# Coefficient sentence example

coefficient
• We may, by a well-known theorem, write the result as a coefficient of z w in the expansion of 1 - z n+1.
• Thus in 3a the coefficient of a is 3.
• The short answer on leather is, leather has the perfect coefficient of friction, and can achieve a very good balance between flex and protection.
• The tranverse electromotive force is equal to KCH/D, where C is the current, H the strength of the field, D the thickness of the metal, and K a constant which has been termed the rotatory power, or rotational coefficient.
• The strength of the induced current is - HScosO/L, where 0 is the inclination of the axis of the circuit to the direction of the field, and L the coefficient of self-induction; the resolved part of the magnetic moment in the direction of the field is equal to - HS 2 cos 2 6/L, and if there are n molecules in a unit of volume, their axes being distributed indifferently in all directions, the magnetization of the substance will be-3nHS 2 /L, and its susceptibility - 3S 2 /L (Maxwell, Electricity and Magnetism, § 838).
• Reference to a geometrical interpretation seems at first sight to throw light on the meaning of a differential coefficient; but closer analysis reveals new difficulties, due to the geometrical interpretation itself.
• The number by which an algebraical expression is to be multiplied is called its coefficient.
• The numerical factor 6 is called the coefficient of a 5 bc 2 (ï¿½ 20); and, generally, the coefficient of any factor or of the product of any factors is the product of the remaining factors.
• (iv.) When the terms of a multinoniial contain various powers of x, and we are specially concerned with x, the terms are usually arranged in descending (or ascending) order of the indices; terms which contain the same power being grouped so as to give a single coefficient.
• If any power is absent, we treat it as present, but with coefficient o.
• The binomial theorem gives a formula for writing down the coefficient of any stated term in the expansion of any stated power of a given binomial.
• The coefficient of A 2 a 3 in the expansion of (A+a) 5 is then the number of terms such as ABcde, AbcDe, AbCde, ...
• The first term is Abcde, in which all the letters are large; and the coefficient of A 2 a 3 is therefore the number of terms which can be obtained from Abcde by changing three, and three only, of the large letters into small ones.
• Then, since nr rl is also a rational integral function of n of degree r, we can find a coefficient c r, not containing n, and such as to make N-c r nr ri contain no power of n higher than n r - 1.
• In ï¿½ 41 (ii.), for instance, the coefficient of A n - r a r in the expansion of (Ada) (A+a) n - 1 has been called (n, .); and it has then been shown that (ii) = (n _ I) d (i).
• The idea is utilized in the elementary consideration :of a differential coefficient; and its importation into the treatment of certain functions as continuous is therefore properly associated with the infinitesimal calculus.
• Thus we arrive at the differential coefficient of f(x) as the limit of the ratio of f (x+8) - f (x) to 0 when 0 is made indefinitely small; and this gives an interpretation of nx n-1 as the derived function of xn (ï¿½ 45)ï¿½ This conception of a limit enables us to deal with algebraical expressions which assume such forms as -° o for particular values of the variable (ï¿½ 39 (iii.)).
• The elementary idea of a differential coefficient is useful in reference to the logarithmic and exponential series.
• He denotes quantities by the letters of the alphabet, retaining the vowels for the unknown and the consonants for the knowns; he introduced the vinculum and among others the terms coefficient, affirmative, negative, pure and adfected equations.
• Its coefficient of linear expansion between 0° and 100° is 0.002717; its specific heat 0.0562; its thermal and electrical conductivities are 145 to 152 and '14.5 to 140.
• The coefficient of expansion is constant for such metals only as crystallize in the regular system; the others expand differently in the directions of the different axes.
• With n=1, the re-entrant walls are given of Borda's mouthpiece, and the coefficient of contraction becomes 2.
• The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds.
• Conversely, if the kinetic energy T is expressed as a quadratic function of x, x x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.
• The coefficient of purity is increased and the viscosity of the juice diminished.
• The coefficient of linear expansion is 0.002,905 for 100° from o° upwards (Fizeau).
• Another steel containing 45% of nickel has, like platinum, the same coefficient of expansion as glass.
• Its high coefficient of thermal expansion, coupled with its low freezing point, renders it a valuable thermometric fluid, especially when the temperatures to be measured are below - 39° C., for which the mercury thermometer cannot be used.
• The coefficient of expansion at constant pressure is equal to the coefficient of increase of pressure at constant volume.
• It is found by experiment that the change of pv with pressure at moderate pressures is nearly proportional to the change of p, in other words that the coefficient d(pv)/dp is to a first approximation a function of the temperature only.
• This coefficient is sometimes called the " angular coefficient," and may be regarded as a measure of the deviations from Boyle's law, 'which may be most simply expressed at moderate pressures by formulating the variation of the angular coefficient with temperature.
• The simplest assumption which suffices to express the small deviations of gases and vapours from the ideal state at moderate pressures is that the coefficient a in the expression for the capillary pressure varies inversely as some power of the absolute temperature.
• The value of the angular coefficient d(pv)/dp is evidently (b - c), which expresses the defect of the actual volume v from the ideal volume Re/p. Differentiating equation (17) at constant pressure to find dv/do, and observing that dcldO= - nc/O, we find by substitution in (is) the following simple expression for the cooling effect do/dp in terms of c and b, Sdo/dp= (n+I)c - b..
• Matthiessen as 73 at 0° C., pure silver being 100; the value of this coefficient depends greatly on the purity of the metal, the presence of a few thousandths of silver lowering it by 10%.
