functional formula

 The final functional formula for the critical value of ku thus becomes ku = f(Bo, Q, q, r, u).

 For large diameter tubes, Bo ®Ґ and the functional dependence on it should disappear.

* * *

уравне́ние ви́да … — an equation of the form …

входи́ть в уравне́ние — appear in the equation, enter into the equation

выводи́ть уравне́ние — derive an equation

запи́сывать уравне́ние относи́тельно, напр. ста́ршей произво́дной — write an equation to solve for, e. g., the highest derivative

уравне́ние име́ет еди́нственное реше́ние — the equation has a unique solution

опро́бовать уравне́ние по о́пытным да́нным — check [test] an equation against experimental data

уравне́ние относи́тельно, напр. х — an equation in, e. g., x

по уравне́нию — according to the equation

подбира́ть уравне́ние, напр. к гипотети́ческому механи́зму реа́кции — fit an equation, e. g., to a postulated reaction mechanism

превраща́ть уравне́ние в то́ждество — reduce an equation to an identity

приводи́ть уравне́ние к сле́дующему ви́ду — reduce an equation to the following form

уравне́ние, разреши́мое относи́тельно, напр. х — an equation solvable for, e. g., x

разреши́ть уравне́ние относи́тельно, напр. ста́ршей произво́дной — re-write an equation to solve for, e. g., the highest derivative; re-write an equation with, e. g., the highest derivative (on the left-hand side)

разреши́ть уравне́ние относи́тельно, напр. х — re-arrange an equation to solve for, e. g., x

реша́ть уравне́ние относи́тельно, напр. х — solve the equation for, e. g., x

реша́ть уравне́ния совме́стно — solve equations simultaneously

реша́ть с по́мощью уравне́ния — solve by equation

уравне́ние с одни́м неизве́стным — an equation in one unknown

соотве́тствовать уравне́нию — fit an equation

составля́ть уравне́ние — formulate [form, set up, write] an equation

уравне́ние сте́пени n — an equation of degree n [of the nth degree], an nth -degree equation

транспони́ровать уравне́ния — transpose equations

удовлетворя́ть уравне́нию — satisfy an equation

уравне́ние адиаба́ты — adiabatic equation

алгебраи́ческое уравне́ние — algebraic equation

уравне́ние бала́нса — balance (equation)

уравне́ние бари́ческой тенде́нции — tendency equation

уравне́ние Берну́лли — Bernoulli's theorem

биквадра́тное уравне́ние — biquadratic, biquadratic [quartic] equation

уравне́ние Бо́льцмана — Boltzmann equation

уравне́ние Ван-дер-Ваа́льса — Van der Waals' equation

веково́е уравне́ние — secular equation

ве́кторное уравне́ние — vector equation

уравне́ние в коне́чных ра́зностях — difference equation

уравне́ние во́дного бала́нса — hydrologic [hydrolicity] equation

волново́е уравне́ние — wave equation

уравне́ние в по́лных дифференциа́лах — total [exact differential] equation

уравне́ние второ́й сте́пени — quadratic [second-degree] equation

уравне́ние Га́мильтона — canonical equation of motion

уравне́ние Ги́ббса-Гельмго́льца — equation of maximum work

уравне́ние горе́ния — combustion equation

уравне́ние да́льности де́йствия рлк ста́нции, основно́е — (radar) range equation

уравне́ние движе́ния — equation of motion

уравне́ние движе́ния жи́дкости — flow equation

уравне́ние дина́мики — equation of motion, dynamic(al) equation

диофа́нтово уравне́ние — Diophantine equation

дифференциа́льное уравне́ние — differential equation

дифференциа́льное уравне́ние в по́лных дифференциа́лах — exact (differential) equation

дифференциа́льное уравне́ние второ́го поря́дка — second-order differential equation

дифференциа́льное уравне́ние в ча́стных произво́дных — partial differential equation

дифференциа́льное, обыкнове́нное уравне́ние — ordinary differential equation

дифференциа́льное уравне́ние пе́рвого поря́дка — first-order differential equation

дифференциа́льное, стохасти́ческое уравне́ние — stochastic differential equation

дифференциа́льное уравне́ние управле́ния — control differential equation

дифференциа́льно-ра́зностное уравне́ние — differential-difference equation

уравне́ние диффу́зии — diffusion equation

интегра́льное уравне́ние — integral equation

интегра́льное уравне́ние пе́рвого ро́да — integral equation of the first kind

интегра́льное уравне́ние Фредхо́льма — Fredholm equation

интегродифференциа́льное уравне́ние — integro-differential equation

исхо́дное уравне́ние — input [original] equation

калибро́вочно-инвариа́нтное уравне́ние — gauge-invariant equation

канони́ческое уравне́ние — canonical equation

квадра́тное уравне́ние — quadratic equation

квадра́тное, непо́лное уравне́ние — pure quadratic (equation), incomplete quadratic (equation)

квадра́тное, по́лное уравне́ние — affected quadratic (equation), general form of a quadratic equation

