U sentence example

u
  • He immediately turned on his lights and made a quick U turn.
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  • A gray brick house dominated the landscape, its ranch style sprawling in a U shape with a garage on one end.
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  • Ja - u  ?I a -a b -u' sh nS2=sh log (Q)=?a - b a - a' b - u' At x where = co, u = o, and q= go, (O n b - a ' a + a -b a' cio) - ?a-a'?b a-a' q In crossing to the line of flow x'A'P'J', b changes from o to m, so that with q = Q across JJ', while across xx the velocity is qo, so that i n = go.
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  • Lind's anemometer, which consists simply of a U tube containing liquid with one end bent into a horizontal direction to face the wind, is perhaps the original form from which the tube class of instrument has sprung.
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  • 5, I ch S2 = u a, sh C2= ' u (I) (I and along the jet APJ, oo > u=aerslc>a, sh S2=i sin 0 =iu=ie zrs/o, (2) PM sin 0 ds = f e ds = 1 = 1 sin 0, (3) cos 272a - cos 2n0 = 2Q - ?ib L a b2 s i n' 27ta u-b A (a- (u -a.u -b') sin 2110 - 2 a-a .u-b  ?l (u -a.u -a') = s in 2na u-b 2n b) A (ab.ba') p l u -bJ (u -a.u -a') sh nS2=i sin 110=i then the radius sin 2170 (30) A', cos nO= i, sin n0=o, n 1 ' b-a' ch nS2= ch log (9) = Va -a' n shnS2= shlog (Q) q _ o> u>a'.
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  • B) - f S,u ds (along The new wave-surface is formed in such a position that the optical distance is constant; and therefore the dispersion, or the angle through which the wave-surface is turned by the change of refrangibility, is found simply by dividing (5) by the distance AB.
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  • Oncken, Athen and Hellas (Leipzig, 1865); U.
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  • Meissner, Alexander u.
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  • ARGENTINE, a city of Wyandotte county, Kansas, U.
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  • 1 p, G r, 2 s, X t and d, V u and o, 8 f, d s (i.e.
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  • See also Didaskalia: Blatter fib' Geist, Gemiith, u.
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  • calities, are psammophytes, as well as plants (such as Cd7luna U:
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  • mes(u), " child," finds more favour, but is not certain.
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  • Bohm.-Bruder (1887); Becker, Zinzendorf and sein Christentum im Verhaltnis zum kirchlichen u.
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  • 1863-1879); Maria Theresa u.
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  • Baentsch, Altorientalischer u.
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  • It possesses the five vowels a, i, u, e, o, both short and long, and one pure diphthong, au.
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  • U.
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  • - Gindely, Geschichte der Bohmischen-Bruder (1858); Goll, Quellen u.
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  • 26 seq., may be supplemented by Willrich, Juden u.
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  • The difference of pressure between the outside air and the smoke-box gases may be measured by the difference of the water levels in the limbs of a U tube, one limb being in communication with the smokebox, the other with the atmosphere.
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  • p. 157 foil.), as well as Wellhausen, and since then by Jacobs and Zapletal (Der Totemismus u.
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  • Baentsch draws attention to this feature in his monograph Altorientalischer u.
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  • 18 foll., and Winckler, Religionsgeschichtlicher u.
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  • Many of the male inhabitants serve in the U.
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  • u.
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  • Meyer, Israeliten u.
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  • Vogelstein, Der Kampf zwischen Priestern u.
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  • 8 T - 9 U
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  • u W.
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  • Of Rhododendron there are i,u soecies.
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  • " u ??
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  • P tO u:..,.. ?
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  • u, Ulnar.
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  • r', u', Radial and ulnar carpal bones; with the three digits I., II., III.
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  • Sachau, Reise in Syrien u.
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  • Mesopotamien (1883), and Am Euphrat u.
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  • In 1841 he published Leber Princip u.
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  • These are Glauben u.
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  • Wissen (1858), Gott u.
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  • From 1847 onward Ulrici edited, jointly with the younger Fichte, the Zeitschrift fiir Philosophie u.
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  • The Arctic voyages of Barents were quickly followed by the establishment of p u a Dutch East India Company; and the Dutch, ousting the Portuguese, not only established factories on the mainland of India and in Japan, but acquired a preponderating influence throughout the Malay Archipelago.
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  • The chief muscular mass, arising from the sternum in the shape of a U, is the pectoralis muscle; its fibres converge into a strong tendon, which is inserted upon the greater tubercle and upper crest of the humerus, which it depresses and slightly rotates forwards during the downstroke.
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  • But the growth of a united Sicilian nation was impossible; the usual style to express the inhabitants of the island is "omnes" or "u n iversi Siciliae populi."
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  • j u ed.
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  • Using this and the temperature 673° in the expression, it will be found that U =185 B.Th.U.
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  • (Munich, 1889); U.
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  • Delitzsch in the notes appended to his first lecture Babel u.
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  • Budde in Die so-genannten Ebed-Jahweh Lieder u.
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  • Tonkunst, p. 161, and Leben u.
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  • Werner's Franz Suarez u.
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  • P. Caraffa u.
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  • CHICKASHA, a city and the county-seat of Grady county, Oklahoma, U.
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  • Texte u.
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  • Gesellschaft (Berlin, 1906) and Religionsgeschichtlicher u.
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  • Jahn, Die Bucher Esra u.
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  • 1 The vowels, which are ten in number (a a e e i i o o u u), were, as usual in the Semitic languages, indicated only partially by the use of consonants as vowel-letters 2 and by means of certain diacritical points, so long as Syriac remained a living language.
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  • Cognetti, U.
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  • von Haller, Papsttum u.
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  • General U.
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  • u, Duct of same.
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  • sb, S u b - intestinal ganglion on the course of the left visceral cord.
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  • u, The otocyst attached to the Ctenidium (gill-plume).
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  • - P l e u r o c e r i d a e.
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  • h, Border of the mantle-flap. u, Cerebral ganglion.
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  • u, Hermaphrodite gonad.
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  • u, v, w, Genitalia.
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  • From the ovo-testis, which lies near the apex of the visceral coil, a common hermaphrodite duct ve proceeds, which receives the duct of the compact white albuminiparous gland, Ed, and then becomes much enlarged, the additional width being due to the development of glandular folds, which are regarded as forming a uterus u.
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  • Kessler's article, "Mandaer," in Herzog-Hauck's Realencyklopddie, and the same author's paper, "Ueber Gnosis u.
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  • A complete bibliography is given in U.
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  • 1914, " U " boats were active in the neighbourhood of Scapa Flow, and on Oct.
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  • The dates of the various parts of the existing ducal palace are indicated on the plan; the rebuilding was carried on in the following order, P, Q, R, S, T, U, V.
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  • In 1868 he was appointed minister to Great Britain and soon after his arrival in England negotiated the Johnson-Clarendon treaty for the settlement of disputes arising out of the Civil War; this, however, the Senate refused to ratify, and he returned home on the accession of General U.
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  • with nouns, and of the final u of the third pers.
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  • ending u, &c).
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  • In Late Latin there was a tendency to this spirant pronunciation which appears as early as the beginning of the 2nd century A.D.; by the 3rd century b and consonantal u are inextricably confused.
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  • When this consonantal u (English w as seen in words borrowed very early from Latin like wall and wine) passed into the sound of English v (labio-dental) is not certain, but Germanic words borrowed into Latin in the 5th century A.D.
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  • 8; 9 a lot „„ Upper - 9 i u 91 Cotton Ships arrived.
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  • Sachau, Am Euphrat u.
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  • Siidenhorst, Syrien u.
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  • I?u FIG.
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  • Archiv fiir Zoologie, ii.; Id., " The Genera of European Nemerteans critically revised," Notes from the Leyden Museum (1879); Id., " Zur Anatomie u.
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  • (1893)+ Id., Fauna u.
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  • Gruppe, U ber d.
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  • On the influence of her cult upon that of the Virgin Mary, see Rusch, Studien u.
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  • ic- b u t civil jurisdiction soon followed.
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  • 1:1)41M .R1 ?Io;u?.
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  • (5) Constitutional History: The Aristotelian " Constitution of Athens "; U.
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  • Ru2C16.4KC1; Ru 2 C1 6.4NH 4 C1, &c. The pure tetrachloride, RuC1 4, has not been isolated, but is chiefly known in the form of its double salts, such as potassium ruthenium chloride, K 2 RuC1 6, which is obtained when finely divided ruthenium is fused with caustic potash and potassium chloride is gradually added to the fused mass (U.
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  • elements received symbols composed of circles, arcs of circles, and lines, while certain class symbols, such as1W for metals, - - foracids, for alkalies, for salts, U for calces, &c., were used.
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  • x u x?X x x HZ which exist between aliphatic and benzenoid compounds make the transformations of one class into the other especially interesting.
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  • The oxidation, which is effected by chromic acid and sulphuric acid, is conducted in a flask provided with a funnel and escape tube, and the carbon dioxide formed is swept by a current of dry air, previously freed from carbon dioxide, through a drying tube to a set of potash bulbs and a tube containing soda-lime; if halogens are present, a small wash bottle containing potassium iodide, and a U tube containing glass wool moistened with silver nitrate on one side and strong sulphuric acid on the other, must be inserted between the flask and the drying tube.
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  • Al, Fe, Cr, Mn; Ce, U (in sesquioxides).
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  • This is occasioned by the y-sound with which u now begins, and is carried further in dialect than in the literary language, sue and suit, for example, being pronounced in Scotland like the Eng.
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  • Kohler, Reichsverwaltung u.