• Its coefficient of expansion for each degree between o° and Ioo C. is 0.000014661, or for gold which has been annealed 0.000015136 (Laplace and Lavoisier).
• Its thermal conductivity is the lowest of all metals, being 18 as compared with silver as 1000; its coefficient of expansion between o° and too° is 0.001341.
• Less accurate formulae are =p W/(W - W 2), the factor involving the density of the air, and the coefficient of the expansion of the solid being disregarded, and 0 =W/(W - W 1), in which the density of water is taken as unity.
• Its advantages rest on its high density and mobility; its main disadvantages are its liability to decomposition, the originally colourless liquid becoming dark owing to the separation of iodine, and its high coefficient of expansion.
• It is almost colourless and has a small coefficient of expansion; its hygroscopic properties, its viscous character, and its action on the skin, however, militate against its use.
• Struve also points out that by attaching a fine scale to the focusing slide of the eye-piece, and knowing the coefficient of expansion of the metal tube, the means would be provided for determining the absolute change of the focal length of the object-glass at any time by the simple process of focusing on a double star.
• This, with a knowledge of the temperature of the screw or scale and its coefficient of expansion, would enable the change of screw-value to be determined at any instant.
• It melts between 2250° and 2300°, its specific heat is 0.0365, coefficient of expansion o 0000079, and specific gravity 16.64.
• The surface tension, on the other hand, is greater than that of pure water and increases with the salinity, according to Kriimmel, in the manner shown by the equation a=77.09+o 0221 S at o° C., where a is the coefficient of surface tension and S the salinity in parts per thousand.
• Buchanan found a mean of 20 experiments made by piezometers sunk in great depths on board the " Challenger " give a coefficient of compressibility K=491 X 107; but six of these experiments made at depths of from 2740 to 3125 fathoms gave K=480Xio 7.
• On this account it is very difficult to know when all the gas is driven out of a sample of sea-water, and a much larger proportion is present than the partial pressure of the gas in the atmosphere and its coefficient of absorption would indicate.
• Conduction has practically no effect, for the coefficient of thermal conductivity in sea-water is so small that if a mass of sea-water were cooled to 0° C. and the surface kept at a temperature of 30° C., 6 months would elapse before a temperature of 15° C. was reached at the depth of 1 3 metres, 1 year at 1 85 metres, and io years at 5.8 metres.
• The maintenance of the conditions of steadiness implied in equation (I) depends upon the constancy of F, and therefore of the coefficient of friction µ between the rubbing surfaces.
• The ratio p is given by e"` e, where e= 2.718; µ is the coefficient of friction and 0 the angle, measured in radians,, subtended by the arc of contact between the rope and the wheel.
• This apparatus was used to find the temperature coefficient of the frequency of forks, the value obtained - .00011 being the same as that found by Koenig.
• C is a constant, equal to the coefficient of viscosity in Helmholtz's theory, but less simple in Kirchhoff's theory.
• The thermal coefficient of expansion of steel and concrete is nearly the same, otherwise changes of temperature would cause shearing stress at the junction of the two materials.
• Then the deviation y= DE of the neutral axis of the bent beam at any point D from the axis OX is given by the relation d 2 y Ml dx 2 = EI' where M is the bending moment and I the amount of inertia of the beam at D, and E is the coefficient of elasticity.
• When the atoms are in motion these strain-forms produce straining and unstraining in the aether as they pass across it, which in its motional or kinetic aspect constitutes the resulting magnetic field; as the strains are slight the coefficient of ultimate inertia here involved must be great.
• The definition of the coefficients is that if (I-2h cos cp+h 2)-i be expanded in ascending powers of h, and if the general term be denoted by P„h', then P is of the Legendrian coefficient of the nth order.
• For the first time we have a correct and convenient expression for Laplace's nth coefficient."
• It is further assumed, as the result of experiment, that the resistance is proportional to the density of the air; so that if the standard density changes from unity to any other relative density denoted by then R= Td 2 p, and is called the coefficient of tenuity.
• Lastly, to allow for the superior centering of the shot obtainable with the breech-loading system, Bashforth introduces a factor a, called the coefficient of steadiness.
• Collecting all the coefficients, into one, we put (I) R = nd 2 p = nd 2 f (v), where and n is called the coefficient of reduction.
• We put and call C the ballistic coefficient (driving power) of the shot, so that (6) At = COT, where (7) AT = Av/gp, and AT is the time in seconds for the velocity to drop Av of the standard shot for which C = I, and for which the ballistic table is calculated.
• The constant may be any arbitrary number, as in using the table the difference only is required of two tabular values for an initial velocity V and final velocity v; and thus (to) T(V) - T(v) = Ev Ov/gp or fvdv/gp; and for a shot whose ballistic coefficient is C (II) t=C[T(V) - T(v)].
• Denoting by S(v) the sum of all the values of AS up to any assigned velocity v, (is) S(v) =E(OS)+ a constant, by which S(v) is calculated from AS, and then between two assigned velocities V and v, V AT, = vAv or rvvdv vgp gp' and if s feet is the advance of a shot whose ballistic coefficient is C, (17) s=C[S(V) - S(v)].
• Given the ballistic coefficient C, the initial velocity V, and a range of R yds.
• +Nxnzk+.., we see that in the term Nx n z k of the development the coefficient N is equal to the number of partitions of n into k parts, with the parts I, 2, 3, 4,, without repetitions.