уравне́ние кинети́ческое уравне́ние — rate [kinetic] equation

уравне́ние Клапейро́на — Clapeyron equation

уравне́ние коли́чества движе́ния — momentum equation

коне́чно-дифференци́руемое уравне́ние — finitely differentiable equation

уравне́ние ко́нтурных то́ков — mesh-current [loop-current] equation

куби́ческое уравне́ние — cubic equation

лине́йное уравне́ние — linear equation

уравне́ние стано́вится лине́йным относи́тельно, напр. вре́мени — the equation becomes linear in, e. g., time

логарифми́ческое уравне́ние — logarithmic equation

уравне́ние Ма́ксвелла — Maxwell's equation

масшта́бное уравне́ние вчт. — transformation equation

материа́льное уравне́ние элк. — constitutive relation

ма́тричное уравне́ние — matrix equation

маши́нное уравне́ние вчт. — machine equation

уравне́ние n [m2]-го поря́дка — equation of the nth order, nth -order equation

уравне́ние n [m2]-й сте́пени — nth -degree equation, equation of degree n

неодноро́дное уравне́ние — inhomogeneous [nonhomogeneous] equation

неопределё́нное уравне́ние — indeterminate equation

уравне́ние непреры́вности — continuity equation

неприводи́мое уравне́ние — irreducible equation

уравне́ние неразры́вности — continuity equation

одноро́дное уравне́ние — homogeneous equation

окисли́тельно-восстанови́тельное уравне́ние — oxidation-reducton equation

опера́торное уравне́ние — operator equation

основно́е уравне́ние — basic equation

уравне́ние параболи́ческого ти́па — parabolic equation

параметри́ческое уравне́ние — parametric equation

уравне́ние пе́рвого поря́дка — first-order equation

уравне́ние пе́рвой сте́пени — simple equation

уравне́ние переме́нного то́ка — equation for an alternating current

уравне́ние перено́са — transport [transfer] equation

уравне́ние пограни́чного сло́я — boundary-layer equation

уравне́ние по́ля — field equation

уравне́ние правдоподо́бия — likelihood equation

уравне́ние преобразова́ния — transformation equation

прове́рочное уравне́ние ( на чётность) — parity(-check) equation

уравне́ние прямо́й в отре́зках — intercept form of [for] the equation of a straight line

уравне́ние прямо́й с угловы́м коэффицие́нтом — slope-intercept form of [for] the equation of a straight line

уравне́ние Пуассо́на — adiabatic equation

уравне́ние равнове́сия — equilibrium equation

уравне́ние радиолока́ции, основно́е — radar equation

уравне́ние радиолока́ции, основно́е, для свобо́дного простра́нства — free-space radar equation

уравне́ние разме́рностей — dimensional equation

ра́зностное уравне́ние — difference equation

уравне́ние регре́ссии — regression equation

проверя́ть уравне́ние регре́ссии на адеква́тность по крите́рию Фи́шера — test the adequacy [validity] of the regression equation on the basis of Fisher's variance ratio

уравне́ние регули́руемого объе́кта автмт. — plant [process] equation

релятиви́стское уравне́ние — relativistic equation

уравне́ние свя́зи — constraint equation

скаля́рное уравне́ние — scalar equation

скоростно́е уравне́ние — rate [kinetic] equation

уравне́ние согласова́ния цвето́в — colour match equation

сопряжё́нное уравне́ние — adjoint equation

уравне́ние состоя́ния — equation of state, characteristic equation

уравне́ние состоя́ния идеа́льного га́за — Clapeyron equation

уравне́ние сохране́ния — conservation equation

уравне́ние сохране́ния моме́нта коли́чества движе́ния — angular momentum [moment-of-momentum] equation

уравне́ние сохране́ния эне́ргии — energy equation

уравне́ние с разделя́ющими(ся) переме́нными — equation with variables separable, separable equation

уравне́ние сте́пени n — an equation of degree n, an nth -degree equation, an equation of the nth degree

степенно́е уравне́ние — exponential equation

стехиометри́ческое уравне́ние — stoichiometric equation

телегра́фное уравне́ние — telegraphers equation

те́нзорное уравне́ние — tensor equation

уравне́ние теплово́го бала́нса — heat balance equation

уравне́ние теплопрово́дности — heat [heat conduction, heat transfer] equation

уравне́ние тече́ния — flow equation

то́чное уравне́ние — exact equation

уравне́ние траекто́рии — path equation

трансценде́нтное уравне́ние — transcendental equation

тригонометри́ческое уравне́ние — trigonometric equation

уравне́ние узловы́х потенциа́лов эл. — nodal-voltage equation

функциона́льное уравне́ние — functional equation

характеристи́ческое уравне́ние — characteristic equation

хими́ческое уравне́ние — chemical equation

уравне́ние хо́да луче́й опт. — ray-tracing equation

цветово́е уравне́ние — trichromatic equation

уравне́ние Шре́дингера — Schrцdinger (wave) equation

уравне́ние Э́йлера для тре́ния кана́та по цили́ндру — capstan equation

уравне́ние Эйнште́йна для вне́шнего фотоэффе́кта — Einstein photoelectric equation

эллипти́ческое уравне́ние — elliptic(al) equation

эмпири́ческое уравне́ние — empirical equation

υἱός, οῦ, ὁ (Hom.+; loanw. in rabb.) prim. ‘son’

① a male who is in a kinship relationship either biologically or by legal action, son, offspring, descendant

