Ab sentence example

ab
  • In nonflowering plants the works usually followed are for ferns, Hooker and Baker's Synopsis filicum; for mosses, Muller's Synopsis muscorum frondosorum, Jaeger & Sauerbeck's Genera et species muscorum, and Engler & Prantl's Pflanzenfamilien; for algae, de Toni's Sylloge algarum; for hepaticae, Gottsche, Lindenberg and Nees ab Esenbeck's Synopsis hepaticarum, supplemented by Stephani's Species hepaticarum; for fungi, Saccardo's Sylloge fungorum, and for mycetozoa Lister's monograph of the group. For the members of large genera, e.g.
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  • A young senator (ab actis senatus) was chosen to draw up these Acta, which were kept in the imperial archives and public libraries.
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  • To explain this result, chemists suppose that both changes can occur simultaneously, and that equilibrium results when the rate at which AB and CD are transformed into AD and CB is the same as the rate at which the reverse change goes on.
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  • Thus, if the molecule of a substance in solution is represented by AB, Grotthus considered a chain of AB molecules to exist from one electrode to the other.
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  • Let AB be the major axis of the orbit, B the pericentre, F the focus or centre of motion, P the position of the body.
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  • Such an expression as a l b 2 -a 2 b i, which is aa 2 ab 2 aa x 2 2 ax1' is usually written (ab) for brevity; in the same notation the determinant, whose rows are a l, a 2, a3; b2, b 2, b 3; c 1, c 2, c 3 respectively, is written (abc) and so on.
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  • This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets a l, a2; b l, b 2 are alternative, are ultimately put equal to one another and (ab) vanishes.
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  • For the cubic (ab) 2 axbx is a covariant because each symbol a, b occurs three times; we can first of all find its real expression as a simultaneous covariant of two cubics, and then, by supposing the two cubics to merge into identity, find the expression of the quadratic covariant, of the single cubic, commonly known as the Hessian.
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  • From the three equations ax = alxl+ a2x2, b.= blxl+b2x2, cx = clxi+c2x2, we find by eliminating x, and x 2 the relation a x (bc)+b x (ca) +c x (ab) =0.
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  • The identities are, in particular, of service in reducing symbolic products to standard forms. A symbolical expression may be always so transformed that the power of any determinant factor (ab) is even.
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  • To obtain the corresponding theorem concerning the general form of even order we multiply throughout by (ab)2' 2c272 and obtain (ab)2m-1(ac)bxc2:^1=(ab)2mc2 Paying attention merely to the determinant factors there is no form with one factor since (ab) vanishes identically.
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  • For two factors the standard form is (ab) 2; for three factors (ab) 2 (ac); for four factors (ab) 4 and (ab) 2 (cd) 2; for five factors (ab) 4 (ac) and (ab) 2 (ac)(de) 2; for six factors (ab) 6, (ab) 2 (bc) 2 (ca) 2, and (ab) 2 (cd) 2 (ef) 2 .
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  • For the quadratic it is the discriminant (ab) 2 and for ax2 the cubic the quadratic covariant (ab) 2 axbx.
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  • If we have a symbolic product, which contains the symbol a only in determinant factors such as (ab), we may write x 2, -x 1 for a 1, a 2, and thus obtain a product in which (ab) is replaced by b x, (ac) by c x and so on.
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  • First observe that with f x =a: = b z = ���,f1 = a l a z ', f 2 = a 2 az-', f x =f,x i +f 2 x i, we find (ab) - (a f) bx - (b f) ax.
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  • The first transvectant, (f,f') 1 = (ab) a x b x, vanishes identically.
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  • This can be verified by equating to zero the five coefficients of the Hessian (ab) 2 axb2.
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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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  • In the a x = aixi+a2x2, observe that a a = a2, ab = aibi +a2b2.
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  • The general form of covariant is therefore (ab) h i (ac) h2 c) (b h3 a i bb2c'e3...abia?2b?3...
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  • X (xa) ki (xb) k2 (xc) k3...axibx2cx3...xx = (AB) hi (AC) h2 (BC) h3...A11 4 13 A1,14131 A B I ?C"' B C "' X (XA) ki (XB) k2 (XC) k3...AXB122cCk...X If this be of order e and appertain to an nie L Eke-/1+2m =e, h i+h2+���+221+ji+j2+���+kl+li =n, hi+h3+..�+222+ji+j3+���+k2+12 = n, h2+h3+���+223+j2+%3+�.�+k3+13 =n; viz., the symbols a, b, c,...
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  • To assist us in handling the symbolic products we have not only the identity (ab) cx + (bc) a x + (ca) bx =0, but also (ab) x x+ (b x) a + (ax) b x = 0, (ab)a+(bc)a s +(ca)a b = 0, and many others which may be derived from these in the manner which will be familiar to students of the works of Aronhold, Clebsch and Gordan.
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  • This is the fundamental system; we may, if we choose, replace (ab) 2 by ab =a, 2, +2a1+a2 since the identity a a b b - a, = (ab) 2 shows the syzygetic relation (74+a 2) 2 - (ao +2a +�2) = 2(aoa2 - ai).
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  • Let the distance of each pole of the rod AB from the centre of the magnetometer needle = d.