• = I -xx2+x5+x7-x12-x15+..., where the only terms are those with an exponent (3n 2 n), and for each such pair of terms the coefficient is (-) n i.
• In the Weston standard cell cadmium and cadmium sulphate are substituted for zinc and zinc sulphate; it has the advantage of a much smaller coefficient of temperature variation than the Clark cell.
• Its coefficient of linear expansion is only 0.0000008 for 1° C. See: Rapport du Yard, Dr Benoit (1896).
• The equations finally arrived at are DX2(A2_ 2) (x2_ A2m)2+g2A2 ' DgA3 (A A l m) 2 +g 2 A2 ' where is the wave-length in free ether of light whose refractive index is n, and A m the wave-length of light of the same period as the electron, is a coefficient of absorption, and D and g are constants.
• This frame or tube is so constructed of iron and brass (one-third iron and two-thirds brass) that its temperature coefficient of linear expansion is the same as that of the platinumsilver alloy.
• Its coefficient of linear expansion by heat is 0.0000222 (Richards) or 0.0000231 (RobertsAusten) per 1° C. Its mean specific heat between o° and ioo° is 0.227, and its latent heat of fusion loo calories (Richards).
• Let A and C be two fixed disks, and B a disk which can be brought at will within a very short distance of either A or C. Let us suppose all the plates to be equal, and let the capacities of A and C in presence of B be each equal to p, and the coefficient of induction between A and B, or C and B, be q.
• The rate of diminution of amplitude expressed by the coefficient a in the index of the exponential is here greater than the coefficient b expressing the retardation of phase by a small term depending on the emissivity h.
• He investigated the expansibility of gases by heat, determining the coefficient for air as 0.003665, and showed that, contrary to previous opinion, no two gases had precisely the same rate of expansion.
• The coefficient of heating of a calorimeter when it is below the temperature of its surroundings is seldom, if ever, the same as the coefficient of cooling at the higher temperature, since the convection currents, which do most of the heating or cooling, are rarely symmetrical in the two cases, and moreover, the duration of the two stages is seldom the same.
• The Result Was To Reduce The Coefficient Of Diminution Of Specific Heat At 15° C. By Nearly One Half, But The Absolute Value At 20° C. Is Practically Unchanged.
• The Resistance R Could Be Deduced From A Knowledge Of The Temperature Of The Calorimeter And The Coefficient Of The Wire.
• In Spite Of The Large Corrections The Results Were Extremely Consistent, And The Value Of The Temperature Coefficient Of The Diminution Of The Specific Heat Of Water, Deduced From The Observed Variation In The Rate Of Rise At Different Points Of The Range 15° To 25°, Agreed With The Value Subsequently Deduced From Rowland'S Experiments Over The Same Range, When His Thermometers Were Reduced To The Same Scale.
• Nickel is used for the manufacture of domestic utensils, for crucibles, coinage, plating, and for the preparation of various alloys, such as German silver, nickel steels such as invar (nickel, 35.7%; steel, 64.3%), which has a negligible coefficient of thermal expansion, and constantan (nickel, 45%; copper, 55%), which has a negligible thermal coefficient of its electrical resistance.
• (II) This relation gives a linear formula for the variation of the total heat, a result which agrees in form with that found by Regnault for steam, and implies that the coefficient of t in his formula should be equal to the specific heat S of steam.
• 1.40, Rankine found S = .385, a value which he used, in default of a better, in calculating some of the properties of steam, although he observed that it was much larger than the coefficient .305 in Regnault's formula for the variation of the total heat.
• 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.
• The mean value, 0.313 of dH/d0, between loo° and 200° agrees fairly well with Regnault's coefficient 0.305, but it is clear that considerable errors in calculating the wetness of steam or the amount of cylinder condensation would result from assuming this important coefficient to be constant.
• Further, by suitably choosing the positions of the deflectors and the coefficient of torsion of the fibre, it is possible to make the temperature coefficient vanish.
• The factor (P+P) cos 0h sin 0 is called the vIrtual coefficient of the two screws which define the types of the wrench and twist, respectively.
• We assume that in limiting equilibrium we have F tsR, everywhere, where u is the coefficient of friction.
• Analytically, it is convenient to put Q~ equal to eirt multiplied by a complex coefficient; owing to the linearity of the equations the factor e~ni will run through them all, and need not always be exhibited.
• Since an, = a,r, the coefficient of Q, in the expression for qr is identical with that of Q,- in the expression for q,.
• Its magnitude is the product of the normal pressure or force which presses the rubbing surfaces together in~ a direction perpendicular to themselves into a specific constant already mentioned in 14, as the coefficient of friction, which depends on the nature and condition of the surfaces of the unguent, if any, with which they are covered.
• The total pressure exerted between the rubbing surfaces is the resultant of the normal pressure and of the friction, and its obliquity, or inclination to the common perpendicular of the surfaces, is the angle of repose formerly mentioned in 14, whose tangent is the coefficient of friction.
• Thus, let N be the normal pressure, R the friction, T the total pressure, f the coefficient of friction, and 4, the angle of repose; then f=tan4, ~8
• The more recent experiments of Lasche (Zeitsch, Verein Deutsche Ingen., 1902, 46, 1881) show that the product of the coefficient of friction, the load on the bearing, and the temperature is approximately constant.
• To express this symbolically, let dii represent the area of a portion of a pair of rubbing surfaces at a distance r from the axis of their relative rotation; p the intensity of the normal pressure at du per unit of area; and f the coefficient of friction.
