# How to use *Coefficients* in a sentence

This can be verified by equating to zero the five

**coefficients**of the Hessian (ab) 2 axb2.To obtain the real form we multiply out, and, in the result, substitute for the products of symbols the real

**coefficients**which they denote.The

**coefficients**of the symbols must then present themselves n 1, n 2, n 3 ...times respectively.In pure algebra Descartes expounded and illustrated the general methods of solving equations up to those of the fourth degree (and believed that his method could go beyond), stated the law which connects the positive and negative roots of an equation with the changes of sign in the consecutive terms, and introduced the method of indeterminate

**coefficients**for the solution of equations.'It is found that the influence of different acids on this action is proportional to their specific

**coefficients**of affinity.AdvertisementHence, if we assume that, in the Daniell's cell, the temperature

**coefficients**are negligible at the individual contacts as well as in the cell as a whole, the sign of the potential-difference ought to be the same at the surface of the zinc as it is at the surface of the copper.Other " Galois " groups were defined whose substitution

**coefficients**have fixed numerical values, and are particularly associated with the theory of equations.Further we find x=aX+a'Y+a"Z, y=bX z= cX+c'Y+ c"Z, and the problem is to express the nine

**coefficients**in terms of three independent quantities.From the differential

**coefficients**of the y's with regard to the x's we form the functional.R is a function of the

**coefficients**which is called the " resultant " or " eliminant " of the k equations, and the process by which it is obtained is termed " elimination."AdvertisementThe resultant being a product of mn root differences, is of degree mn in the roots, and hence is of weight mn in the

**coefficients**of the forms; i.e.Assuming then 01 to have the

**coefficients**B1, B2,...B,, and f l the**coefficients**A 1, A21...A,n, we may equate**coefficients**of like powers of x in the identity, and obtain m+n homogeneous linear equations satisfied by the m+n quantities B1, 2, ...B n, A 1, A 2, ...A m.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.It is the resultant of k polynomials each of degree m-I, and thus contains the

**coefficients**of each form to the degree (m-I)'-1; hence the total degrees in the**coefficients**of the k forms is, by addition, k (m - 1) k - 1; it may further be shown that the weight of each term of the resultant is constant and equal to m(m-I) - (Salmon, l.c. p. loo).If the form, sometimes termed a quantic, be equated to zero the n+I

**coefficients**are equivalent to but n, since one can be made unity by division and the equation is to be regarded as one for the determination of the ratio of the variables.AdvertisementIn the theory of forms we seek functions of the

**coefficients**and variables of the original quantic which, save as to a power of the modulus of transformation, are equal to the like functions of the**coefficients**and variables of the transformed quantic. We may have such a function which does not involve the variables, viz.Invariantive forms will be found to be homogeneous functions alike of the

**coefficients**and of the variables.If we restrict ourselves to this set of symbols we can uniquely pass from a product of real

**coefficients**to the symbolic representations of such product, but we cannot, uniquely, from the symbols recover the real form, This is clear because we can write n-1 n-2 2 2n-3 3 a1a2 =a l a 2, a 1 a 2 = a 1 a2 while the same product of umbrae arises from n n-3 3 2n-3 3 aoa 3 = a l .a a 2 = a a 2 .We write;L 22 = a 1 a 2 .b 1 n-2 b2s 3 n - 3 3 n-3 3 n-3 3 a 3 = a 1 a 2 .b 1 b 2 .c 1 c2, and so on whenever we require to represent a product of real

**coefficients**symbolically; we then have a one-to-one correspondence between the products of real**coefficients**and their symbolic forms. If we have a function of degree s in the**coefficients**, we may select any s sets of umbrae for use, and having made a selection we may when only one quantic is under consideration at any time permute the sets of umbrae in any manner without altering the real significance of the symbolism.For a single quantic of the first order (ab) is the symbol of a function of the

**coefficients**which vanishes identically; thus (ab) =a1b2-a2bl= aw l -a1ao=0 and, indeed, from a remark made above we see that (ab) remains unchanged by interchange of a and b; but (ab), = -(ba), and these two facts necessitate (ab) = o.AdvertisementThere also exist functions, which involve both sets of variables as well as the

