Translations for fragment in French

Here are paragraphs from public domain books translated:
"And over here is what appears to be the fragment of a hotel bill, which interests me deeply." — Et je vois ici un fragment de note d’hôtel, qui m’intéresse extrêmement.
The Adventures of Sherlock Holmes, by Arthur Conan Doyle Les Aventures de Sherlock Holmes, de Arthur Conan Doyle
"On entering the house, however, I examined, as you remember, the sill and framework of the hall window with my lens, and I could at once see that some one had passed out. I could distinguish the outline of an instep where the wet foot had been placed in coming in. I was then beginning to be able to form an opinion as to what had occurred. A man had waited outside the window, some one had brought the gems; the deed had been overseen by your son, he had pursued the thief, had struggled with him, they had each tugged at the coronet, their united strength causing injuries which neither alone could have effected. He had returned with the prize, but had left a fragment in the grasp of his opponent. So far I was clear. The question now was, who was the man, and who was it brought him the coronet? En entrant dans la maison, j’avais examiné, à la loupe, vous vous le rappelez, le rebord et la boiserie de la fenêtre du vestibule. Je pus distinguer le contour d’un pied humide, rentrant. Je pouvais dès lors me former une opinion sur ce qui s’était passé. Un homme s’était posté à la fenêtre, quelqu’un lui avait apporté le diadème ; votre fils avait entendu du bruit et, ayant vu ce qui se passait, avait poursuivi le voleur ; il avait lutté avec lui, chacun tirant de son côté sur le joyau de toutes ses forces, c’est ainsi qu’ils avaient pu réussir à le briser. Ensuite, votre fils était revenu avec sa prise, mais non sans en laisser un fragment entre les mains de son adversaire. Jusque-là rien de plus clair. Restait à savoir qui était l’homme, et qui lui avait apporté le diadème ?
The Adventures of Sherlock Holmes, by Arthur Conan Doyle Les Aventures de Sherlock Holmes, de Arthur Conan Doyle