Values Sentence Examples

The government must reflect the different values these groups have.

You have to stay true to your values while destroying something as well.

You love your country's ideals, goals, values, and aspirations.

The values obtained are shown below.

You behaved as becomes a man who values his honor, perhaps too hastily, but we won't go into that.

They can standardize in a thousand more ways to a world economy, while maintaining their values, traditions, and distinctions.

The largest positive and negative values recorded are met with during disturbed weather.

These will suffice to give a general idea of the mean values met with.

To endow the universal substance with moral attributes, to maintain that it is more than the metaphysical ground of everything, to say it is the perfect realization of the holy, the beautiful and the good, can only have a meaning for him who feels within himself what real not imaginary values are clothed in those expressions.

There is no doubt that under average conditions of atmospheric density, the .005 should be replaced by 003, for many independent authorities using different methods have found values very close to this last figure.

It is important to begin the study of graphics with concrete cases rather than with tracing values of an algebraic function.

Since the maxima occur when u = (m +1)7r it nearly, the successive values are not very different from 4 4 4 &c The application of these results to (3) shows that the field is brightest at the centre =o, =0, viz.

From the general formula (2), if A be the area of aperture, 102 = A2 / x2 f (7) The formation of a sharp image of the radiant point requires that the illumination become insignificant when, n attain small values, and this insignificance can only arise as a consequence of discrepancies of phase among the secondary waves from various parts of the aperture.

The part corresponding to negative values of u is similar, OA being a line of symmetry.

When z has one of the values thus determined, (z)=Jo(z).

The accompanying table is given by Lommel, in which the first column gives the roots of J2(z) =o, and the second and third columns the corresponding values of the functions specified.

The expression for the change of intrinsic energy E between any given limits poOo to po is readily found by substituting these values of the specific heats in equations (II) or (13), and integrating between the given limits.

There are six ancient versions of various values.

My purpose in this chapter will not be to persuade the reader of any political doctrine of trade; please apply your own political and social values as you see fit.

But when we pass to solutions of mineral salts and acids - to solutions of electrolytes in fact - we find that the observed values of the osmotic pressures and of the allied phenomena are greater than the normal values.

Thus in the case of cyanacetic acid, while the volume V changed by doubling from 16 to 1024 litres, the values of k were 0.00 (37 6, 373, 374, 361, 362, 361, 368).

The mean values of k for other common acids were - formic, 0.0000214; acetic, o 0000180; monochloracetic, 0.0.0155; dichloracetic, 0.051; trichloracetic, 1.21; propionic, 0.0000134.

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

For since A 11 A 22 -Ar 2 =,,a 33, with similar relations, we have a number of relations similar to A 11 A 22 =AM 2, and either Ars = +11 (A rr A ss) or - (A r .A ss) for all different values of r and s.

We can prove that if the three equations be satisfied by a system of values of the variable, the same system will also satisfy the Jacobian or functional determinant.

Hence in all there are mn such systems. If, therefore, we have a third equation, and we substitute each system of values in it successively and form the product of the mn expressions thus formed, we obtain a function which vanishes if any one system of values, common to the first two equations, also satisfies the third.

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.

In consequence of it, the values formerly found were systematically too small by an amount which even now it is difficult to estimate with precision.

This last value agrees very closely with a determination made by Gill at the Cape of Good Hope, and most other recent determinations give values exceeding 20.50".

Forces acting on a Small Body in the Magnetic Field.-If a small magnet of length ds and pole-strength m is brought into a magnetic field such that the values of the magnetic potential at the negative and positive poles respectively are V 1 and the work done upon the magnet, and therefore its potential energy, will be W =m(V2-Vi) =mdV, which may be written W =m d s- = M d v= - MHo = - vIHo, ds ds where M is the moment of the magnet, v the volume, I the magnetization, and Ho the magnetic force along ds.

In a similar manner, by giving different values to the resistance 4 F R, any desired number of points R= between a and c in the curve can FIG.

Steinmetz's formula may be tested by taking a series of hysteresis curves between different limits of B,' measuring their areas by a pianimeter, and plotting the logarithms of these divided by 47r as ordinates against logarithms of the corresponding maximum values of B as abscissae.