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  • Denoting the distance AM by d 1, BM by d2, and AB by 1, we have for the force at M due to the magnetism of the rod H P =d 12 - horizontal component (dla - d 2 3).
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  • The city then placed itself under the protection of the emperor .Charles V., created a magistracy of "ten conservators of the liberties of the state" (December 1524), united the different monti in one named the "monte of the reigning nobles," and, rejoicing to be rid of the last of the Petrucci, dated their public books, ab instaurata libertate year I., II., and so on.
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  • Thus a(b+c) and (b+c)a give the same result, though it may be written in various ways, such as abdac, ca+ab, &c. In the same way the associative law is that A(BC) and (AB)C give the same formal result.
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  • Let a+b denote the region made up of a and b together (the common part, if any, being reckoned only once), and let a X b or ab mean the region common to a and b.
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  • It is remarkable that Mobius employs the symbols AB, ABC, Abcd In Their Ordinary Geometrical Sense As Lengths, Areas And Volumes, Except That He Distinguishes Their Sign; Thus Ab = Ba, Abc= Acb, And So On.
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  • The symbol AB, from this point of view, is called the inner product of A and B, as distinguished from the outer product IAB.
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  • In the course of reducing such expressions as (AB)C, (AB){C(DE)} and the like, where a chain of multiplications has to be performed in a certain order, the multiplications may be all progressive, or all regressive, or partly, one, partly the other.
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  • In ordinary algebra we have the disjunctive law that if ab = o, then either a = o or b= o.
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  • By the principle of energy the illumination over the entire focal plane must be equal to that over the diffracting area; and thus, in accordance with the suppositions by which (3) was obtained, its value when integrated from E= co to = -1-x, and from n = - oo to n = -1-oo should be equal to ab.
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  • Let AoBo be a plane wave-surface of the light before it falls upon the prisms, AB the corresponding wave-surface for a particular part of the spectrum after the light has passed the prisms, or after it has passed the eye-piece of the observing telescope.
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  • The path of a ray from the wave-surface AoBo to A or B is determined by the con dition that the optical distance, µ ds, is a minimum; and, as AB is by supposition a wave-surface, this optical distance is the same for both points.
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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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  • If now we suppose the aperture AB to be covered by a great number of opaque strips or bars of width d, separated by transparent intervals of width a, the condition of things in the directions just spoken of is not materially changed.
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  • In another direction, making a small angle with the last, such that the projection of AB upon it amounts to a few wavelengths, it is easy to see that the mode of interference is the same as if there were no ruling.
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  • For example, when the direction is such that the projection of AB upon it amounts to one wave-length, the elementary components neutralize one another, because their phases are distributed symmetrically, though discontinuously, round the entire period.
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  • Let P, Q denote the normal thrust across the sides bc, ca, and R the normal thrust across the base ab.
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  • This is proved by taking any two points A and B at the same level, and considering the equilibrium of a thin prism of liquid AB, bounded by planes at A and B perpendicular to AB.
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  • As gravity and the fluid pressure on the sides of the prism act at right angles to AB, the equilibrium requires the equality of thrust on the ends A and B; and as the areas are equal, the pressure must be equal at A and B; and so the pressure is the same at all points in the same horizontal plane.
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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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  • Sargon, who meanwhile had crushed the confederacy of the northern nations, had taken (717 B.C.) the Hittite stronghold of Carchemish and had annexed the future kingdom of Ecbatana, was now accepted as king by the Babylonian priests and his claim to be the successor of Sargon of Akkad acknowledged up to the time of his murder in 705 B.C. His son Sennacherib, who succeeded Serena- hi m on the 12th of Ab, did not possess the military or cherlb. ?
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  • Another castle, built in the same century, on the east bank, was held direct by the lords of Glamorgan, as the westernmost outpost of their lordship. It was frequently attacked by the Welsh, notably in 1231 when it was taken, and the town demolished by Llewelyn ab Iorwerth.
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  • Radii are drawn from the centre of the quadrant to the points of division of the arc, and these radii are intersected by the lines drawn parallel to BC and through the corresponding points on the radius AB.
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  • The castle was begun, in or about 1109, by Cadwgan ab Bleddyn ab Cynfyn (Cynvyn), and finished by Gwenwynwyn; in 1196 it was besieged, undermined and taken by Hubert, archbishop of Canterbury.
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  • The founda 1 It must be remembered that the Pisans and Florentines dated the beginning of the year ab incarnatione, i.e.
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  • A Catholic commentator of the 16th century, Hieronymus ab Oleastro, seems to have been the first to connect the name " Jehova " with howah interpreting it contritio, sive pernicies (destruction of the Egyptians and Canaanites); Daumer, adopting the same etymology, took it in a more general sense: Yahweh, as well as Shaddai, meant " Destroyer," and fitly expressed the nature of the terrible god whom he identified with Moloch.
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  • Laennec, to whom we are indebted for the practice of auscultation, freely admits that the idea was suggested to him by study of Hippocrates, who, treating of the presence of morbid fluids in the thorax, gives very particular directions, by 1 " Hippocrates Cous, primus quidem ex omnibus memoria dignus, ab studio sapientiae disciplinam hanc separavit, vir et arte et facundia insignis " (Celsus, De medicina).