**coefficients**of u, possessing a like property; such have been termed mixed concomitants, and they, like contravariants, may appertain as well to a system of forms as to a single form.We cannot, however, say that it is an invariant unless it is expressible in terms of the real

**coefficients**.When either of the forms is of an order higher than the first (ab), as not being expressible in terms of the actual

**coefficients**of the forms, is not an invariant and has no significance.For these only will the symbolic product be replaceable by a linear function of products of real

**coefficients**.X1, X 2, u1, /22 being as usual the

**coefficients**of substitution, let x1a ?AdvertisementAn instantaneous deduction from the relation w= 2 n0 is that forms of uneven orders possess only invariants of even degree in the

**coefficients**.The discriminant is the resultant of ax and ax and of degree 8 in the

**coefficients**; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5.Hesse's canonical form shows at once that there cannot be more than two independent invariants; for if there were three we could, by elimination of the modulus of transformation, obtain two functions of the

**coefficients**equal to functions of m, and thus, by elimination of m, obtain a relation between the**coefficients**, showing them not to be independent, which is contrary to the hypothesis.This is of degree 8 in the

**coefficients**, and degree 6 in the variables, and, for the canonical form, has the expression -9m 6 (x 3 +y 3 +z 3) 2 - (2m +5m 4 +20m 7) (x3 +y3+z3)xyz - (15m 2 +78m 5 -12m 8) Passing on to the ternary quartic we find that the number of ground forms is apparently very great.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.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).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.Q 1 The Unreduced Generating Function Which Enumerates The Covariants Of Degrees 0, 0' In The

**Coefficients**And Order E In The Variables.It may denote a simultaneous orthogonal invariant of forms of orders n i, n2, n3,...; degree 0 of the covariant in the

**coefficients**.These researches derive additional importance from having introduced two powerful engines of analysis for the treatment of physical problems, Laplace's

**coefficients**and the potential function.Laplace published in 1779 the method of generating functions, the foundation of his theory of probabilities, and the first part of his Theorie analytique is devoted to the exposition of its principles, which in their simplest form consist in treating the successive values of any function as the

**coefficients**in the expansion of another function with reference to a different variable.A direct and an inverse calculus is thus created, the object of the former being to determine the

**coefficients**from the generating function, of the latter to discover the generating function from the**coefficients**.Exercises in the collection of

**coefficients**of various letters occurring in a complicated expression are usually performed mechanically, and are probably of very little value.We disregard numerical

**coefficients**, so that by the H.C.F.In the same way we have (A-a) 2 =A 2 -2Aa+a 2, (A-a)3 = A 3 -3A 2 a+3Aa 2 -a 3, ..., so that the multinomial equivalent to (A-a)" has the same

**coefficients**as the multinomial equivalent to (A+a)", but with signs alternately + and -.We therefore determine the

**coefficients**by counting the grouped terms individually, instead of adding them.We know that (A+a)" is equal to a multinomial of n+I terms with unknown

**coefficients**, and we require to find these**coefficients**.This is the method of undetermined

**coefficients**.The

**Coefficients**In The Expansion Of (A A) N For Any Particular Value Of N Are Obtained By Reading Diagonally Upwards From Left To Right From The (N 1)Th Number In The First Column.This property enables us to establish, by simple reasoning, certain relations between binomial

**coefficients**.These constructed symbols may be called positive and negative

**coefficients**; or a symbol such as (- p) may be called a negative number, in the same way that we call 3 a fractional number.It may be necessary to introduce terms with zero

**coefficients**.For instance, by equating

**coefficients**of or in the expansions of (I +x) m+n and of (I dx) m .Considered in this way, the relations between the

**coefficients**of the powers of x in a series may sometimes be expressed by a formal equality involving the series as a whole.This accounts for the fact that the same table of binomial

**coefficients**serves for the expansions of positive powers of i+x and of negative powers of i - x.Then a+a = as = a; hence numerical

**coefficients**and indices are not required.His principal discovery is concerned with equations, which he showed to be derived from the continued multiplication of as many simple factors as the highest power of the unknown, and he was thus enabled to deduce relations between the