ⓐ the direct male issue of a person, son τέξεται υἱόν Mt 1:21; GJs 14:2 (cp. Mel., P. 8, 53 ὡς γὰρ υἱὸς τεχθείς). Cp. Mt 1:23 (Is 7:14) and 25; 10:37 (w. θυγάτηρ); Mk 12:6a; Lk 1:13, 31, 57; 11:11; 15:11 (on this JEngel, Die Parabel v. Verlorenen Sohn: ThGl 18, 1926, 54–64; MFrost, The Prodigal Son: Exp. 9th ser., 2, 1924, 56–60; EBuonaiuti, Religio 11, ’35, 398–402); Ac 7:29; Ro 9:9 (cp. Gen 18:10); Gal 4:22 al. W. gen. Mt 7:9; 20:20f; 21:37ab; Mk 6:3; 9:17; Lk 3:2; 4:22; 15:19; J 9:19f; Ac 13:21; 16:1; 23:16; Gal 4:30abc (Gen 21:10abc); Js 2:21; AcPlCor 2:29. Also ἐγὼ Φαρισαῖός εἰμι υἱὸς Φαρισαίων Ac 23:6 is prob. a ref. to direct descent. μονογενὴς υἱός (s. μονογενής 1) Lk 7:12. ὁ υἱὸς ὁ πρωτότοκος (πρωτότοκος 1) 2:7.

ⓑ the immediate male offspring of an animal (Ps 28:1 υἱοὺς κριῶν; Sir 38:25. So Lat. filius: Columella 6, 37, 4) in our lit. only as foal ἐπὶ πῶλον υἱὸν ὑποζυγίου Mt 21:5 (cp. Zech 9:9 πῶλον νέον).

ⓒ human offspring in an extended line of descent, descendant, son Ἰωσὴφ υἱὸς Δαυίδ Mt 1:20 (cp. Jos., Ant. 11, 73); s. 2dα below. υἱοὶ Ἰσραήλ (Ἰσραήλ 1) Mt 27:9; Lk 1:16; Ac 5:21; 7:23, 37; 9:15; 10:36; Ro 9:27; 2 Cor 3:7, 13; Hb 11:22 al.; AcPlCor 2:32. οἱ υἱοὶ Λευί (Num 26:57) Hb 7:5. υἱὸς Ἀβραάμ Lk 19:9. υἱοὶ Ἀδάμ 1 Cl 29:2 (Dt 32:8). υἱοι Ῥουβήλ GJs 6:3.

ⓓ one who is accepted or legally adopted as a son (Herodian 5, 7, 1; 4; 5; Jos, Ant. 2, 263; 20, 150) Ac 7:21 (cp. Ex 2:10).—J 19:26.

② a pers. related or closely associated as if by ties of sonship, son, transf. sense of 1

ⓐ of a pupil, follower, or one who is otherw. a spiritual son (SIG 1169, 12 οἱ υἱοὶ τοῦ θεοῦ=the pupils and helpers [40] of Asclepius; sim. Maximus Tyr. 4, 2c; Just., D. 86, 6 οἱ υἱοὶ τῶν προφητῶν.—Some combination w. παῖδες is the favorite designation for those who are heirs of guild-secrets or who are to perpetuate a skill of some kind: Pla., Rep. 3, 407e, Leg. 6, 769b; Dionys. Hal., Comp. Verbi 22 p. 102, 4 Us./Rdm. ῥητόρων παῖδες; Lucian, Anach. 19, Dial. Mort. 11, 1 Χαλδαίων π.=dream-interpreters, Dips. 5 ἱατρῶν π., Amor. 49; Himerius, Or. 48 [=Or. 14], 13 σοφῶν π.): the ‘sons’ of the Pharisees Mt 12:27; Lk 11:19. Peter says Μᾶρκος ὁ υἱός μου 1 Pt 5:13 (perh. w. a component of endearment; s. Μᾶρκος). As a familiar form of address by a cherished mentor Hb 12:5 (Pr 3:11; ParJer 5:28; 7:24). υἱοὶ καὶ θυγατέρες B 1:1.

ⓑ of the individual members of a large and coherent group (cp. the υἷες Ἀχαιῶν in Homer; also PsSol 2:3 οἱ υἱοὶ Ἰερουσαλήμ; Dio Chrys. 71 [21], 15; LXX) οἱ υἱοὶ τοῦ λαοῦ μου 1 Cl 8:3 (scripture quot. of unknown origin). υἱοὶ γένους Ἀβραάμ Ac 13:26. οἱ υἱοὶ τῶν ἀνθρώπων (Gen 11:5; Ps 11:2, 9; 44:3; TestLevi 3:10; TestZeb 9:7; GrBar 2:4) the sons of men=humans (cp. dγ below) Mk 3:28; Eph 3:5; 1 Cl 61:2 (of the earthly rulers in contrast to the heavenly king).

