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numerical

numerical

numerical Sentence Examples

  • The numerical values are, it is to be noted, exceedingly small.

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  • His treatise on numerical divisions, weights and measures (Distributio) is extant, with the exception of the concluding portion.

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  • In 1905 the value of the wool export regained the £20,000,000 level, and with the rapid recovery of the numerical II.

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  • In 1905 the value of the wool export regained the £20,000,000 level, and with the rapid recovery of the numerical II.

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  • The nature of this problem is illustrated by the numerical example in § g.

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  • He is notorious for his wilful exaggeration, both in narrative and numerical statements.

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  • When a character is said to be transmitted, or to be correlated with another character, the biometricist declares the statement valueless without numerical estimations of the inheritance or correlation.

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  • Owing to the existence of a strong Mussulman minority among its inhabitants, the warlike character of the natives, and the mountainous configuration of the country, which enabled a portion of the Christian population to maintain itself in a state of partial independence, the island has constantly been the scene of prolonged and sanguinary struggles in which the numerical superiority of the Christians was counterbalanced by the aid rendered to the Moslems by the Ottoman troops.

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  • Its symbolic expression, to a numerical factor pres, is (Hbc) (Hbd) (Hcd) (bcd), and it is clearly of degree 6.

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  • Its symbolic expression, to a numerical factor pres, is (Hbc) (Hbd) (Hcd) (bcd), and it is clearly of degree 6.

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  • The small magnet may be a sphere rigidly magnetized in the direction of Ho; if this is replaced by an isotropic sphere inductively magnetized by the field, then, for a displacement so small that the magnetization of the sphere may be regarded as unchanged, we shall have dW = - vIdHo = v I+-, whence W = - 2 I + H2 ° (37) The mechanical force acting on the sphere in the direction of displacement x is 1 Hopkinson specified the retentiveness by the numerical value of the " residual induction " (=47rI).

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  • Of these thirteen sections, the first contains a simple description of the more prominent phenomena, without mathematical symbols or numerical data.

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  • The ultimate value of numerical inquiries must depend on the equivalence of the units on which they are based.

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  • (5) Every letter of a word is reduced to its numerical value, and the word is explained by another of the same quantity.

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  • Remark.-The invariant C is a numerical multiple of the resultant of the covariants i and j, and if C = o, p is the common factor of i and j.

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  • - the same influence, for instance, as the Italians, who had an imposing numerical force.

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  • - the same influence, for instance, as the Italians, who had an imposing numerical force.

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  • Inside a magnetized body, B is the force that would be exerted on a unit pole if placed in a narrow crevasse cut in the body, the walls of the crevasse being perpendicular to the direction of the magnetization (Maxwell, § § 399, 604); and its numerical value, being partly due to the free magnetism on the walls, is generally very different from that of H.

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  • He attended lectures on the numerical solution of equations and on definite integrals by M.

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  • a V(I - a) This constant k gives a numerical value for the chemical affinity, and the equation should represent the effect of dilution on the molecular conductivity of binary electrolytes.

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  • Manchester in 1906, when the Labour party were returned for the first time in numerical force - over 50 in all.

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  • Largesse was especially given on the field of victory, and was, moreover, liberally distributed to stifle sedition and mutiny among the troops, the numerical strength of which was continually increased as the empire enlarged its borders.

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  • The old B retained the numerical value of the Greek i as 2, and no numerical value was given to the new symbol.

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  • In Greek, where I is the twentieth letter of the alphabet, or, if the merely numerical and p are excluded, the eighteenth, another form 1 or S according to the direction of the writing is also widespread.

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  • When, therefore, the latter, on the 22nd, marched southward to reopen his communications by the defeat of the enemy's army, always the surest means of solving this difficulty, he actually reached the neighbourhood of Eckmuhl with a sufficient numerical superiority had he only been prompt enough to seize his opportunity.

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  • Other " Galois " groups were defined whose substitution coefficients have fixed numerical values, and are particularly associated with the theory of equations.

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  • d x the first evectant; and thence 4cxdi the second evectant; in fact the two evectants are to numerical factors pres, the cubic covariant Q, and the square of the original cubic.

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  • There are also cases in which graphics and mensuration are used jointly; a variable numerical quantity is represented by a graph, and the principles of mensuration are then applied to determine related numerical quantities.

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  • The French alphabet, written out with the same numerical values as the Hebrew, in which the first nine letters denote units and the others tens, will have the following significance:

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  • Where the induction is high the lines will be crowded together; where it is weak they will be widely separated, the number per square centimetre crossing a normal surface at any point being always equal to the numerical value of B.

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  • The magnetic susceptibility expresses the numerical relation of the magnetization to the magnetizing force.

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  • The numerical factor 6 is called the coefficient of a 5 bc 2 (� 20); and, generally, the coefficient of any factor or of the product of any factors is the product of the remaining factors.

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  • The discriminant of f is equal to the discriminant of 0, and is therefore (0, 0') 2 = R; if it vanishes both f and 0 have two roots equal, 0 is a rational factor of f and Q is a perfect cube; the cube root being equal, to a numerical factor pres, to the square root of A.

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  • This is rendered possible by the fact that we can use a single letter to represent a single number or numerical quantity, however many digits are contained in the number.

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  • The discriminant of f is equal to the discriminant of 0, and is therefore (0, 0') 2 = R; if it vanishes both f and 0 have two roots equal, 0 is a rational factor of f and Q is a perfect cube; the cube root being equal, to a numerical factor pres, to the square root of A.

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  • In this mode of treating the question the order of the terms is numerical, and though the amount of labour is such as might well have deterred a younger man, yet the details were easy, and a great part of it might be entrusted to a mere computer.

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  • It is (f = (ab) 2 a n-2 r7 2 =Hx - =H; unsymbolically bolically it is a numerical multiple of the determinant a2 f a2f (32 f) 2� It is also the first transvectant of the differxi ax axa x 2 ential coefficients of the form with regard to the variables, viz.

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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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  • In another respect also he aided in establishing an exact science of medicine by the introduction of the numerical or statistical method.

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  • His numerical table, therefore, differs from all others made before or since, prophetically assigning the place of honour to the one ancient scale now recognized as the foundation of the modern tonal system.

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  • Hippocrates, influenced as is thought by the Pythagorean doctrines of number, taught that they were to be expected on days fixed by certain numerical rules, in some cases on odd, in others on even numbers - the celebrated doctrine of "critical days."

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  • Equations (33) and (34) show that when, as is generally the case with ferromagnetic substances, the value of is considerable, the resultant magnetic force is only a small fraction of the external force, while the numerical value of the induction is approximately three times that of the external force, and nearly independent of the permeability.

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  • Curves of magnetization (which express the relation of I to H) have a close resemblance to those of induction; and, indeed, since B = H+47r1, and 47rI (except in extreme fields) greatly exceeds H in numerical value, we may generally, without serious error, put I = B /47r, and transform curves of induction into curves of magnetization by merely altering the scale to which the ordinates are referred.

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  • It seems that confusion and trouble will be best avoided by abstaining from the introduction of the non-evident somites, the ocular and the praegenital, into the numerical nomenclature of the component somites of the three great body regions.

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  • The chamber has the right of initiative in the organization of the annual budget laws and those relative to the numerical strength of the army and navy.

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  • And, their own predominance being assured by their numerical strength and influence, they accorded equal shares of power to the other monti.

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  • The commission also called attention to the numerical insufficiency of magistrates and native commissioners in certain parts of Natal.

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  • According to this reading, William sought to rectify his position by asserting, not the numerical identity of the universal in each individual, but rather its sameness in the sense of indistinguishable similarity.

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  • In the case of division we get two kinds of arithmetical formula, which, however, may be regarded as requiring a single kind of numerical process in order to determine the final result.

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  • We must count with something; and the successive somethings obtained by the addition of successive units are in fact numerical quantities, not numbers.

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  • While mensuration is concerned with the representation of geometrical magnitudes by numbers, graphics is concerned with the representation of numerical quantities by geometrical figures, and particularly by lengths.

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  • Graphic representation thus rests on the principle that equal numerical quantities may be represented by equal lengths, and that a quantity mA may be represented by a length mL, where A and L are the respective units; and the science of graphics rests on the converse property that the quantity represented by pL is pA, i.e.

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  • It must, of course, be remembered (� 23) that this is a statement of arithmetical equality; we call the statement an " identity," but we do not mean that the expressions are the same, but that, whatever the numerical values of a, b and c may be, the expressions give the same numerical result.

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  • The overwhelming numerical superiority of the Sla y s, and the very great differences in ethnical type, belief and mythology between the IndoEuropean and the Ural-Altaic races, may have contributed to the same end.

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  • The principal orders, arranged according to their numerical importance, are as follows: - Leguminosae,Rubiaceae, Orchidaceae, Compositae, Gramineae, Euphorbiaceae, Acanthaceae, Cyperaceae and Labiatae.

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  • The Church of England claims as adherents 39% of the population, and the Roman Catholic Church 22%; next in numerical strength are the Wesleyans and other Methodists, numbering 12% i the various branches of the Presbyterians 11%, Congregationalists 2%, and Baptists 2%.

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  • We disregard numerical coefficients, so that by the H.C.F.

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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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  • It is clear that, since the numerical coefficients of A and of a are each 1, the coefficients in the expansions arise from the grouping and addition of like terms (� 37 (ii.)).

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  • The numerical value is then represented by (P (; thus " (x (< i " means that xis between -1 and +1.1 50.