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  • One of the best methods for doing this is to charge the Ab l condenser by the known voltage of a battery, and then d e t erdischarge it through a galvanometer and repeat this minations.
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  • Then the potential at any point P in this ideal plane PO is equal to q/AP-q/BP=0, whilst the resultant force at P due to the two point charges is 2gAO/AP 3, and is parallel to AB or normal to PO.
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  • A cycle such as ABCD enclosed by parts of two isothermals, BC, AD, and two adiabatics, AB, CD, is the simplest form of cycle for theoretical purposes, since all the heat absorbed, H', is taken in during the process represented by one isothermal at the temperature o', and all the heat rejected, H", is given out during the process represented by the other at the temperature 0".
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  • It is not necessary in this example that AB, CD should be adiabatics, because the change of volume BC is finite.
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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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  • All-important, too, is the order of ceremonial and the formula of prayer: a mistake or omission or an unpropitious interruption may vitiate the whole ritual, and though such misfortunes may occasionally be expiated by the additional offering of a piaculum, in more serious cases the whole ceremony must be recommenced ab initio.
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  • Meanwhile Aquitaine was gradually lost; the defeat of Pembroke off La Rochelle deprived England of the command of the sea, and Sir Owen ap Thomas, a grand-nephew of Llewelyn ab Gruffyd, planned, with French help, an abortive invasion of Wales.
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  • Then MA'B'N is a right trapezium, whose area is equal to that of Cabd; and it is related to the latter in such a way that, if any two lines parallel to AC and BD meet AB, CD, MN, A'B', in E, G, P, E', and F, H, Q, F', respectively, the area of the piece PE'F'Q of the right trapezium 'B.
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  • Take any point P in the latter, and form triangles by joining P to each of the sides AB, BC, ...
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  • Then it may be shown that DE is parallel to AB, and that the area of the figure between chord DE and arc DE is half the sum of the areas DHA and EJB.
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  • For a tetrahedron, two of whose opposite edges are AB and CD, we require the area of the section by a plane parallel to AB and CD.
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  • Let the distance between the parallel planes through AB and CD be h, and let a plane at distance x from the plane through AB cut the edges AC, up -f- .
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  • As AB moves from CD to EF it pushes forward the layer of air in contact with it.
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  • As AB returns from EF towards CD the layer of air next to it follows it as if it D E F were pulled back by AB.
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  • On the whole the air S within ABCD neither gains nor g D loses momentum, so that on the whole it receives as much through AB as it gives up to CD.
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  • If P is the undisturbed pressure and P+w the pressure at AB, the momentum entering through AB per second isJ01(P+w-+pu2)dt.
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  • But the values of 2 which occur successively during the second at AB exist simultaneously at the beginning of the second over the distance U behind AB.
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  • Or if the conditions along this distance U could be maintained constant, and we could travel back along it uniformly in one second, we should meet all the conditions actually arriving at AB and at the same intervals.
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  • If the train of waves is reflected, the value of p at AB will be the sum of the values for the two trains, and will, on the average, be doubled.
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  • Then move AB from right to left with this velocity, and the disturbance remains fixed in space.
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  • Since the conditions in the region PQ remain always the same, the momentum perpendicular to AB entering the region at Q is equal to the momentum perpendicular to AB leaving the region at P. But, since the motion at Q is along AB, there is no momentum there perpendicular to AB.
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  • The material between A and B, though continually changing, is always in the same condition, and therefore the momentum within AB is constant.
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  • Set off these distances along AB and let Y1, y2,.
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  • Thus we shall call the first inclined line on the left hand the line AG, the line representing the first force on the top left-hand joint AB, the first horizontal member at the top left hand the line BH, &c; similarly each point requires at least three letters to denote it; the top first left-hand joint may be called Abhg, being the point where these four spaces meet.
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  • The line XZ consists of a series of lengths, as XA, AB ...
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  • All real connexion or relation, therefore, and with it all possibility of an objective system, disappears; it is, in fact, excluded by Hume ab initio, for " the mind never perceives any real connexion among distinct existences."
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  • There is no evidence of any settlement on the site of the present town prior to about 1092, when Bernard Newmarch, after defeating Bleddin ab Maenarch, built here a castle which he made his residence and the chief stronghold of his new lordship. For this purpose he utilized what remained of the materials of the Roman fort, 3 m.
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  • The second class, referred to in inscriptions by the name of praefecti ab decurionibus creati lege Petronia, seem to have been appointed by the local senate in case of a complete absence of higher magistrates, such as would have led in Rome to the appointment of an interrex.
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  • The line AB is termed the " radical axis."
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  • To prove this let AB, AB' be the tangents from any point on the line AX.
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  • Since PD = PD', it follows that AB =AB'.
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  • Take AB equal to one-fourth of the given line; on AB describe a square ABCD; join AC; in AC produced find, by a known process, a point C 1 such that, when C 1 B 1 is drawn perpendicular to AB produced and C 1 D 1 perpendicular to BC produced, the rectangle BC,.
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  • The diameter sought is the straight line from A to the limiting position of the series of B's, say the straight line AB co.
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  • Denoting the number of units in AB by 4c, we can express BB 1j B 1 B 2,.
