(21) a formula giving the coefficient of transmission in terms of the refraction, and of the number of particles per unit volume.
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 this rise is equivalent to half a vibration (o.5) per degree Fahr.
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.
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).
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.
This ratio, termed by Guye the critical coefficient, has the following approximate values: C. H.
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.
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.
By division we obtain n 3 = 2.121/K i, or n=(2.121/K i) i, the coefficient of association being thus determined.
It is then possible to assign to each body a specific coefficient of affinity.
Arrhenius has pointed out that the coefficient of affinity of an acid is proportional to its electrolytic ionization.
The temperature coefficient of conductivity has approximately the same value for most aqueous salt solutions.
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.
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.
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.
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.
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 .
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.
= ...+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 2 3 We have found above that the coefficient of (x 1 1 x 12 x 13...) i n the product XmiXm2X m3 ...
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!!
Suppose we wish to find the coefficient of (52413) in the product (20(2' 4)(0).
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.
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.
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.
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 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-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.
We may, by a well-known theorem, write the result as a coefficient of z w in the expansion of 1 - z n+1.
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 the coefficients will give us the complete asyzygetic system of the given degree and weight.
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 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.
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.
+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.
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.
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.
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.
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.
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.
Ashworth, 9 who showed that the temperature coefficient of permanent magnets might be reduced to zero (for moderate ranges of temperature) by suitable adjustment of temper and dimension ratio; also by R.
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.