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fachach m. [ˈfaxəx], -[ax], -[ɔx], gen. fachaich -[iç], ‘Manx shearwater, Puffinus puffinus, or great shearwater, Puffinus gravis; (occasionally) puffin, Fratercula arctica’. Christiansen (1938, 4: (pl.) fachich [sic], 16: fachidh [in error]) notes Dwelly’s (1911) fachach ‘puffin, coulterneb’ and compares Scots (Orkney) feckie (Marwick 1929, s.v.), which corresponds with ON vákr m. ‘a type of hawk’ (NO), but concludes that this would hardly seem appropriate for the puffin.
McDonald (2009, 426; 2015a, 131) considers the loan uncertain.
Ir. fach, abhach, ábhach ‘lair’
While O’Reilly’s (1817; 1864) Irish dictionary lists Ir. fach ‘hole’, he cites Shaw’s (1780) Scottish Gaelic dictionary as his source. However, Dinneen’s (1947) Irish dictionary lists fach f. ‘a hole in which a lobster is found’, along with a cross-reference to abhthach (Aran Islands), and aice f. ‘a hole in a sea rock in which a crab or lobster hides’ (Donegal), along with a cross-reference to fach ‘idem’ (Connaught); Ó Dónaill’s (1977) Irish dictionary, on the other hand, equates fach with ábhach m. ‘hole, recess’. O’Reilly’s Scottish Gaelic borrowing aside, Ir. fach may be an aphaerised form of second-syllable-stressed abhach 
For the second-syllable stress in this instance in this dialect, see O’Rahilly 1976, 109–10.
For the lengthening, cf. EG aḃac ‘dwarf, sprite’ > Ir. (Cork) abhac /auk/ (Ó Cuiv 1988, 28), (Donegal) ábhach /æːwɑx/ (Wagner 1979, 78), ?under the influence of words in adhbh- > ábh- (Ó Maolalaigh 2006a, 58).
Ir. aice, SG aice, faic(e), fochd ‘lair’
In addition to the sense ‘place for crustaceans to hide in’, Ir. aice has the more general sense ‘habitat’ (Ó Dónaill) and appears to go back to EG aicce ‘nearness, proximity’; 
Cf. O’Brien (1768; 1832) and O’Reilly (1817; 1864) aice ‘tribe’.
AFB˄: Benbecula, Skye, Jura.
SG fach, faiche ‘lair’
Ir. fach ‘lair’ seems to have been borrowed as SG fach (Shaw 1780; so Armstrong 1825 and Dwelly 1911) and is also attested in the normalised oblique form faiche (MacBain 1911; MacLennan 1925), recorded in Barra, Skye, Mull (AFB˄), Harris, South Uist, Tiree and Islay (ibid.; Faclan bhon t-Sluagh˄).
SG fach, fachach, fachadh, fathach ‘shearwater; puffin’
Monosyllabic SG fach (Faclan bhon t-Sluagh˄: Harris) and disyllabic fachach (Shaw 1780; Armstrong 1825; HSS 1828; Forbes 27, 254, 323, 332, 334; Dwelly 1911; MacLennan 1925; McDonald 1972; Cunningham 1990, 34–35: Outer Hebrides; Garvie 1999, 58, 63, 66; Faclan bhon t-Sluagh˄: [fɑxɑx], Lewis), fachadh (Fergusson 1886, 81, 91; Faclan bhon t-Sluagh˄: fachadh bàn, with the adjective bàn ‘white’, South Uist and Skye) and fathach (Mackenzie 1905, 141–42: St Kilda) occur in the sense ‘shearwater; puffin’. Fachach, fachadh and fathach appear to be derivatives of fach ‘lair’ + the suffix -ach, or with a dental variant, in the sense ‘burrower’. Monosyllabic fach may be a truncation of fachach, or a back-formation via conflation with famh etc. ‘mole’ (see below).
SG fadh, fabh, famh, fath; fachach ‘mole’
The cognate of W gwadd, MC goth, OBret. guod, Bret. goz ‘mole’ (GPC˄) is SG fadh (Lhuyd 1707; Shaw 1780, so Armstrong 1825; HSS 1828; Forbes 1905, 10, 187). A number of variants occur, perhaps partly promoted in compounds, e.g. with talmhainn, gen. of talamh ‘earth, ground’. Besides fadh, there are fabh (Lhuyd 1700: fav, Argyll, unless this represents famh, below; Lhuyd 1707: fadhbh, 
Angus Matheson (≈CG VI, 70, s.v. famh) writes that, ‘[t]he vowel of famh seems to be long, judging by Iain Gobha’s lines: Thilg e mach chum nam fàmh iad | S ghleidh e Rāchel dha féin [Henderson 1896 I, 186, stanza xiii]. Cf. Lhuyd’s spelling fadhbh. The vowel would, of course, be shortened in a compound like famh-thalmhainn.’ The relevant stanza reads:
Is e bh’ ann fuighleach a’ naduir
Bh’ ann an Rachel i fein,
Fhuair i ’n leasraidh a parant
Dh’ fhàs na ’ghrain dhi na dheigh:
An uair a lorgadh le Iacob
Iad cha b’ fhabhar dhoibh e,
Thilg e mach chum nan fàmh iad
S ghleidh e Rachel dha fein.
The only lengthmark supplied by the poet’s editor George Henderson is over fàmh (and that is also how it appears in the glossary), but the stanza contains aicill rhyme throughout, with assonance between nàduir : Ràchel, pàrant : ghràin, Iàcob : fhàbhar, fàmh : Ràchel, hence Matheson’s Rāchel. It seems likely that long fàmh here is a poetic expedient for the sake of rhyme. Matheson’s reference to Lhuyd is abridged: the latter gives ‘fadh & fadhbh “a mole” Sc.’. The spelling fadhbh (also fathbh) may be etymological in the sense of retaining a connection with fadh, while still indicating final [v]. On the other hand, it may simply be the result of conflation with fadhb ‘lump, knob; knot’ (Dwelly 1911), cf. Ir. fadhb (Ó Dónaill 1977: ‘knot in timber; callosity; lump from blow; lumpy object’) ‘mole, knob, bunch’ (O’Brien 1768, who separately lists fadh & fadhb ‘a mole’) and ‘knob, knot, bunch, excrescence’ (O’Reilly 1817, who separately lists fadhb ‘a mole’), and †fadhban ‘molehill’ (Dwelly), cf. Ir. fadhbán (Ó Dónaill 1977: ‘(small) knot, lump’) ‘mole-hillock’ (sic) (Lhuyd 1707; O’Brien 1768; O’Reilly 1817), in which the sense of mole is ‘blemish, fault’ but has also been extended to ‘mole (the animal)’.
Citing MacDomhnuill 1741 as its source, although the latter gives famhthórr ‘molehill’ (p. 7), but fath ‘mole’ (p. 79).
Alexander Carmichael’s Carmina Gadelica gives a number of related forms: CG VI, 70: famh, amh, lamh, labh, fathan, fath-thalmhainn; see also p. 234, in the Index: famh, fomh, famh-thalmhan, famh-ùir, famh-bhual, famh-fhual, lamh-fhual, labh-alan, bad-alan.
In summary, SG fachach ‘shearwater; puffin’ appears to be a derivative of SG fach, faiche ‘lair’, a loan from Ir. fach (abhach, ábhach), itself from EG aḋḃa, aḋḃaḋ ‘abode, dwelling place; lair’; SG faic(e) etc. has a different origin.