I consider myself something of an authority on apologies because a quick temper has provided me with plenty of opportunities to make them. In one of my earliest memories, my mother is telling me, “Don’t look at the ground when you say, ‘I’m sorry’. Hold your head up and look the person in the eye, so he’ll know you mean it”.
My mother thus conveyed the first principle of successful apology: it must be direct. You must never pretend to be doing something else. You do not leaf through a sheaf of correspondence while apologising to a subordinate after blaming her for a mistake that turned out to be your fault. You do not apologise to a hostess, whose guest of honour you insulted, by sending flowers the next day without mentioning your bad behaviour.