If H l and H2 be the values of 47rinll and 47ri' - 'Z/ l for the 2 2 same induction B, it can be shown that the true magnetizing force is H = H l - (H 2 - H 1).

The values assigned to H were calculated from H= 2ni/r, and ranged from 3.9 to 585, but inasmuch as no account was taken of any 2 Since in most practicable experiments H 2 is negligible in comparison with B 2, the force may be taken as B 2 /87r without sensible= error.

He showed that there was, on the whole, a fair agreement between the values determined ballistic ally and those given by the formula B = 871-F.

Thus any desired number of corresponding values of H and B can be easily and quickly found.

Values of I are derived from (B -H)/477and of from B/H.

For fields of moderate intensity the first term of the expression is the more important, but when the value of H exceeds 12,000 or thereabouts, the second preponderates, and with the highest values that have been actually obtained, HI is several times greater than 21rI 2.

Ann., 18 95, 54,655), who found the limiting values of to be 7.5 to 9.5 for iron, and 11.2 to 13.5 for steel, remaining constant up to H = 06; by P. Culmann (Elekt.

Further, if the alternations take place so slowly that the full maximum and minimum values of the magnetization are reached in the intervals between the reversals, there will again be no dissipation of energy.

Honda subjected tubes of iron, steel and nickel to the simultaneous action of circular and longitudinal fields, and observed the changes of length when one of the fields was varied while the other remained constant at different successive values from zero upwards.

Bidwell 2 accordingly found upon trial that the Wiedemann twist of an iron wire vanished when the magnetizing force reached a certain high value, and was reversed when that value was exceeded; he also found that the vanishing point was reached with lower values of the magnetizing force when the wire was stretched by a weight.

The values of the permeability corresponding to the highest and lowest temperatures are given in the following table.

These high values render hardened tungsten-steel particularly suitable for the manufacture of permanent magnets.

The first column contains the symbols of the various elements which were added to the iron, and the second the percentage proportion in which each element was present; the sample containing 0.03% of carbon was a specimen of the best commercial iron, the values obtained for it being given for comparison.

Annexed are values of Io 6 K for the different salts examined, w being the weight of the salt per c.c. of the solution.

The values assigned to the atomic susceptibilities of most of the known elements are appended.

His determination 2 of maximum and minimum values for the slowly varying planetary eccentricities was the earliest attempt to deal with the problem.

To illustrate the comparative productiveness and relationship of these sources of national wealth and industry, the following official returns of export for the years 1905 and 1906 are arranged in the four general classes previously discussed, the values being in Brazilian gold milreis, worth 2s.

In this period the increase in the sterling equivalents would be proportionately greater than that of the currency values.

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.

Bodin showed a more rational appreciation than many of his contemporaries of the causes of this revolution, and the relation of the variations in money to the market values of wares in general as well as to the wages of labour.

In particular, the equality or inequality of values of two functions is more readily grasped by comparison of the lengths of the ordinates of the graphs than by inspection of the relative positions of their bounding lines.

In the first class come equations in a single unknown; here the function which is equated to zero is the Y whose values for different values of X are traced, and the solution of the equation is the determination of the points where the ordinates of the graph are zero.

The values of the first ten of Bernoulli's numbers are B1= t, B2= 1, B3 =412, B4 =30, B5 =6 = 6 9 1 B7 = l, B =3 =4 4 fl, IV.

We cannot, for instance, say that the fraction C _2 I is arithmetically equal to x+I when x= I, as well as for other values of x; but we can say that the limit of the ratio of x 2 - I to x - I when x becomes indefinitely nearly equal to I is the same as the limit of x+ On the other hand, if f(y) has a definite and finite value for y = x, it must not be supposed that this is necessarily the same as the limit which f (y) approaches when y approaches the value x, though this is the case with the functions with which we are usually concerned.

Thus there appear to be discontinuities in the values of y.

It could not escape notice that one and the same symbol, such as -1 (a - b), or even (a - b), sometimes did and sometimes did not admit of arithmetical interpretation, according to the values attributed to the letters involved.

The values of x and y are different, unless V (qq o) = o.

These values are assumed to be independent, so we have 2n(n - I) derived units of the second species or order.

