In "The Adventure of the Speckled Band," what motives might Dr. Roylott have for killing Julia Stoner?

When Helen Stoner comes to see Sherlock Holmes at Baker Street in the early morning, she tells him, significantly:

“When Dr. Roylott was in India he married my mother, Mrs. Stoner....My sister Julia and I were twins, and we were only two years old at the time of my mother's re-marriage. She had a considerable sum of money—not less than £1000 a year—and this she bequeathed to Dr. Roylott entirely while we resided with him, with a provision that a certain annual sum should be allowed to each of us in the event of our marriage."

A bit further on in her long back story, Helen tells the detective and his friend Dr. Watson about her and her sister's unhappy lives at Stoke Moran, but adds:

"We had, however, an aunt, my mother's maiden sister...and we were occasionally allowed to pay short visits at this lady's house. Julia went there at Christmas two years ago, and met there a half-pay major of marines, to whom she became engaged....but within a fortnight of the day which had been fixed for the wedding, the terrible event occurred which has deprived me of my only companion.” 

Before Holmes and Watson go down to Stoke Moran to examine the premises, Holmes does some research in the civil records and learns that the deceased mother's will provided that the girls' stepfather would be legally obligated to pay either of them one-third of the proceeds of the capital she had left him if either girl married. Holmes naturally suspects that Dr. Roylott must have been responsible for Julia's death two years ago to spare himself from having to pay her such a large sum annually once she was married. Holmes ascertains that:

"The total income, which at the time of the wife's death was little short of £1100, is now...not more than £750. Each daughter can claim an income of £250, in case of marriage. It is evident, therefore, that if both girls had married, this beauty would have had a mere pittance, while even one of them would cripple him to a very serious extent." 

Dr. Roylott would have a strong motive to kill Julia in order to avoid paying her and her husband £250 a year indefinitely. Now Helen Stoner has become engaged to be married within two or three months. Dr. Roylott has found an excuse for moving Helen into Julia's former room, which is right next to his own. He pretends that some repair work was necessary on Helen's own room, which is separated from Roylott's by Julia's. Evidently her stepfather wanted her to be sleeping in Julia's old room for some reason. The main mystery in "The Adventure of the Speckled Band" is how Dr. Roylott could have killed Julia when she was sleeping in a locked room which seemed impenetrable from the outside.

Holmes and Watson travel down to Stoke Moran and hire a trap at Leatherhead Station to take them out to Roylott's crumbling country mansion. There Holmes examines the room where Julia died two years ago and then examines Dr. Roylott's room next door. Holmes observes that there is a ventilator between the two rooms. In Helen's room there is a dummy bell-rope by the bed. The bed cannot be moved because it has been bolted to the floor. He and Watson stay in Julia's room that same night, and Holmes drives the "speckled band" back up the bell-rope and through the ventilator, where it bites Dr. Roylott and kills him instantly. Later Holmes explains his deductions to Watson:

"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."

So Sherlock Holmes not only saves the life of his client Helen Stoner but solves the two-year-old mystery of the death of her sister Julia. Dr. Roylott had only one motive for murdering Julia with his poisonous snake. He was in desperate financial trouble and simply could not afford to pay Julia £250 a year out of his dwindling  assets. By killing her, he solved that problem. But then Helen became engaged and he had the same problem all over again. He moved Helen into the room next to his and was planning to kill her by the same means. But she became terrified on the second night when at around three o'clock she heard the same low whistle which her sister had described to her shortly before she died an agonizing death. Helen waited until daybreak and then hurried to London to consult Sherlock Holmes, whom she had heard of from a friend.