So the recipe says lime juice and syrup, but the blurb at the end mentions a Raymond Chandler reference to "Rose's lime juice". I would very strongly suspect that the original actually would have said "Rose's lime cordial", which I happen to know you can still get because there is some in my mother's fridge at this moment. So you wouldn't use juice + syrup, you'd just use the cordial which is already syrup-ish.