How did Holmes catch Dr. Roylott in "The Adventure of the Speckled Band"?

Holmes caught Dr. Grimesby Roylott in the act of attempting to murder his niece Helen Stoner by using a deadly poisonous snake he had obtained from India. Holmes had instructed Helen not to sleep in the bedroom next to her stepfather's that night. Instead, he and Watson climbed through the window after Roylott had retired and spent some hours waiting in total silence in the dark. Holmes had deduced what was going to happen, and he knew that Dr. Roylott would have to act quickly because Helen was engaged to be married very soon, in which case Roylott would have been forced to give her an annual payment of approximately one-third of the income from her deceased mother's estate. Holmes assumed that Roylott had murdered Helen's sister Julia two years ago for the same reason: to get out of paying her what amounted to one-third of his total annual income. Holmes explains his deductions to his friend Dr. Watson after Roylott is dead and the case is all wrapped up.

My attention was speedily drawn, as I have already remarked to you, to this ventilator, and to the bell-rope which hung down to the bed. The discovery that this was a dummy, and that the bed was clamped to the floor, instantly gave rise to the suspicion that the rope was there as a bridge for something passing through the hole and coming to the bed. The idea of a snake instantly occurred to me, and when I coupled it with my knowledge that the doctor was furnished with a supply of creatures from India, I felt that I was probably on the right track. The idea of using a form of poison which could not possibly be discovered by any chemical test was just such a one as would occur to a clever and ruthless man who had had an Eastern training. The rapidity with which such a poison would take effect would also, from his point of view, be an advantage. It would be a sharp-eyed coroner, indeed, who could distinguish the two little dark punctures which would show where the poison fangs had done their work. Then I thought of the whistle. Of course he must recall the snake before the morning light revealed it to the victim. He had trained it, probably by the use of the milk which we saw, to return to him when summoned. He would put it through this ventilator at the hour that he thought best, with the certainty that it would crawl down the rope and land on the bed. It might or might not bite the occupant, perhaps she might escape every night for a week, but sooner or later she must fall a victim.

The snake had no natural inclination to bite the victim, and Dr. Roylott had not trained it to bite anyone. What Holmes doesn't say, because of Victorian inhibitions, is that the snake, coming from a warm tropical country, would seek warmth when it reached the bed and would most likely crawl right under the covers with the sleeping girl, where it would curl up beside her. Watson and Holmes have both mentioned several times that the weather is very cold. This would explain why the snake would not try to escape from the house, which was just barely possible if it could manage to squeeze underneath the bedroom door into the corridor. The girl in the bed would not be bitten unless she were to turn over in the night and roll right on top of the snake. Then it would certainly bite her. That was what Roylott was expecting. Everybody turns over in his or her sleep during the night, so it was just a matter of time. Julia had told Helen that she had heard the strange whistling for three nights before she was fatally bitten on the fourth night; so she was sleeping with a deadly snake curled up beside her for four nights. Helen tells Holmes:

"In her right hand was found the charred stump of a match, and in her left a match-box.”

So what had happened to Julia was not that she turned over in her sleep but that she had heard the whistling again on the fourth night and reached for the box of matches. In doing so, she had rolled right on top of the snake and had been bitten through her nightgown.

Holmes does not realize the snake is there in the room with them until he hears the whistling. He quickly strikes a match and sees the "speckled band" climbing up the dummy bell-rope as it had been trained to do. Holmes hits it with his cane and drives the angered swamp adder back through the ventilator more quickly than Dr. Roylott had expected. That was how he came to be bitten and to die almost instantly.