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  • and of the numerical value of i+b i x+b 2 x 2 +...

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  • (vii.) The comparison of the numerical value of I-Fn(1)x +n(242+...+n(r)xr, when n is fractional, with that of (i+x)n, involves advanced methods (� 64).

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  • In order that numerical results obtained by summing the first few terms of a series may be of any value, it is usually necessary that the series should converge to a limit; but there are exceptions to this rule.

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  • Then a+a = as = a; hence numerical coefficients and indices are not required.

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  • A point A in space may be associated with a (real, positive, or negative) numerical quantity a, called its weight, and denoted by the symbol aA.

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  • In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion.

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  • a n are scalars, and in particular applications may be restricted to real or complex numerical values.

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  • The Greeks had succeeded in solving an isolated example; the Arabs accomplished the general solution of numerical equations.

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  • The component vibrations at P due to the successive zones are thus nearly equal in amplitude and opposite in phase (the phase of each corresponding to that of the infinitesimal circle midway between the boundaries), and the series which we have to sum is one in which the terms are alternately opposite in sign and, while at first nearly constant in numerical magnitude, gradually diminish to zero.

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  • If, on the other hand, the point be well immersed in the geometrical shadow, the earlier zones are altogether missing, and, instead of a series of terms beginning with finite numerical magnitude and gradually diminishing to zero, we have now to deal with one of which the terms diminish to zero at both ends.

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  • The denominator sin a is the quantity well known (after Abbe) as the " numerical aperture."

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  • But no vertebrate advance in force took place until comparatively late in the afternoon, and by evening the attacking side, although enjoying a great numerical superiority, had only reached the foot of the hills that lay to the E.

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  • The great numerical superiority which they had at first possessed was gone by the 9th, and their task had come to be the ejection of an almost equal enemy from a naturally formidable position.

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  • The Turks, therefore, now possessed a huge numerical preponderance in the theatre of war.

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  • The absolute unit of force is employed here, and not the gravitation unit of hydrostatics; in a numerical application it is assumed that C.G.S.

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  • vertical, where Wh tan 0 = N = (c 2 - cl) c2 g2 tan 0, (6) (7) in which we have put k' 2 = ek 2, where E is a numerical factor depending on the shape.

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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) .

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  • In other species there is a definite arrangement of the leaves, some with and others without tendrils opposite to them, the numerical order remaining constant or nearly so.

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  • When it was over the Liberal party was just short of the numerical strength which was requisite to defeat the combination of Tories and Parnellites.

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  • " Now for the interpolation of the rest, I considered that the denominators I, 3, 5, &c., were in arithmetical progression; and that therefore only the numerical coefficients of the numerators were to be investigated.

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  • Since the establishment of the privileged province they have lost the Ottoman support which used to compensate for their numerical inferiority as compared with the Christians; and they are fast losing also their old habits and distinctiveness.

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  • The foreign-born are more numerously represented among the criminal, defective and dependent classes than their numerical strength would justify.

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  • The number is precisely that of the total numerical value of the consonants of the name "Eliezer" (Gen.

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  • He devised an approximate numerical solution of equations of the second and third degrees, wherein Leonardo of Pisa must have preceded him, but by a method every vestige of which is completely lost.

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  • In the numerical expression of absolute densities it is necessary to specify the units of mass and volume employed; while in the case of relative densities, it is only necessary to specify the standard substance, since the result is a mere number.

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  • The latter's enormous numerical superiority was neutralized by Sulla's judicious choice of ground and the steadiness of his legionaries; the Asiatics after the failure of their attack were worn down and almost annihilated.

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  • The attempt to apply numerical methods to the comparison of geometrical quantities led to the doctrine of incommensurables, and to that of the infinite divisibility of space.

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  • From this point of view the atomic doctrine might be regarded as a relic of the old numerical way of conceiving magnitude, and the opposite doctrine of the infinite divisibility of matter might appear for a time the most scientific. The atomists, on the other hand, asserted very strongly the distinction between matter and space.

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  • As the same writer puts it: " There is such a thing as numerical difference, different from conceptual difference," 2 or expressing the same thing in other words " there are relations not grounded in the nature of the related terms."

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  • Idealism has, of course, no quarrel with numerical difference.

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  • All difference has its numerical aspect: two different things are always two both in knowledge and in reality.

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  • He might thus compensate for his numerical inferiority by superior mobility and superior leadership.

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  • Here they would be in contact, and ready to act united against Napoleon with a numerical superiority of two to one.

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  • He had to delay the French advance for 24 hours and give time for Blucher's concentration, at the same time retaining his own freedom of manoeuvre, and this in spite of the great length of the summer day, the short distance that he lay 'in front of Fleurus, the tremendous numerical superiority of the French and Napoleon's personal presence at their head.

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  • mensura, a measure), the science of measurement; or, in a more limited sense, the science of numerical representation of geometrical magnitudes.

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  • The measurement is numerical, i.e.

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  • It might, therefore, be described as that branch of mathematics which deals with formulae for calculating the numerical measurements of curved lengths, areas and volumes, in terms of numerical data which determine these measurements.

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  • Mensuration, then, is mainly concerned with quadratureformulae and cubature formulae, and, to a not very clearly defined extent, with the methods of obtaining such formulae; a quadrature-formula being a formula for calculating the numerical representation of an area, and a cubature-formula being a formula for calculating the numerical representation of a volume, in terms, in each case, of the numerical representations of particular data which determine the area or the volume.

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  • On the other hand, it is worth noticing that the words " quadrature " and " cubature " are originally due to geometrical rather than numerical considerations; the former implying the construction of a square whose area shall be equal to that of a given surface, and the latter the construction of a cube whose volume shall be equal to that of a given solid.

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  • The failure seems (§ 2) to be due to difficulty in realizing the numerical expression of an area or a solid in terms of a specified unit, while the same difficulty does not arise in the case of linear measure or liquid measure, where the number of units can be ascertained by direct counting.

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  • The main object to be aimed at, therefore, in the study of elementary mensuration, is that the student should realize the possibility of the numerical expression of areas and volumes.

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  • (iii) Association of solid measure with liquid measure, presenting numerical measurement in a different aspect, should be retained by testing volumes as found from linear dimensions with the volumes of the same bodies as found by the use of measures of capacity.

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  • The numerical result obtained by applying a formula to particular data will generally not be exact.

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  • Let u and x be the numerical expressions of the magnitudes of E and F.

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  • The general principle is that the numerical data from which a particular result is to be deduced are in general not exact, but are given only to a certain degree of accuracy.

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  • In the case of mathematical functions certain conditions of continuity are satisfied, and the extent to which the value given by any particular formula differs from the true value may be estimated within certain limits; the main inaccuracy, in favourable cases, being due to the fact that the numerical data are not absolutely exact.

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  • On the other hand, the numerical increase had not kept pace with the increase of population.

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  • When war broke out the Ottoman forces in Europe numbered less than 250,000 men, dispersed over Macedonia and Thrace; they were thus at great numerical disadvantage.

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  • His strong opposition to "dissenting churches" was nowhere so clearly shown as in a pamphlet published in 1816 to dissuade all Episcopalians from joining the American Bible Society, which he thought the Protestant Episcopal Church had not the numerical or the financial strength to control.

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  • The second volume (1817) relates to the Eulerian integrals, and to various integrals and series, developments, mechanical problems, &c., connected with the integral calculus; this volume contains also a numerical table of the values of the gamma function.

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  • Under these circumstances, and also because of their numerical weakness and the rigour of the weather, the Germans advanced but slowly.

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  • The equations (66) - (71) are Siacci's, slightly modified by General Mayevski; and now in the numerical applications to high angle fire we can still employ the ballistic table for direct fire.

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  • It has been said that the blockade of the Confederate coast became in the end practically impenetrable, and that every attempt of the Confederate naval forces to break out was checked at once by crushing numerical preponderance.

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  • System; Unites et Etalons, Guillaume (Paris, 1890); Lupton's Numerical Tables, 1892; Metric Equivalent Cards, 1901; Dictionary of Metric Measures, L.

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  • For engineering and manufacturing purposes the more important linear gauges are, however, now used, adjusted to some fundamental unit of measure as the inch; although in certain trades, as for wires and flat metals, gauges continue to be used of arbitrary scales and of merely numerical sizes, having no reference to a legal unit of measure; and such are rarely accurate.

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  • The numerical value of this modulus is o 4342944 81 9 03251 82765 11289..

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  • The quantity denoted by e is the series, I I I+ I + i.2 +1.2.3+1.2.3.4 + the numerical value of which is, 2.71828 18284 59 0 45 2 353 6 02874.

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  • One of the chief deductions is that there are special dangers in numerical diminution of herds, which may arise from a chief or original cause and be followed by a conspiracy of other causes which are cumulative in effect.

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  • In Great Britain, whither they began to straggle over during the revolutionary troubles at the close of the 18th century, and where, practically unaffected by the clause directed against them in the Emancipation Act of 1829, their chief settlement has been at Stonyhurst in Lancashire, an estate conferred on them by Thomas Weld in 1795, they have been unmolested; but there has been little affinity to the order in the British temperament, and the English province has consequently never risen to numerical or intellectual importance in the Society.

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  • The arrival of a Spanish squadron of twelve ships of the line in June gave a great numerical superiority to the allies, and Rodney retired to Gros Islet Bay in Santa Lucia.

