I am quoting from The Memoirs of Sherlock Holmes, "The Stockbroker's Clerk", by Arthur Conan Doyle. Does "a connection" mean "a clinic" or "a business" or something else? I looked it up in 3 online dictionaries but couldn't find anything.
The narrator of a Sherlock Holmes story by Arthur Conan Doyle is Doctor Watson. In those days doctors were private practitioners who had to set themselves up in business (there being no public employment of doctors as in our present UK National Health Service). Watson did this by buying a share (in this case 100%) in the practice (the business) of an older doctor, a Mr Farquhar. This would have been normal at the time, with Watson getting access to the patient list of Fraquhar and thus making a living from them, while Farquhar received money for his old age. The practice was in the Paddington district.
At that time, the way of describing the purchase of part of the business was to say you had bought a connection (to the business). This phraseology has fallen out of use: if you look at the google ngram for "buy a connection" you will find that it was in vogue from about 1890 to 1910.
The story is The Stockbroker's Clerk and it may be found in