**coefficients**and various functions of the roots.If we put for shortness 7 for the quantity under the last circular function in (I), the expressions (i), (2) may be put under the forms u sin T, v sin (T - a) respectively; and, if I be the intensity, I will be measured by the sum of the squares of the

**coefficients**of sin T and cos T in the expression u sin T +v sin (T - a), so that I =u 2 +v 2 +2uv cos a, which becomes on putting for u, v, and a their values, and putting f =Q .The resolution of the questions concerning the motion of fluids was effected by means of Euler's partial differential

**coefficients**.Bornemann re-examined all these results with great care, and gave formulae expressing the variation of the

**coefficients**of discharge in different conditions (Civil Ingenieur, 1880).Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; n ap, dy, P d z, or X, Y, Z (4) are the partial differential

**coefficients**of some function P, =fdplp, of x, y, z; so that X, Y, Z must be the partial differential**coefficients**of a potential -V, such that the force in any direction is the downward gradient of V; and then dP dV (5) ax + Tr=0, or P+V =constant, in which P may be called the hydrostatic head and V the head of potential.The hypothesis that the state was steady, so that interchanges arising from convection and collisions of the molecules produced no aggregate result, enabled him to interpret the new constants involved in this law of distribution, in terms of the temperature and its spacial differential

**coefficients**, and thence to express the components of the kinetic stress at each point in the medium in terms of these quantities.The results coincide with Maxwell's so far as above stated, though the numerical

**coefficients**do not agree.He knew the connexion existing between the positive roots of an equation (which, by the way, were alone thought of as roots) and the

**coefficients**of the different powers of the unknown quantity.These measurements were utilized in combination with appropriate elastic

**coefficients**of the material to find the horse-power transmitted from the engines along the shaft to the propeller.Ow are any momenta or functions of the co-ordinates and momenta or co-ordinates alone which are subject only to the condition that they do not enter into the

**coefficients**a 1, a 2, &c.Since the volume at constant pressure is exactly proportional to the absolute temperature, it follows that the

**coefficients**of expansion of all gases ought, to within the limits of error introduced by the assumptions on which we are working, to have the same value 1/273.Writing m = 2p, and grouping the

**coefficients**of the successive differences, we shall find area = 2ph up+ 2 652up + 3 p4365p2 84up 3p,6 - 21p4 28p2 15120 If u is of degree 2f or 2f + i in x, we require to go up to b 2f u p, so that m must be not less than 2f.In what follows it will be assumed that the conditions of continuity (which imply the continuity not only of u but also of some of its differential

**coefficients**) are satisfied, subject to the small errors in the values of u actually given; the limits of these errors being known.The most simple case is that in which the trapezette tapers out in such a way that the curve forming its top has very close contact, at its extremities, with the base; in other words, the differential

**coefficients**u', u", u"',.The following are the results (for the formulae involving chordal areas), given in terms of differential

**coefficients**and of central differences.In the first of these, entitled " Recherches sur l'attraction des spheroides homogenes," published in the Memoires of the Academy for 1785, but communicated to it at an earlier period, Legendre introduces the celebrated expressions which, though frequently called Laplace's

**coefficients**, are more correctly named after Legendre.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.The introduction of the

**coefficients**now called Laplace's, and their application, commence a new era in mathematical physics.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.Schlomilch defines these functions as the

**coefficients**of the power of t in the expansion of exp 2p(t - t1).The difference between the

**coefficients**o 97 and 1 17 arises from the refraction of the ray, but for which they would be equal.In spite of this difficulty, however, the values of the correlation

**coefficients**so far obtained cluster fairly well round the mean value of all of them, which is almost exactly 2.He has given by means of it a simple proof of the existence of n roots, and no more, in every rational algebraic equation of the nth order with real

**coefficients**.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.It is generally called Dupre's formula in continental text-books, but he did not give the values of the

**coefficients**in terms of the difference of specific heats of the liquid and vapour.It was employed as a purely empirical formula by Bertrand and Barus, who calculated the values of the

**coefficients**for several substances, so as to obtain the best general agreement with the results of observation over a wide range, at high as well as low pressures.The value of c is determined by the throttling experiments, so that all the

**coefficients**in the formula with the exception of A are determined independently of any observations of the saturationpressure itself.In the application to mechanics these