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  • At Alexandria the state cult of him seems to have been instituted by the second Ptolemy, when his body was laid in the Sema (Otto, Priester u.
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  • 51 sqq.; Otto, Priester u.
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  • Kan, "Onze geographische kennis der Keij-Eilanden," in Tijdschrift Aardrijkskundig Genootschap (1887); Martin, "Die Kei-inseln u.
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  • The introduction of additional diacritical marks, such as - and used to express quantity, and the diaeresis, as in ai, to express consecutive vowels, which are to be pronounced separately, may prove of service, as also such letters as a, o and ii, to be pronounced as in German, and in lieu of the French ai, eu or u.
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  • u m.
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  • /e y.7?Y- .?C ?Tosc::: ? ?;, echeleb P. D u ?
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  • (3) Italian protectorate - Somalia italiana,1885-1895 (official " Green Book "); C. Rossetti, Somalia italiana settentrionale, with map (Rome, 1906); U.
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  • 3, we have the following impossible sentence, where Esau is referred t0: vEKpbs i v iipet /Lap, Kai 7ropevbµEVOS 'Avovipa,u airiaavev.
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  • Io - I I) admonishes his sons: Hpbs Ta y Ta7rELYWVEL Kaptlas u,I.LWV 'isa SE577vOs EUXoyLav Ea roD u7-6,l,Garoi abroli.
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  • ake A U /l '.?
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  • " Alexandria"; U.
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  • U, ulna; R, radius; c, cuneiform; 1, lunar; s, scaphoid; u, unciform; m, magnum; td, trapezoid; tm, trapezium.
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  • Astrales im Weltbilde des Thalmud u.
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  • - In addition to books mentioned under PALESTINE see the following: - U.
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  • a a i h am (SC?U?l'Hh~1 r :N 1 l ?
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  • 53161IICS APaHonaA) h' u dAx?
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  • In J u d y 1909, however, the Greek flag was hoisted in Canea and Candia, and it was only lowered again after the war-ships of the protecting powefs had been reinforced and had landed an international force.
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  • Husn u 'A s4 (Beauty and Love), as his great poem is called, is an allegorical romance full of tenderness and imaginative power.
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  • Dess u 0 W.
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  • N Sch [Idea u Grosser;hayn ' Konl Ra eburg "a.a Battle of ° Leipzig c8t3 to one of his unaccountable attacks of apparent intellectual paralysis.
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  • He graduated from the U.
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  • The excavations at the Hieron have been recorded as they went on in the Ilpaeroat of the Greek Archaeological Society, especially for 1881-1884 and 1889, and also in the 'E4»u€pis 'ApxatoXoynoi, especially for 1883 and 1885; see also Kavvadias, Les Fouilles d'Epidaure and Tb r03'A?KX iv 'E7rukbpq, eat 9Epa7reta7'CJY Defrasse and Lechat, Epidaure.
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  • He wrote Die siebzig Jahre des Jeremias u.
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  • die siebzig Jahrwochen des Daniel (1836); Geschichte des Aufruhrs in den Cevennen (1837); Lehrbuch der Weltgeschichte fitr Gymnasien (1839), which became a text-book in the Protestant gymnasia of Bavaria; Weissagung u.
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  • Erfiillung im alten u.
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  • In 734 their king Sanip(b)u was a vassal of Tiglathpileser IV., and his successor, P(b)udu-ilu, held the same position under Sennacherib and Esarhaddon.
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  • Tubes are generally made up around mandrels, and allowed throughout the curing to remain imbedded i n p u lverized French chalk, which affords a useful support for many articles that tend to lose their shape during the process.
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  • Zhukovskis U,nar Khayym and the Wandering Quatrains, translated from the Russian by E.
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  • so that A breaks u p into a sum of determinants, and we also obtain a theorem for the addition of determinants which have rows in common.
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  • For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=�.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.
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  • Further, if m '=' p, we obtain by differentiation 7+x =m (u;1-2xl.
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  • + v ?xl 1 + u l U1+v1V 1 + w1W1) � or T x0a,?_ (m-I) J+m(o .
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  • If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x, 2, z 2, yz, zx, xy from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails.
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  • U 2.1 U2) 1.2 0 3) /3..., j1!j2!j3!...
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  • = 11(1 +�alxl+� g al x2+�3a1x3+...), the product on the right involving a factor for each of the quantities a l, a2, a3..., and u being arbitrary.
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  • A 1, A2 � Ai, A 1 A 2, A2 and then Ao = al Ai+2a1a2AIA2+a2 A2 - (a1A1+a2A2) 2 = a?, A l = (a 1 A 1 +a2A2) (al�l +a2�2) = aAa�, A 2 = (al�l +a2/-12) 2 = aM; so that A = aa l +2a A a u 152+aM5 2 = (aA6+a,e2)2; whence A1, A 2 become a A, a m, respectively and ?(S) = (a21+a,E2) 2; The practical result of the transformation is to change the umbrae a l, a 2 into the umbrae a s = a1A1 +a2A2, a � = a1/�1 + a21=2 respectively.
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  • If u, a quantic in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and if, n,, ..., be quantities contragredient to x, y, z, ...; there are found to exist functions of, n, ?, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of E, H, Z, ...
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  • of the transformed coefficients of u; such functions are called contravariants of u.
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  • There also exist functions, which involve both sets of variables as well as the coefficients of u, possessing a like property; such have been termed mixed concomitants, and they, like contravariants, may appertain as well to a system of forms as to a single form.
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  • uo; 4, 1=0=f.; where u 0 =1, u1=o, assume that tfik = (af) k ay -k = f.
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  • u k =�y.
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  • ukx(n-2) � Taking the first polar with regard to y (n - k) (a f) xa x -k-l ay+ k (af) k-l ay -k (ab) (n -1) b12by n kn-2k-1 n-1 k(n-2) =k(n- 2)a u x u5+nax ayux and, writing f 2 and -f l for y1 and 3,21 (n-k)(a f) k+ta i k-1 + k (n - 1)(ab)(a f) k-1 (b f)4 1 k by-2 = (uf)u xn-2k-1?
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  • Every covariant is rationally expressible by means of the forms f, u 2, u3,...
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  • u n since, as we have seen uo =I, u 1 =o.
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  • y1 = x 15+f2n; f� y2 =x2-f?n, f .a b = ax+ (a f) n, l; n u 2 " 2 22 2 +` n) u3 n-3n3+...+U 2jn� 3 n Now a covariant of ax =f is obtained from the similar covariant of ab by writing therein x i, x 2, for yl, y2, and, since y?, Y2 have been linearly transformed to and n, it is merely necessary to form the covariants in respect of the form (u1E+u2n) n, and then division, by the proper power of f, gives the covariant in question as a function of f, u0 = I, u2, u3,...un.
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  • we find that Di must be equal to p x g x for then t x (p x) 3 +, u (g x) 3, Hence, if px, qx be the linear factors of the Hessian 64, the cubic can be put into the form A(p x) 3 +�(g x) 3 and immediately solved.
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  • We have A +k 1 f =0 2, O+k 2 f = x2, O+k3f =4) 2, and Cayley shows that a root of the quartic can be xpressed in the determinant form 1, k, 0.1y the remaining roots being obtained by varying 1, k, x the signs which occur in the radicals 2 u The transformation to the normal form reduces 1, k 3, ?
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  • He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the contragredient line variables-it is completely determinate if its leading coefficient be known.
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  • The general form of perpetuant is (4 K 3 A 2"`) and the generating function 1-z2.1-z3.1-z4 In general when 0 is even and =20, the condition is a l a 2 ...U 24 II(v 1 +a 2)II(a l +a 2 +cr 3)...II(Q 1 +a 2 -}-...
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  • Solving the equation by the Ordinary Theory Of Linear Partial Differential Equations, We Obtain P Q 1 Independent Solutions, Of Which P Appertain To S2Au = 0, Q To 12 B U =0; The Remaining One Is Ab =Aobl A 1 Bo, The Leading Coefficient Of The Jacobian Of The Two Forms. This Constitutes An Algebraically Complete System, And, In Terms Of Its Members, All Seminvariants Can Be Rationally Expressed.
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  • then of course (AB) = (ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A i = Xa i +,ia2, A2= - �ay + X a2, AA =A?-}-A2= (X2 +M 2)(a i+ a z) =aa; A B =AjBi+A2B2= (X2 +, U2)(albi+a2b2) =ab; (XA) = X i A2 - X2 Ai = (Ax i + /-Lx2) (- /-jai + Xa2) - (- / J.x i '+' Axe) (X a i +%Ga^2) = (X2 +, u 2) (x a - = showing that, in the present theory, a a, a b, and (xa) possess the invariant property.
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  • Holtzmann's "Urchristentum u.
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  • [partial] 1899); Das Realgymnasium u.
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  • Protestantismus (1899); Schopenhauer, Hamlet u.
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  • Mephistopheles (1900); Philosophia militans (1900, 1901); Parteipolitik u.
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  • The borate, Pb 2 B 6 0 1 u 4H20, is obtained as a white precipitate by adding borax to a lead salt; this on heating with strong ammonia gives PbB2044H2.
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  • Wilhelmsage u.
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  • From the equation K=(µ - I)/47r, it follows that the magnetic susceptibility of a vacuum (where µ = I) is o, that of a diamagnetic substance (where, u I) is positive.
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  • u Proc. Roy.
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  • Without a more accurate knowledge of the masses of the planets than was then possessed a satisfactory solution was impossible; but the upper limits assigned by him agreed closely with those obtained later by U.
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  • It was restored to sacred uses in 1887, and has been carefully liberated from later alterations (U.