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  • The town derives its name from the river Avon (corrupted from Avan), which also gave its name to a medieval lordship. On the Norman conquest at Glamorgan, Caradoc, the eldest son of the defeated prince, Lestyn ab Gwrgan, continued to hold this lordship, and for the defence of the passage of the river built here a castle whose foundations are still traceable in a field near the churchyard.
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  • He also brought out editions of the Byzantine historians, John Cinnamus and John Zonaras, as Joannis Cinnami historiarum de rebus gestis a Joanne et Manuele Comnenis (Paris, 1670) and Joannis Zonarae Annales ab exordio mundi ad mortem Alexii Comneni (Paris, 1686).
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  • The diagonal AD of the parallelogram, of which AB and AC are adjacent sides, will represent, both in direction and magnitude, the motion of the rain as apparent to the observer.
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  • At A we 66 have the freezing point of pure water, which is lowered by the gradual addition of 46 ferric chloride in the manner shown by the curve AB.
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  • To explain the universe Spinoza proceeds to argue that God, though undetermined ab extra, is capable of infinite self-determination.
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  • If, ignoring temporarily and for simplicity the fact that part of the carbon may exist in the state of graphite, we consider the behaviour of iron in cooling from the molten state, AB and BC give the temperature at which, for any given percentage of carbon, solidification begins, and Aa, aB, and Bc that at which it ends.
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  • As freezing progresses, at each successive temperature reached the frozen austenite has the carbon-content of the point on Aa which that temperature abscissa cuts, and the still molten part or " mother-metal " has the carbon-content horizontally opposite this on the line AB.
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  • In other words, the composition of the frozen part and that of the mother-metal respectively are p and q at the beginning of the freezing, and r and t' at the end; and during freezing they slide along Aa and AB from p to r and from q to t'.
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  • Thus, we find Sumerian ab, " dwelling," " sea "; ab, " road," and -ab, a grammatical suffix, which words, with many others of a similar character, were perhaps originally uttered with different voice-tones.
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  • In the logic of Mill, e.g., we find much of a special character that has no counterpart in Hume, much that is introduced ab extra, from general considerations of scientific procedure, but, so far as the groundwork is concerned, the System of Logic is a mere reproduction of Hume's doctrine of knowledge.
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  • Working on these lines, and attaching special importance to common descent, naturalists applied the term with more and more precision, until Linnaeus, in his Philosophia botanica, gave the aphorism, "species tot sunt diversae, quot diversae formae ab initio sunt creatae" - "just so many species are to be reckoned as there were forms created at the beginning."
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  • We thus obtain the differential equation gk(d 2 0/dx 2) =cgdo/dt+hpo, which is satisfied by terms of the type =c" sin where a 2 -b 2 = hp/qk, and ab = urnc/k.
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  • He was also the author of volumes entitled De satisfactione Christi disputationes (Geneva, 1666) and De necessaria secessione nostra ab ecclesia romana (Geneva, 1687).
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  • Bernard himself initiated this policy by building a castle at Talgarth on the Upper Wye, but in 1091 he moved southwards, defeated the regulus of Brycheiniog, Bleddyn ab Maenarch, and his brother-in-law Rhys ap Tewdwr, the prince of south-west Wales, and with materials obtained from the Roman fort of Caer Bannau, built a castle at Brecon, which he made his caput baroniae.
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  • Then if a pencil be placed along B C so as to keep the string taut, and the limb AB be slid along the directrix, the A pencil will trace out the parabola.
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  • Some words he quoted from St Augustine influenced Newman profoundly: " Quapropter securus judicat orbis terrarum bonos non esse qui se dividunt ab orbe terrarum."
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  • His orations and letters were published in 1492; but his title to any measure of fame he possesses rests upon his history of Venice, De origine urbis Venetiarum rebusque ab ipsa gestis historia (1492), which was translated into Italian by Domenichi in 1545, and which at the time of its appearance was undoubtedly the best work upon the subject of which it treated.
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  • The axis of rotation AB bears a rigidly attached rod DBC inclined to it at an angle equal to the sun's polar distance.
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  • A large fire-grate (ab), having a cave (c) to facilitate stoking and stepped back at (d), is bounded on one side by a fire-bridge (e); on the other side of this, separated by an air-channel (g), there is first the proper fluxing bed (h), and behind this the "back-bed "(i) for pre-heating the charge.
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  • Thus the figure represents a section the (ideally simplified) uni verse cut perpendicular to C P' D the planes AB and CD between which the stars are contained, 1 This number is the 3/2th power of the ratio of the brightness of stars differing by a unit magnitude.
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  • Imagine this stratum to be uniformly filled with stars (of course in the actual universe instead of sharply defined boundaries AB and CD, we shall have a gradual thinning out of the stars) it follows that in the two directions SP and SP' the fewest stars will be seen; these then are the directions of the galactic poles.
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  • An army sent to the rescue by Ibn Zobair under the command of his brother Mus`ab was beaten in Palestine by `Amr Ashdaq.
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  • Mokhtar was now at the zenith of power, but Ibn Zobair, determined to get rid at all costs of so dangerous an enemy, named his brother Mus`ab governor of Basra and ordered him to march against Kufa.
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  • Mohallab then marched with Mus`ab against Kufa.
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  • Ibn Zobair, however, was occupied at Mecca with the rebuilding of the Ka`ba, and Mus`ab was harassed not only by the Kharijites, but also by a noble freebooter, Obaidallah b.
