Both Uses

consequence
in
An Ideal Husband
(Edited)

It really is of no consequence, Lady Chiltern.†
Act 2 *

consequence = importance
He always seems to think that he is addressing the House, and consequently whenever he discusses the state of the agricultural labourer, or the Welsh Church, or something quite improper of that kind, I am obliged to send all the servants out of the room.
Act 2 *

consequently = resultantly (as a result)

Definitions:

(1) (consequence as in: a direct consequence of)
a result of something (often an undesired side effect)
(2) (consequence as in: of little consequence)
importance or relevance
(3) (meaning too rare to warrant focus)
In classic literature, consequential may refer to someone with too much feeling of self-importance as when Dickens wrote "Because he's a proud, haughty, consequential, turned-up-nosed peacock."

Self-consequence was used in a similar manner, but is more easily understood by modern readers since important is one of the modern senses of consequence.

Another classic sense of consequent that is similar to importance or significance refers to "material wealth or prominence" as when Jane Austen wrote: "They had each had money, but their marriages had made a material difference in their degree of consequence."