When cycling to Doctor Birkin's office, there are two shots of the bicycle Murdoch is riding - prominently featuring a fancy electric lamp for riding at night; the next two shots of him pulling up to the building plainly show him on a completely different bike with no headlamp.
When Inspector Thomas Brackenreid is cut by a scalpel at the end, he is sliced in the gut and clearly shown holding his hand to his vest (below his pocket watch chain) where a bloody incision opens from his abdomen and through the vest. When Brackenreid returns to the station from the hospital he is inexplicably wearing a sling on his arm, and there is no evidence that the wound was to his abdomen - but rather it's implying that his arm was injured. Especially since he tells Murdoch he's in need of "a good right hand... or left-hand man to keep me out of trouble." Before Brackenreid is stabbed, Dr Birkin attacks him, and Brackenreid throws his arms up to protect himself.
Inspector Murdoch walks into Dr. Ogden's office as she is listening to a Gilbert & Sullivan melody on her Edison Diamond Disc phonograph. The show at this time is part of the late 1890's; Edison did not create his flat-disc phonograph until 1914. Prior to that, Edison's musical output came from cylinders.
Early in the story a Leyden Jar is referred to as a "capacitor". That term did not come into general use until the latter part of the 20th Century. In the late 19th Century the device would have been referred to as a "condenser".
Inspector Brackenreid uses the term "brown-nose", which didn't originate until the 1930s.
One of the Medal Ribbons worn by Chief Stockton is for the General Service Medal issued from 1964 to 2000. This was a UK Medal that was also issued to Canadian, Australian and New Zealand armed forces.
Murdoch prevents the inspector from walking into a tripwire, and he uses the inspector's walking stick to trigger the wire. A device had been set and a knife shoots across the room. Murdoch had no way of knowing that the tripwire wasn't connected to an explosive device.