Even in this case the series converges sufficiently to give the value of the root with considerable accuracy, while for higher values of m it is all that could be desired.

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 .

By separation of real and imaginary parts, C =M cos 27rv 2 +N sin 27rv2 1 S =M sin 27rv 2 - N cos 27rv2 where 35+357.9 N _ 7rv 3 7r 3 v 7 + 1.3 1.3.5.7 1.3.5.7.9.11 These series are convergent for all values of v, but are practically useful only when v is small .

Taking the refractive index of water for the red rays as 0;, and for the violet rays as 1 r, we can calculate the following values for the minimum deviations corresponding to certain assigned values of n.

It may be readily deduced that the directions of minimum deviation for a pencil of parallel rays lie on the surface of cones, the semi-vertical angles of which are equal to the values given in the above table.

Primary, secondary and spurious bows were formed, and their radii measured; a comparison of these observations exhibited agreement with Airy's analytical values.

The total foreign trade in 1908 amounted to $9,778,810 imports and $14,560,830 exports, the values being in U.S. gold.

In the absence of statistical returns it is impossible to give the values of this branch of trade.

Hero's expressions for the areas of regular polygons of from 5 to 12 sides in terms of the squares of the sides show interesting approximations to the values of trigonometrical ratios.

All metals are elastic to this extent that a change of form, brought about by stresses not exceeding certain limit values, will disappear on the stress being removed.

These values may vary within certain limits for different specimens.

Negative values of n must be interpreted by a streaming motion on a parallel plane at a level slightly different, as on a double Riemann sheet, the stream passing from one sheet to the other across a cut SS' joining the foci S, S'.

In the case of texts from the oldest historical periods we encounter hundreds of names that are genuinely Sumerian, and here in view of the multiplicity of the phonetic values attaching to the signs used it is frequently difficult definitely to determine the reading of the names.

Sayce's phonetic values and interpretations of determinatives are his best assured achievements.

The coinage began before the Roman conquest was completed; the monetary system resembles that of the Roman republic, with values analogous to denarii and quinarii.

The values of the coinage are pieces of 5 and 10 centimes in bronze, of 50 centimes, 1 franc and 2 francs in silver, of 10 francs and 20 francs in gold.

In some foreign words like cicala the ch- (tsh) value is given to c. In the transliteration of foreign languages also it receives different values, having that of tsh in the transliteration of Sanskrit and of is in various Slavonic dialects.

So perfectly does the modern Japanese embroiderer elaborate his scheme of values that all the essential elements of pictorial effects chiaroscuro, aerial perspective and atmosphere are present in his work.

In its modern form the theorem, which is true for all values of n, is written as (x +a) n -1+ I.

Newton gave no proof, and it was in the Ars Conjectandi (1713) that James Bernoulli's proof for positive integral values of the exponent was first published, although Bernoulli must have discovered it many years previously.

A rigorous demonstration was wanting for many years, Leonhard Euler's proof for negative and fractional values being faulty, and was finally given by Niels Heinrik Abel.

There are very considerable differences between the values assigned by different observers, sometimes no doubt due to differences in method, but in most cases unquestionably depending on variations in the quality of the specimens examined.

Dividing by (0' - e"), and writing dp/do and dL/do for the limiting values of !the ratios (p' - p")/(o' - o") and (L' - L")/(o' - o"), we obtain the important relations s' - s"+dL/do= (v" - v')dp/do=L/o,..

To find the total heat of a substance in any given state defined by the values of p and 0, starting from any convenient zero of temperature, it is sufficient to measure the total heat required to raise the substance to the final temperature under a constant pressure equal to p. For instance, in the boiler of a steam engine the feed water is pumped into the boiler against the final pressure of the steam, and is heated under this constant pressure up to the temperature of the steam.

We may therefore reasonably assume that the limiting values of the specific heats at zero pressure do not vary with the temperature, provided that the molecule is stable and there is no dissociation.

Denoting by So, so, these constant limiting values at p=o, we may obtain the values at any pressure by integrating the expressions (27) and (28) from co to v and from o to p respectively.

If J', J" represent the values of the function for unit mass of the substance of specific volumes v' and v" in the two states at temperature 0 and pressure and if a mass m is in the state v', and 1-m in the.