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  • When, in the year (69 A.H.) 689 Abdalmalik had at last encamped at Botnan Habib in the vicinity of Kinnesrin (Qinnasrin),1 with the purpose of marching against Mus`ab, his cousin `Amr Ashdaq, to whom by the treaty of Jabia, before the battle of Merj Rahit, the succession to Merwan had been promised, took advantage of his absence to lay claim to the supreme power, and to have himself proclaimed caliph by his partisans.
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  • Mus`ab was encamped at Bajomaira in the neighbourhood of Takrit.
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  • Meanwhile, Mus`ab had to curb a violent revolt in Basra, brought about by agents of Abdalmalik, and called after a place in the city the revolt of the Jofrites.
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  • Mus`ab's best troops were fighting under Mohallab against the Kharijites; many Basrians were secretly favourable to the Omayyads, nor were the Kufian soldiers to be trusted.
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  • Abdalmalik, therefore, wrote secretly to the chiefs of Mus`ab's army, and persuaded them to desert to him, with the exception of Ibrahim b.
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  • His death, in the beginning of the battle, decided the fate of Mus`ab, who was slain sword in hand by a Shiite of Kuf a.
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  • As soon as the expedition to Irak against Mus`ab had terminated, the holy war against the Greeks was renewed.
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  • That the most important section, the list of forms of combination, was never achieved - this too was after the Baconian example while the mode of symbolization was crude with a= ab and the like - matters little.
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  • Tradition ascribes the death of Aaron to the first day of Ab.
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  • It is therefore adequately represented, for mathematical purposes, by a straight line AB drawn in the direction in question, of length proportional (on any convenient scale) to the magnitude of the force.
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  • Since the four co-ordinates (Cartesian or other) of these two points are connected by the relation which expresses the invariability of the length AB, it is plain that virtually three inde pendent elements are re quired and suffice to specify the position of the lamina.
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  • For suppose that in consequence of the displacement a point of the lamina is brought from A to B, whilst the point of the lamina which was originally at B is brought to C. Since AB, BC, are two different positions of the same line in the B C lamina they are equal, and it is evident that the rotation could have been effected by a rotation about J,
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  • As a special case A the three points A, B, C may be in a straight line; J is then at infinity and the displacement is equivalent to FIG 10 a pure translation, since every point of the lamina is now displaced parallel to AB through a space equal to AB.
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  • Hence the locus of J relative to AB, and the locus relative to CD are equal ellipses of which A, B and C, D are respectively the foci.
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  • These formulae give a means of constructing the resultant by means of any transversal AB cutting the lines of action.
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  • When P, Q have opposite signs the point C divides AB externally on the side of the greater force.
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  • Again, suppose we have a bar AB resting with its ends on twc smooth inclined planes which face each other.
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  • If F, Q, R, be any three forces acting along BC, CA, AB, respectively, the line of action of the resultant is determined by the consideration that the sum of the moments about any point on it must vanish.
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  • Thus in the figure the sum of the moments about 0 is P. OAP. OB or P. AB, which is independent of the position of o 0.
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  • The force corresponding to AB may be replaced by the two components marked x, y; and a force corresponding to BA may be represented by the two components marked x, y.
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  • Thus ii AB, BC, CD represent the given loads, in the force-diagram, we construct the sides corresponding to OA, OB, OC, OD in the funicular; we then draw the closing line of the funicular polygon, and a parallel OE to it in the force diagram.
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  • A graphical method can also be applied to find the moment of a force, or of a system of forces, about any assigned point P. Let F be a force represented by AB in the force-diagram.
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  • To find the pressure exerted by a bar AB on the pin A we compound with the force in AB given by the diagram a force equal to P. Conversely, to find the pressure of the pin A on the bar AB we must compound with the force given by the diagram a force equal and opposite to P. This question arises in practice in the theory of three-jointed structures; for the purpose in hand such a structure is sufficiently represented by two bars AB, BC. The right-hand figure represents a portion of the force-diagram; in particular ZX represents the pressure of AB on B
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  • The spherical A isosceles triangles AJB, BJC are con gruent, and we see that AB can be brought into the position BC by a rotation about the axis OJ through an FIG.
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  • This is the axis of the required screw; the amount of the translation is measured by the projection of AB or BC or CD on the axis; and the angle of rotation is given, by the inclination of the aforesaid bisectors.
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  • An infinitely small rotation about any axis is conveniently represented geometrically by a length AB measures along the axis and proportional to the angle of rotation, with the convention that the direction from A to B shall be related to the rotation as is the direction of translation to that of rotation in a righthanded screw.
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  • If AB, AC represent infinitesimal rotations about intersecting axes, the consequent displacement of any point 0 in the plane BAC will be at right angles to this plane, and will be represented by twice the sum of the areas OAB, OAC, taken with proper signs.
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  • Let AB be the shortest distance between the lines of action, and let AA, BB
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  • This product is the same whatever point on the line of action be taken, since the lengthwise components of the dispJacements of any two points A, B on a line AB are equal, to the first order of small quantities.
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  • It is to be noticed that All need not be in the same plane with AB, AC.
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  • Imagine the bar CF to be removed, and consider a deformation in which AB is fixed.