ⓒ of one whose identity is defined in terms of a relationship with a person or thing

α. of those who are bound to a personality by close, non-material ties; it is this personality that has promoted the relationship and given it its character: son(s) of: those who believe are υἱοὶ Ἀβραάμ, because Abr. was the first whose relationship to God was based on faith Gal 3:7. In a special sense the devout, believers, are sons of God, i.e., in the light of the social context, people of special status and privilege (cp. PsSol 17:27; Just., D, 124, 1; Dio Chrys. 58 [75], 8 ὁ τοῦ Διὸς ὄντως υἱός; Epict. 1, 9, 6; 1, 3, 2; 1, 19, 9; Sextus 58; 60; 135; 376a; Dt 14:1; Ps 28:1; 72:15; Is 43:6 [w. θυγατέρες μου]; 45:11; Wsd 2:18; 5:5; 12:21 al.; Jdth 9:4, 13; Esth 8:12q; 3 Macc 6:28; SibOr 3, 702) Mt 5:45; Lk 6:35; Ro 8:14, 19 (‘Redeemer figures’ EFuchs, Die Freiheit des Glaubens, ’49, 108; against him EHommel in ThViat 4, ’52, 118, n. 26); 9:26 (Hos 2:1); 2 Cor 6:18 (w. θυγατέρες, s. Is 43:6 cited above); Gal 3:26 (cp. PsSol 17:27); 4:6a, 7ab (here the υἱός is the κληρονόμος and his opposite is the δοῦλος); Hb 2:10 (JKögel, Der Sohn u. die Söhne: Eine exeget. Studie zu Hb 2:5–18, 1904); 12:5–8 (in vs. 8 opp. νόθος, q.v.); Rv 21:7; 2 Cl 1:4; B 4:9. Corresp. there are sons of the devil (on this subj. cp. Hdb. on J 8:44) υἱὲ διαβόλου Ac 13:10. οἱ υἱοὶ τοῦ πονηροῦ (masc.) Mt 13:38b. τοῦ υἱοῦ τοῦ ἐν Ἅιδου ApcPt Rainer. In υἱοί ἐστε τῶν φονευσάντων τοὺς προφήτας Mt 23:31 this mng. is prob. to be combined w. sense 1c. The expr. υἱοὶ θεοῦ Mt 5:9 looks to the future (s. Betz, SM ad loc.; cp. KKöhler, StKr 91, 1918, 189f). Lk 20:36a signifies a status akin to that of angels (Ps 88:7; θεῶν παῖδες as heavenly beings: Maximus Tyr. 11, 5a; 12a; 13, 6a.—Hierocles 3, 424 the ἄγγελοι are called θεῶν παῖδες; HWindisch, Friedensbringer-Gottessöhne: ZNW 24, 1925, 240–60, discounts connection w. angels and contends for the elevation of the ordinary followers of Jesus to the status of Alexander the Great in his role as an εἰρηνηποιός [cp. Plut., Mor. 329c]; for measured critique of this view s. Betz, SM 137–42.).

β. υἱός w. gen. of thing, to denote one who shares in it or who is worthy of it, or who stands in some other close relation to it, oft. made clear by the context; this constr. is prob. a Hebraism in the main, but would not appear barbaric (B-D-F §162, 6; Mlt-H. 441; Dssm., B p. 162–66 [BS 161–66]; PASA II 1884, no. 2 υἱὸς πόλεως [time of Nero; on this type of formulation SEG XXXIX, 1864]; IMagnMai 167, 5; 156, 12) οἱ υἱοὶ τοῦ αἰῶνος τούτου (αἰών 2a) Lk 16:8a (opp. οἱ υἱοί τοῦ φωτός vs. 8b); 20:34. τῆς ἀναστάσεως υἱοί (to Mediterranean publics the functional equivalent of ἀθάνατοι ‘immortals’; cp. ἀνάστασις 2b) 20:36b. υἱοὶ τῆς ἀνομίας (ἀνομία 1; cp. CD 6:15) Hv 3, 6, 1; ApcPt 1:3; τῆς ἀπειθείας (s. ἀπείθεια) Eph 2:2; 5:6; Col 3:6; τῆς ἀπωλείας ApcPt 1:2. ὁ υἱὸς τῆς ἀπωλείας of Judas the informer J 17:12 (cp. similar expressions in Eur., Hec. 425; Menand., Dyscolus 88f: s. FDanker, NTS 7, ’60/61, 94), of the end-time adversary 2 Th 2:3. υἱοὶ τῆς βασιλείας (βασιλεία 1bη; s. SEG XXXIX, 1864 for related expressions) Mt 8:12; 13:38a. υἱοὶ βροντῆς Mk 3:17 (s. Βοανηργές). υἱὸς γεέννης (s. γέεννα) Mt 23:15; τ. διαθήκης (PsSol 17:15) Ac 3:25; εἰρήνης Lk 10:6. υἱοὶ τοῦ νυμφῶνος (s. νυμφών) Mt 9:15; Mk 2:19; Lk 5:34. υἱὸς παρακλήσεως Ac 4:36 (s. Βαρναβᾶς). υἱοὶ (τοῦ) φωτός (Hippol., Ref. 6, 47, 4 in gnostic speculation) Lk 16:8b (opp. υἱοὶ τοῦ αἰῶνος τούτου); J 12:36. υἱοὶ φωτός ἐστε καὶ υἱοὶ ἡμέρας 1 Th 5:5 (EBuonaiuti, ‘Figli del giorno e della luce’ [1 Th 5:5]: Rivista storico-critica delle Scienze teol. 6, 1910, 89–93).