If G' and G" are the values of the function G for the two states in equilibrium at the same pressure and temperature, we must have G' =G".

The values of the corresponding functions for the liquid or solid cannot be accurately expressed, as the theoretical variation of the specific heat is unknown, but if we take the specific heat at constant pressure s to be approximately constant, and observe the small residual variation dh of the total heat, we may write F'=s'D+dh+B'.

To find the border curve of equilibrium between the two states, giving the saturation pressure as a function of the temperature, we have merely to equate the values of G and G".

The annual output of Victoria declined until the year 1892, when it began to increase rapidly, but not to its former level, the values for 1900 and 1905 being £3,142,000 and £3,138,000.

Halting just short of this analysis, the Assyrian ascribed syllabic values to the characters of his script, and hence, instead of finding twenty odd characters sufficient, he required about five hundred.

There was a further complication in that each one of these characters had at least two different phonetic values; and there were other intricacies of usage which, had they been foreknown by inquirers in the middle of the 19th century, might well have made the problem of decipherment seem an utterly hopeless one.

Working with some inscriptions from Persepolis which were found to contain references to Darius and Xerxes, Grotefend had established the phonetic values of certain of the Persian characters, and his successors were perfecting the discovery just about the time when the new Assyrian finds were made.

Assuming, then, that the proper names found in the Persian portion of the Behistun inscription occurred also in the Assyrian portion, retaining virtually the same sound in each, a clue to the phonetic values of a large number of the Assyrian characters was obviously at hand.

Phonetic values known, Assyrian was found to be a Semitic language cognate to Hebrew.

In the following table are given the values of the diameters of the molecules of six substances with which it is easy to experiment in the gaseous state, these values being calculated in different ways from formulae supplied by the kinetic theory.

The agreement of the values obtained for the same quantity by different methods provides valuable confirmation of the truth of the molecular theory and of the validity of the methods of the kinetic theory of gases.

This and the difficulty of obtaining accurate experimental results fully account for the differences inter se in the values of the quantities calculated.

Let us suppose that an infinite number of exactly similar systems start simultaneously from all possible values of pi, q1,

Thus after a time dt the values of the coordinates and momenta of the small group of systems under consideration will lie within a range such that pi is between pi +pidt and pi +dp,+(pi+ap?dpi) dt „ qi +gidt „ qi+dqi+ (qi +agLdgi) dt, Thus the extension of the range after the interval dt is dp i (i +aidt) dq i (I +?gidt).

Since the values of the co-ordinates and momenta at any instant during the motion may be treated as " initial " values, it is clear that the " extension " of the range must remain constant throughout the whole motion.

This result at once disposes of the possibility of all the systems acquiring any common characteristic in the course of their motion through a tendency for their co-ordinates or momenta to concentrate about any particular set, or series of sets, of values.

Let us imagine that the systems had the initial values of their co-ordinates and momenta so arranged that the number of systems for which the co-ordinates and momenta were within a given range was proportional simply to the extension of the range.

Then the result proves that the values of the coordinates and momenta remain distributed in this way throughout the whole motion of the systems. Thus, if there is any characteristic which is common to all the systems after the motion has been in progress for any interval of time, this same characteristic must equally have been common to all the systems initially.

Thus the contribution to the total impulsive pressure exerted on the area dS in time dt from this cause is mu X udtdS X (11 3 m 3 /,r 3)e hm (u2+v2+w2 )dudvdw (I o) The total pressure exerted in bringing the centres of gravity of all the colliding molecules to rest normally to the boundary is obtained by first integrating this expression with respect to u, v, w, the limits being all values for which collisions are possible (namely from - co too for u, and from - oo to + oo for v and w), and then summing for all kinds of molecules in the gas.

Clearly the integral is the sum of the values of mu g for all the molecules of the first kind in unit volume, thus p=v mu l +v'm'u 2 +...

This equation is found experimentally to be capable of representing the relation between p, v, and T over large ranges of values.

In the case of corporations realty and machinery are taxed generally by the local authorities, and stock values by the commonwealth.

Seubert (Ber., 1888, 21, p. 1839) from the analysis of potassium and ammonium osmichlorides, the values obtained being approximately 191.