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  • He became secretary of the American Oriental Society and editor of its Journal, to which he contributed many valuable papers, especially on numerical and temporal categories in early Sanskrit literature.

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  • The representatives of the extreme monotheistic view, which while regarding Christ as Redeemer, clung tenaciously to the numerical unity of the Deity, were called Monarchians, a term brought into general use by Tertuliian.

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  • Various chemists had traced numerical sequences among the atomic weights of some of the elements and noted connexions between them and the properties of the different substances; but it was left to him to give a full expression to the generalization, and to treat it not merely as a system of classifying the elements according to certan observed facts, but as a "law of nature" which could be relied upon to predict new facts and to disclose errors in what were supposed to be old facts.

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  • In character the Indians are, as a rule, peaceable, though conscious of their numerical superiority and at times driven to join in the revolutions which so often disturb the course of local politics; they are often intensely religious, but with a few exceptions are thriftless, indolent and inveterate gamblers.

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  • Thus the numerical superiority of the Baptists and Methodists in the two Southern divisions is complementary to that of the Roman Catholics in the other three divisions of the country.

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  • This correlative numerical problem and the two purely geometrical problems are inseparably connected historically.

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  • In his earlier years he devoted himself to chemistry, both theoretical and applied, publishing papers on the preparation of gold and platinum, numerical relations between the atomic weights of analogous elements, the formation of aventurine glass, the manufacture of illuminating gas from wood, the preservation of oil-paintings, &c. The reaction known by his name for the detection of bile acids was published in 1844.

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  • " In the paper quoted the numerical values of p are given for different widths of slit, and a table shows to what extent the loss of purity due to a widening of the slit is accompanied by a gain in luminosity.

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  • The efforts which were consequently made in the early days of spectroscopy to discover some numerical relationship between the different wave lengths of the lines belonging to the same spectrum rather disregard the fact that even in acoustics the relationship of integer numbers holds only in special and very simple cases.

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  • These forms all agree in making the frequency negative when s falls below a certain value sp. Rydberg's second law states that if the main branch series is taken, the numerical value of np_I corresponding to sp_ I is equal.

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  • The widening of lines does not lend itself easily to accurate measurements; more precise numerical data are obtainable by the study of the displacements consequent on increased density which were discovered and studied by W.

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  • Down to 1848, and even still later, " Democracy " was used to cover the whole mass of the people, pre-eminently represented by the broad strata of the bourgeoisie; in 1900 the Democratic party itself meant by this term the rule of the labouring class organized as a nation, which, by its numerical superiority, thrust aside all other classes, including the bourgeoisie, -and excluded them from participation in its rule.

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  • Henry Cavendish had before 1773 discovered that glass, wax, rosin and shellac have higher specific inductive capacities than air, and had actually determined the numerical ratios of these capacities, but this was unknown both to Faraday and to all other electricians of his time, since Cavendish's Electrical Researches remained unpublished till 1879.

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  • It is found that this gives a consistent result; that is to say, if by an experiment with two particles A and B we get the ratio of their masses, and by an experiment with B and a third particle C we get the ratio of the masses of B and C, and thus the ratio of the masses of A and C, we should get the same ratio by a direct experiment with A and C. For the numerical measure of mass that of some standard body is chosen as a unit, and the masses of other bodies are obtained by comparison with this.

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  • Before the Seven Years' War the Quakers dominated the government, but from that time until the failure of the Whisky Insurrection (17g4) the more belligerent Scotch-Irish (mostly Presbyterians) were usually in the ascendancy, the reasons being the growing numerical strength of the Scotch-Irish and the increasing dissatisfaction with Quaker neglect of means of defending the province.

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  • The physical investigation of osmotic pressure, and its correlation by Van't Hoff with the pressure of a gas, brought forward a new aspect of the phenomena, and suggested an identity of physical modus operandi as well as of numerical value.

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  • often a numerical asymmetry on the two sides.

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  • If we were to say that on his view the essential step must be the establishment of identities or equivalences, we should probably be doing justice to his doctrine of numerical reasoning, but should have some difficulty in showing the application of the method to geometrical reasoning.

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  • Ang3trom's value for iron, when corrected for obvious numerical errors, and for the probable variation of c, becomes Iron, k =0.164 (1-0.0013 0), but this is very doubtful as c was not measured.

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  • Since the plebeian element in the state had an immense numerical preponderance over the patrician these disabilities were not widely spread, and seem generally to have been cheerfully borne as the price of belonging to the families still recognized as the oldest and noblest in Rome.

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  • At Oxford the marks are not numerical, but the papers are judged as of this or that supposed " class," and various degrees of merit are indicated by the symbols a, (3, -y, S, to which the signs + or - may be prefixed, according as they are above or below a certain standard within each class.

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  • At Cambridge, numerical marks are used.

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  • The advantage of numerical marks is that they are more easily manipulated than symbols; the disadvantage, that they produce the false impression that merit can be estimated with mathematical accuracy.

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  • The element of chance, which prevails in the region on either side of the border between pass and failure, obviously prevails equally on either side of the border between " classes," where candidates are classified; it has been suggested by Dr Schuster that numerical order should accompany classification so as to avoid the creation of an artificial gap between the last candidate in one class and the highest in the next.

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  • Latham (The Action of Examinations, 1877, p. 490) describes other numerical adjustments used to meet this difficulty, especially that used in English civil service examinations.

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  • For while Hegel, depending on a numerical proportion suggested by Plato, hinted in a single sentence that it might be a mistake to look for a planet between Mars and Jupiter, Giuseppe Piazzi (q.v.) had already discovered the first of the asteroids (Ceres) on the ist of January 1801.

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  • The numerical preponderance of the works of Philodemus led some people to believe that this had been the library of that philosopher.

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  • The numerical value of the mechanical equivalent necessarily depends on the particular units of heat and work employed in the comparison.

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  • This appears to be the explanation of Stumpe's and Porter's results; they both divided their proper motions into groups according to their numerical amount, and found that the declination of the solar apex progressively increased as the size of the motions used diminished.

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  • The ideal is progressively to determine a universe of discourse till true infimae species are reached, when no further distinction in the determinate many is possible, though there is still the numerical difference of the indefinite plurality of particulars.

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  • Owing to the variation in the value of the dielectric constant of glass with the temperature and with the frequency of the applied electromotive force, and also owing to electric glow discharge from the edges of the tin foil coatings, the capacity of an ordinary Leyden jar is not an absolutely fixed quantity, but its numerical value varies somewhat with the method by which it is measured, and with the other circumstances above mentioned.

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  • But the numerical factor appears to be yz'+zy', while it is the quantity yz' - zy' which really vanishes.

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  • Hamilton seems never to have been quite satisfied with the apparent heterogeneity of a quaternion, depending as it does on a numerical and a directed part.

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  • The results of these two kinds of multiplication correspond respectively to the numerical and the directed parts of Hamilton's quaternion product.

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  • And we now see that the square of V is the negative of Laplace's operator; while V itself, when applied to any numerical quantity conceived as having a definite value at each point of space, gives the direction and the rate of most rapid change of that quantity.

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  • It is evident that this is a very delicate method of determining the wetness z, but, since with dry saturated steam at low pressures this formula always gives negative values of the wetness, it is clear that Regnault's numerical coefficients must be wrong.

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  • Rep. 1897) to steam near Ioo° C. He found the specific heat smaller than 0.475, but no numerical results were given.

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  • Admitting the value S =0.497 for the specific heat at 108° C., it is clear that the form of Regnault's equation (io) must be wrong, although the numerical value of the coefficient 0.305 may approximately represent the average rate of variation over the range (loo° to 190° C.) of the experiments on which it chiefly depends.

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  • By assuming suitable forms of the characteristic equation to represent the variations of the specific volume within certain limits of pressure and temperature, we may therefore with propriety deduce equations to represent the saturation-pressure, which will certainly be thermodynamically consistent, and will probably give correct numerical results within the assigned limits.

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  • Taking for ice and water the following numerical data, L = 674.7, 6 74.7, L 1 =595.2, L r = 79.5, R = o 11 03 cal./deg., po = 4.61 mm., s-S = 519 cal./deg., and assuming the specific heat of ice to be equal to that of steam at constant pressure (which is sufficiently approximate, since the term involving the difference of the specific heats is very small), we obtain the following numerical formulae, by substitution in (23), Ice..

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  • They give the following numerical values The error of the formula for water is less than t mm.

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  • The symbols in the Greek alphabet from Y to S2, or in the numerical alphabet to 7), are not found in the Phoenician alphabet.

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  • Sliding Contact (direct): Teeth of Wheels, their Number anc PitchThe ordinary method of connecting a pair of wheels, or wheel and a rack, and the only method which ensures the exaci maintenance of a given numerical velocity ratio, is by means of i series of alternate ridges and hollows parallel or nearly parallel t(the successive lines of contact of the ideal smooth wheels whon velocity ratio would be the same with that of the toothed wheels The ridges are called teeth; the hollows, spaces.

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  • As the mortality amongst boys, especially during the first year, is considerably above that of the other sex, numerical equilibrium between the two is established in early youth, and in most cases girls outnumber boys, except for a few years between twelve and sixteen.

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  • Thus, though the generally higher proportion of females in the community may seldom be enough to depress more than slightly the death-rate as a whole, it has a substantial effect upon it at the ages where women are in more marked numerical predominance, as in later life, and in places where the number of domestic servants is unusually great.