ⓓ in various combinations as a designation of the Messiah and a self-designation of Jesus

α. υἱὸς Δαυίδ son of David of the Messiah (PsSol 17:21) Mt 22:42–45; Mk 12:35–37; Lk 20:41–44; B 12:10c. Specif. of Jesus as Messiah Mt 1:1a; 9:27; 12:23; 15:22; 20:30f; 21:9, 15; Mk 10:47f; Lk 18:38f.—WWrede, Jesus als Davidssohn: Vorträge u. Studien 1907, 147–77; WBousset, Kyrios Christos² 1921, 4, Rel.³ 226f; ELohmeyer, Gottesknecht u. Davidssohn ’45, esp. 68; 72; 77; 84; TNicklin, Gospel Gleanings ’50, 251–56; WMichaelis, Die Davidsohnschaft Jesu usw., in D. histor. Jesus u. d. kerygm. Christus, ed. Ristow and Matthiae, ’61, 317–30; LFisher, ECColwell Festschr. ’68, 82–97.

β. ὁ υἱὸς τοῦ θεοῦ, υἱὸς θεοῦ (the) Son of God (for the phrase s. JosAs 6:2 al. Ἰωσὴφ ὁ υἱὸς τοῦ θεοῦ; there is no undisputed evidence of usage as messianic title in pre-Christian Judaism [s. Dalman, Worte 219–24, Eng. tr. 268–89; Bousset, Kyrios Christos² 53f; EHuntress, ‘Son of God’ in Jewish Writings Prior to the Christian Era: JBL 54, ’35, 117–23]; cp. 4Q 246 col. 2, 1 [JFitzmyer, A Wandering Aramean ’79, 90–93; JCollins, BRev IX/3, ’93, 34–38, 57]. Among polytheists on the other hand, sons of the gods in a special sense [s. Just., A I, 21, 1f] are not only known to myth and legend, but definite historical personalities are also designated as such. Among them are famous wise men such as Pythagoras and Plato [HUsener, Das Weihnachtsfest² 1911, 71ff], and deified rulers, above all the Roman emperors since the time of Augustus [oft. in ins and pap: Dssm., B 166f=BS 166f, LO 294f=LAE 346f; Thieme 33]. According to Memnon [I B.C./ I A.D.]: 434 Fgm. 1, 1, 1 Jac., Clearchus [IV B.C.] carried his boasting so far as Διὸς υἱὸν ἑαυτὸν ἀνειπεῖν. Also, persons who were active at that time as prophets and wonder-workers laid claim to the title υἱὸς τοῦ θεοῦ, e.g. the Samaritan Dositheus in Origen, C. Cels. 6, 11; sim. an Indian wise man who calls himself Διὸς υἱός Arrian, Anab. 7, 2, 3; cp. Did., Gen. 213, 18 ὁ Ἀβρὰμ υἱὸς θεοῦ διὰ δικαιοσύνην. S. GWetter, ‘Der Sohn Gottes’ 1916; Hdb. exc. on J 1:34; s. also Clemen² 76ff; ENorden, Die Geburt des Kindes 1924, 75; 91f; 132; 156f; EKlostermann, Hdb. exc. on Mk 1:11 [4th ed. ’50]; M-JLagrange, Les origines du dogme paulinien de la divinité de Christ: RB 45, ’36, 5–33; HPreisker, Ntl. Zeitgesch. ’37, 187–208; HBraun, ZTK 54, ’57, 353–64; ANock, ‘Son of God’ in Paul. and Hellen. Thought: Gnomon 33, ’61, 581–90 [=Essays on Religion and the Anc. World II, ’72, 928–39]—originality in Paul’s thought): Ps 2:7 is applied to Jesus υἱός μου εἶ σύ, ἐγὼ σήμερον γεγέννηκά σε Lk 3:22 D; GEb 18, 37.—Ac 13:33; Hb 1:5a; 5:5; 1 Cl 36:4. Likew. Hos 11:1 (w. significant changes): Mt 2:15, and 2 Km 7:14: Hb 1:5b. The voice of God calls him ὁ υἱός μου ὁ ἀγαπητός (s. ἀγαπητός 1) at his baptism Mt 3:17; Mk 1:11; Lk 3:22; GEb 18, 37 and 39 and at the Transfiguration Mt 17:5; Mk 9:7; Lk 9:35 (here ἐκλελεγμένος instead of ἀγαπ.); 2 Pt 1:17. Cp. J 1:34. The angel at the Annunciation uses these expressions in referring to him: υἱὸς ὑψίστου Lk 1:32; GJs 11:3 and υἱὸς θεοῦ Lk 1:35 (Ar. 15, 1 ὁ υἱὸς τοῦ θεοῦ τοῦ ὑψίστου. Cp. Just., A I, 23, 2 μόνος ἰδίως υἱὸς τῷ θεῷ γεγέννηται). The centurion refers to him at the crucifixion as υἱὸς θεοῦ Mt 27:54; Mk 15:39; GPt 11:45; cp. vs. 46 (CMann, ET 20, 1909, 563f; JPobee, The Cry