• Let Ti be the tension of the free part of the band at that side towards which it tends to draw the pulley, or from which the pulley tends to draw it; 1, the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; 0 the ratio of the length of that arc to the radius of the pulley; do the ratio of an indefinitely small element of that arc to the radius; F=TiT2 the total friction between the band and the pulley; dF the elementary portion of that friction due to the elementary arc do; f the coefficient of friction between the materials of the band and pulley.
• Its specific heat is o 0899 at 0° C. and 0.0942 at 10o; the coefficient of linear expansion per 1° C. is o o01869.
• The coefficient (q) of the time in the exponential term (e at) may be considered to measure the degree of dynamical instability; its reciprocal 1 /q is the time in which the disturbance is multiplied in the ratio I: e.
• The diameter of the orifice was 3 millims., from which that of the jet is deduced by the introduction of the coefficient 8.
• Its specific heat is 0.05701 (Regnault) or 0.0559 (Bunsen); its coefficient of linear expansion is 0.0000-1921.
• It is better then to define the coefficient of absorption as a quantity k such that kln of the light is absorbed in i/nth part of a centimetre, where n may be taken to be a very large number.
• There is another coefficient of absorption (K) which occurs in Helmholtz's theory of dispersion (see Dispersion).
• It is closely related to the coefficient k which we have just defined, the equation connecting the two being k= 4lrK/X,X being the wavelength of the incident light.
• The small amount of red transmitted is at first quite overpowered by the green, but having a smaller coefficient of absorption, it becomes finally predominant.
• The extension of a spiral spring is given by the formula: Extension =W4n1VÃ† r 4, in which W = weight causing extension, in lbs; n = number of coils; R = radius of spring, from centre of coil to centre of wire, in inches; r = radius of wire of which the spring is made, in inches; E = coefficient of elasticity of wire, in lbs per square inch.
• Neglecting quantities of the second order, the pressure on the pulley is TdO, and the friction is MTd9 where p, is the coefficient of friction between the belt and the pulley.
• For a single pulley of diameter D, turning on a fixed pin of diameter d, the relation of the effort E to the load W, where f is the coefficient of friction, is expressed by E/W = (D-pfd)/(D - fd) _ 1 +2fd/D approximately.
• The resistance of the rope to bending causes an additional resistance, which experiment shows can be expressed in the form Wd 2 /cD where c is a coefficient.
• Considering the equations ax +by +cz =d, a'x +b'y +c' z =d', a"x+b"y+cnz=d" and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = o, and the whole coefficient of z becomes = o; the factors in question are b'c" - b"c', b"c - be", bc' - b'c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value a(b'c" - b"c') +a'(b"c - bc") +a'(bc' - b'c), and this function, represented in the form a, b,c, a' b'c', a" b" c" is said to be a determinant; or, the number of elements being 32, it is called a determinant of the third order.
• It is easy, by induction, to arrive at the general results: A determinant of the order n is the sum of the 1.2.3...n products which can be formed with n elements out of n 2 elements arranged in the form of a square, no two of the n elements being in the same line or in the same column, and each such product having the coefficient = unity.
• By what precedes it appears that there exists a function of the n 2 elements, linear as regards the terms of each column (or say, for shortness, linear as to each column), and such that only the sign is altered when any two columns are interchanged; these properties completely determine the function, except as to a common factor which may multiply all the terms. If, to get rid of this arbitrary common factor, we assume that the product of the elements in the dexter diagonal has the coefficient + 1, we have a complete definition of the determinant, and it is interesting to show how from these properties, assumed for the definition of the determinant, it at once appears that the determinant is a function serving for the solution of a system of linear equations.
• Any determinant I a,' b, I formed out of the elements of the original determinant, by selecting the lines and columns at pleasure, is termed a minor of the original determinant; and when the number of lines and columns, or order of the determinant, is n - I, then such determinant is called a first minor; the number of the first minors is = n 2, the first minors, in fact, corresponding to the several elements of the determinant - that is, the coefficient therein of any term whatever is the corresponding first minor.
• With the latest accepted diminution of the eccentricity, the coefficient is 5.91".
• Its specific gravity is given variously from 5.395 to 5'959; its specific heat is 0.083, and its coefficient of linear expansion 0.0000-0559 (at 40° C.).
• According to Plucker, the coefficient of cubical dilatation at moderately low temperatures is 0.0001585.
• - The limiting value, dE/dt, of the coefficient, p, for an infinitesimal difference, dt, between the junctions is called the Thermoelectric Power of the couple.
• In using the above table to find the value of E or dE/dt at any temperature or between any limits, denoting by p the value of dE/dt at 50° C., and by 2C the constant value of the second coefficient, we have the following equations :- dE/dt = p+2c(t -50), at any temperature t, Cent.
• The Peltier coefficient may also be expressed in volts or microvolts, and may be regarded as the measure of an E.M.F.
• If the quantity of heat absorbed and converted into electrical energy, when unit quantity of electricity (one ampere-second) flows from cold to hot through a difference of temperature, dt, be represented by sdt, the coefficient s is called the specific heat of electricity in the metal, or simply the coefficient of the Thomson effect.
• Like the Peltier coefficient, it may be measured in joules or calories per ampere-second per degree, or more conveniently and simply in microvolts per degree.
• 4, which is given as an illustration, the cold junctions are supposed to be at o° C. and the hot junctions at 100° C. Noll's values (Table I.) are taken for the E.M.F., and it is supposed that the coefficient of the Thomson effect is zero in lead, i.e.