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  • Vischer, Erinnerungen u.
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  • E acrpi u rns (insc. of Miletus, Sitzungsber.
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  • The grammatical forms are expressed, as in Turkish, by means of affixes modulated according to the high or low vowel power of the root or chief syllables of the word to which they are appended-the former being represented by e, o, S, ii, i l l, the latter by a, d, o, 6, u, it; the sounds e, i, i are regarded as neutral.
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  • If, for instance, we are told that 15= 4 of (x- 2), what is meant is that (I) there is a number u such that x=u+2, (2) there is a number v such that u=4 times v, and (3) 15=3 times v.
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  • From these statements, working backwards, we find successively that v= 5, u = 20, X = 22.
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  • (a) The statement is that (I) there is a number u such that x= 2u,(2) there is a number v such that x= 3v, and (3) u+v= Io.
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  • Hence u is the greatest common measure of p and q.
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  • The standard form is usually taken to be ax2+bx+c =0, from which we find, by transformation, (2 ax+b) 2 =b 2 - 4ac, 4 (}b 2 -4ac} -b and thence x = 2a (ii.) In an equation of the form Q=V, the expressions P, Q, U, V are usually numerical.
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  • (viii.) If u, denotes the term involving a r in the expansion of (A-{-a) n, then ur/ur - 1 = {(n-r+ i)/r}.a/A.
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  • (a) Let S r denote the sum u o+ u 1+.
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  • +u,, this sum being taken so as to include the greatest term (or terms); and let u r+1 /u r = 0, so that 0
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  • u r.
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  • we obtain I, 4, 9, � � � This suggests that, if u n is the sum of the first n odd numbers, then u n = n 2 .
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  • Assume this true for u 1, u 2, ., u,,.
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  • Then u,,+1=un+(2n-+-I)=n2+(2n+I)=(n+I)2, so that it is true for u n+1.
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  • But it is true for u 1 .
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  • (i.) Denote n (r) x n - r h r by u r, and uo+u i +...
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  • Schroder, Lehrbuch der Arithmetik u.
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  • His travels and mercantile experience had led E t u eopre him to conclude that the Hindu methods of computing were in advance of those then in general use, and in 1202 he published his Liber Abaci, which treats of both algebra and arithmetic. In this work, which is of great historical interest, since it was published about two centuries before the art of printing was discovered, he adopts the Arabic notation for numbers, and solves many problems, both arithmetical and algebraical.
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  • i' ro m u s Coelomocoela.
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  • Norden, Das Papsttum u.
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  • Sagmtiller, Die Thcitigkeit u.
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  • It vanishes when u =mlr, m being any whole number other than zero.
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  • When u = o, it takes the value unity.
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  • The maxima occur when u=tan u, (4), and then sin 2 u/u 2 = cos 2 u (5).
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  • To calculate the roots of (5) we may assume u=(m+1)7r-y= U-y, (3), where y is a positive quantity which is small when u is large.
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  • Substituting this, we find cot y = U-y, whence 5 7 y U(1+/-1-2:-+2...) - y3 -15-315' This equation is to be solved by successive approximation.
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  • It will readily be found that u =U -y =U-U-1-U-a-15U 5 105U- -.
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  • In the first quadrant there is no root after zero, since tan u> u, and in the second quadrant there is none because the signs of u and tan u are opposite.
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  • Since the maxima occur when u = (m +1)7r it nearly, the successive values are not very different from 4 4 4 &c The application of these results to (3) shows that the field is brightest at the centre =o, =0, viz.
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  • 15, p. 315) to determine the absolute intensity of a secondary wave, may be at once effected by means of the known formula isin 2 u f sin u du = du =7-.
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  • u 2 u It will be observed that, while the total intensity is proportional to ab, the intensity at the focal point is proportional to a 2 b 2.
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  • The part corresponding to negative values of u is similar, OA being a line of symmetry.
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  • The obliquity, corresponding to u =7, is such that the phases of the secondary waves range over a complete period, i.e.
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  • Arago, the index, u for air at t° C. and at atmospheric pressure is given by 00029 iu 1-1 + 0037t.
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  • The sine of an angle can never be greater than unity; and consequently under the most favourable circumstances only 1/m 2 ir 2 of the original light can be obtained in the m u ' spectrum.
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  • Then, if Q be any radiant point and Q' its image (primary focus) in the spherical mirror AP, we have 1 1 2cos4) v l + u 'a ' ' where v 1 = AQ', u =AQ, a =OA, =angle of incidence QAO, equal to the angle of reflection Q'AO.
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  • To find the former, we have, if OAQ=4), AOP=w, QP 2 =u 2 +4a 2 sin 2 2w - 4au sin la) sin (2w-4)) = (u +a sin 4) sin w) 2 -a 2 sin 2 4)sin 2 c0+4a sin 2 2w(a-u cos 0).
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  • If we put for shortness 7 for the quantity under the last circular function in (I), the expressions (i), (2) may be put under the forms u sin T, v sin (T - a) respectively; and, if I be the intensity, I will be measured by the sum of the squares of the coefficients of sin T and cos T in the expression u sin T +v sin (T - a), so that I =u 2 +v 2 +2uv cos a, which becomes on putting for u, v, and a their values, and putting f =Q .
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  • Taking 2 rv2 = u (9), we may write (17r oueiudu ?/ 2) u Again, by a known formula, 1 1 °° -1 uu = 1/ 7r?r o Substituting this in (to), and inverting the order of integration, we get uc 2?
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  • dx ru i-x) C-Fi S= „ 7 o?I x o 1 fGO dx eu(ti-x) - 1 2Jo x i - x Thus, if we take _ 1 `°el 1 ('°° e uxdx G 7r12 Jo 1+ x 2 ' H 7r-N/2Jo -Vx.(1-i-x2)' C = 2-G cos u+ H sin u, S =1---G sin u-H cos u.
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  • Comparing the expressions for C, S in terms of M, N, and in terms of G, H, we find that G = z (cos u+sin u)-M, H = z (cos u-sin u) +N.
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  • (15), formulae which may be utilized for the calculation of G, H when u (or v) is small.
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  • For example, when u = o, M = o, N =o, and consequently G =H = 2.
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  • Descending series of the semi-convergent class, available for numerical calculation when u is moderately large, can be obtained from (12) by writing x=uy, and expanding the denominator in powers of y.
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  • See U.
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  • He graduated at the U.
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  • URANIUM [[[symbol]] U, atomic weight 238.5 (0=16)], a metallic chemical element.
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  • Peligot's results, though called in question by Berzelius, have been amply confirmed by all subsequent investigators; only now, on theoretical grounds, first set forth by Mendeleeff, we double Peligot's atomic weight, so that U now signifies 240 parts of uranium, while UO 3 stands as the formula of the yellow oxide, and UO 2 as that of Berzelius's metal.
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  • Its specific gravity has the high value 18.7; its specific heat is 0.02765, which, according to Dulong and Petit's law, corresponds to U = 240: It melts at bright redness.
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  • The powdery metal when heated in air to 150° or 170° C. catches fire and burns brilliantly into U 3 0 8; it decomposes water slowly at ordinary temperatures, but rapidly when boiling.
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  • Dilute sulphuric acid attacks it but slowly; hydrochloric acid, especially if strong, dissolves it readily, with the formation, more immediately, of a hyacinthcoloured solution of U 2 C1 6, which, however, readily absorbs oxygen from the air, with the formation of a green solution of UC1 4, which in its turn gradually passes into one of yellow uranyl salt, U02.
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  • The solution in sulphuric acid deposits green crystals of the sulphate, U(S04)2.8H20, on evaporation.
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  • Dilute sulphuric acid precipitates uranium yellow, Na 2 U 2 0 7.6H 2 O, from the solution so obtained.
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  • Studien u.
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  • 4 E I u 4.
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  • Hommel, Aufsatze u.
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  • pteronura) and one of the skunk; two species of bear (Ursus ornatus and U.
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  • Buchheister, " Die Elbe u.
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  • Willisen, Die Feldzge 1859 u.
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  • Strobl, Trautenau (Vienna, 1901); KUhne, Kritische u.
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  • Kriegsereignisse zwischen Preussen u.
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  • Falckenstein u.
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  • Hertwig, Die Zelle u.
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  • Morphologic u.
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  • j.," u being changed to v and j to i according to the ordinary rules of the game.
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  • II?:,?: ??U? ?I?h'?a' .? ?
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  • The integral equation of continuity (I) may now be written l f fdxdydz+ff (lpu+mpv+npdso, (4) which becomes by Green's transformation (dt +d dz dy dx (p u) + d (p v) + d (p w) l I dxdydz - o, dp leading to the differential equation of continuity when the integration is removed.
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  • Eliminating H between (5) and (6) DS du dv dw (du dv d1zv dt u dx n dx udx' 5 -, dzi =°' and combining this with the equation of continuity Dp du dv dw p iit dx+dy+ dz = °' (10) D i du n dv dw_ dt (p p dx p dx p dx - o, with two similar equations.
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  • Taking the axis of x for an instant in the normal through a point on the surface H = constant, this makes u = o, = o; and in steady motion the equations reduce to dH/dv=2q-2wn = 2gco sin e, (4) where B is the angle between the stream line and vortex line; and this holds for their projection on any plane to which dv is drawn perpendicular.
