He demonstrates that in both the Old and the New Testaments, the word was used to refer to pagan gods, not fallen angels (as the Early Fathers claimed), nor supernatural (but non-Divine), entities of evil (as modern Christians claim).
In the following quotes from Elliott, Hebrew words in Elliott’s original text have not been transcribed or transliterated. Their absence has been noted in each case. All emphasis in bold has been added.
Next as to daimonion, daemon. This is a word used both in the Septuagint and New Testament, alike in the plural as the singular, in two senses.
In the Septuagint, its first and clearest signification is as a simple designative of the imaginary heathen gods.
So in Psalm xcvi. 5; oi theoi twn ethnwn daimoniaeisin 'the gods of the heathen are daemons'; also in Deuteronomy xxxii, 17; ethusan daimoniois, ka ou thew 'they sacrificed to daemons, and not to God:' and again Psalm cvi. 37; ethusan tas thugateras autwn daimoniois.
In these passages the Hebrew words corresponding to daimonia are [Hebrew word in original text] and [Hebrew word in original text]: the one, according to Gesenius, signifying vanities; the other, lords or rulers. 
So that there is nothing in them to fix on these spirits the character of devilish, or satanic; as the word satanim, or some indubitable equivalent, would have done. 
Nor, though the tone of the two latter statements be deemed objurgatory, does there need any such explanation of the word to account for it.
It is sufficiently explained, on the hypothesis of its simple meaning, by multitudes of parallel Scriptural passages: in the which Israel's sin is depicted as made up of two evils; viz., 1st, forsaking God; 2ndly, forsaking Him (not for devilish or satanic spirits, but) for them that were no gods, but profitless idol vanities. (Deut. xxxii. 21, &c.)
Thus, there being nothing implied of devilish, or satanic, in the original Hebrew, so neither, we may reasonably infer, as it seems to me, in the daimonia of the Septuagint translation.
It is plain that the Alexandrine translators used the word in its popular meaning, simply to signify the gods or daemons of heathen mythology; Alexandria being a place where the Platonic philosophy had necessarily made that meaning most familiar to them.
 In 1 Chron xvi, 26 the former of these two Hebrew words also occurs; but in the Septuagint it is rendered eidwla, instead of daimonia. Buxtorf derives [Hebrew word in original text] from [Hebrew word in original text], vastavit: whence the word in Psalm xci. 6, noted in the next page.
 Compare too 2 Chron. xi. 15; where it is said of Jeroboam, katesthsen eautw iereis twn upshlwn, kai tois eidwlois, kai tois mataiois, kai tois mosxoios, a epoisen, answering to our authorized version, 'priests for the devils, (daemons,) and for the calves which he had made.' Heb. [Hebrew word in original text] the same word as in Is. xiii. 22, xxxiv. 14. referred to overleaf.
Elliott, 'Horae Apocalypticae', volume 2, pages 498-9, 5th edition, 1862