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  • The simplest case is that of a frame of three bars, when the three joints A, B, C fall into a straght line; a small displacement of the joint B at right angles to AC would involve changes in the lengths of AB, BC which are only of the second order of small quantities.
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  • If E be the point in which the line of the string meets AB, we have pjCP, pz=EP. Many contrivances for actually drawing the resulting curves have been devised.
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  • The line T on the surface bbb has for the instant no velocity it a direction perpendicular to AB; becau2e for the instant it touches, without sliding, the line T on the fixed surface aaa.
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  • The line T on the surface bbb has also for the instant no velocity in the plane AB; for it has just ceased to move towards the fixed surface aaa, and is just about to begin to move away from that surface.
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  • Set off ab = ac = 1/2p. Draw radii bd, Ce; draw fb, cg, making angles of e 753/4 with those radii.
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  • The links AB and CD are equal in length and are centred respectively at A and C. The ends D and B are joined by a link DB.
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  • But tile point B must also move at right angles to AB in the case under consideration.
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  • Hence draw a line through O in the velocity diagram at right angles to AB to cut cbi in b.
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  • The year of Ab Saids death is most likely that of the first great didactic mathnawi, the Rshan.
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  • There exists a mid14th-century Historia do Santo Graal, and an unprinted Josep' ab Aramadia, while, though the MS. is lost, we have abundant evidence of the existence of a primitive Portuguese prose redaction of Amadis de Gaula anterior to the present Spanish text.
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  • If, in the rectangle Aacc, Aa= a, and AC = b, its area is S = ab, and its energy Tab.
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  • Hence if F is the force by which the slip BB is pulled towards AA, F =d b T ab =Ta,..
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  • For if in the triangle abc the side ab is taken so as to represent on a given scale the tension of the surface of contact of the fluids a and b, and if the other sides be and ca are taken so as to represent on the same scale the tensions of the surfaces between b and c and between c and a respectively, then the condition of equilibrium at 0 for the corresponding tensions R, P and Q is that the angle ROP shall be the supplement of abc, POQ of bca, and, therefore, QOR of cab.
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  • If four fluids, a, b, c, d, meet in a point 0, and if a tetrahedron AB CD is formed so that its edge AB represents the tension of the surface of contact of the liquids a and b, BC that of b and c, and so on; then if we place this tetrahedron so that the face ABC is normal to the tangent at 0 to the line of concourse of the fluids abc, and turn it so that the edge AB is normal to the tangent plane at 0 to the surface of contact of the fluids a and b, then the other three faces of the tetrahedron will be normal to the tangents at 0 to the other three lines of concourse of the liquids, an the other five edges of the tetrahedron will be normal to the tangent planes at 0 to the other five surfaces of contact.
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  • But once the idea of supernatural grace going along with office as such (of which we have already a trace in the Ignatian bishop, though without the notion of actual apostolic succession) arose in connexion with successio ab apostolis, the full development of the doctrine was but a matter of time.4 Literature.
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  • Thus Herbert sought to do for the religion of nature what his friend Grotius was doing for natural law, - making a new application of the standard of Vincent of Lerins, Quod semper, quod ubique, quod ab omnibus.
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  • Its proper title was Ab urbe condita libri (also called historiae and annales).
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  • Over a mile south of Mold, on the right of the road to Nerquis, is the "Tower" (15th century, but perhaps restored in the 18th), where, in 1465 or 1 475, the royal chieftain, Rheinallt ab Gruffyd ad Bleddyn, hanged Robert Byrne, mayor of Chester, and subsequently burned alive some 200 Chester folk who tried to 'arrest him.
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  • It is generated as follows: Let ABC be a circle having AB for a diameter.
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  • Draw any line DE perpendicular to AB and meeting the circle in E, and take a point P on DE such that the line DP =arc BE; then the locus of P is the companion to the cycloid.
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  • Not, indeed, a preliminary criticism of our faculties or conceptions such as Kant himself proposed to institute, in order to determine the limits of their application; such a criticism ab extra of the nature of our experience is essentially a thing impossible.
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  • Among his other works are his Annotationes in Biblia (1607), of which an English translation (Pious and Learned Annotations upon the Holy Bible) was published in London in 1648, and various polemical treatises, such as De fictitio Pontificiorum Purgatorio (1619); De justa secessione Reformatorum ab Ecclesia Romana (1628); De Antichristo, &c. He also published French translations of Sarpi's History of the Council of Trent, and of Edwin Sandys's Account of the State of Religion in the West.
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  • There is considerable distortion of the clay, resulting from combined shearing and tensile stress, above each of the steps of rock, and reaching its maximum at and above the highest rise ab, where it has proved sufficient to produce a dangerous line of weakness ac, the tension at a either causing actual rupture, or such increased porosity as to permit of percolation capable of keeping open the wound.
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  • It is clear that the greatest crushing pressure must occur, either, with the reservoir empty, near the lower part of the water face ab.
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  • We have hitherto considered only the horizontal overturning pressure of the water; but if from originally defective construction, or from the absence of vertical pressure due to weight of masonry towards the water edge of any horizontal bed, as at ab in fig.
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  • Line ab is in such a position that the stresses.