• Taking the lead-iron couple as an example, the value of dE/dt at the hot junction too° C. is 10.305 microvolts per degree, and the value of the Peltier coefficient P = TdE/dT is +3844 microvolts.
• In this case, however, in order to account for the phenomenon of the Peltier effect at the junctions, it is necessary to suppose that there is a real convection of heat by an electric current, and that the coefficient P or pT is the difference of the quantities of heat carried by unit quantity of electricity in the two metals.
• Assuming the correctness of these, friction is generally measured in terms simply of the total pressure between the surfaces, by multiplying it by a "coefficient of friction" depending on the material of the surfaces and their state as to smoothness and lubrication.
• Both at very high and very low pressures the coefficient of friction is affected by the intensity of pressure, and, just as with velocity, it can only be regarded as independent of the intensity and proportional simply to the total load within more or less definite limits.
• These experiments distinctly point to the conclusion, although without absolutely proving it, that in such cases the coefficient of kinetic friction gradually increases as the velocity becomes extremely small, and passes without discontinuity into that of static friction.
• The relation then between the work expended and the actual cooling work performed denotes the efficiency of the process, and this is expressed by Qt/(Q2-Q1); but as in a perfect refrigerating machine it is understood that the whole of the heat Q i is taken in at the absolute temperature T 11 and the whole of the heat Q2, is rejected at the absolute temperature T2, the heat quantities are proportional to the temperatures, and the expression T,/(T 2 -T,) gives the ideal coefficient of performance for any stated temperature range, whatever working substance is used.
• If, however, the heat is to be rejected at loo°, then the coefficient is reduced to 4.6.
• The ideal coefficient of performance is about 1, but the actual coefficient will be about 8 i after allowing for the losses incidental to working.
• He dealt with the coefficient of performance as a common basis of comparison for all machines, and showed that the compression vapour machine more nearly reached the theoretic maximum than any other (Bayerisches Industrie and Gewerbeblatt, 1870 and 1871).
• An ideal machine, working between 5° below zero and 75° above, has a coefficient of about 5.7, or nearly six times that of an ideal compressed-air machine of usual construction performing the same useful cooling work.
• Therefore, coefficient alpha will be equal to zero.
• Also plotted is a curve for the minimum buckling stress coefficient.
• The correlation coefficient between adjacent pairs of axes can be viewed by clicking an axis with the right mouse button.
• The last stage can be used to estimate a stress diffusion coefficient.
• The regression coefficient for a straight line through origin is.
• Alternatively, the absorption coefficient maps can be calculated based on an adequate model of the sample.
• We therefore suggest that the linear attenuation coefficient be treated as a separate unit.
• The quantity t is known as the Kendall rank correlation coefficient tau.
• To calculate the intracluster correlation coefficient, and thereby refine the estimation of the sample size needed for the trial.
• The degree of relationship between three or more sample populations may be quantified using the multiple correlation coefficient.
• One factor that had a high correlation coefficient was the rate of unemployment.
• A robust measure of numerical model parameter sensitivity was employed, namely the rank order correlation coefficient.
• Correlation Coefficient The r-squared correlation coefficient The r-squared correlation coefficient should also remain high.
• Two voxel-based similarity measures, the linear correlation coefficient and the entropy correlation coefficient, are used.
• The ratio K o = s ' he ratio K o = s ' h / s ' z is known as the coefficient of earth pressure at rest.
• For wings, 92% of the data for the pitching moment coefficient increment are predicted to within ± 0.02.
• The effect of wing area extension on the profile drag coefficient increments may be neglected provided that the area extension is small.
• The recent analysis now included in the Data Items shows that the lift coefficient increment at maximum lift coefficient has a Reynolds number dependence.
• The bars represent the best fit coefficient of the suitably scaled wavelet to the data over the interval around the bar.
• (21) a formula giving the coefficient of transmission in terms of the refraction, and of the number of particles per unit volume.
• For very accurate work it is desirable that the base-plate, the slide and the scale should be of nickel steel, having the same thermal coefficient of expansion as glass.
• For the pressure coefficient per degree, between oÃ‚° and Ice C., they give the value 0036-6255, when the initial pressure is 700 mm.
• The coefficient of this rise is equivalent to half a vibration (o.5) per degree Fahr.
• 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).
• Ramsay and Shields found from investigations of the temperature coefficient of the surface energy that Tin the equation y(Mv) 3 = KT must be counted downwards from the critical temperature T less about 6Ã‚°.
• By division we obtain n 3 = 2.121/K i, or n=(2.121/K i) i, the coefficient of association being thus determined.
• If we regard the thermal effect at each junction as a measure of the potential-difference there, as the total thermal effect in the cell undoubtedly is of the sum of its potentialdifferences, in cases where the temperature coefficient is negligible, the heat evolved on solution of a metal should give the electrical potential-difference at its surface.
• Ã¯¿½ Oxl d 2x 77n If we have new variables z such that zs=4s(yl, Y2,...yn), we have also z s =1 Y 8(x1, x2,Ã¯¿½Ã¯¿½Ã¯¿½xn), and we may consider the three determinants which i s 7xk, the partial differential coefficient of z i, with regard to k .
• Auxiliary Theorem.-The coefficient of l l i xl2x13...
• Let a covariant of degree e in the variables, and of degree 8 in the coefficients (the weight of the leading coefficient being w and n8-2w = Ã¯¿½), be Coxl -}- ec l l 1 x 2 -{-...