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  • Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it.-A stream-function 4, must be determined to satisfy the conditions v24 =o, throughout the liquid; (I) I =constant, over any fixed boundary; (2) d,t/ds = normal velocity reversed over a solid boundary, (3) so that, if the solid is moving with velocity U in the direction Ox, d4y1ds=-Udy/ds, or 0 +Uy =constant over the moving cylinder; and 4,+Uy=41' is the stream function of the relative motion of the liquid past the cylinder, and similarly 4,-Vx for the component velocity V along Oy; and generally 1,1'= +Uy -Vx (4) is the relative stream-function, constant over a solid boundary moving with components U and V of velocity.
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  • In the general case 4'=11.+Uy-Vx+2R(x 2 +y 2) is the relative stream function for velocity components, U, V, R.
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  • Consider the motion given by w=U(z+a2/z), (I) 4,=U(r+- r) cos 0= U + a1x, so that (2) = U (r-)sin 0= U(i -¢) y.
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  • Over a concentric cylinder, external or internal, of radius r=b, 4,'=4,+ Uly =[U I - + Ui]y, (4) and 4" is zero if U 1 /U = (a 2 - b2)/b 2; (5) so that the cylinder may swim for an instant in the liquid without distortion, with this velocity Ui; and w in (I) will give the liquid motion in the interspace between the fixed cylinder r =a and the concentric cylinder r=b, moving with velocity U1.
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  • When b = o, U 1 =00; and when b = oo, U 1 = -U, so that at infinity the liquid is streaming in the direction xO with velocity U.
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  • If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by w = - Uz, we are left with w = Ua2/z, (6) =U(a 2 /r) cos 0= Ua2x/(x2+y2), (7) 4, = -U(a 2 /r) sin 0= -Ua2y/( x2+y2), (8) giving the motion due to the passage of the cylinder r=a with velocity U through the origin 0 in the direction Ox.
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  • 208.) If 01 denotes the velocity function of the liquid filling the cylinder r = b, and moving bodily with it with velocity Ul, 41 = -U1x, (12) and over the separating surface r =b 4, = I U (+- a2) a2 +b2 (13) 1 Ul b2 - a 2 ...
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  • When the cylinder r =a is moved with velocity U and r =b with velocity U 1 along Ox, = U b e - a,1 r +0 cos 0 - U ib2 - 2 a, (r +Q 2 ') cos 0, = - U be a2 a2 (b 2 - r) sin 0 - Uib2 b1)a, (r - ¢2 sin 0; b and similarly, with velocity components V and V 1 along Oy a 2 b2 ?= Vb,_a,(r+r) sin g -Vi b, b2 a, (r+ 2) sin 0, (17) = V b, a2 a, (b2 r) cos 0+Vi b, b, a, (r- ¢ 2) cos h; (18) and then for the resultant motion z 2zz w= (U 2 + V2)b2a a2U+Vi +b a b a2 U z Vi -(U12+V12) b2 z a2b2 Ui +VIi b 2 - a 2 U1 +Vii b 2 - a 2 z The resultant impulse of the liquid on the cylinder is given by the component, over r=a (§ 36), X =f p4 cos 0.ad0 =7rpa 2 (U b z 2 + a 2 Uib.2bz a2); (20) and over r =b Xi= fp?
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  • bdo 7rpb 2 (u, b 2 a2 Uibb +¢z), and the difference X-X 1 is the component momentum of the liquid in the interspace; with similar expressions for Y and Y1.
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  • (22) But if the outside cylinder is moved with velocity U1, and the inside cylinder is solid or filled with liquid of density v, 2 U i 2pb2 and the inside cylinder starts forward or backward with respect to the outside cylinder, according as p> or < v.
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  • Taking two planes x = =b, and considering the increase of momentum in the liquid between them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is pdy (U - Ua 2 r2 cos 20 +mr i sin 0) (Ua 2 r 2 sin 2 0+mr 1 cos 0) + pdy (- U+Ua 2 r 2 cos 2 0 +mr1 sin 0) (Ua 2 r 2 sin 2 0 -mr 1 cos 0) =2pdymUr '(cos 0 -a 2 r 2 cos 30), (8) and with b tan r =b sec this is 2pmUdo(i -a 2 b2 cos 30 cos 0), (9) and integrating between the limits 0 = 27r, the resultant, as before, is 27rpmU.
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  • Over any ellipse n, moving with components U and V of velocity, =i+Uy-Vx=[msh(n-a) cos (3+Ucshn] sin k -[msh(n-a) sin (3+Vcchn] cos h; (7) so that ' =o, if U c sh n cos R, V = c ch n sin a, (8) m sh(n - a) m sh(n - a).
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  • An ellipse interior to n = a will move in a direction opposite to the exterior current; and when n = o, U = oo, but V = (m/c) sh a sin 13.
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  • U -U?
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  • (p -a)(b2 -a2) U i p(b2 +a2) +0-(b2-a2)' (23) (19) (21) (14) (15) (16) X =-7rUa U, U -p(bz +a2) +0_(b2-a2), 2g 2g, (3) so that cavitation will take place, unless the head at a great distance exceeds this loss.
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  • The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a resultant thrust on the whole cylinder is 2mU sin Olga, and its thrust is 27rpmU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r=a with velocity U reversed, and the cylinder is surrounded by a vortex.
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  • Similarly, the streaming velocity V reversed will give rise to a thrust 27rpmV in the direction xC. Now if the cylinder is released, and the components U and V are reversed so as to become the velocity of the cylinder with respect +m /a) 2 - U2 The components of the liquid velocity q, in the direction of the normal of the ellipse n and hyperbola t, are -mJi sh(n--a)cos(r-a),mJ2 ch(n-a) sin (E-a).
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  • The velocity past the surface of the sphere is dC r sin 0 dy 2U (2r+ a 2) r sin g z U sin e, when r =a; (20) so that the loss of head is (!
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  • sin e 0 - i) U 2 /2g, having a maximum a U 2 /2g, (21) which must be less than the head at infinite distance to avoid cavitation at the surface of the sphere.
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  • Flow, Circulation, and Vortex Motion.-The line integral of the tangential velocity along a curve from one point to another, defined by s v as + u'a s) ds =f (udx+vdy-}-zdz), (I) is called the " flux " along the curve from the first to the second point; and if the curve closes in on itself the line integral round the curve is called the " circulation " in the curve.
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  • So far these theorems on vortex motion are kinematical; but introducing the equations of motion of § 22, Du + dQ =o, Dv+dQ =o, Dw + dQ dt dx dt dy dt dz and taking dx, dy, dz in the direction of u, v, w, and dx: dy: dz=u: v: w, (udx + vdy + wdz) = Du dx +u 1+..
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  • The components of velocity of the moving origin are denoted by U, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u', v', w' the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we have u =U +u' - yR +zQ, v =V +v -zP +xR, w=W +w -xQ +yP.
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  • u '= - dx -md x, ' - dy -m dy, w = - dz-mdz' as in § 25 (I), a first integral of the equations in (5) may be written dp V + 2q 2 - d - n dt +14-14) (dx + m dz) +(v-v') (+m) +(w - w) (+m) =F(t), (7) in which d4, do, d?
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  • dt-(u)dy- (w-w) dz = d - (U-yR+zQ) dy - (V-zP+xR)d -(W-xQ+yP) d z (8) is the time-rate of change of 49 at a point fixed in space, which is left behind with velocity components u-u', v-v', w-w'.
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  • I, ' 2 dx (y dx) +dy U dy) so that § 34 (4) is satisfied, with f' (W') =1.0 a2, f (Y") = 2 U'a2; and (ro) reduces to `)(() P +v-3 U j _ S = constant; (16) this gives the state of motion in M.
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  • As an application of moving axes, consider the motion of liquid filling the ellipsoidal case 2 y 2 z2 Ti + b1 +- 2 = I; (1) and first suppose the liquid be frozen, and the ellipsoid l3 (4) (I) (6) (9) (I o) (II) (12) (14) = 2 U ¢ 2, (15) rotating about the centre with components of angular velocity, 7 7, f'; then u= - y i +z'i, v = w = -x7 7 +y (2) Now suppose the liquid to be melted, and additional components of angular velocity S21, 522, S23 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function 2224_ - S2 b c 6 a 5 x b2xy, as may be verified by considering one term at a time.
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  • If u', v', w' denote the components of the velocity of relative to the axes, = u +yR - zQ =a2+ b2S23y - c2 a2 ?
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  • u, du +v, du +w, du =o,...,..., (9) ax '?
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  • Uniplanar motion alone is so far amenable to analysis; the velocity function 4 and stream function 1G are given as conjugate functions of the coordinates x, y by w=f(z), where z= x +yi, w=4-Plg, and then dw dod,y az = dx + i ax - -u+vi; so that, with u = q cos B, v = q sin B, the function - Q dw u_vi=g22(u-}-vi) = Q(cos 8+i sin 8), gives f' as a vector representing the reciprocal of the velocity in direction and magnitude, in terms of some standard velocity Q.
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  • u - b), u =ae ?, (5) where u =a, a' at the edge A, A l.; u at a corner B; u=o across xx' where 4 = oo; and u = oo, 43-= co across the end J J' of the jet, bounded by the curved lines APJ, A'P'J', over which the skin velocity is Q.
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  • - n)= l b - au - ' (8) a - a u - b (9) dS2 I A I (b-a.b-a') dw m du = 21/(U - b)- ‘ 1 (u-a.0-a')' du -, r u' Io) the formulas by which the conformal representation is obtained.
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  • For the 2 polygon has a right angle at a=a, a', and a zero angle at u = b, where 0 changes from o to 27/n and 1 - 2 increases by 21-rr/n; so that dSt A (b -a.b -a') a - (u -b),/ (u - a.u - a where A= tn (II) And the w polygon has a zero angle at u =o, oo, where 4, changes from o to m and back again, so that w changes by im, and du =B, where B=-.