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  • The vertical distance above the line ab of any point in the dotted line dc is proportional to the vertical component of the compressive stress on the line ale assumed to vary uniformly from face to face, and similarly the vertical distance of any point in the3-dot-and-dash line ae above the line ab is proportional to the vertical component of the stress determined experimentally.
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  • In order to increase the sensitiveness of a balance, the line AB joining the points of suspension and the centre of gravity of the balance must be brought nearer to each other.
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  • Io AB is the steelyard.
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  • Deinde ab amicis curam ejus susceptam, domoque clauso remedia volenti nolenti adhibita, quibus jam sanitatem recuperavit ut jam rursus librum suum Principiorum Philosophiae Mathematicorum intelligere incipiat."
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  • It is constructed by the following method: Let AQB be a semicircle of diameter AB, produce MQ the ordinate of Q to P so that MQ: MP :: AM: AB.
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  • The cartesian equation, if A be taken as origin and AB (= 2a) for the axis of x, is xy 2 =4a2(2a - x).
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  • If a line S2 cut an arc aa at b, so that the two segments ab, ba lie on opposite sides of the line, then projecting the figure so that the line Sl goes off to infinity, the tangent at b is projected into the asymptote, and the arc ab is projected into a hyperbolic leg touching the asymptote at one extremity; the arc ba will at the same time be projected into a hyperbolic leg touching the same asymptote at the other extremity (and on the opposite side), but so that the two hyperbolic legs may or may not belong to one and the same branch.
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  • Next, if the line S2 touch at b the arc aa so that the two portions ab, ba lie on the same side of the line Sl, then projecting the figure as before, the tangent at b, that is, the line S2 itself, is projected to infinity; the arc ab is projected into a parabolic leg, and at the same time the arc ba is projected into a parabolic leg, having at infinity the same direction as the other leg, but so that the two legs may or may not belong to the same branch.
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  • To this new period belong Einleitung zur historic der vornehmsten Reiche and Staaten, also the Commentarium de rebus suecicis, libri XXVI., ab expeditione Gustavi Adolphi regis in Germaniam ad abdicationem usque Christinae and De rebus a Carolo Gustavo gestis.
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  • Bellarmine (De jejunio) distinguishes jejunium spirituale (abstinentia a vitiis), jejunium morale (parsimonia et temperantia cibi et potus), jejunium naturale (abstinentia ab omni prorsus cibo et potu, quacunque ratione sumpto), and jejunium ecclesiasticum.
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  • Thus, for three columns, it appears by either rule that 123, 231, 312 are positive; 213, 321, 132 are negative; and the developed expression of the foregoing determinant of the third order is =ab'c" - ab "c'+a'b "c - a'bc" - a"bc' - a"b'c. 3.
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  • But his enemies increased in power, and about 1194 he was driven from Wales by the partisans of his half-brother Llewelyn ab Iorwerth.
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  • Take a rod LMN bent at right angles at M, such that MN= AB; let the leg LM always pass through a fixed point 0 on AB produced such that OA = CA, where C is the middle point of AB, and cause N to travel along the line perpendicular to AB at C; then the midpoint of MN traces the cissoid.
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  • The cartesian equation, when A is the origin and AB = 2a, is y 2 (2a - x) =x 8; the polar equation is r= 2a sin 0 tan 0.
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  • The form is " A B C D, &c., are P; AB CD are all M; thus all M are P."
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  • But that the origins of such mythical beings were, ab initio, either solar or vegetable, or, for that matter, animal, it would often be hard to prove.
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  • He also wrote or edited various Chinese works on geography, the celestial and terrestrial spheres, geometry and arithmetic. And the detailed history of the mission was drawn out by him, which after his death was brought home by P. Nicolas Trigault, and published at Augsburg, and later in a complete form at Lyons under the name De Expeditione Christiana apud Sinas Suscepta, ab Soc. Jesu, Ex P. Mat.
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  • In analytical geometry, r the equation axe+2hxy+bye+2gx+2fy+ c = o represents an ellipse when ab > h 2; if the centre of the curve be the origin, the equation is a 1 x 2 +2h 1 xy+b i y 2 =C 1, and if in addition a pair of conjugate diameters are the axes, the equation is further simplified to Ax e +By 2 = C. The simplest form is x 2 /a 2 +y 2 /b 2 = 1, in which the centre is the origin and the major and minor axes the axes of co-ordinates.
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  • Kal`ab: i-nau and Rindan are among the more northerly inhabited villages.
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  • A lowresistance galvanometer is connected by a very fine wire (2 to 3 mils) to the centre C of the experimental wire AB, and also to the middle point D of the parallel wire so as to form a Wheatstone bridge.
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  • The balance is adjusted by shunting either AD or BD with a box, S, containing 20 to Ioo ohms. All the wires in the quadrilateral must be of the same metal as AB, to avoid accidental thermoelectric effects which would obscure the result.
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  • If the current flows from A to B there will be heat absorbed in AC and evolved in CB by the Thomson effect, if the specific heat of electricity in AB is positive as in copper.
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  • This is shown by the curve AB, and is the useless work represented by the expression (q 2 -q 1)/r.
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  • The GAMESS-UK Benchmark is designed to represent the typical range of calculations commonly performed by the ab initio quantum chemist.