Let KA and LB be the positions of NP corresponding to the extreme values of x.

If the planes of one set divide it into m slabs of thickness h, and those of the other into n slabs of thickness k, so that H =mh, K = nk, then the values of x and of y for any ordinate may be denoted by xo+Oh and yo+Ok, and the length of the ordinate by uo, 0.

Generally, if the area of a trapezette for which u is an algebraical function of x of degree 2n is given correctly by an expression which is a linear function of values of u representing ordinates placed symmetrically about the mid-ordinate of the trapezette (with or without this mid-ordinate), the same expression will give the area of a trapezette for which u is an algebraical function of x of degree 2n + 1.

In the case of the sphere, for instance, whose radius is R, the area of the section at distance x from the centre is lr(R 2 -x 2), which is a quadratic function of x; the values of So, Si, and S2 are respectively o, 7rR 2, and o, and the volume is therefore s.

The process is simplified by writing down the general formula first and then substituting the values of u.

In the case of a trapezette, for instance, the data are the magnitudes of certain ordinates; the problem of interpolation is to determine the values of intermediate ordinates, while that of mensuration is to determine the area of the figure of which these are the ordinates.

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.

It is only necessary to consider the trapezette and the briquette, since the cases which occur in practice can be reduced to one or other of these forms. In each case the data are the values of certain equidistant ordinates, as described in §§ 43-45.

The second consists in taking a comparatively simple expression obtained in this way, and introducing corrections which involve the values of ordinates at or near the boundaries of the figure.

Combining this with the first equation, we obtain the values of P, Q, R, ..

If we do not know values of u outside the figure, we must use advancing or receding differences.

Then, if the true moments are denoted by Pi, v 2, ..., their values are given by vi ?Pl P2 - 11.

The expressions in square brackets are in each case to be taken as relating to the extreme values x =xo and x=xm, as in §§ 75 and 76.

The Euler-Maclaurin formula (§ 75) assumes that the bounding values of u', u"',..

One method is to construct a table for interpolation of x in terms of u, and from this table to calculate values of x corresponding to values of u, proceeding by equal intervals; a quadrature-formula can then be applied.

Even where u is an explicit function of x, so that f x udx may be expressed in terms of x, it is often more convenient, for construction of a table of values of such an integral, to use finite-difference formulae.

Attention must be given to the possible accumulation of errors due to the small errors in the values of u.

The use of quadrature-formulae is important in actuarial work, where the fundamental tables are based on experience, and the formulae applying these tables involve the use of the tabulated values and their differences.

Among the incidental operations are (a) the valuation of the bullion by weighing and assaying it; (b) " rating" the bullion, or calculating the amount of copper to be added to make up the standard alloy; (c) recovering the values from ground-up crucibles, ashes and floor sweepings (the Mint " sweep "); (d) assaying the melted bars; (e) " pyxing " the finished coin or selecting specimens to be weighed and assayed; (f) " telling " or counting the coin.

Undoubtedly also commercial confidence was restored by the reconstruction in 1895 of the Bank of New Zealand, and activity has been stimulated by large public loans, while more cautious banking and the systems of taxation and rating on land values, adopted in 1891 and 1896, have done something to check land speculation.

Values can therefore be accumulated only by being reproduced in the course or, as often happens, by the very act of consumption; hence his distinction between reproductive and unproductive consumption.

It may be noted that the elasticity E is only constant for small volume changes or for small values of dy/dx.

But the values of 2 which occur successively during the second at AB exist simultaneously at the beginning of the second over the distance U behind AB.

Substituting these values and neglecting powerssof dy/dx above the second we get CU P= 17.

If the train of waves is reflected, the value of p at AB will be the sum of the values for the two trains, and will, on the average, be doubled.

Now we may resolve these trains by Fourier's theorem into harmonics of wave-lengths X, 2X, 3A, &c., where X=2AB and the conditions as to the values of y can be shown to require that the harmonics shall all have nodes, coinciding with the nodes of the fundamental curve.

Each section then vibrates, and its amplitude goes through all its values in time given by 21rUT/A =2r, or T =A/U, and the frequency is U/A.

