# Coefficient Sentence Examples

- Suppose we wish to find the
**coefficient**of (52413) in the product (20(2' 4)(0). - By the x process of Aronhold we can form the invariant S for the cubic ay+XH:, and then the
**coefficient**of X is the second invariant T. - Arrhenius has pointed out that the
**coefficient**of affinity of an acid is proportional to its electrolytic ionization. - This ratio, termed by Guye the critical
**coefficient**, has the following approximate values: C. H. - 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. - We may, by a well-known theorem, write the result as a
**coefficient**of z w in the expansion of 1 - z n+1. - 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. - 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. - 1 2 3 We have found above that the
**coefficient**of (x 1 1 x 12 x 13...) i n the product XmiXm2X m3 ... - 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. - The Number Of Linearly Independent Seminvariants Of The Given Type Will Then Be Denoted By (W; 0, P; 0', Q) (W; 0, P; 0', Q); And Will Be Given By The
**Coefficient**Of A E B E 'Z W In L Z 1 A. - 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. - 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°. - 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
of the three equations, and since it is at once seen to be of degree mn in the**coefficients****coefficient**of the third equation, by symmetry it must be of degrees np and pm in theof the first and second equations respectively.**coefficients** - 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. - The orders of the quantic and covariant, and the degree and weight of the leading
**coefficient**; calling these 'n, e,' 0, w respectively we can see that they are not independent integers, but that they are invariably connected by a certain relation n9 -2w = 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 theof the form; the remaining two may be regarded as operators which cause the vanishing of the covariant.**coefficients** - Let a covariant of degree e in the variables, and of degree 8 in the
(the weight of the leading**coefficients****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. - From the invariant a2 -2a 1 a 3 -2aoa4 of the quartic the diminishing process yields ai-2a 0 a 21 the leading
**coefficient**of the Hessian of the cubic, and the increasing process leads to a3 -2a 2 a 4 +2a i a 5 which only requires the additional term-2aoa 6 to become a seminvariant of the sextic. A more important advantage, springing from the new form of S2, arises from the fact that if x"-aix n- +a2x n-2. - 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 literalmust 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).**coefficients** - 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. - A irl aï¿½ 2 a a3 ...Ev 1 02 2 ?3 3 ...; and, if we express Ea l v2 2 0-3 3 in terms of A2, A3 i ..., and arrange the whole as a linear function of products of A2, A3,..., each
**coefficient**will be a seminvariant, and the aggregate of thewill give us the complete asyzygetic system of the given degree and weight.**coefficients** - The existence of such a relation, as 0-1+0-2+.,.+cr2=0, necessitates the vanishing of a certain function of the
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**coefficients****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. - Hence, whatever the position of the body, if the field be resolved into three components parallel to the 1 For all except ferromagnetic substances the
**coefficient**is sensibly equal to See W. - Ber., 1886, 94, 560) have found that the rotational
**coefficient**of tellurium is more than fifty times greater than that of bismuth, its sign being positive. - 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. - 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. - 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. - Suppose the
**coefficient**of association be n, i.e. - It is then possible to assign to each body a specific
**coefficient**of affinity. - Abrupt alterations, take place in its density, specific heat, thermo-electric quality, electrical conductivity, temperature-
**coefficient**of electrical resistance, and in some at least of its mechanical properties. - The temperature
**coefficient**of conductivity has approximately the same value for most aqueous salt solutions. - Nevertheless, in certain cases, the temperature
**coefficient**of conductivity becomes negative at high temperatures, a solution of phosphoric acid, for example, reaching a maximum conductivity at 75° C. - 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 . - On the right-hand side is such that the
**coefficient**of x n ix n Zx n 3... - = ...+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 +. - And take therein the
**coefficient**of the function (mi t tm7t t m 31 t ...), we obtain the same result as if we formed the separation function in regard to the specification (mï¿½ It t'tm2 32tm"`l3t...), multiplied by Alt!! - And took ï¿½1a ï¿½ therein the
**coefficient**of the function (mis mï¿½ 2s Es m 3s 3s ...). - 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. - 1 Bz4' That Is, By The
**Coefficient**Of Z W In Zp '.