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  • Most ab initio groups quickly become very close-knit since students see each other almost every day.
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  • Unsure about my new contraceptive coil Will ab toner affect the coil?
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  • They describe an English speaking ab initio student of German who exhibited an unusually strong tendency to use word coinage and lexical innovation.
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  • Doing abdominal crunches on the AB Revolutionizer is a perfect abdominal workout.
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  • The calories expended during ab exercises are very low.
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  • Passed by voice not obligated to must be willing ab to require.
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  • Still problems with pt retrieval Some success with constituent retrievals using AB microwindows (not all MWs processed ).
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  • Patients were further subdivided into those with or without elevated quantitative serum Chlamydia trachomatis IgG antibody (Ab) titres.
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  • Instant abs 6 second workout and lateral thigh stepper exercises, fitness & ab lounge 2.. .
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  • Quintus Curtius was largely used for the Alexandreis (c. 1180) of Gaultier de Lille or de Chatillon (Galtherus ab Insulis or de Castellione).
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  • Calling the foci S, S', the real vertices A, A', the extremities of the conjugate axis B, B' and the centre C, the positions of B, B' are given by AB = AB' = CS.
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  • Analytically the hyperbola is given by ax2+2hxy+by2+2gx+ 2fy+c=o wherein ab>h 2 .
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  • Ita Cornua Ammonis, quae ex nautilorum numero habeantur, passim et forma et magnitudine (nam et pedali diametro aliquando reperiuntur) ab omnibus illis naturis discrepare dicunt, quas praebet mare.
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  • Censures were as follows: (i) Suspension from attending divine offices or ab ingressu ecclesiae, more appropriate for a layman.
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  • It was published (1589) by Giacomo Castelvetri, who had married his widow, with the title Explicatio gravissimae quaestionis utrum excommunicatio, quatenus religionem intelligentes et amplexantes, a sacramentorum usu, pro pier admissum facinus arcet, mandato nitatur divino, an excogitata sit ab hominibus.
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  • This belief is, of course, not specifically Christian; it has been held at all times and everywhere by men of the most various races and creeds; and, if there be any validity in the contention that that is true which has been held semper, ubique, et ab omnibus, no fact is better established.
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  • First observe that with f x =a: = b z = ���,f1 = a l a z ', f 2 = a 2 az-', f x =f,x i +f 2 x i, we find (ab) - (a f) bx - (b f) ax.
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  • X (xa) ki (xb) k2 (xc) k3...axibx2cx3...xx = (AB) hi (AC) h2 (BC) h3...A11 4 13 A1,14131 A B I ?C"' B C "' X (XA) ki (XB) k2 (XC) k3...AXB122cCk...X If this be of order e and appertain to an nie L Eke-/1+2m =e, h i+h2+���+221+ji+j2+���+kl+li =n, hi+h3+..�+222+ji+j3+���+k2+12 = n, h2+h3+���+223+j2+%3+�.�+k3+13 =n; viz., the symbols a, b, c,...
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  • For the quadratic aoxi +2a i x i x 2 +a 2 x, we have (i.) ax = 7/1x1+2aixix2-I-7/24, (ii.) xx=xi+xzi (ab) 2 =2(aoa2 - ai), a a = a o+712, _ (v.) (xa)ax= i'?- (a2 - ao)xix2 - aix2.
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  • This is the fundamental system; we may, if we choose, replace (ab) 2 by ab =a, 2, +2a1+a2 since the identity a a b b - a, = (ab) 2 shows the syzygetic relation (74+a 2) 2 - (ao +2a +�2) = 2(aoa2 - ai).
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  • The commutative law in arithmetic, for instance, states that adb and b+a, or ab and ba, are equal.
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  • It had probably not been completed, though it was already defensible, when Prince Llewelyn ab Griffith, incensed by its construction and claiming its site as his own, laid siege to it in 1271 and refused to retire except on conditions.
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  • He advanced to the arm-chair prepared for him in the centre of the writing-table, laid his hat on the left-hand corner; his snuff-box was deposited on the same side beside the quire of paper placed in readiness for his use, and dipping the pen twice into the ink-bottle, then bringing it to within an inch of his nose to make sure it was properly filled, he broke silence: ` We have said that the chord AB,' &c. For three-quarters of an hour he continued his demonstration, making short notes as he went on, to guide the listener in repeating the problem alone; then, taking up another cahier which lay beside him, he went over the written repetition of the former lesson.
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  • If we bisect AB and CD in P and Q respectively, and describe the arc PQ of a circle with centre 0, the length of this arc is 2(b+a)0; and b - a =AB.
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  • Then (see Parabola) Tc = Cv, Av = Vb, and K ab is parallel to AB, so that aC = Cb.
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  • As AB again moves from CD towards EF another compression or push is sent out, as it returns from EF towards CD another extension or pull, and so on.
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  • Thus waves are propagated along OX, each wave consisting of one push and one pull, one wave emanating from each complete vibration to and fro of the source AB.
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  • The marriage was declared invalid ab initio either on the ground of Anne's precontract with Lord Percy or more probably on the ground of the affinity established between Henry and Anne by Henry's previous relations with Mary Boleyn.
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  • Furthermore it is seen that AB is perpendicular to the line joining the centres, and divides it in the ratio of the squares of the radii.
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