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  • under the same influence, those passing from their prime into the second period of danger acquire a numerical preponderance which throws its weight upon the general death-rate and tends to raise it.

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  • A study of his works reveals an unusual combination of skill and originality in the mathematical treatment of many of the most difficult problems of astronomy, an unfailing patience and sagacity in dealing with immense masses of numerical results, and a talent for observation of the highest order.

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  • It represents the Koppa of the earliest Greek alphabets surviving in that form of the Ionic alphabet, which ultimately superseded all others, merely as the numerical symbol for 90.

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  • In the Phoenician alphabet a sibilant Zade (Tzaddi) stands between q and p. Hence Q is the nineteenth letter in the Phoenician alphabet, the eighteenth in the Greek numerical alphabet, which alone contains it, the sixteenth (owing to the omission of 8 and E) in the Latin, and (from the addition of J) the seventeenth in the English alphabet.

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  • Maxwell made a comparison between the optical refractive index and the dielectric constant of paraffin wax, and the approximation between the numerical values of the square of the first and that of the last was sufficient to show that there was a basis for further work.

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  • The accumulating determinations of the numerical value of the electromagnetic velocity (v) from the earliest made by Lord Kelvin (Sir W.

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  • Lastly, statistical research has shown that the children of the married British clergy have been distinguished far beyond their mere numerical proportion.8/n==Authorities== - Henry Charles Lea, History of Sacerdotal Celibacy (3rd ed., 1907, 2 vols), is by far the fullest and best work on this subject, though a good deal of important matter omitted by Dr Lea may be found in Die Einfiihrung der erzwungenen Ehelosigkeit by the brothers Johann Anton and Augustin Theiner, which was put on the Roman Index, though Augustin afterwards became archivist at the Vatican (Altenburg, 1828, 2 vols.).

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  • The remarkable numerical progress of Baptists in South Carolina from 1787 to 1812 (from 1620 members to 11,325) was due to the "Separate" Baptist movement under Stearns and Marshall far more than to the activity of the churches of the Charleston Association.

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  • By 1812 there were in the United States 173,972 Baptist church members, the denominational numerical strength having considerably more than doubled since the beginning of the 19th century.

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  • xxvi.) "the numerical orders of the image," are consequently only odd powers; the condition for the formation of an image of the mth order is that in the series for and the coefficients of the powers of the 3rd, 5th.

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  • Lagrange used simple continued fractions to approximate to the solutions of numerical equations; thus, if an equation has a root between two integers a and a+1, put x=a+I/y and form the equation in y; if the equation in y has a root between b and b+i, put y = b + I /z, and so on.

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  • He used them to represent square roots, but only for particular numerical examples, and appears to have had no theory on the subject.

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  • So condensed and coarsened is the indwelling air-current of inorganic bodies that no trace of elasticity or life remains; it cannot even afford them the power of motion; all it can do is to hold them together Su va us), and, in technical language, Pneuma is present in stone or metal as a retaining principle (g t;ts=hold), explaining the attributes of continuity and numerical identity (rvvE X i) «al 1)vw J .zEva) which even these natural substances possess.

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  • de Chimie et de Physique, 1866-1868) applied the principles of thermodynamics to capillary phenomena, and the experiments of his son Paul were exceedingly ingenious and well devised, tracing the influence of surfacetension in a great number of very different circumstances, and deducing from independent methods the numerical value of the surface-tension.

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  • Hence the numerical value of the intensity of the surface-tension is equal to the numerical value of the surface-energy per unit of surface.

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  • In all these cases the internal pressure exceeds the external by 2T/a where a is the semi-transverse axis of the conic. The resultant of the internal pressure and the surface-tension is equivalent to a tension along the axis, and the numerical value of this tension is equal to the force due to the action of this pressure on a circle whose diameter is equal to the conjugate axis of the ellipse.

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  • Army was still cramped and out of touch with its neighbours went far to neutralize the numerical disadvantage.

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  • we have memoirs relating to the proof of the theorem that every numerical equation has a real or imaginary root, the memoir on the Hypergeometric Series, that on Interpolation, and the memoir Determinatio attractionis - in which a planetary mass is considered as distributed over its orbit according to the time in which each portion of the orbit is described, and the question (having an implied reference to the theory of secular perturbations) is to find the attraction of such a ring.

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  • Summary of Numerical Data.

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  • Of his extensive investigations into the solution (especially by numerical approximation) of certain classes of differential equations which constantly occur in the treatment of physical questions, only a few items have been published, at intervals, in the Philosophical Magazine.

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  • representation for each political party in proportion to its numerical strength, by providing for first and second choice in voting - the system of preferential voting adopted in Idaho in 1909; and the "recall," by which the voters may remove from office after six months' service by a special election any local official.4 Judiciary.

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  • This, however, leads to a troublesome numerical solution.

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  • a pcO / rtKi, sc. TExvn, the art of counting, from apiO,u6, number), the art of dealing with numerical quantities in their numerical relations.

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  • is only an arithmetical statement in so far as each of the three expressions denotes a numerical quantity (§ II).

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  • The division into abstract and concrete, for instance, is logical, if the former is taken as relating to number and the latter to numerical quantity (§ I I).

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  • (i) If we consider that we are dealing with numerical quantities, we must recognize the fact that, while addition and subtraction might in the first instance be limited to such quantities, multiplication and division necessarily introduce the idea of pure number.

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  • Numerical Quantities

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  • Notation of Numerical Quantities 2.6 (i) Vigesimal Scale 2.7 (ii) Roman System 2.8 21.

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  • If the term number is confined to number in the abstract, then number in the concrete may be described as numerical quantity.

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  • The £I is termed the unit, A numerical quantity, therefore, represents a certain unit, taken a certain number of times.

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  • Thus arithmetical processes deal with numerical quantities by dealing with numbers, provided the unit is the same throughout.

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  • This question of direction is of importance in reference to the development of useful number-forms (§ 23); and the existence of the two methods mentioned under (iii) and (iv) above produces confusion in comparing numerical tabulation with graphical representation.

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  • Over a large part of the civilized world the introduction of the metric system (§ 118) has caused the notation of all numerical quantities to be in the denary scale.

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  • (i) A set of written symbols is sometimes read in more than one way, while on the other hand two different sets of symbols (at any rate if denoting numerical quantities) may be read in the same way.

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  • Numerical quantities, to be added or subtracted, must be in the same denomination; we cannot, for instance, add S5 shillings and loo pence, any more than we can add 3 yards and 2 metres.

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  • The application of the above principles, and of similar principles with regard to multiplication and division, to numerical quantities expressed in any of the diverse British denominations, presents no theoretical difficulty if the successive denominations are regarded as constituting a varying scale of notation (§17).

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  • are the names given to certain numerical processes which have to be performed in order to find the result of certain arithmetical operations.

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  • Multiplication occurs when a certain number or numerical quantity is treated as a unit (§ II), and is taken a certain number of times.

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  • The pair of compartments on either side may, as here, contain numerical quantities, or may contain numbers.

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  • may denote either numbers or numerical quantities, while m and n.

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  • The positive quantity or number obtained from a negative quantity or number by omitting the " - " is called its numerical value.

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  • It should be observed that the numerical value of the error is to be subtracted from or added to the stated value according as the error is positive or negative.

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  • of a numerical quantity is usually called reduction, so that this term covers, e.g., the expression of £ 1 53, 7s.

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  • In §§ 33 4 2 We Have Dealt With The Parallelism Of The Original Number Series With A Series Consisting Of The Corresponding Multiples Of Some Unit, Whether A Number Or A Numerical Quantity; And The Relations Arising Out Of Multiplication, Division, &C., Have Been Exhibited By Diagrams Comprising Pairs Of Corresponding Terms Of The Two Series.

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  • Correspondence of Numerical Quantities.-Again, in § 89, we have considered cases of multiple-tables of numerical quantities, where each quantity in one series is equivalent to the corresponding quantity in the other series.

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  • A B C of numerical quantities, merely correspond with each other, the correspondence being the result of some relation.

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  • It is desirable, wherever possible, to perform operations on numbers or numerical quantities from the left, rather than from the right.

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  • There is no difference in principle between addition (or subtraction) of numbers and addition (or subtraction) of numerical quantities.

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  • We need therefore consider numerical quantities only, our results being applicable to numbers by regarding the digits as representing multiples of units in different denominations.

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  • In other cases we are concerned with a proportion stated as a numerical percentage, or as a money percentage (i.e.

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  • In calculating an average it should be observed that the addition of any numerical quantity (positive or negative) to each of the measurements produces the addition of the same quantity to the average, so that the calculation may often be simplified by taking some particular measurement as a new zero from which to measure.

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  • In 1829 he discovered the theorem, regarding the determination of the number of real roots of a numerical equation included between given limits, which bears his name (see Equation, V.), and in the following year he was appointed professor of mathematics at the College Rollin.

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  • The individual teeth of each group are enumerated from before backwards, and by such a formula as the following: i I, 12, 23, c, p 1, p 2, p3, p 4,m I,m2,m3 i I, 22,13,c, p 1, p2, p3, p4,m I,m2,m3 a special numerical designation is given by which each one can be indicated.

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  • This contains a reprint of the First Notions, an elaborate development of his doctrine of the syllogism, and of the numerical definite syllogism, together with chapters of great interest on probability, induction, old logical terms and fallacies.

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  • Numerical computation of this kind might almost be described as his pastime.