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  • 2n u (a -b.b -a')' not requiring the integration of (II) and (12) If 0=a across the end J J' of the jet, where u = oo, q= Q, b-a' a-b ch 7/2,=cos na = I, , sh 162 = i sin na = i,, a - a a-a Then a-a'+(a+a) cos 2na-[a+a'+(a-a) cos 2najcos 2110 (a-a') sin' 2na X cos 211a - cos 2110 Along the wall AB, cos nO =o, sin n0= I, a> u> b, ch n2= i sh log (q) n= i?
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  • m _ c Q du dO dt - rrqu 2r qu AB _ Q du L bq (u -a') +V (b -a')1,l (a -u)11/ndu) L 1,1 (a-a/),,1 (u-b), f u Along the wall Bx, cos n0 =I, sin n0 =o, b >u>o ch nSt = ch log () n=, , fb-a?, ?
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  • Thus with a' =o, a stream is split symmetrically by a wedge of angle ' zr/n as in Bobyleff's problem; and, by making a = oo, the wedge extends to infinity; then chnS2= u, sh nS2= b n u.
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  • ', In Bobyleff's problem of the wedge of finite breadth, ch nS2 = ?a b' s h n S 2 = V b a a 1 u u b, (6)?
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  • b a-b (7) cos = a, sin na = 1j a, nd along the free surface APJ, q =Q, 4) =o, u =e-.
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  • 4l m =aeffsle, e rs /° - I u =ae-"' - (a -b)w', (io) w'+ ch nS2 =, sh nS2 = w,, (II) I n which we may write w' =4)+41i.
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  • 7), and so must be excluded from the boundary of u; the conformal re presentation is made now with du= (b-a.b-a') du - (u-b) A l (u-a.0-a) (I) dw m I m' du = 7r u-j - u -j' _ m+m' u-b it u' j.0-j" b = mj i m'j m+m', taking u = co at the source where FIG.7.
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  • 0-00, u = b at the branch point B, u = j, j at the end of the two diverging streams where = -oo; while ¢=0 along the stream line which divides at B and passes through A, A'; and 4 ' =m, -m' along the outside boundaries, so that m/Q, m'/Q is the final breadth of the jets, and (m+m')/Q is the initial breadth, c, of the impinging stream.
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  • u -b' s h 2S2 =1 1a-a ?
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  • u -b' Along a jet surface, q=Q, and ch S2= cos 0 =cos a-i sin2a(a-a')/(u-b), (5) if 0 =-a at the source x of the jet xB, where u = co; and supposing 0=0,13 at the end of the streams where u =j, j', u-b i sin 2 a u - j cos 0-cos /3 i a -a cos a sin a -cos 0' aa' - 2 (cos a -cos (3) (cos a-cos 0)' u-j' 1 2 cos 0-cos, (6) a -a' - 2 S i n a (cos a -cos (3') (cos a -cos B)' and 4' being constant along a stream line d4 - dw ds _d8 d4 _ dw du du du' d- -dud0' 7rQ ds_ it ds (cos a-cos /3) (cos a -cos (3') sin 0 m+m' dB c d0 - (cos a-cos B) (cos 0-cos /3) (cos 0 -cos /3')' _ sin 0 cos a-cos 13 sin 0 - cos a-cos B + cos 0-cos (3' cos 0-cos 13 cos a -cos $ sin 6 cos (3-cos /3' cos 0-cos 0" giving the intrinsic equation of the surface of a jet, with proper attention to the sign.
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  • From A to B, a>u >b, 0=0, ch S2= ch log Q=cos a-i sin 2a a-b I sh S2= sh log Q= I (a u-b-a/) s i n a Q = (u-b) cos a-2(a-a') sin 2 a+1,/ (a-u.u- a')sin a (8) u-b ds _ ds d4 _ Q dw Q du - Q d 4) du q du (u-b) cos a-2(a- a') sin 2 a (a-u.0 - a') sin a (9) it j- -j' AB _f a(2b - a - a')(u-b)-2(a-b)(b-a')+2V (a - b.
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  • a - u.
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  • u - a') du, () - a - a' .j - u.0 - j' IO a with a similar expression for BA'.
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  • Beginning with a single body in liquid extending to infinity, and denoting by U, V, W, P, Q, R the components of linear and angular velocity with respect to axes fixed in the body, the velocity function takes the form = Ucb1+V42+W43+ P xi+Qx2+Rx3, (I) where the 0's and x's are functions of x, y, z depending on the shape of the body; interpreted dynamically, C -p0 represents the impulsive pressure required to stop the motion, or C +p4) to start it again from rest.
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  • The terms of 0 may be determined one at a time, and this problem is purely kinematical; thus to determine 4)1, the component U alone is taken to exist, and then 1, m, n, denoting the direction cosines of the normal of the surface drawn into the exterior liquid, the function 01 must be determined to satisfy the conditions v 2 0 1 = o, throughout the liquid; (ii.) ' = -1, the gradient of 0 down the normal at the surface of the moving solid; 1 =0, over a fixed boundary, or at infinity; similarly for 02 and 03.
    0
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  • l ' so that over the surface of an ellipsoid where X and ¢ are constant, the normal velocity is the same as that of the ellipsoid itself, moving as a solid with velocity parallel to Ox U = -q, - 2 (a2+X) dtP, and so the boundary condition is satisfied; moreover, any ellipsoidal surface X may be supposed moving as if rigid with the velocity in (I I), without disturbing the liquid motion for the moment.
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  • The continuity is secured if the liquid between two ellipsoids X and X 11 moving with the velocity U and 15 1 of equation (II), is squeezed out or sucked in across the plane x=o at a rate equal to the integral flow of the velocity I across the annular area a l.
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  • Denoting the effective inertia of the liquid parallel to Ox by aW' the momentum aW'U = 4)0W' (24) _ U i -AO' 25) in this way the air drag was calculated by Green for an ellipsoida pendulum.
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  • When the liquid is bounded externally by the fixed ellipsoid A = A I, a slight extension will give the velocity function 4 of the liquid in the interspace as the ellipsoid A=o is passing with velocity U through the confocal position; 4 must now take the formx(1'+N), and will satisfy the conditions in the shape CM abcdX ¢ = Ux - Ux a b x 2+X)P Bo+CoB I - C 1 (A 1 abcdX, I a1b1cl - J o (a2+ A)P and any'confocal ellipsoid defined by A, internal or external to A=A 1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox BA+CA-B 1 -C1 W'.
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  • ZUy2BB0 Bll; reducing, when the liquid extends to infinity and B 3 =0, to = xA o' _ - zUy 2B o so that in the relative motion past the body, as when fixed in the current U parallel to xO, A 4)'=ZUx(I+Bo), 4)'= zUy2(I-B o) (6) Changing the origin from the centre to the focus of a prolate spheroid, then putting b 2 =pa, A = A'a, and proceeding to the limit where a = oo, we find for a paraboloid of revolution P B - p (7) B = 2p +A/' Bo p+A y2 i =p+A'- 2x, (8) p+?
    0
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  • Two equal spheres, intersecting at 120°, will require - I U j x _ a 3 a4(a 7 2 x) a3 a4(a+2x)] (II) 2 - _ 2 y a 271 3 271 +2Y2 3 2720 ' with a similar expression for cylinders; so that the plane x=o may be introduced as a boundary, cutting the surface at 60°.
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  • = constant, _ ff 00 NdA N BA-AA X - JA (a' +X) (b 2 +A)P - abc' a2 -b2 ' and at the surface A = o, I I N Bo-A 0 N I R - (a2+b2) abc a 2 -b 2 abc a2b2 I /b 2 N = R I /b2 - I /a2 abc I 1 I Bo - AO' a 2 b 2 - a2 b2 a 2 b2 = R (a 2 - b 2) /(a 22 + /b2) 2 - r (B o - Ao) U Bo+Co - B I - CI' Since - Ux is the velocity function for the liquid W' filling the ellipsoid A = o, and moving bodily with it, the effective inertia of the liquid in the interspace is Ao+B1+C1 Bo+Co - B1 - C, If the ellipsoid is of revolution, with b=c, - 2 XBo - - C BI' and the Stokes' current function 4, can be written down (I) is (5) (7) (6) The velocity function of the liquid inside the ellipsoid A=o due to the same angular velocity will be = Rxy (a2 - b2)/(a2 + b2), (7) and on the surface outside _ N Bo -Ao c1)0xy abc 2 62' so that the ratio of the exterior and interior value of at the surface is ?o= Bo-Ao (9) 4)1 (a 2 -6 2)/(a2 + b) - (Bo - Ao)' and this is the ratio of the effective angular inertia of the liquid, outside and inside the ellipsoid X = o.
    0
    0
  • But supposing them determined for the motion of a body through a liquid, the kinetic energy T of the system, liquid and body, is expressible as a quadratic function of the components U, V, W, P, Q, R.
    0
    0
  • Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a quadratic function of xi, 'x2, x3, yi, Y2, Y3, U = d, V= dx, ' w= ax dT Q_ dT dT dy 1 dy2 dy The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow X = dt x2 dy +x3 d Y = ..., Z ..., (3) = dt1 -y2?y - '2dx3+x3 ' M =..
    0
    0
  • origin up to the moving origin 0, so that dy x=y=z=o, but dt U, dt= ' dG _ dyl =l (- yi y3Q x2w+xiv) +m (dY2yP+Yrxu+xw) +n (?