**coefficients**are the masses of particles situate at the respective points, and are therefore all positive.The signs of the

**coefficients**follow the same rule as in the case of (4).The images of the Gauss theory being of the third order, the next problem is to obtain an image of 5th order, or to make the

**coefficients**of the powers of 3rd degree zero.The aberrations can also be expressed by means of the "characteristic function " of the system and its differential

**coefficients**, instead of by the radii, &c., of the lenses; these formulae are not immediately applicable, but give, however, the relation between the number of aberrations and the order.The value of such a fraction is the positive root of a quadratic equation whose

**coefficients**are real and of which one root is negative.If r be the number of quotients in the recurring cycle, we can by writing down the relations connectin g the successive p's and q's obtain a linear relation connecting p nr +m, t'(n-1)r +m, +m in which the

**coefficients**are all constants.The experiments on solids lead to certain laws of elasticity expressed in terms of

**coefficients**, the values of which can be determined only by experiments on each particular substance.By far the simplest supposition open to us is that the functions are the same in all cases, the attractions differing merely by

**coefficients**analogous to densities in the theory of gravitation.A curve of the order na has for its equation (*1 x, y, 1)m=o; and when the

**coefficients**of the function are arbitrary, the curve is said to be the general curve of the order in.We can only use the general equation (*fix, y, z) m = o, say for shortness u= o, of a curve of the mth order, which equation, so long as the

**coefficients**remain arbitrary, represents a curve without nodes or cusps.Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§ 8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second differential

**coefficients**of u in regard to the variables (x, y, z); H= o is thus a curve of the order 3 (m - 2), and the number of inflections is =3m(m-2).It may be noticed that the nine inflections of a cubic curve represented by an equation with real

**coefficients**are three real, six imaginary; the three real inflections lie in a line, as was known to Newton and Maclaurin.If, however, the geometrical property requires two or more relations between the

**coefficients**, say A = o, B = o,&c., then we must have between the new**coefficients**the like relations, A' = o, B' = o, &c., and the two systems of equations must each of them imply the other; when this is so, the system of equations, A = o, B = o, &c., is said to be invariantive, but it does not follow that A, B, &c., are of necessity invariants of u.Here the

**coefficients**of 1 and l' may separately take all integral values, though as a general rule the**coefficients**a, b, c, &c. diminish rapidly when these**coefficients**become large, so that only small values have to be considered.The expressions for the longitude, latitude `and parallax appear as an infinite trigonometric series, in which the

**coefficients**of the sines and cosines are themselves infinite series proceeding according to the powers of the above small numbers.Now Fresnel's formulae were obtained by assuming that the incident, reflected and refracted vibrations are in the same or opposite phases at the interface of the media, and since there is no real factor that converts cos T into cos (T+p), he inferred that the occurrence of imaginary expressions for the

**coefficients**of vibration denotes a change of phase other than 7r, this being represented by a change of sign.Formulae for metallic reflection may be obtained from Fresnel's expressions by writing the ratio sin i / sin r equal to a complex quantity, and interpreting the imaginary

**coefficients**in the manner explained above.The

**coefficients**, P and P', are called**coefficients**of the Peltier effect, and may be stated in calories or joules per ampere-second.Consider an elementary couple of two metals A and B for which s has the values s and s" respectively, with junctions at the temperature T and T+dT (absolute), at which the

**coefficients**of the Peltier effect are P and P+dP. Equating the quantity of heat absorbed to the quantity of electrical energy generated, we have by the first law of thermodynamics the relation dE/dT =dP/dT+(s' - s").The signs of the Peltier and Thomson effects will be the same as the signs of the

**coefficients**given in Table I., if we suppose the metal s to be lead, and assume that the value of s may be taken as zero at all temperatures.In analytical geometry the conic is represented by an algebraic equation of the second degree, and the species of conic is solely determined by means of certain relations between the

**coefficients**.The subject of analytical geometry which he virtually created enabled him to view the conic sections as algebraic equations of the second degree, the form of the section depending solely on the

**coefficients**.These

**coefficients**for a number of cases met with in practice are given in the following table.The angular integral of the reduced matrix element is expressed in terms of recoupling

**coefficients**and**coefficients**of fractional parentage.This method offers the option of using results from the base calculation to calculate the