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  • This is Spinoza's theory of the infinitely infinite," the limiting notion of infinity being of a numerical, quantitative series, each term of which is a qualitative determination itself quantitatively little, e.g.

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  • It was conceived by Huxley, among others, that the same number of cephalic somites would be found to be characteristic of all the diverse classes of Arthropoda, and that the somites, not only of the head but of the various regions of the body, could be closely compared in their numerical sequence in classes so distinct as the Hexapods, Crustaceans and Arachnids.

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  • The range of modification of which the rami or limb-branches of the limbs of Arthropoda are capable is very large, and in allied orders or even families or genera we often find d z what is certainly the palp of the same appendage (as determined by numerical position of the segments) - in one case antenniform, in another chelate, in another pediform, and in another reduced to a mere stump or absent altogether.

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  • An essay composed by him in 1673 on the true and apparent diameters of the planets furnished Newton with data for the third book of the Principia, and he fitted numerical elements to J.

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  • To the days Fabre d'Eglantine gave names which retained the idea of their numerical order, calling them Primedi, Duodi, &c., the last day of the ten, the day of rest, being named Decadi.

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  • The numerical quantities by which the distance and direction, and therefore the position, are defined, are termed co-ordinates of the point.

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  • This numerical proportion, as being a necessary consequence of the law of gravitation, must prevail in every system under its sway.

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  • It consisted, first, in the identification, by strict numerical comparisons, of terrestrial gravity with the mutual attraction of the heavenly bodies; secondly, in the following out of its mechanical consequences throughout the solar system.

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  • Other numerical particulars relating to the moon are: Mean distance from the earth (earth's radius as I) ..

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  • The Numerical Determination of the Elements of the Moon's Motion.

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  • - In order that the two capital discoveries just mentioned should have the highest scientific value, it was essential that the numerical values of the elements involved in these complicated motions should be fixed with precision.

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  • Most of the elements are small numerical fractions: e, the eccentricity of the moon's orbit, about 0.055; e', the eccentricity of the earth's orbit, about o 017: y, the sine of half the inclination of the moon's orbit, about 0.046; m, the ratio of the mean motions of the moon and earth, about 0.075.

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  • The third method seeks to avoid the difficulty by using the numerical values of the elements instead of their algebraic symbols.

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  • This method has the advantage of leading to a more rapid and certain determination of the numerical quantities required.

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  • Among applications of the third or numerical method, the most successful yet completed is that of P. A.

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  • During the twenty years following he devoted a large part of his energies to the numerical computation of the lunar inequalities, the redetermination of the elements of motion, and the preparation of new tables for computing the moon's position.

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  • It was therefore necessary that the numerical inequalities should be again determined by an entirely different method.

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  • The numerical computations were worked out by Hill only for the first approximation.

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  • The principal feature of his work was that the quantity m, which is regarded as constant, appears only in a numerical form, so that the uncertainties arising from development in a series accruing to its powers is done away.

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  • Subsequent investigations naturally modified the numerical values upon which this nomenclature was based, but without altering the order of the periods.

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  • But as this surface is obtained by assuming that the interfering streams follow the same route in the crystal, and by neglecting the refraction out of the crystal, it does not lend itself to accurate numerical calculations.

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  • are in fact the most natural and most convenient method of recording the numerical data, more particularly in cases where they do not admit of being adequately represented by a formula.

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  • a, b, c = Numerical constants inIformulae.

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  • the nation was well able to make good the numerical losses involved by a serious war; that its numbers tend to increase steadily; and that the rate of increase has hitherto shown a marked acceleration in periods of commercial expansion.

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  • The function n sin u = A, for the microscope, has been called by Abbe the numerical aperture.

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  • As the semiaperture of a pencil proceeding from an object point cannot exceed 90°, the numerical aperture of a dry-system cannot be greater than I.

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  • On the other hand, in immersion-systems the numerical aperture can almost amount to the refractive index, for A=n sin u

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  • From the Fraunhofer formula I =X/n sin a one can immediately deduce the limit to the diffraction constant I, so that the banding by an objective of fixed numerical aperture can be perceived.

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  • The value n sin u equals the numerical aperture A, where n is the refractive index of the immersion-liquid, and u is the semi-aperture on the object-side.

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  • If microscopic preparations are observed by diffused daylight or by the more or less white light of the usual artificial sources, then an objective of fixed numerical aperture will only represent details of a definite fineness.

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  • With the best immersion-system, having a numerical aperture of 1.6, details of the size o.

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  • Then the denominator of the fraction, the numerical aperture, must be correspondingly increased, in order to ascertain the real resolving power.

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  • In this way a monochromat for glycerin of a numerical aperture 1.25 gives a relative numerical aperture of 2.50.

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  • From the section Regulation of the Rays (above) it is seen that the resolving power is opposed to the depth of definition, which is measured by the reciprocal of the numerical aperture, I/A.

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  • The structure of a modern system of this type, with a numerical aperture of I.

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  • 36 shows an apochromat of a numerical aperture of 1.40.

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  • The resolving power of an objective depends on its numerical aperture.

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  • The numerical aperture can be determined in two ways.

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  • The sine of this angle is the numerical aperture for dry lenses.

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  • The angular or numerical aperture can then be read off.

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  • If the numerical aperture be known the resolving power is easily found.

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  • It is noteworthy that the difference between Albrecht's numerical results and Chandler's formulae is generally less than o 05".

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  • Symmetry, then, in botanical language, has reference to a certain definite numerical relation of parts.

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  • of Denmark that the enemies' harbours could be wrested from them only by a successful offensive war on land; and, while quite alive to the risks of such an enterprise in the face of two large armies, Tilly's and Wallenstein's, each of them larger than his own, he argued that the vast extent of territory and the numerous garrisons which the enemy was obliged to maintain, more than neutralized his numerical superiority.

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  • At the beginning of 1632, in order to bring about the general peace he so earnestly desired, he proposed to take the field with an overwhelming numerical majority.

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  • Verbal reasoning, numerical reasoning and abstract reasoning are timed tests, each lasting 20 minutes.

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  • The test is very similar to those used by major employers and covers verbal, numerical and diagrammatic reasoning.

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  • tracer advection is a longstanding and notoriously difficult problem with numerical models, and a severe limitation on their validity.

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  • By being presented as Grid services, numerical optimization algorithms can be consumed with a number of message interactions.

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  • Archimedes the numerical analyst Here we summarize the main points in the paper by Phillips with the above title.

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  • numerical aperture The numerical aperture of a microscope objective is a measure of its ability to resolve fine specimen detail.

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  • Further numerical analysis to deal with the effects of sea ice data assimilation.

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  • Furthermore, they can replace heavy 3D numerical calculations (for example finite element calculations) with high accuracy.

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  • You are advised to bring a simple, numerical, non-programmable, pocket calculator.

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  • figure captions should be listed in numerical order at the end of the article.

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  • Put Ronnie's collection of toenail clippings into numerical order.

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  • codexguistic or numerical listing was employed and the storage media were clay tablets, papyrus codices, leather scrolls or hieroglyphics.

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  • numerical cognition without words: evidence from the Amazonia.

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  • They all require a certain amount of numerical computation.

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  • He is a numerical analyst who specializes in the development and analysis of algorithms in matrix computations.

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  • Numerical Geometry Ltd. Games Writing games is a good way of exploring strongly interactive computer graphics.

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  • concludein numerical findings are presented in the second part of this paper, leading on to the concluding remarks in the final section.

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  • A very commonly used dry condenser has a numerical aperture (NA) of 0.9.

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  • Priority values define a numerical rank or ordering of the threads that a thread scheduler can use to resolve simultaneous contention for processing resources.

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  • Project 26: From convection to chaos The early investigations into chaos are intimately tied into the numerical study of problems in thermal convection.

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  • The function installs a forward type converter to convert string values to numerical representation type values.

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  • A robust measure of numerical model parameter sensitivity was employed, namely the rank order correlation coefficient.

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  • decreases in numerical abundance.

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  • The professional user can adjust any operation via numerical input and can precisely monitor the results using an online densitometer.

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  • However, returning to the problem in hand which is to compute the accurate numerical derivative of a given external function.

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  • The UM uses a full compressible equation set, no turbulence closure (numerical diffusion only) and a hybrid terrain following co-ordinate system.

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  • interesting digression: There's a school of thought that believes that states of drunkeness can be assigned numerical values.

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  • It has numerical gain 8.4 with respect to a practical half-wave dipole.

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  • The magnitude of the numerical uncertainity of the results is controlled through the input parameter epsilon.

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  • numerical experiments already now are able to simulate reliable vertical profiles of dissolved gas content at various times of the storm development.

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  • One involves experimental and numerical research into the behavior of liquid gallium under the combined action of a temperature gradient and a magnetic field.

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  • This report is intended primarily for numerical modelers in civil engineering hydraulics.

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  • numerical hydrodynamics can be a useful tool in predicting diagnostics of dynamically active regions.

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  • Will be followed by an numerical indiction of how far the progress has gone.

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  • initiated that involves experiments and numerical simulations.

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  • Numerical Model Simulates the impact of alternative operating policies or changed water and sediment inputs on future sedimentation.

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  • In case the software cannot calculate surface integrals analytically it offers the possibility to proceed with a numerical evaluation of the corresponding terms.

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  • numerical methods Newton's method for finding a root; numerical integration of a function.

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  • For Hamiltonian systems, one can exactly preserve the symplectic structure of the system using numerical methods the have become known as symplectic integrators.

