Allerton was the criminal), he could have gone over the side (very improbable) or he could have gone into a cabin. Jacqueline’s cabin was just two away from Dr Bessner’s. She had only to throw down the revolver, bolt into the cabin, ruffle her hair and fling herself down on the bunk. It was risky, but it was the only possible chance.’

There was a silence, then Race asked:

‘What happened to the first bullet fired at Doyle by the girl?’

‘I think it went into the table. There is a recently made hole there. I think Doyle had time to dig it out with a penknife and fling it through the window. He had, of course, a spare cartridge, so that it would appear that only two shots had been fired.’

Cornelia sighed.

‘They thought of everything,’ she said. ‘It’s – horrible!’

Poirot was silent. But it was not a modest silence. His eyes seemed to be saying: ‘You are wrong. They didn’t allow for Hercule Poirot.’

Aloud he said,

‘And now, Doctor, we will go and have a word with your patient…’