If L 1 is the internodal distance and U 1 the velocity in a gas, L and U being the corresponding values for air, we have U 1 /U =L1/L.

Wohler's and Bauschinger's experiments give values of t, u, and s, for some materials.

Putting the values of F in (1) and solving for f max ., we get for the breaking stress of a bar subjected to repetition of varying stress, f max.

But if the load is distributed to the bracing intersections by rail and cross girders, then the shear at C' will be greatest when the load extends to N, and will have the values wXADN and -wXNEB.

Professor Claxton Fidler (Treatise on Bridge Construction, 1887) has made a very careful theoretical analysis of the weights of bridges of different types, and has obtained the following values for the limiting spans.

There is a want of depth in Christian experience, in the power of realizing relative spiritual values in the light of the master principle involved in the distinctively Christian consciousness, such as could raise Clement above a verbal eclecticism, rather than comprehensiveness, in the use of Apostolic language.

The general question of food values is discussed in the article Dietetics; see also Nutrition.

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

Starting with the experimental values of p, for a standard projectile, fired under standard conditions in air of standard density, we proceed to the construction of the ballistic table.

Denoting the value of T at any velocity v by T (v), then (8) T(v) = sum of all the preceding values of AT plus an arbitrary constant, expressed by the notation (9) T(v) =Z(Av)/gp+ a constant, or fdv/gp+ a constant, in which p is supposed known as a function of v.

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)].

The differences OD and DI are thus calculated, while the values of D(v) and I (v) are obtained by summation with the arithmometer, and entered in their respective columns.

The initial values of T, S, D, I, A must be accepted as belonging to the anterior portion of the table.

There are reasons to believe that these values were considerably understated.

Heinrich Eduard Heine has shown that the functions of higher orders may be considered as limiting values of the associated functions; this relation was discussed independently, in 1878, by Lord Rayleigh.

Other important manufactures, with their product values in 1900 and in 1905, are iron and steel ($5,004,572 in 1900; $6,167,542 in 1905); railway cars ($4,248,029 in 1900; $5,739,071 in 1905); packed meats ($5, 1 77, 16 7 in 1900; $5, 6 93,73 1 in 1905); foundry and machine shop products ($4,434,610 in 1900; $4, 6 99,559 in 1905); planing mill products, including sash, doors and blinds ($1,891,517 in 1900; $4,593, 2 5 1 in 1905-an increase already remarked); carriages and wagons ($2,849,713 in 1900; $4,059,438 in 1905); tanned and curried leather ($3,757,016 in 1900; $3,952,277 in 1905); and malt liquors ($3,186,627 in 1900; $3,673,678 in 1905).

In 1898 there began an increased activity in the mining of fluorspar, and Crittenden, Fayette and Livingston counties produced in 1902, 29,030 tons (valued at $143,410) of this mineral, in 1903 30,835 tons (valued at $153,960) and in 1904 19,096 tons (valued at $111,499), amounts (and values) exceeding those produced in any other state for these years; but in 1907 the quantity (21,058 tons) was less than the output of Illinois.

Altogether, we see that weights have descended from original varieties with so little intercomparison that no rectification of their values has been made, and hence there is as much variety in any one place and time as in all together.

It should be borne in mind that in early times the larger values, such as minae, would be transmitted by commerce, while after the introduction of coinage the lesser values of shekels and drachmae would be the units; and this needs notice, because usually a borrowed unit was multiplied or divided according to the ideas of the borrowers, and strange modifications thus arose.

The early observance of the relative values may be inferred from Num.

The values here given are from some Persian buildings (25), which indicate 21.4, or slightly less; Oppert's value, on less certain data, is 21.52.

There is great uncertainty as to the exact values of all ancient standards of volume -- the only precise data being those resulting from the theories of volumes derived from the cubes of feet and cubits.

The values are 1.5 times the Attic (Athenaeus, Theophrastus, &c.) (2, 18), or more closely 11 to 12 times (1/8)th of Attic. Hence, the Attic cotyle being 17.5 cub.

Turning now to its usual trade values in Greece (44), the mean of 113 gives 67.15; but they vary more than the Egyptian examples, having a sub-variety both above and below the main body, which itself exactly coincides with the Egyptian weights.

