Lewis & Short

H, h, the eighth letter of the Latin alphabet and the weakest guttural. The sign is borrowed from the Greek, in which H was the old form of the spiritus asper, corresp. to the Latin H-sound (HEKATON, ἑκατόν, ΗΟΣ, ὅς, etc.). Even some of the ancients doubted whether the Latin H was properly a letter: si H littera est, non nota, Quint. 1, 5, 19; cf.: H litteram, sive illam spiritum magis quam litteram dici oportet, etc., Gell. 2, 3, 1. Before the fall of the republic, the sound of H before vowels became so weak that it was frequently omitted in writing; and this weakness became more marked in many words in the time of the empire; cf.: aheneus and aeneus; cohors and coörs; prehendo and prendo; vehemens and vemens, etc. (v. Corss. Ausspr. 1, 96 sqq.).As an initial and medial, H may be combined with any vowel, but the orthography, in this respect, was inconstant: thus we have herus and erus; honus, honera, and onus, onera; harundo and arundo; and even hac for ac (Inscr. Orell. 23); aruspex and haruspex; ercisco, erctum, and hercisco, herctum; aheneus and aëneus; Annibal and Hannibal; Adria and Hadria, etc.; v. Gell. l. l.
As a sign for the aspiration of the consonants c, p, r, and t (as in Greek the aspirates χ, φ, θ were originally designated by KH, HH, TH), H first came into use in the seventh century of Rome; cf. Cic. Or. 48, 160; and v. the letter C.
Medial h is often dropped.
As a final, h occurs only in the interjections ah and vah.In the formation of words, h was changed into c before t, as tractum from traho; vectum from veho; and coalesced with s into x, as traxi, vexi; cf. also onyx from onych-s; v. the letter X.As an abbreviation, H. denotes hic, haec, hoc, hujus, etc.; habet, heres, honor, etc. HH. heredes. H. AQ. hic acquiescit. H. B. M. heredes bene merenti. H. C. Hispania citerior or hic condiderunt. H. E. T. heres ex testamento. H. F. C. heres faciundum curavit. H. L. hunc locum. H. L. ET. M. H. N. S. hic locus et monumentum heredem non sequitur. H. M. S. D. M. hoc monumentum sine dolo malo. H. S. E. hic situs est. H. S. F. hoc sibi fecit, etc.; v. Inscr. Orell. II. p. 461 sq.
Note: The abbreviation HS. for sestertium does not strictly belong here, because H is not the letter of that shape, but the numeral II. crossed; v. sestertius init.