of the Centurion, A Cry of Defeat: CFDMoule Festschr. ’70, 91–102; EJohnson, JSNT 31, ’87, 3–22 [an indefinite affirmation of Jesus]). The high priest asks εἰ σὺ εἶ ὁ Χριστὸς ὁ υἱὸς τοῦ θεοῦ Mt 26:63 (DCatchpole, NTS 17, ’71, 213–26). Passers-by ask him to show that he is God’s Son 27:40; sim. the devil 4:3, 6; Lk 4:3, 9. On the other hand, evil spirits address him as the Son of God Mt 8:29; Mk 3:11; 5:7; Lk 4:41; 8:28; and disciples testify that he is Mt 14:33; 16:16. S. also Mk 1:1 (s. SLegg, Ev. Sec. Marc. ’35).—Jesus also refers to himself as Son of God, though rarely apart fr. the Fourth Gosp.: Mt 28:19 (the Risen Lord in the trinitarian baptismal formula); Mt 21:37f=Mk 12:6 (an allusion in the parable of the vinedressers).—Mt 27:43; Mk 13:32; Rv 2:18. The main pass. is the so-called Johannine verse in the synoptics Mt 11:27=Lk 10:22 (s. PSchmiedel, PM 4, 1900,1–22; FBurkitt, JTS 12, 1911, 296f; HSchumacher, Die Selbstoffenbarung Jesu bei Mt 11:27 [Lk 10:22] 1912 [lit.]; Norden, Agn. Th. 277–308; JWeiss, Heinrici Festschr. 1914, 120–29, Urchristentum 1917, 87ff; Bousset, Kyrios Christos² 1921, 45ff; EMeyer I 280ff; RBultmann, Gesch. d. synopt. Trad.² ’31, 171f; MDibelius, Die Formgeschichte des Evangeliums² ’33, 259; MRist, Is Mt 11:25–30 a Primitive Baptismal Hymn? JR 15, ’35, 63–77; TArvedson, D. Mysterium Christi: E. Studie zu Mt 11:25–30, ’37; WDavies, ‘Knowledge’ in the Dead Sea Scrolls and Mt 11:25–30, HTR 45, ’53, 113–39; WGrundmann, Sohn Gottes, ZNW 47, ’56, 113–33; JBieneck, Sohn Gottes als Christusbez. der Synopt. ’51; PWinter, Mt 11:27 and Lk 10:22: NovT 1, ’56, 112–48; JJocz, Judaica 13, ’57, 129–42; OMichel/OBetz, Von Gott Gezeugt, Beih. ZNW [Jeremias Festschr.] 26, ’60, 3–23 [Qumran]).—Apart fr. the synoptics, testimony to Jesus as the Son of God is found in many parts of our lit. Oft. in Paul: Ro 1:3, 4, 9; 5:10; 8:3, 29, 32; 1 Cor 1:9; 15:28; 2 Cor 1:19; Gal 1:16; 2:20; 4:4; Eph 4:13; Col 1:13; 1 Th 1:10. Cp. Ac 9:20. In Hb: 1:2, 8; 4:14; 5:8; 6:6; 7:3, 28; 10:29. In greatest frequency in John (cp. Herm. Wr. 1, 6 the Λόγος as υἱὸς θεοῦ. Likew. Philo, Agr. 51 πρωτόγονος υἱός, Conf. Lingu. 146 υἱὸς θεοῦ.—Theoph. Ant. 2, 1 [p. 154, 12] ὁ λόγος ὁ τοῦ θεοῦ, ὅς ἐστιν καὶ υἱὸς αὐτοῦ; Iren. 3, 12, 2 [Harv. II 55, 2]): J 1:49; 3:16–18 (s. μονογενής 2), 35f; 5:19–26; 6:40; 8:35f; 10:36; 11:4, 27; 14:13; 17:1; 19:7; 20:31; 1J 1:3, 7; 2:22–24; 3:8, 23; 4:9f, 14f; 5:5, 9–13, 20; 2J 3, 9.—B 5:9, 11; 7:2, 9; 12:8; 15:5; Dg 7:4; 9:2, 4; 10:2 (τὸν υἱὸν αὐτοῦ τὸν μονογενῆ; also ApcEsdr 6:16 p. 31, 22 Tdf.; ApcSed 9:1f); IMg 8:2; ISm 1:1; MPol 17:3; Hv 2, 2, 8; Hs 5, 2, 6 (ὁ υἱὸς αὐτοῦ ὁ ἀγαπητός); 8; 11; 5, 4, 1; 5, 5, 2; 3; 5; 5, 6, 1; 2; 4; 7 (on the Christology of the Shepherd s. Dibelius, Hdb. on Hs 5, also ALink and JvWalter [πνεῦμα 5cα]); Hs 8, 3, 2; 8, 11, 1. Cp. 9, 1, 1; 9, 12, 1ff.—In trinitarian formulas, in addition to Mt 28:19, also IMg 13:1; EpilMosq 5; D 7:1, 3.—The deceiver of the world appears w. signs and wonders ὡς υἱὸς θεοῦ D 16:4 (ApcEsdr 4:27 p. 28, 32 Tdf. ὁ λέγων• Ἐγώ εἰμι ὁ υἱὸς τοῦ θεοῦ [of Antichrist]).—EKühl, Das Selbstbewusstsein Jesu 1907, 16–44; GVos, The Self-disclosure of Jesus 1926.—EBurton, ICC Gal 1921, 404–17; TNicklin, Gospel Gleanings ’50, 211–36; MHengel, The Son of God (tr. JBowden) ’76; DJones, The Title υἱὸς θεοῦ in Acts: SBLSP 24, ’85, 451–63.