• 1-az2....1-azn' Hence (w; 0, n) - (w - I; 0, n) is given by the coefficient of aez'Ã‚° in the fraction 1-z 1 -a.1-az.
• In general the coefficient, of any product A n A m A 7, 3 ..., will have, as coefficient, a seminvariant which, when expressed by partitions, will have as leading partition (preceding in dictionary order all others) the partition (Tr1lr2lr3Ã¯¿½..).
• +bx 2, every leading coefficient of a simultaneous covariant vanishes by the operation of a+Sib=aoda +alda.2+...+a7,-1d a P+bod b Observe that we may employ the principle of suffix diminution to obtain from any seminvariant one appertaining to a (p-I)i c and a q - I ie, and that suffix augmentation produces a portion of a higher seminvariant, the degree in each case remaining unaltered.
• Taking The First Generating Function, And Writing Az P, Bz4, 2 For A, B And Z Respectively, We Obtain The Coefficient Of Aobe'Zpo 0' 2W That Is Of A E B E 'Z Ã¯¿½, In 1 Z 2 1 Azp. 1 Azp 2....1 A2 P 2.1 Az P .
• Moreau (C. R., 1900, 130, pp. 122, 412, 562) that if K is the coefficient of the Hall effect (I) and K' the analogous coefficient of the Nernst effect (i.) (which is constant for small values of H), then K' = Ka/p, v being the coefficient of the Thomson effect for the metal and p its specific resistance.
• The numerical factor 6 is called the coefficient of a 5 bc 2 (Ã¯¿½ 20); and, generally, the coefficient of any factor or of the product of any factors is the product of the remaining factors.
• (ii.) To verify this, let us denote the true coefficient of An-rar by (,), so that we have to prove that (;`.) = n(r), where n(r) is defined by (I); and let us inspect the actual process of multiplying the expansion of (A+ a) n -' by A+a in order to obtain that of (A+a)".
• In Ã¯¿½ 41 (ii.), for instance, the coefficient of A n - r a r in the expansion of (Ada) (A+a) n - 1 has been called (n, .); and it has then been shown that (ii) = (n _ I) d (i).
• Thus we arrive at the differential coefficient of f(x) as the limit of the ratio of f (x+8) - f (x) to 0 when 0 is made indefinitely small; and this gives an interpretation of nx n-1 as the derived function of xn (Ã¯¿½ 45)Ã¯¿½ This conception of a limit enables us to deal with algebraical expressions which assume such forms as -Ã‚° o for particular values of the variable (Ã¯¿½ 39 (iii.)).
• Its coefficient of linear expansion between 0Ã‚° and 100Ã‚° is 0.002717; its specific heat 0.0562; its thermal and electrical conductivities are 145 to 152 and '14.5 to 140.
• The coefficient of linear expansion is 0.002,905 for 100Ã‚° from oÃ‚° upwards (Fizeau).
• Its high coefficient of thermal expansion, coupled with its low freezing point, renders it a valuable thermometric fluid, especially when the temperatures to be measured are below - 39Ã‚° C., for which the mercury thermometer cannot be used.
• Matthiessen as 73 at 0Ã‚° C., pure silver being 100; the value of this coefficient depends greatly on the purity of the metal, the presence of a few thousandths of silver lowering it by 10%.
• Its coefficient of expansion for each degree between oÃ‚° and Ioo C. is 0.000014661, or for gold which has been annealed 0.000015136 (Laplace and Lavoisier).
• Its thermal conductivity is the lowest of all metals, being 18 as compared with silver as 1000; its coefficient of expansion between oÃ‚° and tooÃ‚° is 0.001341.
• It melts between 2250Ã‚° and 2300Ã‚°, its specific heat is 0.0365, coefficient of expansion o 0000079, and specific gravity 16.64.
• The surface tension, on the other hand, is greater than that of pure water and increases with the salinity, according to Kriimmel, in the manner shown by the equation a=77.09+o 0221 S at oÃ‚° C., where a is the coefficient of surface tension and S the salinity in parts per thousand.
• Conduction has practically no effect, for the coefficient of thermal conductivity in sea-water is so small that if a mass of sea-water were cooled to 0Ã‚° C. and the surface kept at a temperature of 30Ã‚° C., 6 months would elapse before a temperature of 15Ã‚° C. was reached at the depth of 1 3 metres, 1 year at 1 85 metres, and io years at 5.8 metres.
• If the two materials are disposed symmetrically, the amount of load carried by each would be in direct proportion to the coefficient of elasticity and inversely as the moment of inertia of the cross section.
• The resistance of the air is reduced considerably in modern projectiles by giving them a greater length and a sharper point, and by the omission of projecting studs, a factor called the coefficient of shape, being introduced to allow for this change.
• Its coefficient of linear expansion is only 0.0000008 for 1Ã‚° C. See: Rapport du Yard, Dr Benoit (1896).
• Its coefficient of linear expansion by heat is 0.0000222 (Richards) or 0.0000231 (RobertsAusten) per 1Ã‚° C. Its mean specific heat between oÃ‚° and iooÃ‚° is 0.227, and its latent heat of fusion loo calories (Richards).
• The coefficient of correlation is 0.5699, which indicates that the standard deviation of an array is equal to that of the leaves in general multiplied by 1 / I - (0.5699) 1; and performing this multiplication, we find 1.426 as the standard deviation of an array.