**coefficients**for the larger data set.They should be familiar with the series expressions for the virial

**coefficients**for a Lennard-Jones potential.Also, you will need to know very basic combinatorial concepts, in particular binomial

**coefficients**.Very few analytic models require that these links be defined as biological links, as is required to legitimately calculate inbreeding

**coefficients**.P0, P1, P2, ..., Pn are the polynomial

**coefficients**.Developed new technique for the determination of the thermal heat transfer

**coefficients**[2] .Typical friction

**coefficients**for these types of bearings are between 0.001 - 0.01.C When one-way surface conflation is active it takes no action, thus preserving C the convection

**coefficients**calculated by adaptive convection algorithm.Even with the modification the blanket estimator gave low correlation

**coefficients**.The

**coefficients**are obtained by solving the spin Hamiltonian.A facility to allow the punching of both transformed integrals and CI

**coefficients**from the full CI code has been introduced.This powerful method is applicable to linear odes or systems of linear ODEs with constant

**coefficients**.Some sensitivity

**coefficients**can be calculated by taking partial derivatives of the equation defining the hardness value.The sum of these two

**coefficients**is always one. a+r = 1 Light colored smooth and shiny surfaces tend to have a higher reflectance.These studies involve measuring reaction rate

**coefficients**at the low temperatures characteristic of the mesosphere/lower thermosphere region.Another peculiarity of Malay (and likewise of Chinese, Shan, Talaing, Burmese and Siamese) is the use of certain classwords or

**coefficients**with numerals, such as orang (man),when speaking of persons, ekor (tail) of animals, keping (piece) of flat things, biji (seed) of roundish things; e.g.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.An important fact, discovered by Cayley, is that these

**coefficients**, and also the complete covariants, satisfy certain partial differential equations which suffice to determine them, and to ascertain many of their properties.The expressions designated by Dr Whewell, Laplace's

**coefficients**(see Spherical Harmonics) were definitely introduced in the memoir of 1785 on attractions above referred to.When, for instance, we find that the quotient, when 6+5x+7x2+13x1+5x4 is divided by 2+35+5 2, is made up of three terms+3, - 2x, and +5x 2, we are really obtaining successively the values of co, c 1, and c 2 which satisfy the identity 6+ 5x+ 7x 2 + 13x 3 + 5x4 = (co+c i x+c 2 x 2) (2+3x+5 2); and we could equally obtain the result by expanding the right-hand side of this identity and equating

**coefficients**in the first three terms, the**coefficients**in the remaining terms being then compared to see that there is no remainder.He was one of the early founders of the theory of determinants; in particular, he invented the functional determinant formed of the n 2 differential

**coefficients**of n given functions of n independent variables, which now bears his name (Jacobian), and which has played an important part in many analytical investigations (see Algebraic Forms).The introduction of the

**coefficients**now called Laplace's, and their application, commence a new era in mathematical physics."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.By forming a device with reflection

**coefficients**of -1 and 1, a bipolar MLS diffuser produces better scattering.Ceramic tiles designed by the calculations of scattering

**coefficients**have been installed for the sidewalls of a 400-seat concert hall.Concurrent validity was established by calculating correlation

**coefficients**between Deleted Essential Test scores and scores on other integrative and holistic tests.From the resulting overall volumetric heat transfer

**coefficients**the total volume curve for 2, 3 and 4-6 stream units.Also the phase difference between the same two wavelet

**coefficients**gives the phase of the cross-spectral density between these signals.The mental and motor scales have high correlation

**coefficients**(.83 and .77 respectively) for test-retest reliability.In general his object is to reduce the final equation to a simple one by making such an assumption for the side of the square or cube to which the expression in x is to be equal as will make the necessary number of

**coefficients**vanish.Multiplying out the right-hand side and comparing

**coefficients**X1 = (1)x1, X 2 = (2) x2+(12)x1, X3 = (3)x3+(21)x2x1+ (13)x1, X4 = (4) x 4+(31) x 3 x 1+(22) x 2+(212) x2x 1 +(14)x1, Pt P2 P3 P1 P2 P3 Xm=?i(m l m 2 m 3 ...)xmlxm2xm3..., the summation being for all partitions of m.