Pearl Reed is in a random Southern city, at the train station. After you talk to her, you will have to obtain a medicine for the sister, before the time limit ends. If Pearl dies, the game does not end and you will soon receive a contract from your other brother.
To obtain the medicine, you will have to go to any drugstore, buy them and get them back to the girl. The nearest one is in Philadelphia. Once there, you only have to buy a small phial of the medicine - all depends on how much time you have left, a larger bottle will ensure her a longer life. This is because, in the second step, you will have to obtain special healing herbs.
For the herbs, you will have to go to Louisville. There, talk to Matsuwa. He wants you to bring him the herbs and shoes. Unfortunately, the shoemaker's is as far as in Boston, which is why it is better to buy them before you set off for Louisville. It's enough to buy the cheapest shoes.
To get the herbs, you have to go to Cleveland. There, while talking to Marguet pick the second dialogue option - for your politeness, you will be rewarded with herbs, free of charge. However, this is only one of the two jobs to perform.
You can buy the other herb - sweet grass - from a woman (ab. $700), go to the next destination, or go to an Native American Camp - in nearly all of them, you can buy the herb for around $100. Finally, return to Matsuwa and give him a ride, in the passenger carriage, to Pearl, and you will have completed the mission.