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  • irreducible numerical semigroups containing it.

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  • The numerical method used is multisection for eigenvalues [16] and repeated inverse iteration and orthogonalization for eigenvectors [17] .

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  • All controllers are easily programmed via the numerical keypad to create custom setups in minutes.

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  • In numerical linear algebra he developed backward error analysis methods.

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  • The approach will be to apply 3-D numerical modeling methods to existing ship-board magnetometer survey data.

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  • And so I think there is a whole range of variation in the numerical field as there is in the purely mathematical relativity area.

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  • mergeuse a combined observational and numerical approach to study interacting and merging galaxies in the local Universe.

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  • The candidate should ideally have experience in statistical data analysis and/or numerical modeling and in mountain meteorology.

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  • numerical simulations.

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  • numerical semigroup.

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  • numerical computation.

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  • numerical aperture.

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  • numerical modeling.

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  • numerical approximation.

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  • E.g. br / attribute values must be quoted even when purely numerical.

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  • The work of Barbara Thiering, tho not numerical, depends upon word codes and time codes, and seems just as incredible.

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  • The problem was not simply numerical but also political.

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  • They are in alphabetical and then numerical order but the actual process of cataloging them all will be a massive task.

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  • You ask the questions in a variety of formats, including true or false, multiple choice or even numerical.

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  • Your site should combine information in text, graphical and possibly numerical format.

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  • The project has been funded to investigate the causes of cardio vascular problems and cancer using numerical modeling.

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  • You will also conduct comparisons between the numerical models and satellite observations of flows around sunspots and coronal activity above active regions.

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  • observed in direct numerical simulations, these computations are often expensive and lack generality.

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  • The numerical method then must be able to handle discontinuities without introducing non-physical, spurious oscillations.

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  • We will integrate aspects of these standards and extend them for numerical computing on massively parallel systems.

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  • The method is illustrated by two numerical examples, namely a single pendulum on a cart and a parametrically driven inverted double pendulum.

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  • Wyn Williams uses powerful computer facilities for intensive numerical modeling of the underlying physics of magnetism in mineral grains.

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  • Numerical values can be integer, real, or double precision.

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  • Efforts are underway to compare the numerical predictions of the response of gravity key walls with those obtained for flexible key walls.

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  • A numerical prefix says how many places to rotate the text.

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  • The project will combine analytical studies and numerical modeling tools for simulating wave propagation in fractured porous rock.

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  • The default numerical method used by NEURON produces values which have an error proportional to dt.

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  • Solution method: Recurrence relations, power series expansion, numerical quadrature.

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  • numerical relativity makes use of methods which are conceptually very different.

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  • There is qualitative sameness, and then there is numerical sameness.

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  • So there can be qualitative sameness without numerical sameness.

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  • Numerical studies like this can help to understand wave propagation phenomena observed on field recordings in volcano seismology.

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  • Series A number of items arranged in numerical or alphabetical sequence.

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  • simulateoject will combine analytical studies and numerical modeling tools for simulating wave propagation in fractured porous rock.

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  • numerical simulations of the fully nonlinear equation verify all former statements.

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  • singularityew chapters describe numerical experiments on two- and three-dimensional rotors, and phase singularities in the heart wall.

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  • The project aim is to develop a new process-based numerical model to predict rates and styles of periglacial solifluction.

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  • A numerical evaluation of sparse direct solvers for the solution of large sparse direct solvers for the solution of large sparse.. .

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  • Dynamo models of our Galaxy as a flattened spheroid have been investigated, using asymptotic analytic methods and high resolution numerical techniques.

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  • The numerical substructure may be either linear or non-linear, the latter using a new, fast non-linear analysis algorithm which we have developed.

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  • Each reduction rule contributes to the final summation formula by a numerical factor or by an additional summation variable.

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  • The attacker is likely to have numerical superiority to offset the defender being hidden.

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  • In particular the paper links to the use of numerical taxonomy within the natural sciences where similar techniques have met with considerable success.

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  • For the extension of a squid tentacle, our numerical results are in excellent agreement with existing experimental results.

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  • toenail clippings into numerical order.

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  • For the future it would be beneficial to have fortnightly tutorials to go over numerical examples.

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  • tweaking numerical parameters of robots, a non-trivial exercise.

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  • discrete numerical variates have only a fixed set of possible values - typically integer values, arising from counting things.

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  • Numerical Equations These are based on the diffusion equation in one dimension, where is the eddy viscosity.

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  • Thus we deduce one of the fundamental rules for successful work with numerical workbenches: prefer compiled functions over interpreted code.

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  • His treatise on numerical divisions, weights and measures (Distributio) is extant, with the exception of the concluding portion.

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  • The Church of England claims as adherents 39% of the population, and the Roman Catholic Church 22%; next in numerical strength are the Wesleyans and other Methodists, numbering 12% i the various branches of the Presbyterians 11%, Congregationalists 2%, and Baptists 2%.

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  • It is a fundamental result in dynamics that, if a body be projected vertically upwards in vacuo, with a velocity of v centimetres per second, it will rise to a height of v 2 /2g centimetres, where g represents the numerical value of the acceleration produced by gravity in centimetre-second units.

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  • If the two circuits are in tune so that the numerical product of capacity and inductance of each circuit is the same or C L, = C L +CL and if k is the coefficient of coupling then the natural frequency of each circuit is n = I /2w / (CL), and when coupled two oscillations are set up in the secondary circuit having frequencies n and n2 such that n = n0/ (i - k) and n = nh,/ (I +k).

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  • He is notorious for his wilful exaggeration, both in narrative and numerical statements.

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  • When a character is said to be transmitted, or to be correlated with another character, the biometricist declares the statement valueless without numerical estimations of the inheritance or correlation.

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  • The ultimate value of numerical inquiries must depend on the equivalence of the units on which they are based.

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  • The height or length of the arm of a human being, for instance, is the result of many factors, some inherent, some due to environment, and until these have been sifted out, numerical laws of inheritance or of correlation can have no more than an empirical value.

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  • It is among the Invertebrata that epidemics of destruction are referred to, though we should bear in mind that it is only the difference in numerical proportion that prevents our speaking of an epidemic of elephants or of rabbits, though we use the term when speaking of blight insects; there is little consistency in the matter, as it is usual to speak of an invasion or scourge of locusts, caterpillars, &c. Insect injuries are very varied in degree and in kind.

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  • The numerical predominance of certain families or their absence affords criteria for marking out boundary lines and tracing relationships.

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  • It consisted essentially in the adoption of Delauny's final numerical expressions for longitude, latitude and parallax, with a symbolic term attached to each number, the value of which was to be determined by substitution in the equations of motion.

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  • In this mode of treating the question the order of the terms is numerical, and though the amount of labour is such as might well have deterred a younger man, yet the details were easy, and a great part of it might be entrusted to a mere computer.

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  • When, therefore, the supreme God is called by the name Af3pava or A0paas, which contains the numerical value 365, it is worthy of remark that the name of the Persian god Mithras (Mciepas) also was known in antiquity to contain this numerical value (Jerome in Amos 3; Opp. Vallarsi VI.

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  • The overwhelming numerical superiority of the Sla y s, and the very great differences in ethnical type, belief and mythology between the IndoEuropean and the Ural-Altaic races, may have contributed to the same end.

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  • The principles by means of which the magnitude and position of balance weights are worked out are given in the article Mechanics (Applied Mechanics), and the whole subject of locomotive balancing is exhaustively treated with numerous numerical examples in The Balancing of Engines by W.

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  • The nature of this problem is illustrated by the numerical example in § g.

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  • Manchester in 1906, when the Labour party were returned for the first time in numerical force - over 50 in all.

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  • Owing to the existence of a strong Mussulman minority among its inhabitants, the warlike character of the natives, and the mountainous configuration of the country, which enabled a portion of the Christian population to maintain itself in a state of partial independence, the island has constantly been the scene of prolonged and sanguinary struggles in which the numerical superiority of the Christians was counterbalanced by the aid rendered to the Moslems by the Ottoman troops.

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  • The principal orders, arranged according to their numerical importance, are as follows: - Leguminosae,Rubiaceae, Orchidaceae, Compositae, Gramineae, Euphorbiaceae, Acanthaceae, Cyperaceae and Labiatae.

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  • Next in numerical importance to the Mongolians are the races which have been called by Professor Huxley Melanochroic and Xanthochroic. The former includes the dark-haired people of southern Europe, and extends over North Africa, Asia Minor, Syria to south-western Asia, and through Arabia and Persia to India.

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  • Carpenter (1899, 1902-1904) has lately endeavoured to show an exact numerical correspondence in segmentation between the Hexapoda, the Crustacea, the Arachnida, and the most primitive of the Diplopoda.

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  • The old B retained the numerical value of the Greek i as 2, and no numerical value was given to the new symbol.

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  • In Greek, where I is the twentieth letter of the alphabet, or, if the merely numerical and p are excluded, the eighteenth, another form 1 or S according to the direction of the writing is also widespread.

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  • (5) Every letter of a word is reduced to its numerical value, and the word is explained by another of the same quantity.

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  • 1 Thus Canon V., according to which a word is reduced to its numerical value and interpreted by another word of the same value, is recognized in the New Testament (cf.

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  • Largesse was especially given on the field of victory, and was, moreover, liberally distributed to stifle sedition and mutiny among the troops, the numerical strength of which was continually increased as the empire enlarged its borders.