For the purpose of thus simplifying the operations of arithmetic, the base is taken to be Io, and use is made of tables of logarithms in which the values of x, the logarithm, corresponding to values of m, the number, are tabulated.

This equation defines log x for positive values of x; if o the formula ceases to have any meaning.

A connexion with the circular functions, however, appears later when the definition of log x is extended to complex values of x.

It is positive for all values of y and increases steadily from o toward as y increases from -co towards + oo.

The definitions of the logarithmic and exponential functions may be extended to complex values of x.

Thus even when the argument is real log x has an infinite number of values; for putting 71 =o and taking positive, in which case a = o, we obtain for log the infinite system of values log +2n7ri.

It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation log(I +x) =x -",, x2 +3x 3 -4x 4 + is true only when the analytical modulus of x is less than unity.

The exponential function, which may still be defined as the inverse of the logarithmic function, is, on the other hand, a uniform function of x, and its fundamental properties may be stated in the same form as for real values of x.

Corresponding to the argument log x it gives the values of log (I -Fx - 1) and log (1+x).

By means of these formulae Adams calculated the values of log e 2, log e 3, log e s, and loge?

In 1846 Peter Gray constructed a new table to 12 places, in which the factors were of the form I - (oi) r n, so that n had the values I, 2,.

If suitable values are chosen for these constants, the formula can be made to represent the dispersion of ordinary transparent media within the visible spectrum very well, but when extended to the infra-red region it often departs considerably from the truth, and it fails altogether in cases of anomalous dispersion.

As A m is a wave-length corresponding to an absorption band, this formula can be used to find values of A m which satisfy the observed values of n within the region of transparency, and so to determine where the absorption bands are situated.

Aeuer (Ber., 1904, 37, p. 2 53 6; Ann., 1904, 337, p. 362), who converted pure ethyl iodide into hydriodic acid and subsequently into silver iodide, which they then analysed, obtained the value 126.026 (H =1); a discussion of this and other values gave as a mean 126.97 (0=16).

It is probable that this estimate is generous according to the values of that time.

But even supposing $1,000,000,000 to be a juster estimate according to present-day values, it is probable that the increase of this since 1790 has been more than a hundredfold and since 1850 (since when such data have been gathered by the census) about fifteenfold.

The same conditions have made of importance general theories, such as the single tax theory of Henry George, for taxing landed values.

The following table gives the values of this constant and several expressions involving it Useful fractional approximations are 22/7 and 355/113.

Equating The Above Two Values Of X, We Have 15 N' 3 =19 M' 16; Whence N' =M' 4'N1 13.

In Consequence Of The Solar And Lunar Equations, It Is Evident That The Epact Or Moon'S Age At The Beginning Of The Year, Must, In The Course Of Centuries, Have All Different Values From One To Thirty Inclusive, Corresponding To The Days In A Full Lunar Month.

P= 54 E 1=57 E, Or (57 E) 7 R' By Substituting One Or Other Of These Values Of P And 1, According A; The Case May Be, In The Formula P = P (L L), We Shall Have P, Orthe Number Of Days From The 21St Of March To Easter Sunday.

So That J W=6 (Y) R 3 (113 0 3) R (Rejecting Sevens), The Values Of Which Obviously Circulate In A Period Of 7 Times 30 Or 210 Years.

Great importance was attached to the mystical values of letters and numbers, especially the numbers 18 and 19 (" the number of the unity ") and 19 2 = 361 (" the number of all things ").

He finds a remarkable agreement between the theoretical and experimental values, which it would be important to confirm with the more suitable instruments which are now at our disposal, as we might in this way get an estimate of the energy of translatory motion of the luminous molecules.

Balmer, who showed that the four hydrogen lines in the visible part of the spectrum may be represented by the equation n = A(i - 4/s2), where n is the reciprocal of the wave-length and therefore proportional to the wave frequency, and s successively takes the values 3, 4, 5, 6.

Rydberg favours the former view, but he does not attempt to obtain any very close approximation between the observed and calculated values of the frequencies.

The present writer, therefore, in his Theory of Optics, adopted different names, and called the series respectively the " Trunk," the " Main Branch " and the " Side Branch," the main branch being identical with the second subordinate series; the limit of frequency for high values of s is called the " root " of the series, and it is found in all cases that the two branches have a common root at some point in the trunk.

