Both Uses

consequence
in
A Death in the Family
(Edited)

Sometimes on these evenings his father would hum a little and the humming would break open into a word or two, but he never finished even a part of a tune, for silence was even more pleasurable, and sometimes he would say a few words, of very little consequence, but would never seek to say much, or to finish what he was saying, or to listen for a reply; for silence again was even more pleasurable.†
Chpt 1 *

consequence = importance
She would remember, every ten minutes or so, to inquire courteously whether he was tired, but he was seldom tired in her company; with her, he never felt embarrassment in saying he had to go to the bathroom, for she never seemed annoyed, but in consequence he seldom found it necessary to go when they came together on these downtown trips.†
Chpt 7 *

consequence = 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."