Thompson nodded.

‘That’s usually the case. In non-medical terms it couldn’t have been put better. You know something about such things, M. Poirot. Don’t you agree?’

I don’t think that Crome liked Thompson’s appeal to Poirot. He considered that he and he only was the expert on this subject.

‘It is as Inspector Crome says,’ agreed Poirot.

‘Paranoia,’ murmured the doctor.

Poirot turned to Crome.

‘Are there any material facts of interest in the Bexhill case?’

‘Nothing very definite. A waiter at the Splendide at Eastbourne recognizes the dead girl’s photograph as that of a young woman who dined there on the evening of the 24th in company with a middle-aged man in spectacles. It’s also been recognized at a roadhouse place called the Scarlet Runner halfway between Bexhill and London. They say she was there about 9 pm on the 24th with a man who looked like a naval officer. They can’t both be right, but either of them’s probable. Of course, there’s a host of other identifications, but most of them not good for much. We haven’t been able to trace the A B C.’

‘Well, you seem to be doing all that can be done, Crome,’ said the Assistant Commissioner. ‘What do you say, M. Poirot? Does any line of inquiry suggest itself to you?’

Poirot said slowly:

‘It seems to me that there is one very important clue—the discovery of the motive.’

‘Isn’t that pretty obvious? An alphabetical complex. Isn’t that what you called it, doctor?’

‘C¸a, oui,’ said Poirot. ‘There is an alphabetical complex. But why an alphabetical complex? A madman in particular has always a very strong reason for the crimes he commits.’

‘Come, come, M. Poirot,’ said Crome. ‘Look at Stoneman in 1929. He ended by trying to do away with anyone who annoyed him in the slightest degree.’

Poirot turned to him.

‘Quite so. But if you are a sufficiently great and important person, it is necessary that you should be spared small annoyances. If a fly settles on your forehead again and again, maddening you by its tickling—what do you do? You endeavour to kill that fly. You have no qualms about it. You are important—the fly is not. You kill the fly and the annoyance ceases. Your action appears to you sane and justifiable. Another reason for killing a fly is if you have a strong passion for hygiene. The fly is a potential source of danger to the community—the fly must go. So works the mind of the mentally deranged criminal. But consider now this case—if the victims are alphabetically selected, then they are not being removed because they are a source of annoyance to the murderer personally. It would be too much of a coincidence to combine the two.’

‘That’s a point,’ said Dr Thompson. ‘I remember a case where a woman’s husband was condemned to death. She started killing the members of the jury one by one. Quite a time before the crimes were connected up. They seemed entirely haphazard. But as M. Poirot says, there isn’t such a thing as a murderer who commits crimes at random. Either he removes people who stand (however insignificantly) in his path, or else he kills by conviction. He removes clergymen, or policemen, or prostitutes because he firmly believes that they should be removed. That doesn’t apply here either as far as I can see. Mrs Ascher and Betty Barnard cannot be linked as members of the same class. Of course, it’s possible that there is a sex complex. Both victims have been women. We can tell better, of course, after the next crime—’