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  • When, therefore, the latter, on the 22nd, marched southward to reopen his communications by the defeat of the enemy's army, always the surest means of solving this difficulty, he actually reached the neighbourhood of Eckmuhl with a sufficient numerical superiority had he only been prompt enough to seize his opportunity.

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  • The numerical values are, it is to be noted, exceedingly small.

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  • a V(I - a) This constant k gives a numerical value for the chemical affinity, and the equation should represent the effect of dilution on the molecular conductivity of binary electrolytes.

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  • When a homogeneous polynomial is transformed by general linear substitutions as hereafter explained, and is then expressed in the original form with new coefficients affecting the new variables, certain functions of the new coefficients and variables are numerical multiples of the same functions of the original coefficients and variables.

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  • Other " Galois " groups were defined whose substitution coefficients have fixed numerical values, and are particularly associated with the theory of equations.

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  • and by elimination we obtain the resultant ao 0 bo 0 0 al ao b1 bo 0 a 2 a i b 2 b 1 bo a numerical factor being disregarded.

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  • af Expression in Terms of Roots.-Since x+y y =mf, if we take cx any root x 3, y1, ofand substitute in mf we must obtain, y 1 C) zaZ1 �; hence the resultant of and f is, disregarding numerical factors, y,y2...y,,.

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  • = ...+O(s i s 2 s 3 ...)xl1x12x13...+..., where 0 is a numerical coefficient, then also O ?2 0.3 P1 P2 P3 Al A2 A3 +.

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  • Since, If F = An, 4) = By, 1 = I (Df A4) Of A?) Ab A"'^1Bz 1=, (F, Mn Ax I Ax 2 Axe Ax1) J The First Transvectant Differs But By A Numerical Factor From The Jacobian Or Functional Determinant, Of The Two Forms. We Can Find An Expression For The First Transvectant Of (F, �) 1 Over Another Form Cp. For (M N)(F,4)), =Nf.4Y Mfy.4), And F,4, F 5.4)= (Axby A Y B X) A X B X 1= (Xy)(F,4))1; (F,Ct)1=F5.D' 7,(Xy)(F4)1.

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  • It is (f = (ab) 2 a n-2 r7 2 =Hx - =H; unsymbolically bolically it is a numerical multiple of the determinant a2 f a2f (32 f) 2� It is also the first transvectant of the differxi ax axa x 2 ential coefficients of the form with regard to the variables, viz.

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  • d x the first evectant; and thence 4cxdi the second evectant; in fact the two evectants are to numerical factors pres, the cubic covariant Q, and the square of the original cubic.

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  • The quartic will have two pairs of equal roots, that is, will be a perfect square, if it and its Hessian merely differ by a numerical factor.

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  • Remark.-The invariant C is a numerical multiple of the resultant of the covariants i and j, and if C = o, p is the common factor of i and j.

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  • Of these thirteen sections, the first contains a simple description of the more prominent phenomena, without mathematical symbols or numerical data.

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  • Inside a magnetized body, B is the force that would be exerted on a unit pole if placed in a narrow crevasse cut in the body, the walls of the crevasse being perpendicular to the direction of the magnetization (Maxwell, § § 399, 604); and its numerical value, being partly due to the free magnetism on the walls, is generally very different from that of H.

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  • Where the induction is high the lines will be crowded together; where it is weak they will be widely separated, the number per square centimetre crossing a normal surface at any point being always equal to the numerical value of B.

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  • The magnetic susceptibility expresses the numerical relation of the magnetization to the magnetizing force.

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  • Equations (33) and (34) show that when, as is generally the case with ferromagnetic substances, the value of is considerable, the resultant magnetic force is only a small fraction of the external force, while the numerical value of the induction is approximately three times that of the external force, and nearly independent of the permeability.

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  • The small magnet may be a sphere rigidly magnetized in the direction of Ho; if this is replaced by an isotropic sphere inductively magnetized by the field, then, for a displacement so small that the magnetization of the sphere may be regarded as unchanged, we shall have dW = - vIdHo = v I+-, whence W = - 2 I + H2 ° (37) The mechanical force acting on the sphere in the direction of displacement x is 1 Hopkinson specified the retentiveness by the numerical value of the " residual induction " (=47rI).

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  • Curves of magnetization (which express the relation of I to H) have a close resemblance to those of induction; and, indeed, since B = H+47r1, and 47rI (except in extreme fields) greatly exceeds H in numerical value, we may generally, without serious error, put I = B /47r, and transform curves of induction into curves of magnetization by merely altering the scale to which the ordinates are referred.

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  • It seems that confusion and trouble will be best avoided by abstaining from the introduction of the non-evident somites, the ocular and the praegenital, into the numerical nomenclature of the component somites of the three great body regions.

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  • He attended lectures on the numerical solution of equations and on definite integrals by M.

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  • The chamber has the right of initiative in the organization of the annual budget laws and those relative to the numerical strength of the army and navy.

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  • And, their own predominance being assured by their numerical strength and influence, they accorded equal shares of power to the other monti.

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  • The commission also called attention to the numerical insufficiency of magistrates and native commissioners in certain parts of Natal.

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  • According to this reading, William sought to rectify his position by asserting, not the numerical identity of the universal in each individual, but rather its sameness in the sense of indistinguishable similarity.

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  • Races," in the article Austria), the census returns of 1880, 1890 and 1900, exhibiting the numerical strength of the different nationalities, are of great interest.

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  • In the case of division we get two kinds of arithmetical formula, which, however, may be regarded as requiring a single kind of numerical process in order to determine the final result.

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  • We must count with something; and the successive somethings obtained by the addition of successive units are in fact numerical quantities, not numbers.

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  • This is rendered possible by the fact that we can use a single letter to represent a single number or numerical quantity, however many digits are contained in the number.

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  • While mensuration is concerned with the representation of geometrical magnitudes by numbers, graphics is concerned with the representation of numerical quantities by geometrical figures, and particularly by lengths.

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  • An important development, covering such diverse matters as the equilibrium of forces and the algebraic theory of complex numbers (� 66), has relation to cases where the numerical quantity has direction as well as magnitude.

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  • There are also cases in which graphics and mensuration are used jointly; a variable numerical quantity is represented by a graph, and the principles of mensuration are then applied to determine related numerical quantities.

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  • Graphic representation thus rests on the principle that equal numerical quantities may be represented by equal lengths, and that a quantity mA may be represented by a length mL, where A and L are the respective units; and the science of graphics rests on the converse property that the quantity represented by pL is pA, i.e.

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  • The numerical factor 6 is called the coefficient of a 5 bc 2 (� 20); and, generally, the coefficient of any factor or of the product of any factors is the product of the remaining factors.

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  • It must, of course, be remembered (� 23) that this is a statement of arithmetical equality; we call the statement an " identity," but we do not mean that the expressions are the same, but that, whatever the numerical values of a, b and c may be, the expressions give the same numerical result.

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  • We disregard numerical coefficients, so that by the H.C.F.

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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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  • It is clear that, since the numerical coefficients of A and of a are each 1, the coefficients in the expansions arise from the grouping and addition of like terms (� 37 (ii.)).

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  • The numerical value is then represented by (P (; thus " (x (< i " means that xis between -1 and +1.1 50.

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  • in arithmetic; numerical coefficients of the factor as a whole being ignored (cf.

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  • If, therefore, we take r i /(i+x) as equal to I -xdx2-...+ (-) r x r, there is an error whose numerical magnitude is I xr+1/ (I +x) I; and, if I x

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  • and of the numerical value of i+b i x+b 2 x 2 +...

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  • (vii.) The comparison of the numerical value of I-Fn(1)x +n(242+...+n(r)xr, when n is fractional, with that of (i+x)n, involves advanced methods (� 64).

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  • In order that numerical results obtained by summing the first few terms of a series may be of any value, it is usually necessary that the series should converge to a limit; but there are exceptions to this rule.

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  • Then a+a = as = a; hence numerical coefficients and indices are not required.

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  • A point A in space may be associated with a (real, positive, or negative) numerical quantity a, called its weight, and denoted by the symbol aA.

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  • In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion.

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  • a n are scalars, and in particular applications may be restricted to real or complex numerical values.

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  • The Greeks had succeeded in solving an isolated example; the Arabs accomplished the general solution of numerical equations.

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  • The component vibrations at P due to the successive zones are thus nearly equal in amplitude and opposite in phase (the phase of each corresponding to that of the infinitesimal circle midway between the boundaries), and the series which we have to sum is one in which the terms are alternately opposite in sign and, while at first nearly constant in numerical magnitude, gradually diminish to zero.

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  • If, on the other hand, the point be well immersed in the geometrical shadow, the earlier zones are altogether missing, and, instead of a series of terms beginning with finite numerical magnitude and gradually diminishing to zero, we have now to deal with one of which the terms diminish to zero at both ends.

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  • The denominator sin a is the quantity well known (after Abbe) as the " numerical aperture."

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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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  • His numerical table, therefore, differs from all others made before or since, prophetically assigning the place of honour to the one ancient scale now recognized as the foundation of the modern tonal system.

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  • Hippocrates, influenced as is thought by the Pythagorean doctrines of number, taught that they were to be expected on days fixed by certain numerical rules, in some cases on odd, in others on even numbers - the celebrated doctrine of "critical days."

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  • In another respect also he aided in establishing an exact science of medicine by the introduction of the numerical or statistical method.