• The Result Was To Reduce The Coefficient Of Diminution Of Specific Heat At 15Ã‚° C. By Nearly One Half, But The Absolute Value At 20Ã‚° C. Is Practically Unchanged.
• In Spite Of The Large Corrections The Results Were Extremely Consistent, And The Value Of The Temperature Coefficient Of The Diminution Of The Specific Heat Of Water, Deduced From The Observed Variation In The Rate Of Rise At Different Points Of The Range 15Ã‚° To 25Ã‚°, Agreed With The Value Subsequently Deduced From Rowland'S Experiments Over The Same Range, When His Thermometers Were Reduced To The Same Scale.
• Admitting the value S =0.497 for the specific heat at 108Ã‚° 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 (looÃ‚° to 190Ã‚° C.) of the experiments on which it chiefly depends.
• The mean value, 0.313 of dH/d0, between looÃ‚° and 200Ã‚° agrees fairly well with Regnault's coefficient 0.305, but it is clear that considerable errors in calculating the wetness of steam or the amount of cylinder condensation would result from assuming this important coefficient to be constant.
• Its specific heat is o 0899 at 0Ã‚° C. and 0.0942 at 10o; the coefficient of linear expansion per 1Ã‚° C. is o o01869.
• The extension of a spiral spring is given by the formula: Extension =W4n1VÃƒâ€  r 4, in which W = weight causing extension, in lbs; n = number of coils; R = radius of spring, from centre of coil to centre of wire, in inches; r = radius of wire of which the spring is made, in inches; E = coefficient of elasticity of wire, in lbs per square inch.
• The number of coefficients is 2(m+ r) (m+2); but there is no loss of generality if the equation be divided by one coefficient so as to reduce the coefficient of the corresponding term to unity, hence the number of coefficients may be reckoned as 1(m-1-- 1) (m+2) - r, that is, Zm(m+3); and a curve of the order in may be made to satisfy this number of conditions; for example, to pass through Zrn(m+3) points.
• Its specific gravity is given variously from 5.395 to 5'959; its specific heat is 0.083, and its coefficient of linear expansion 0.0000-0559 (at 40Ã‚° C.).
• In using the above table to find the value of E or dE/dt at any temperature or between any limits, denoting by p the value of dE/dt at 50Ã‚° C., and by 2C the constant value of the second coefficient, we have the following equations :- dE/dt = p+2c(t -50), at any temperature t, Cent.
• 4, which is given as an illustration, the cold junctions are supposed to be at oÃ‚° C. and the hot junctions at 100Ã‚° C. Noll's values (Table I.) are taken for the E.M.F., and it is supposed that the coefficient of the Thomson effect is zero in lead, i.e.
• Taking the lead-iron couple as an example, the value of dE/dt at the hot junction tooÃ‚° C. is 10.305 microvolts per degree, and the value of the Peltier coefficient P = TdE/dT is +3844 microvolts.
• If, however, the heat is to be rejected at looÃ‚°, then the coefficient is reduced to 4.6.
• An ideal machine, working between 5Ã‚° below zero and 75Ã‚° above, has a coefficient of about 5.7, or nearly six times that of an ideal compressed-air machine of usual construction performing the same useful cooling work.
• Every extra moment that is calculated allows another recursion coefficient to be found.
• The rate of movement of this boundary gives the sedimentation coefficient s.
• Explain what is meant by the Peltier coefficient and thermoelectric power.
• 6 1.86 3.01 o 88 1.03 Since at the boiling-point under atmospheric pressure liquids are in corresponding states, the additive nature of the critical coefficient should also be presented by boiling-points.
• Arrhenius has pointed out that the coefficient of affinity of an acid is proportional to its electrolytic ionization.
• In general the coefficient, of any product A n A m A 7, 3 ..., will have, as coefficient, a seminvariant which, when expressed by partitions, will have as leading partition (preceding in dictionary order all others) the partition (Tr1lr2lr3ï¿½..).
• The generating function is I - z2' 52 For 0 =3, (alai +a2a2+a3a3) 10; the condition is clearly a1a2a3 = A3 = 0, and since every seminvariant, of proper degree 3, is associated, as coefficient, with a product containing A3, all such are perpetuants.
• The coefficient K/(i +171-K) is positive for ferromagnetic and paramagnetic substances, which will therefore tend to move from weaker to stronger parts of the field; for all known diamagnetic substances it is negative, and these will tend to move from stronger to weaker parts.
• But in 3.4a the coefficient of 4a is 3, while the coefficient of a is3.4.
• In the case of the inductive mode of exciting the oscillations an important quantity is the coefficient of coupling of the two oscillation circuits.
• Viscosity increases with density, but oils of the same density often vary greatly; the coefficient of expansion, on the other hand, varies inversely with the density, but bears no simple relation to the change of fluidity of the oil under the influence of heat, this being most marked in oils of paraffin base.
• We can calculate the heat of formation from its ions for any substance dissolved in a given liquid, from a knowledge of the temperature coefficient of ionization, by means of an application of the well-known thermodynamical process, which also gives the latent heat of evaporation of a liquid when the temperature coefficient of its vapour pressure is known.
• The earliest formulation of the subject, due to Lord Kelvin, assumed that this relation was true in all cases, and, calculated in this way, the electromotive force of Daniell's cell, which happens to possess a very small temperature coefficient, was found to agree with observation.
• Now by the theory of symmetric functions, any symmetric functions of the mn values which satisfy the two equations, can be expressed in terms of the coefficient of those equations.
