Meaning of the valediction "Yours, &c."
Solution 1:
John S. Locke states in The Art of Correspondence (1884):
In closing a letter never subscribe yourself Yours, &c. &c. is an abbreviation of etc., which is an abbreviation from the Latin words et cetera, meaning and others, or, and so forth; forth means onward or forward. Hence there can be no propriety in saying I am yours and others, or I am respectfully yours and onward. This awkward expression is very common in business letters. Printers frequently abbreviate the complimentary closing of a letter, and to save type or time they print Yours, &c, which implies that the letter is closed in the usual form. Doubtless some thoughtless writer seeing this, concluded that it looked well and gave the letter an air of business despatch; hence it came into its present improper use.
If Mr. Locke is correct, the appearances of &c. in Pride and Prejudice may have been nothing more than a way to reduce the required typesetting.
Even though he states that &c. indicates that the closing is of the "usual form," there is no one usual form. For business letters he lists many possibilities:
Yours truly; Truly yours; Respectfully yours; Yours respectfully; or, if it is desired to give more emphasis, the adverb very may be added, as, Very truly yours; Yours very respectfully.
Solution 2:
According to Fowler's Dictionary of Modern English Usage, yours &c was a common letter form that preceded one's signature in a letter. There seems to be no clear answer on what it expands to, but this Google thread offers some direction:
I believe this reflects much earlier usage, when in Europe letters were signed under formal expressions or respect to the person addressed in the letter - just as the opening included more resepctful terms than just "Dear ...".
This formality arose in times when social hierarchy were more rigid and written communications were between persons of at the upper levels and very conscious of their relative status, so when signing a letter to a higher ranked person it was common (and appropriate, and the omission a breach of etiquette) to sign with something like "with greatest respect, your (most) humble and obedient servant ..."
Later in that thread, a user writes:
I happened on this page after a rereading of Jane Austen's Pride and Prejudice. After seeing multiple instances of "yours, etc." in one form or another appearing at the end of correpsondence, I would have to agree with myoarin-ga's assessment of its origins. Although no expert on the matter, it would appear that "yours, etc." and its variants allude to a set of standard salutations in correspondence. I have found a few examples which seem to point to this intepretation. Darcy ends a letter to Lady Catherine de Bourgh with "YOURS SINCERELY, ETC." A letter from Lydia to Elizabeth towards the end of the novel ends with "YOURS, ETC."
Maybe meaningless, but observed nonetheless, is that the characters who share a more casual relationship (Lydia to Lizzie, Elizabeth with her aunt and uncle) ended their letters with "Yours, etc." Maybe this could imply that formality and the length of these endings are directly related.
So in Austen's time, this valediction "&c" could have stood for anything that described the writer's relation to the recipient. However, it does not seem to expand into any one other valediction.
Solution 3:
I am not particularly well-versed in this subject, but I do think I should chance an answer here.
The 'et cetera' or '&c' part of the valediction 'Yours &c' is a contraction for a common valedictory statement used during earlier eras of English, as stated before, when relationships between the sender and recipient were more clearly defined--the contraction being of the formal
"I remain, sir, your most loyal and faithful servant"
Used, perhaps, even in correspondence between peers as a respectable gesture of putting oneself in the service of one's correspondent.
But how did we come to 'Yours, etc' from here?
You'll notice that the form used above is 'YourS' and not 'Your', which is used in the expanded valediction.
This is because the 's' is contraction for 'servant' and as such 'Yours etc.' meant 'Your servant, etc (most humble, loyal, faithful, sincere. . . .)' and thus even the more common 'Yours faithfully' and 'Yours sincerely' used today meant 'I remain, sir, your most faithful servant' and 'I remain, sir, your most sincere servant'--the former of which is used when the sender does not know the recipient in person.
And thus more archiac version of 'Yours, &c' would be 'I remain, &c' which you will still find appended to some letters today.
Hope this is satisfactory.
Solution 4:
The last paragraph in the Google thread referred to in previous answer says: "I suppose most of what I just wrote is merely conjecture. Still, one thing we can be sure about is that "Yours, etc." has been in use since 1813, when Pride and Prejudice was published, and one can assume that it was used further back than that."
Indeed, it's used in one of the earliest English novels, Samuel Richardson's Pamela, or Virtue Rewarded. In 1739, at the request of friends who were booksellers, Richardson began writing Pamela as a "letter-writer", that is, as a volume of model letters for "country readers" "unable to indite for themselves".[1] Of course, he ultimately turned the volume into a novel.
Phrases like "I am, &c.", "From, &c.", and "Yours, &c." appear at the ends of a handful of letters in Pamela, while dozens of times as many letters have valedictions like "Your most dutiful DAUGHTER.", "Your dutiful daughter,", "Your dutiful DAUGHTER till death.", "Your honest as well as dutiful DAUGHTER.", "Your ever dutiful DAUGHTER.", "Your most afflicted DAUGHTER.", "Your dutiful and honest daughter,", "Your careful, but loving Father and Mother,", "Your truly loving, but careful, FATHER and MOTHER.", and so forth.
In Pamela the "... &c" valedictions appear without explanation, they are relatively rare, and the correspondents are mostly Pamela and her parents, so it's difficult to draw real conclusions. It does seem clear that prolific correspondents would have little trouble interpreting "I am, &c." in accord with formula, but that is not to say they "will understand exactly what has been abridged" since there is much variability within the framework. I imagine that consulting a contemporaneous letter-writing manual is the only way to answer your question definitively.
[1] Introduction, W. M. Sale, in 1958 Norton Library edition of Pamela, or Virtue Rewarded