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  • But no vertebrate advance in force took place until comparatively late in the afternoon, and by evening the attacking side, although enjoying a great numerical superiority, had only reached the foot of the hills that lay to the E.

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  • The great numerical superiority which they had at first possessed was gone by the 9th, and their task had come to be the ejection of an almost equal enemy from a naturally formidable position.

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  • The Turks, therefore, now possessed a huge numerical preponderance in the theatre of war.

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  • Conductivity.-Conductivity, whether thermic or electric, is very differently developed in different metals; and, as an exact knowledge of these conductivities is of great importance, much attention has been given to their numerical determination (see Electric conduction; and Conduction Of Heat).

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  • The absolute unit of force is employed here, and not the gravitation unit of hydrostatics; in a numerical application it is assumed that C.G.S.

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  • vertical, where Wh tan 0 = N = (c 2 - cl) c2 g2 tan 0, (6) (7) in which we have put k' 2 = ek 2, where E is a numerical factor depending on the shape.

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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) .

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  • In other species there is a definite arrangement of the leaves, some with and others without tendrils opposite to them, the numerical order remaining constant or nearly so.

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  • In this he goes so far as to deny any historical connexion between the two, maintaining with all the devices of an extravagant allegorism, including the Rabbinic Gematria based on the numerical values of letters (ix.

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  • When it was over the Liberal party was just short of the numerical strength which was requisite to defeat the combination of Tories and Parnellites.

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  • The great numerical superiority of the Federals enabled Sherman to press back the Confederates without a pitched battle, but the severity of the skirmishing may be judged from the casualties of the two armies (Sherman's about 26,000 men, Johnston's over io,000), and the obstinate steadiness of Johnston by the fact that his opponent hardly progressed more than one mile a day.

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  • " Now for the interpolation of the rest, I considered that the denominators I, 3, 5, &c., were in arithmetical progression; and that therefore only the numerical coefficients of the numerators were to be investigated.

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  • A particular trick of his is also to divide his arguments after the manner of the preachers of his day into heads and subheads, with actual numerical signs affixed to them.

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  • Since the establishment of the privileged province they have lost the Ottoman support which used to compensate for their numerical inferiority as compared with the Christians; and they are fast losing also their old habits and distinctiveness.

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  • Coulomb proved the proportionality of electric surface force to density, but the above numerical relation E= 42ra was first established by Poisson.

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  • This assumption represents qualitatively the theoretical isothermal of James Thomson (see Vaporization) and the phenomena of the critical state (see Condensation Of Gases); but the numerical results to which it leads differ so widely from experiment that it is necessary to suppose the constant, a, to be a function of the temperature.

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  • For such purely abstract purposes, the possibility of numerical evaluation of the function is of secondary importance, and it is often possible to make qualitative deductions with regard to the general nature of a transformation without any knowledge of the actual form of the function.

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  • The foreign-born are more numerously represented among the criminal, defective and dependent classes than their numerical strength would justify.

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  • The number is precisely that of the total numerical value of the consonants of the name "Eliezer" (Gen.

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  • The results coincide with Maxwell's so far as above stated, though the numerical coefficients do not agree.

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  • He devised an approximate numerical solution of equations of the second and third degrees, wherein Leonardo of Pisa must have preceded him, but by a method every vestige of which is completely lost.

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  • Thoughout, Ephraim played a distinctive and prominent part; it probably excelled Manasseh in numerical strength, and the name became a synonym for the northern kingdom of Israel.

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  • In the numerical expression of absolute densities it is necessary to specify the units of mass and volume employed; while in the case of relative densities, it is only necessary to specify the standard substance, since the result is a mere number.

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  • The latter's enormous numerical superiority was neutralized by Sulla's judicious choice of ground and the steadiness of his legionaries; the Asiatics after the failure of their attack were worn down and almost annihilated.

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  • The attempt to apply numerical methods to the comparison of geometrical quantities led to the doctrine of incommensurables, and to that of the infinite divisibility of space.

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  • From this point of view the atomic doctrine might be regarded as a relic of the old numerical way of conceiving magnitude, and the opposite doctrine of the infinite divisibility of matter might appear for a time the most scientific. The atomists, on the other hand, asserted very strongly the distinction between matter and space.

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  • As the same writer puts it: " There is such a thing as numerical difference, different from conceptual difference," 2 or expressing the same thing in other words " there are relations not grounded in the nature of the related terms."

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  • Idealism has, of course, no quarrel with numerical difference.

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  • All difference has its numerical aspect: two different things are always two both in knowledge and in reality.

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  • He might thus compensate for his numerical inferiority by superior mobility and superior leadership.

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  • Here they would be in contact, and ready to act united against Napoleon with a numerical superiority of two to one.

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  • He had to delay the French advance for 24 hours and give time for Blucher's concentration, at the same time retaining his own freedom of manoeuvre, and this in spite of the great length of the summer day, the short distance that he lay 'in front of Fleurus, the tremendous numerical superiority of the French and Napoleon's personal presence at their head.

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  • mensura, a measure), the science of measurement; or, in a more limited sense, the science of numerical representation of geometrical magnitudes.

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  • The measurement is numerical, i.e.

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  • It might, therefore, be described as that branch of mathematics which deals with formulae for calculating the numerical measurements of curved lengths, areas and volumes, in terms of numerical data which determine these measurements.

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  • Mensuration, then, is mainly concerned with quadratureformulae and cubature formulae, and, to a not very clearly defined extent, with the methods of obtaining such formulae; a quadrature-formula being a formula for calculating the numerical representation of an area, and a cubature-formula being a formula for calculating the numerical representation of a volume, in terms, in each case, of the numerical representations of particular data which determine the area or the volume.

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  • On the other hand, it is worth noticing that the words " quadrature " and " cubature " are originally due to geometrical rather than numerical considerations; the former implying the construction of a square whose area shall be equal to that of a given surface, and the latter the construction of a cube whose volume shall be equal to that of a given solid.

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  • The failure seems (§ 2) to be due to difficulty in realizing the numerical expression of an area or a solid in terms of a specified unit, while the same difficulty does not arise in the case of linear measure or liquid measure, where the number of units can be ascertained by direct counting.

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  • The main object to be aimed at, therefore, in the study of elementary mensuration, is that the student should realize the possibility of the numerical expression of areas and volumes.

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  • (iii) Association of solid measure with liquid measure, presenting numerical measurement in a different aspect, should be retained by testing volumes as found from linear dimensions with the volumes of the same bodies as found by the use of measures of capacity.

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  • The numerical result obtained by applying a formula to particular data will generally not be exact.

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  • Let u and x be the numerical expressions of the magnitudes of E and F.

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  • The general principle is that the numerical data from which a particular result is to be deduced are in general not exact, but are given only to a certain degree of accuracy.

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  • In the case of mathematical functions certain conditions of continuity are satisfied, and the extent to which the value given by any particular formula differs from the true value may be estimated within certain limits; the main inaccuracy, in favourable cases, being due to the fact that the numerical data are not absolutely exact.

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  • On the other hand, the numerical increase had not kept pace with the increase of population.

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  • When war broke out the Ottoman forces in Europe numbered less than 250,000 men, dispersed over Macedonia and Thrace; they were thus at great numerical disadvantage.

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  • The same result will be ob tained with a stationary fork and a movable glass plate; and, if the time occupied by the plate in moving through a given distance can be ascertained and the number of complete undulations exhibited on the plate for that distance, which is evidently the number of vibrations of the fork in that time, is reckoned, we shall have determined the numerical vibration value of the note yielded by the fork.

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  • His strong opposition to "dissenting churches" was nowhere so clearly shown as in a pamphlet published in 1816 to dissuade all Episcopalians from joining the American Bible Society, which he thought the Protestant Episcopal Church had not the numerical or the financial strength to control.

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  • Mayer's real merit consists in the fact that, having for himself made out, on inadequate and even questionable grounds, the conservation of energy, and having obtained (though by inaccurate reasoning) a numerical result correct so far as his data permitted, he applied the principle with great power and insight to the explanation of numerous physical phenomena.

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  • The second volume (1817) relates to the Eulerian integrals, and to various integrals and series, developments, mechanical problems, &c., connected with the integral calculus; this volume contains also a numerical table of the values of the gamma function.

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  • Under these circumstances, and also because of their numerical weakness and the rigour of the weather, the Germans advanced but slowly.

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  • The equations (66) - (71) are Siacci's, slightly modified by General Mayevski; and now in the numerical applications to high angle fire we can still employ the ballistic table for direct fire.

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  • It has been said that the blockade of the Confederate coast became in the end practically impenetrable, and that every attempt of the Confederate naval forces to break out was checked at once by crushing numerical preponderance.

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  • System; Unites et Etalons, Guillaume (Paris, 1890); Lupton's Numerical Tables, 1892; Metric Equivalent Cards, 1901; Dictionary of Metric Measures, L.

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  • For engineering and manufacturing purposes the more important linear gauges are, however, now used, adjusted to some fundamental unit of measure as the inch; although in certain trades, as for wires and flat metals, gauges continue to be used of arbitrary scales and of merely numerical sizes, having no reference to a legal unit of measure; and such are rarely accurate.

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  • The numerical value of this modulus is o 4342944 81 9 03251 82765 11289..

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  • The quantity denoted by e is the series, I I I+ I + i.2 +1.2.3+1.2.3.4 + the numerical value of which is, 2.71828 18284 59 0 45 2 353 6 02874.

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