    0
    0
  • are the components of a constant vector having a fixed direction; while (4) shows that the vector resultant of y, y, y moves as if subject to a couple of components x Wx V, x Ux W, x V-x U, (Io) and the resultant couple is therefore perpendicular to F, the resultant of x, x, x, so that the component along OF is constant, as expressed by (iii).
    0
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  • Consider, for example, a submarine boat under water; the inertia is different for axial and broadside motion, and may be represented by (1) c 1 =W+W'a, c2=W+W'/3' where a, R are numerical factors depending on the external shape; and if the C.G is moving with velocity V at an angle 4) with the axis, so that the axial and broadside component of velocity is u = V cos 0, v =V sin 4), the total momentum F of the medium, represented by the vector OF at an angle 0 with the axis, will have components, expressed in sec. Ib, F cos 0 =c 1 - = (W +W'a) V cos 43, F sin 0 = c 2.11 = (W +W'/3) V sin 4) .
    0
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  • (6) The least admissible value of p is that which makes the roots equal of this quadratic in µ, and then ICI s ec 0,, u= z - p (7) the roots would be imaginary for a value of p smaller than given by Cip 2 - 4(c 2 -c i)c2C 2 u 2 =o, (8) p2 = 4(c 2 -c l)cl C2.
    0
    0
  • which is the ratio of the linear velocity of rotation 2dp to u, the velocity of advance, -T2 d2 C 22 tans = n 2 = 4 = (c 2 - Ct) cg C12 2 W!
    0
    0
  • As the ring is moved from 0 to 0' in time t, with velocity Q, and angular velocity R, the components of liquid momentum change from aM'U +E and SM'V along Ox and Oy to aM'U'+ and /3M'V' along O'x' and O'y', (I) the axis of the ring changing from Ox to O'x'; and U = Q cos 0, V = Q sin 0, U' =Q cos (o - Rt), V' =Q sin (0 - Rt), (2) so that the increase of the components of momentum, X 1, Y 1, and N1, linear and angular, are X 1 = (aM'U'+ 0 cos Rt - aM'U - - 1 3M'V' sin Rt =(a - (3)M'Q sin_(0 - Rt) sin Rt - ver Rt (3) Y 1 = (aM'U'+) sin Rt-[-13M'V' cos Rt - (3M'V = (a - (3) M'Q cos (0 - Rt) sin Rt +t sin RT, N1=[ - (aM'U'+E) sin (0 - Rt)+ 1 3M'V' cos (o - Rt)]OO' = [- (a - 1 3) M'Q cos (o - Rt) sin (o - Rt) - sin (o - Rt) ]Qt.
    0
    0
  • ZI /t = - (a - s) M'Q 2 sine cos ° - EQ sin() =[ - (a - (3)M'U+E]V (8) Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic reaction of the liquid - aM' (Ç_vR), - (3M' (-- E -FUR), - EC' dR, (9) so that its equations of motion are M (Ç - vR) _ - aM' (_vR) - (a - $) M'VR, (io) M (Ç+uR) = - OM' (dV+U R) - (a - ()M'UR - R, '(II) C dR = dR + (a - Q)M'UV+0V; (12) and putting as before M+aM'=ci, M+13M' = c2, C+EC'=C3, ci dU - c2VR=o, dV +(c1U+E)R=o, c 3 dR - (c 1 U+ - c 2 U)V =o; showing the modification of the equations of plane motion, due to the component E of the circulation.
    0
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  • The integral of (14) and (15) may be written ciU+E=Fcoso, c 2 V= - Fsino, dx F cost o F sinz o 71 = U cos o - V sin o = cl + c c ic os o, chi = U sine +V coso= (F - F) sin cos o - l sino, (19) c i 2 2 2 sin o cos o - l ?
    0
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  • Colossal \ l U D.
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  • In his valiant attempt to fill these gaps Rad„ u was obliged to invent kings and even dynasties, 13 the existence f which is now definitely disproved.
    0
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  • Thus, to take a classic example, the name of the famous king Nebuchadrezzar occurs written in the following different manners: - (a) Na-bi-urn-ku-du-ur-ri-u-su-ur,(b)AK-DU u-su-ur, (c) AK-ku-dur-ri-Shes, and (d) PA-GAR-DU-Shes, from which we are permitted to conclude that PA or AK (with the determinative for deity AN) = Na-bi-um or Nebo, that GAR-DU or DU alone = kudurri, and that Shes = u.
    0
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  • atu $u?
    0
    0
  • It is most probably distinct from the 'Aran&Autkes 'Af3pa6 u used by the gnostic Sethites (Epiphanius, Haer.
    0
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  • 199, 200; Littmann u.
    0
    0
  • Extraction of cane juice by diffusion (a process more fully described under the head of beetroot sugar manufacture) is adopted in a few plantations in Java and Cuba, in Louisiana Etr cti o n and the Hawaiian Islands, and in one or two factories y f i in Egypt; b u t hitherto, except under exceptional conditions (as at Aska, in the Madras Presidency, where the local price for sugar is three or four times the London price), it would not seem to offer any substantial advantage over double or triple crushing.
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  • The mud collects at the bottom of the u, and allows the upper part of the bag to filter for a longer time than would be the case if the bottom end were closed and if the bag hung straight like the letter I.
    0
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  • Relazione dell' U „ Jicio Regionale per la conservazione dei monumenti del Veneto, Venice, 1895, p. 111).
    0
    0
  • For authorities see U.
    0
    0
  • A good bibliography of general works and monographs on the archaeology and the history of the town and diocese of Coutances is given in U.
    0
    0
  • (From Braun, in Bronn`s Klassen u.
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  • - (I) Leuckart, The Parasites of Man (Edinburgh, 1886); (2) Braun, The Animal Parasites of Man (London, 1906); (3) Id., " Cestodes " in Braun's Klassen u.
    0
    0
  • In the same year appeared the first volume of the Hebrelisches u.
    0
    0
  • Other works: De Pentateuchi Samaritani origine, indole, et auctoritate (1815), supplemented in 1822 and 1824 by the treatise De Samaritanorum theologia, and by an edition of Carmina Samaritana; Palaographische Studien fiber phdnizische u.
    0
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  • Seeliger, Die soziale u.
    0
    0
  • cs i.;1(f)has'a ksQ, 1/4,..)d G U 1 f 450 aongitude East 36° of Greenwich Kamar Bay.
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  • This in its simplest form gave rise to the rajaz verses, where each half-line ends in the same rhyme and consists of three feet of the measure - u -.
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  • In the United States the president is empowered to pardon offences against the United States, except in cases of impeachments (U.
    0
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  • The violent tone of some of his printed manifestoes about this time, especially of his Lob des Konigs u.
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  • Of his works the more important are: - Die Composition der Genesis kritisch untersucht (1823), an acute and able attempt to account for the use of the two names of God without recourse to the document-hypothesis; he was not himself, however, permanently convinced by it; De metris carminum Arabicorum (1825); Das Hohelied Salomo's Ubersetzt u.
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  • Sprachlehre fur Anfanger (4th ed., 1874); Uber einige dltere Sanskritmetra (1827); Liber Vakedii de Mesopotamiae expugnatae historia (1827); Commentarius in Apocalypsin Johannis (1828); Abhandlungen zur biblischen u.
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  • erkldrt (1850); Ober das dthiopische Buch Henoch (1854); Die Sendschreiben des Apostels Paulus iibersetzt u.
    0
    0
  • erkldrt (1857); Die Johanneischen Schriften iibersetzt u.
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    0
  • erkldrt (1861-1862); Uber das vierte Esrabuch (1863); Sieben Sendschreiben des neuen Bundes (1870); Das Sendschreiben an die Hebraer u.
    0
    0
  • Jakobos' Rundschreiben (1870); Die Lehre der Bibel von Gott, oder Theologie des alien u.
    0
    0
  • the evidence tabulated in The New Testament in the Apostolic 6 To the details furnished in the present writer's Historical New Testament (2nd ed., 1901, pp. 6 346 35) may be added references to Volter's Paulus u.
    0
    0
  • Weiss in his Der Philipperbrief ausgelegt u.
    0
    0
  • C.) V This letter was originally, like Y, only one of the earlier forms of the letter U.
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  • Gabelentz, " Baskisch u.
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  • Poetae Latini, u.
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    0
  • The skull is preserved in the U.
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  • EDWY (EADWIG), "THE Fair" (c. 94 o -959), king of the English, was the eldest son of King Edmund and lElfgif u, and succeeded his uncle Eadred in 955, when he was little more than fifteen years old.
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  • (Padua, 1739); Buder, Leben u.
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  • (Xioo.) reproductive system; C, Cirrus; H, hooks on the ventral sucker; I, small piece of the intestine to show its connexion with the reproductive organs by the narrow duct that passes from it to the union of the vaginae; M, mouth; 0, ovary; S, oral sucker; SC, sucker; SH, shell-gland; T, Testis; U, uterus; V, vaginal pore; Y, yolkgland.
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  • p, Lips of redia; q, collar; r, processes serving as rudimentary feet; s, embryos; 1, trabecula crossing body-cavity of redia; u, glandular cells; v, birth-opening; w, w', morulae; y, oral sucker; y', ventral sucker; z, pharynx.
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  • Braun, "Trematodes," Klassen u.
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    0
  • - Adam Badeau's Military History of U.
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  • Wilson's Life and Public Services of U.
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  • U S.
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  • (26) Seeliger, "Larven u.
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  • Among swimming birds the most numerous are the gull (kamome), of which many varieties are found; the cormorant (u)which is trained by the Japanese for fishing purposesand multitudinous flocks of wild-geese (gan) and wild-ducks (kanjo), from the beautiful mandarinduck (oshi-dori), emblem of cunjugal fidelity, to teal (koga,no) and widgeon (hidori-ganto) of several species.
