Hi and welcome to RootsChat. How wonderful to be living in Ruk. I am quite envious. Thank you for the link to the photos.
I don't know exactly who cared for Beulah while she was recuperating in Australia. There was a lengthy article in the Hawaiian Gazette [23 February 1900] which I can send to you if you would like to read it. In summary, it says that Beulah and two servants were transported from Ruk to Australia on the SS Archer, arriving there on 31 January. She was well cared for on board by Captain Cargill. John T Arundel, Esq. in Sydney [I don't know who he was] arranged for her to be treated by Dr. Jenkins, a "leading expert on spinal troubles," who estimated that she would need three months recovery before she could travel. A room was engaged for her at "Miss Duffy's private hospital." She was visited there by several missionary friends of Mr. Arundel. At that time, she had not yet been informed of her mother's death, which had occurred in December 1899. The article also states that she had originally injured her back when the missionary schooner 'Robert W. Logan' was wrecked on a reef and she fell when she jumped from it into a smaller boat.
And there is this note concerning her return to the United States from Australia. It was published in 'The Friend' [Volume 58, Number 6, 1 June 1900 Edition 01]:
"On the 9th of May Miss Beulah Logan arrived with her two native attendants from Sydney, enfeebled and somewhat helpless by a fall she had sustained on shipboard, which had brought on again a little of the old trouble caused by the wreck of the Robert Logan on the island of Satoan, one of the Mortlocks. Good medical aid in Sydney probably saved her life and the aid of two medical practitioners, fellow voyagers on the steamship, probably again saved her life. It is her wish and expectation to return to her friends in Buffalo, New York. She hopes to take passage on the China. While in the city she is making her home with Mr. and Mrs. Bouenat Makiki."
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Mary Fenn Logan died on 1 December 1899 in Creston, Ohio.