• Hence, finally, the resultant is expressed in terms of the coefficients of the three equations, and since it is at once seen to be of degree mn in the coefficient of the third equation, by symmetry it must be of degrees np and pm in the coefficients of the first and second equations respectively.
• 1 2 3 We have found above that the coefficient of (x 1 1 x 12 x 13...) i n the product XmiXm2X m3 ...
• It is definied as having four elements, and is written the coefficient of a0 o a1 a2 2 ...
• The number of partitions of a biweight pq into exactly i biparts is given (after Euler) by the coefficient of a, z xPy Q in the expansion of the generating function 1 - ax.
• Solving the equation by the Ordinary Theory Of Linear Partial Differential Equations, We Obtain P Q 1 Independent Solutions, Of Which P Appertain To S2Au = 0, Q To 12 B U =0; The Remaining One Is Ab =Aobl A 1 Bo, The Leading Coefficient Of The Jacobian Of The Two Forms. This Constitutes An Algebraically Complete System, And, In Terms Of Its Members, All Seminvariants Can Be Rationally Expressed.
• At barometric pressures such as exist between 18 and 36 kilometres above the ground the mobility of the ions varies inversely as the pressure, whilst the coefficient of recombination a varies approximately as the pressure.
• The coefficient of friction is a variable quantity depending upon the state of the rails, but is usually taken to be This is the fundamental equation between the forces acting, however the torque may be applied.
• But the temperature coefficient of conductivity is now generally less than before; thus the effect of temperature on ionization must be of opposite sign to its effect on fluidity.
• In order to obtain the seminvari ants we would write down the (w; 0, n) terms each associated with a literal coefficient; if we now operate with 52 we obtain a linear function of (w - I; 8, n) products, for the vanishing of which the literal coefficients must satisfy (w-I; 0, n) linear equations; hence (w; 8, n)-(w-I; 0, n) of these coefficients may be assumed arbitrarily, and the number of linearly independent solutions of 52=o, of the given degree and weight, is precisely (w; 8, n) - (w - I; 0, n).
• It is shown in the article on Combinatorial Analysis that (w; 0,n) is the coefficient of a e z w in the ascending expansion of the fraction 1-a.
• 1 -az 74 - 2.1 -azn-4....1 - azn+4.1 - az n+2.1 - az-n in which we have to take the coefficient of aezne-2', the expansion.
• The existence of such a relation, as 0-1+0-2+.,.+cr2=0, necessitates the vanishing of a certain function of the coefficients A2, A 3, ...A 9, and as a consequence one product of these coefficients can be eliminated from the expanding form and no seminvariant, which appears as a coefficient to such a product (which may be the whole or only a part of the complete product, with which the seminvariant is associated), will be capable of reduction.
• Taking The First Generating Function, And Writing Az P, Bz4, 2 For A, B And Z Respectively, We Obtain The Coefficient Of Aobe'Zpo 0' 2W That Is Of A E B E 'Z ï¿½, In 1 Z 2 1 Azp. 1 Azp 2....1 A2 P 2.1 Az P .
• The internal force F is opposite to the direction of the magnetization, and equal to NI, where N is a coefficient depending only on the ratio of the axes.
• For small bodies other than spheres the coefficient will be different, but its sign will always be negative for diamagnetic substances and positive for others; hence the forces acting on any small body will be in the same directions as in the case of a sphere' Directing Couple acting on an Elongated Body.
• For the pressure coefficient per degree, between o° and Ice C., they give the value 0036-6255, when the initial pressure is 700 mm.
• If L and N are the inductances of any two circuits which have a coefficient of mutual inductance M, then M/-/ (LN) is called the coefficient of coupling of the circuits and is generally expressed as a percentage.
• Two circuits are said to be closely coupled when this coefficient is near unity and to be loosely coupled if it is very small.
• Ramsay and Shields found from investigations of the temperature coefficient of the surface energy that Tin the equation y(Mv) 3 = KT must be counted downwards from the critical temperature T less about 6°.
• Suppose the coefficient of association be n, i.e.
• The Partial Differential Equations.--It will be shown later that covariants may be studied by restricting attention to the leading coefficient, viz.
• He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the contragredient line variables-it is completely determinate if its leading coefficient be known.
• Two of these show that the leading coefficient of any covariant is an isobaric and homogeneous function of the coefficients of the form; the remaining two may be regarded as operators which cause the vanishing of the covariant.
• Let a covariant of degree e in the variables, and of degree 8 in the coefficients (the weight of the leading coefficient being w and n8-2w = ï¿½), be Coxl -}- ec l l 1 x 2 -{-...
• It is for this reason called a seminvariant, and every seminvariant is the leading coefficient of a covariant.
• It is then possible to assign to each body a specific coefficient of affinity.
• The temperature coefficient of conductivity has approximately the same value for most aqueous salt solutions.
• 1-az2....1-azn' Hence (w; 0, n) - (w - I; 0, n) is given by the coefficient of aez'° in the fraction 1-z 1 -a.1-az.
• ï¿½ Oxl d 2x 77n If we have new variables z such that zs=4s(yl, Y2,...yn), we have also z s =1 Y 8(x1, x2,ï¿½ï¿½ï¿½xn), and we may consider the three determinants which i s 7xk, the partial differential coefficient of z i, with regard to k .
• = ...+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 +.
• 1 Bz4' That Is, By The Coefficient Of Z W In Zp '.