    0
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  • U - Conder, in the Proceedings of the Royal institute of British Architects.)
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  • Th third road, the OshOkaidO runs northward from Yedo o h~k 1d~ (now Tokyo) to Aomori on the extreme north of the S U 5 O~ main island, a distance of 445 iii., and several lesser highways give access to other regions.
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  • Trobridge (London, 1907); also Emanuel Swedenborg, the Spiritual Columbus, a Sketch, by U.
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  • This arises from the pronunciation of u as yu, and does not affect the English dialects which have not thus modified the u sound.
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  • The importance of his achievement may be judged from the fact that, while the visible spectrum includes rays having wave-lengths of from about o 4 p to 0.76, u, and no invisible heat-rays were known before 1881 having a wave-length greater than 1.8 µ, he detected rays having a wave-length of 5.3 A.
    0
    0
  • Bury, The Later Roman Empire (London, 1889), u.
    0
    0
  • U.)
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    0
  • u, u, Upper labials.
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  • The corresponding changes in the case of the mixture Tuvw are easily understood - the first halt at U, due to the crystallization of pure B, will probably occur at a different temperature, but the second halt, due to the simultaneous crystallization of A and B, will always occur at the same temperature whatever the composition of the mixture.
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  • On the tribute see also U.
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  • Berliner Akademie (1869) and U.
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    0
  • 2 Consider then an ellipsoidal shell the axes of whose bounding surfaces are (a, b, c) and (a+da), (b+db), (c+dc), where da/a =db b =dc/c =,u.
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  • This mass is equal to 47rabcp,u; therefore Q = A47rabcp s and b =pp, where p is the length of the perpendicular let fall from the centre of the ellipsoid on the tangent plane.
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  • Then if U is the potential outside the surface due to this electric charge inside alone, and V that due to the opposite charge it induces on the inside of the metal surface, we must have U+V =O or U = - V at all points outside the earthed metal surface.
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  • (Leipzig, 1885); Paulig, Friedrich Wilhelm II., sein Privatleben u.
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  • If we consider any short length of the stream bounded by two imaginary cross-sections A and B on either side of the plug, unit mass of the fluid in passing A has work, p'v', done on it by the fluid behind and carries its energy, E'+ U', with it into the space AB, where U' is the kinetic energy of flow.
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  • In passing B it does work, p"v", on the fluid in front, and carries its energy, E"+ U", with it out of the space AB.
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  • If there is no external loss or gain of heat through the walls of the pipe, and if the flow is steady, so that energy is not accumulating in the space AB, we must evidently have the condition E'+U'+p'v' =E'+ U"+p"v" at any two cross-sections of the stream.
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  • It is easy to arrange the experiment so that U is small and nearly constant.
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  • In the limiting case of a long fine tube, the bore of which varies in such a manner that U is constant, the state of the substance along a line of flow may be represented by the line of constant total heat, d(E+pv) = o; but in the case of a porous plug or small throttling aperture, the steps of the process cannot be followed, though the final state is the same.
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  • U, Kinetic energy of flow of fluid.
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    0
  • LITERATu U'E.
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  • u Gramm.
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  • His first work on Philo (Philo u.
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  • westfrankischen Karolinger (Freiburg, 1848); but those on the pseudo-Isidorian Decretals (Untersuchung 'Ober Alter, Ursprung, u.
    0
    0
  • Scholz, Hubert Languet als kursachsischer Berichterstatter u.
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    0
  • Hahn, Bibliothek der Symbole u.
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    0
  • None of them, in point of fact, has held its ground, and even his proposal to denote unknown quantities by the vowels A, E, I, 0, u, Y - the consonants B, c, &c., being reserved for general known quantities - has not been taken up. In this denotation he followed, perhaps, some older contemporaries, as Ramus, who designated the points in geometrical figures by vowels, making use of consonants, R, S, T, &c., only when these were exhausted.
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  • 98) gives the translation, u yas ap7)tos, and considers the name as a compound of Xerxes, showing thereby that he knew nothing of the Persian language; the later Persian form is Ardashir, which occurs in the form Artaxias (Artaxes) as the name of some kings of Armenia.
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  • (X 8.) have existed since the Early U, ulna; R, radius; c, cuneiform; Eocene period.
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  • The os magnum 1, lunar; sc, scaphoid; u, unciform; of the carpus articulates freely m, magnum; td, trapezoid; tm, with the scaphoid.
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  • Sca, through,, u rpov, measure), in geometry, a line passing through the centre of a circle or conic section and terminated by the curve; the "principal diameters of the ellipse and hyperbola coincide with the "axes" and are at right angles; " conjugate diameters " are such that each bisects chords parallel to the other.
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  • Let u represent the volume of air in the cup before the body was inserted, v the volume of the body, a the area of the horizontal FIG.
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    0
  • Then, by Boyle's law (u - v+al) (h - k) = (u - v)h, and therefore v=u - al(h - k)/k.
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    0
  • The volume u may be determined by repeating the experiment when only air is in the cup. In this case v =o, and the equation becomes (u --al l) (h - k') =uh, whence u = al' (h - k l) /k'.
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  • In one, applicable only to liquids which do not mix, the two liquids are poured into the limbs of a U tube.
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    0
  • After Howard, Year Book U.
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    0
  • Esra u.
    0
    0
  • Jahn (Esra u.
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    0
  • The whole micrometer-box is moved by u? ??'
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    0
  • Most of his few remaining letters are printed by Mosheim; his letter from Louvain was despatched in duplicate (to evade capture), but both were seized; one is in the Record Office (U.
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  • 30.3: St' EKEivov av ip0977 Tou Kai H y 'AlroKaXokt y EcepaKOTOS o15 E 'yap 7r0 7roXXOJ xpovov EWpeter/, aXXa U X Esov E7r1 T'91 iii.IETEpas ')'EUEas, ran TW ?Exec T?)s AoµETtavou - Apx7)s.
    0
    0
  • u a i o i a i i of u,  ?
    0
    0
  • U, Ulna.
    0
    0
  • u, Unciform.
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    0
  • S, B `u' E FIG.
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  • London, 1889); of the Vorlesungen U.
    0
    0
  • la u,?
    0
    0
  • (mu 2 +mv 2 +mw 2 + aie12 + x2022 + 2 E(m'u' 2 +%nY 2 +m'w' 2 +0101 2 +0202'+
    0
    0
  • are very large, then, for all states except an infinitesimal fraction of the whole number, the values of u, v, w lie within ranges such that (i) the values of u (and similarly of v, w) are distributed among the s molecules of the first kind according to the law of trial and error; and similarly of course for the molecules of other kinds: (ii) E2mu2 E2mv 2 E2mw2 ?2aie12 s S s s s s - s E s' S' s' - - s' ' See Jeans, Dynamical Theory of Gases (1904), ch.
    0
    0
  • have velocities of which the components lie between u and u+du, v and v+dv, w and w+dw, while the corresponding number of molecules of the second kind is, similarly, s' y (h3m'3/73)e hm'(u2-1-v2+"' 2)dudvdw.
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    0
  • The number of molecules of the first kind of gas, whose components of velocity lie within the ranges between u and u+du, v and v+dv, w and w+dw, will, by formula (5), be v?l (h 3 m 3 /7 3)e hm (u2+v2+w2)dudvdw (9) per unit volume.
    0
    0
  • The cylinder is of volume u dt dS, so that the product of this and expression (9) must give the number of impacts between the area dS and molecules of the kind under consideration within the interval dt.
    0
    0
  • Thus the contribution to the total impulsive pressure exerted on the area dS in time dt from this cause is mu X udtdS X (11 3 m 3 /,r 3)e hm (u2+v2+w2 )dudvdw (I o) The total pressure exerted in bringing the centres of gravity of all the colliding molecules to rest normally to the boundary is obtained by first integrating this expression with respect to u, v, w, the limits being all values for which collisions are possible (namely from - co too for u, and from - oo to + oo for v and w), and then summing for all kinds of molecules in the gas.
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    0
  • The aggregate amount of these pressures is clearly the sum of the momenta, normal to the boundary, of all molecules which have left dS within a time dt, and this will be given by expression (pp), integrated with respect to u from o to and with respect to v and w from - oo to +oo, and then summed for all kinds of molecules in the gas.
    0
    0
  • Clearly the integral is the sum of the values of mu g for all the molecules of the first kind in unit volume, thus p=v mu l +v'm'u 2 +...
    0
    0
  • 17ropa&Es, from air€ip€u', to sow), the islands scattered about the Greek Archipelago, as distinguished from the Cyclades, which are grouped round Delos, and from the islands attached, as it were, to the mainlands of Europe and Asia.
    0
    0
  • Pressel, Leben u.
    0
    0
  • Doflein, Die Protozoen als Parasiten u.
    0
    0
  • Biitschli in Bronn's Klassen u.
    0
    0
  • Warner, Picturesque Berkshire (also Franklin, Hampden, Hampshire, Northampton, 1890-1893); U.
    0
    0
  • See Hermann-Thumser, Griechische Staatsaltertiimer (6th ed., Freiburg, 18 9 2), 3 6 5-37 1, 3 8 7-39 1, 788; U.
    0
    0
  • Miller, Asien u.
    0
    0
  • Alt, Israel u.
    0
    0
  • Stark, Gaza u.
    0
    0
  • §§ 1728; see also the article "Benedictinerorden" in Wetzer u.
    0
    0
  • Survey, U.
    0
    0
  • 451), preserved in Arabic (see Iselin, Texte u.