γ. ὁ υἱὸς τοῦ ἀνθρώπου lit. ‘the son of the man’ (the pl. form οἱ υἱοὶ τῶν ἀνθρώπων appears freq. in the LXX to render בְּנֵי אָדָם = mortals, e.g. Gen 11:5; Ps 10:4; 11:2; cp. ὁ υἱὸς τῆς ἀπολείας J 17:12 [s. 2cβ]) ‘the human being, the human one, the man’ in our lit. only as a byname in ref. to Jesus and in an exclusive sense the Human One, the Human Being, one intimately linked with humanity in its primary aspect of fragility yet transcending it, traditionally rendered ‘the Son of Man.’ The term is found predom. in the gospels, where it occurs in the synoptics about 70 times (about half as oft. if parallels are excluded), and in J 12 times (s. EKlostermann, Hdb. exc. on Mk 8:31). In every case the title is applied by Jesus to himself. Nowhere within a saying or narrative about him is it found in an address to him: Mt 8:20; 9:6; 10:23; 11:19; 12:8, 32, 40; 13:37, 41; 16:13, 27f; 17:9, 12, 22; 18:10 [11] v.l.; 19:28; 20:18, 28; 24:27, 30, 37, 39, 44; 25:13 v.l., 31; 26:2, 24ab, 45, 64; Mk 2:10, 28; 8:31, 38; 9:9, 12, 31; 10:33, 45; 13:26; 14:21ab, 41, 62; Lk 5:24; 6:5, 22; 7:34; 9:22, 26, 44, 56 v.l., 58; 11:30; 12:8, 10, 40; 17:22, 24, 26, 30; 18:8, 31; 19:10; 21:27, 36; 22:22, 48, 69; 24:7.—John (FGrosheide, Υἱὸς τ. ἀνθρ. in het Evang. naar Joh.: TSt 35, 1917, 242–48; HDieckmann, D. Sohn des Menschen im J: Scholastik 2, 1927, 229–47; HWindisch, ZNW 30, ’31, 215–33; 31, ’32, 199–204; WMichaelis, TLZ 85, ’60, 561–78 [Jesus’ earthly presence]) 1:51; 3:13, 14; 5:27 (BVawter, Ezekiel and John, CBQ 26, ’64, 450–58); 6:27, 53, 62; 8:28; 9:35; 12:23, 34; 13:31. Whether the component of fragility (suggested by OT usage in ref. to the brief span of human life and the ills to which it falls heir) or high status (suggested by traditions that appear dependent on Da 7:13, which refers to one ‘like a human being’), or a blend of the two dominates a specific occurrence can be determined only by careful exegesis that in addition to extra-biblical traditions takes account of the total literary structure of the document in which it occurs. Much neglected in the discussion is the probability of prophetic association suggested by the form of address Ezk 2:1 al. (like the OT prophet [Ezk 3:4–11] Jesus encounters resistance).—On Israelite thought contemporary w. Jesus and alleged knowledge of a heavenly being looked upon as a ‘Son of Man’ or ‘Man’, who exercises Messianic functions such as judging the world (metaph., pictorial passages in En 46–48; 4 Esdr 13:3, 51f) s. Bousset, Rel.³ 352–55; NMessel, D. Menschensohn in d. Bilderreden d. Hen. 1922; ESjöberg, Kenna 1 Henok och 4 Esra tanken på den lidande Människosonen? Sv. Ex. Årsb. 5, ’40, 163–83, D. Menschensohn im äth. Hen. ’46. This view is in some way connected w. Da 7:13; acc. to some it derives its real content fr. an eschatological tradition that ultimately goes back to Iran (WBousset, Hauptprobleme der Gnosis 1907, 160–223; Reitzenstein, Erlösungsmyst. 119ff, ZNW 20, 1921, 18–22, Mysterienrel.³ 418ff; Clemen² 72ff; CKraeling, Anthropos and Son of Man: A Study in the Religious Syncretism of the Hellenistic Orient 1927); acc. to this tradition the First Man was deified; he will return in the last times and usher in the Kingdom of God.—Outside the gospels: Ac 7:56 (v.l. τοῦ θεοῦ; GKilpatrick, TZ 21, ’65, 209); Rv 1:13; 14:14 (both after Da 7:13; sim. allusion to Da in Just., D. 31, 1). The quot. fr. Ps 8:5 in Hb 2:6 prob. does not belong here, since there is no emphasis laid on υἱὸς ἀνθρώπου. In IEph 20:2 Jesus is described as υἱὸς ἀνθρώπου καὶ υἱὸς θεοῦ. Differently B 12:10 Ἰησοῦς, οὐχὶ υἱὸς ἀνθρώπου ἀλλὰ υἱὸς τοῦ θεοῦ Jesus, not a man’s son, but Son of God.—HLietzmann, Der Menschensohn 1896; Dalman, Worte 191–219 (Eng. tr., 234–67); Wlh., Einl.