    0
    0
  • Harnack's edition in Texte u.
    0
    0
  • This discourse, from its explanatory character, and from the easy conversational manner of its delivery, was for a long time called o 1 u Xia rather than Aoyos: it was regarded as part of 1 See Philo, Quod omnis probes liber, sec. 12 (ed.
    0
    0
  • Theologie u.
    0
    0
  • Let u and x be the numerical expressions of the magnitudes of E and F.
    0
    0
  • Then we may, ignoring the units G and H, speak of ON and NP as being equal to x and u respectively.
    0
    0
  • To illustrate the importance of the mensuration of graphs, suppose that we require the average value of u with regard to x.
    0
    0
  • The ordinate of the trapezette will be denoted by u, and the abscissa of this ordinate, i.e.
    0
    0
  • the distance of its foot from a certain fixed point or origin 0 on the base (or the base produced), will be denoted by x, so that u is some function of x.
    0
    0
  • If there are m of these strips, and if the breadth of each is h, so that H =mh, it is convenient to write x in the form xo+Oh, and to denote it by x 0, the corresponding value of u being ue.
    0
    0
  • u m _ i, u m of the strips, or their mid-ordinates 44.
    0
    0
  • In the case of the briquette the position of the foot of the ordinate u is expressed by co-ordinates x, y, referred to a pair of axes parallel to a pair of sides of the base of the briquette.
    0
    0
  • ., or (iii) the " mid-ordinates " u1,2, u, ...
    0
    0
  • u I, ...
    0
    0
  • u_ 2, u_ l, uo, u 1
    0
    0
  • U m, u m+1 14m+2.
    0
    0
  • (b) - 49(a), where 4)(x) is any function of x, by [c P(x)]; the area of the trapezette whose bounding ordinates are uo and u m may then be denoted by [Ax.
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  • u] x - xo or [A..
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  • u 8= u, instead of by fxo udx.
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  • In the same way the volume of a briquette between the planes x = xo, y = yo, x= a, y = b may be denoted by [[Vx,y ]y=yo] u 'x' =xo.
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  • The statement that the ordinate u of a trapezette is a function of the abscissa x, or that u=f(x), must be distinguished from u =f(x) as the equation to the top of the trapezette.
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  • The simplest case is that in which u is constant or is a linear function of x, i.e.
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  • The next case is that in which u is a quadratic function of x, i.e.
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  • If we take these to be uo and u 2, and u 1, so that m = 2, we have area = 6H(uo + 4u1 + u2) = 'h(uo + 4 u 1 + 142).
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  • If instead of uo, u 1, and u 2, we have four ordinates uo, ul, u2, and u 3, so that m = 3, it can be shown that area = 8h(uo + 3/41 + 3u2 - Fu3).
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  • Denoting the areas of the three strips by A, B, and C, and introducing the middle ordinate ug, we can express A + B; B -{- C; A + B -FC; and B in terms of uo, u 1, u 2; u 1, u2, u3; uo, u, u 3; and u 1, ug, u 2 respectively.
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  • Simpson's two formulae also apply if u is of the form px 3 - }- 5x 2 + rx -}- s.
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  • Generally, if the area of a trapezette for which u is an algebraical function of x of degree 2n is given correctly by an expression which is a linear function of values of u representing ordinates placed symmetrically about the mid-ordinate of the trapezette (with or without this mid-ordinate), the same expression will give the area of a trapezette for which u is an algebraical function of x of degree 2n + 1.
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  • When u is of degree 4 or 5 in x, we require at least five ordinates.
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  • If m = 4, and the data are ul, u2, Us, U4, we have area = h (7 u o + 3 2u 2 -112/42 + 3 2u 3 + 7u4).
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  • The breadth of the trapezette being mh, it may be shown that its area is 2 2 N 4 4 iv I mh ug m + 24 m h u gm + 1920 m h u g m + 322560 no/tourgm -}- m3l1gu i„?
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  • 9289-7280 gm where u gm, u g m, u g m, ..
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  • denote the values for x = xim ' of the successive differential coefficients of u with regard to x; the series continuing until the differential coefficients vanish.
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  • (i) If m is even, ug m will be onejof the given ordinates, and we can express h 2 u, m, 4 u" m, ...
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  • Writing m = 2p, and grouping the coefficients of the successive differences, we shall find area = 2ph up+ 2 652up + 3 p4365p2 84up 3p,6 - 21p4 28p2 15120 If u is of degree 2f or 2f + i in x, we require to go up to b 2f u p, so that m must be not less than 2f.
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  • (b2 u g m - s + S2u lm+ g),.
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  • The general formulae of § 54 (p being replaced (i) by 2m) may in the same way be applied to obtain formulae giving the area of the trapezette in terms of the mid-ordinates of the strips, the series being taken up to b 2f ul m or /th 2J ug m at least, where u is of degree 2f or 2f + I in x.
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  • This is a particular case of a general theorem, due to Gauss, that, if u is an algebraical function of x of degree 2p or 2p + I, the area can be expressed in terms of p -}- i ordinates taken in suitable positions.
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  • - Since all points on any ordinate are at an equal distance from the axis of u, it is easily shown that the first moment (with regard to this axis) of a trapezette whose ordinate is u is equal to the area of a trapezette whose ordinate is xu; and this area can be found by the methods of the preceding sections in cases where u is an algebraical function of x.
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  • moments of a trapezette with regard to the axis of u.
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  • If u is an algebraical function of x of degree not exceeding p, and if the area of a trapezette, for which the ordinate v is of degree not exceeding p+q, may be expressed by a formula Aovo-1--yivi+..
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  • To extend these methods to a briquette, where the ordinate u is an algebraical function of x and y, the axes of x and of y being parallel to the sides of the base, we consider that the area of a section at distance x from the plane x = o is expressed in terms of the ordinates in which it intersects the series of planes, parallel to y=o, through the given ordinates of the briquette (§ 44); and that the area of the section is then represented by the ordinate of a trapezette.
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  • Suppose, for instance, that u is of degree not exceeding 3 in x, and of degree not exceeding 3 in y, that it contains terms in x3y3, x 3 y 2, x2y3, &c.; and suppose that the edges parallel to which x and y are measured are of lengths 2h and 3k, the briquette being divided into six elements by the plane x=xo+h and the planes y = yo+k, y = yo+2k, and that the 12 ordinates forming the edges of these six elements are given.
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  • The area of the section by a plane at distance x from the edge 0 is a function of x whose degree is the same as that of u.
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  • The process is simplified by writing down the general formula first and then substituting the values of u.
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  • The formula, in the above case, is 3h{ *k(uo,o + 3 where u 0, 0 denotes the ordinate for which x=xo+Oh, y=yo+c¢k The result is the same as if we multiplied lk(vo 3v1+3v2 + v 3) by lh(uo 4u1 +u2), and then replaced uovo, uov1, ..
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  • The volume of the briquette for which u is a function of x and y is found by the operation of double integration, consisting of two successive operations, one being with regard to x, and the other with regard to y; and these operations may (in the cases with which we are concerned) be performed in either order.
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  • - We have next to consider the extension of the preceding methods to cases in which u is not necessarily an algebraical function of x or of x and y.
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  • In what follows it will be assumed that the conditions of continuity (which imply the continuity not only of u but also of some of its differential coefficients) are satisfied, subject to the small errors in the values of u actually given; the limits of these errors being known.
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  • Thus a quadrature-formula is a formula for expressing [A x .24] or fudx in terms of a series of given values of u, while a cubature-formula is a formula for expressing [[Vx, 0 .
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  • u]] or ffudxdy in terms of the values of u for certain values of x in combination with certain values of y; these values not necessarily lying within the limits of the integrations.
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  • If the data are uo, u 1,.
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  • If the data are u;, U I, ...
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  • u m _ 4, we can form a series of trapezia by drawing the tangents at the extremities of these ordinates; the sum of the areas of these trapezia will be h(u 4 .+u 2 +...
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  • _f)_ 111040+ 4u1 + 2u2 + 4 u 3 + 2464 +.
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  • + 214m-2 + 4 u m-i + um) This is Simpson's rule.
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  • O16u3) If we replace 440136u3 in this expression by g405 6 u 3, the method of § 68 gives A -Q AIL h (uo + 5 u 1 + u2 + 6u3 + u4 + 5 14 6 + us); the expression on the right-hand side being an approximate expression for B, and differing from it only by s1eH5 6 u 3.
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  • + um-2+ um -1 + 2 u m) 3 h (2140+ u2 + u.
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  • + u m-p + zum), which may be denoted by Cp. With this notation, the area as given by Simpson's rule may be written in the form sC l - 3 C2 or CI+ 1 3 `-(C1C2).
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  • The justification of the above methods lies in certain properties of the series of successive differences of u.
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  • The fundamental assumption is that each group of strips of the trapezette may be replaced by a figure for which differences of u, above those of a certain order, vanish (§ 54).
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  • (iv) In order to find what formula may be applied, it is necessary to take the successive differences of u; and it is then just as easy, in most cases, to use a formula which directly involves these differences and therefore shows the degree of accuracy of the approximation.
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  • Hence, if the angle which the tangent at the extremity of the ordinate u 0 makes with the axis of x is denoted by fie, we have area from uo to u1= 2h(uo + ui) - -- i i h 2 (tan y l - tan t u 2 = Wu ' + u2) - 1 Tih 2 (tan 4,2 - tan um-1 t0 26 m, - 2 h(um-1 + um) i h (tan 4, m - and thence, by summation, A =C I - i i h 2 (tan - tan 1,1/o).
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