² 123–30; PFiebig, Der Menschensohn 1901; NSchmidt, The Prophet of Nazareth 1905, 94–134, Recent Study of the Term ‘Son of Man’: JBL 45, 1926, 326–49; FTillmann, Der Menschensohn 1907; EKühl, Das Selbstbewusstsein Jesu 1907, 65ff; HHoltzmann, Das messianische Bewusstsein Jesu, 1907, 49–75 (lit.), Ntl. Theologie² I 1911, 313–35; FBard, D. Sohn d. Menschen 1908; HGottsched, D. Menschensohn 1908; EAbbott, ‘The Son of Man’, etc., 1910; EHertlein, Die Menschensohnfrage im letzten Stadium 1911, ZNW 19, 1920, 46–48; JMoffatt, The Theology of the Gospels 1912, 150–63; WBousset, Kyrios Christos² 1921, 5–22 (the titles of the works by Wernle and Althaus opposing his first edition [1913], as well as Bousset’s answer, are found s.v. κύριος, end); DVölter, Jesus der Menschensohn 1914, Die Menschensohnfrage neu untersucht 1916; FSchulthess, ZNW 21, 1922, 247–50; Rtzst., Herr der Grösse 1919 (see also the works by the same author referred to above in this entry); EMeyer II 335ff; HGressmann, ZKG n.s. 4, 1922, 170ff, D. Messias 1929, 341ff; GDupont, Le Fils d’Homme 1924; APeake, The Messiah and the Son of Man 1924; MWagner, Der Menschensohn: NKZ 36, 1925, 245–78; Guillaume Baldensperger, Le Fils d’Homme: RHPR 5, 1925, 262–73; WBleibtreu, Jesu Selbstbez. als der Menschensohn: StKr 98/99, 1926, 164–211; AvGall, Βασιλεία τοῦ θεοῦ 1926; OProcksch, D. Menschensohn als Gottessohn: Christentum u. Wissensch. 3, 1927, 425–43; 473–81; CMontefiore, The Synoptic Gospels² 1927 I 64–80; ROtto, Reich Gottes u. Menschensohn ’34, Eng. tr. The Kgdm. of God and the Son of Man, tr. Filson and Woolf² ’43; EWechssler, Hellas im Ev. ’36, 332ff; PParker, The Mng. of ‘Son of Man’: JBL 60, ’41, 151–57; HSharman, Son of Man and Kingdom of God ’43; JCampbell, The Origin and Mng. of the Term Son of Man: JTS 48, ’47, 145–55; HRiesenfeld, Jésus Transfiguré ’47, 307–13 (survey and lit.); TManson, ConNeot 11, ’47, 138–46 (Son of Man=Jesus and his disciples in Mk 2:27f); GDuncan, Jesus, Son of Man ’47, 135–53 (survey); JBowman, ET 59, ’47/48, 283–88 (background); MBlack, ET 60, ’48f, 11–15; 32–36; GKnight, Fr. Moses to Paul ’49, 163–72 (survey); TNicklin, Gospel Gleanings ’50, 237–50; TManson (Da, En and gospels), BJRL 32, ’50, 171–93; TPreiss, Le Fils d’Homme: ÉThR 26/3, ’51, Life in Christ, ’54, 43–60; SMowinckel, He That Cometh, tr. Anderson, ’54, 346–450; GIber, Überlieferungsgesch. Unters. z. Begriff des Menschensohnes im NT, diss. Heidelb. ’53; ESjöberg, D. verborgene Menschensohn in den Ev. ’55; WGrundmann, ZNW 47, ’56, 113–33; HRiesenfeld, The Mythological Backgrd. of NT Christology, CHDodd Festschr. ’56, 81–95; PhVielhauer, Gottesreich u. Menschensohn in d. Verk. Jesu, GDehn Festschr. ’57, 51–79; ESidebottom, The Son of Man in J, ET 68, ’57, 231–35; 280–83; AHiggins, Son of Man- Forschung since (Manson’s) ‘The Teaching of Jesus’: NT Essays (TW Manson memorial vol.) ’59, 119–35; HTödt, D. Menschensohn in d. synopt. Überl. ’59 (tr. Barton ’65); JMuilenburg, JBL 79, ’60, 197–209 (Da, En); ESchweizer, JBL 79, ’60, 119–29 and NTS 9, ’63, 256–61; BvIersel, ‘Der Sohn’ in den synopt. Jesusworten, ’61 (community?); MBlack, BJRL 45, ’63, 305–18; FBorsch, ATR 45, ’63, 174–90; AHiggins, Jesus and the Son of Man, ’64; RFormesyn, NovT 8, ’66, 1–35 (barnasha=‘I’); SSandmel, HSilver Festschr. ’63, 355–67; JJeremias, Die älteste Schicht der Menschensohn-Logien, ZNW 58, ’67, 159–72; GVermes, MBlack, Aram. Approach³, ’67, 310–30; BLindars, The New Look on the Son of Man: BJRL 63, ’81, 437–62; WWalker, The Son of Man, Some Recent Developments CBQ 45, ’83, 584–607; JDonahue, Recent Studies on the Origin of ‘Son of Man’ in the Gospels, CBQ 48, ’86, 584–607; DBurkitt, The Nontitular Son of Man, A History and Critique: NTS 40, ’94 504–21 (lit.); JEllington, BT 40, ’89, 201–8; RGordon, Anthropos: 108–13.—B. 105; DELG. M-M. EDNT. TW. Sv.