The other method starts from the observed values of the periods, and establishes a differential equation from which these periods may be derived.

If this is the case it is obvious that an equation of the form n=A - +a does, for small values of s, becomes identical with Deslandres' equation, a representing a constant which is large compared with unity.

If we only assign positive values to n and a, the band fades away from the head, the lines at first increasing in distance.

If we adopt Thiele's view that each band is accompanied by a second branch for which s has negative values the complication is still further increased, but there does not seem to be sufficient reason to adopt this view.

Close to and on either side of the absorptive band µ 2 has large positive and negative values, and if the above expression remains correct the change of frequency would, close to the centre of absorption, be 2 k-2"+3, which for n =3 and k= Io is 1/2000, or 500 times greater than the observed shifts, but this represents now the maximum displacement and not the displacement of the most intense portion of the radiation.

This division of values brings us to the second point in his philosophy, his theory of what he called " vital series," by which he assayed to explain all life, action and thought.

In 1769, in a letter to Dr Franklin, he wrote some observations on the expectation of lives, the increase of mankind, and the population of London, which were published in the Philosophical Transactions of that year; in May 1770 he communicated to the Royal Society a paper on the proper method of calculating the values of contingent reversions.

Following the first chart of lines of equal variation compiled by Edmund Halley in 1700, charts of similar type have been published from time to time embodying recent observations and corrected for the secular change, thus providing seamen with values of the variation accurate to about 30' of arc. Possessing these data, it is easy to ascertain by observation the effects of the iron in a ship in disturbing the compass, and it will be found for the most part in every vessel that the needle is deflected from the magnetic meridian by a horizontal angle called the deviation of the compass; in some directions of the ship's head adding to the known variation of the place, in other directions subtracting from it.

If, instead of one phase, we have r phases, we must find out the values of r(n - I) quantities before we know the composition of the whole system.

Substituting these values, we find that the relative lowering of vapour pressure in a very dilute solution is equal to the ratio of the numbers of solute and solvent molecules, or (p - p')/p = n/N.

Although even good membranes of copper ferrocyanide are rarely perfectly semi-permeable, and in other membranes such as indiarubber, &c., which have been used, the defects from the theoretical values of the equilibrium pressure are very great, yet, in the light of the exact verification of theory given by the experiments described above, it is evident that such failures to reach the limiting value in no wise invalidate the theory of osmotic equilibrium.

Dilute solutions of substances such as cane-sugar, as we have seen, give experimental values for the connected osmotic properties - pressure, freezing point and vapour pressure - in conformity with the theoretical values.

The osmotic pressure (defined as the difference in the hydrostatic pressures of the solution and solvent when their vapour pressures are equal and they are consequently in equilibrium through a perfect semi-permeable membrane) may also depend on the absolute values of the hydrostatic pressures, as may the vapour pressure of the liquids.

The probable error in neglecting any variation of specific heat is small, and we may calculate L from the values of Lo - (s - s') (To - T), where s - s' is about 0.5 calories.

Putting in these values and integrating we have, neglecting terms involving 0', P=12.06 0-0.021 O s where P is the osmotic pressure in atmospheres.

In the limit of dilution when n is very small compared with N this gives Raoult's experimental law that the relative lowering is n/N, which we deduced from the osmotic law, and conversely from which the osmotic law follows, while for more concentrated solutions agreement is obtained by assigning arbitrary values to a, which, as we have seen, is 5 in the case of cane-sugar.

This separation causes a difference of potential, which can be calculated and is found to agree with the values obtained experimentally.

If a more dilute acid than this be distilled, water passes over in excess and the residue in the retort reaches the above composition and boiling point; on distillation of a stronger acid, excess of acid passes into the distillate and the boiling point rises until the values of the constant boiling mixture are reached.

The relative values of hand and machine work are often discussed.

These are (1) the origin of the cuneiform signs, (2) the etymology of the phonetic values, and (3) the elucidation of the many and varied primitive sign-meanings.

We arrive thus at two distinct and opposite uses and values of fur.

The Hudson's Bay Co.'s sales take place before the others, and, as no reserves are placed on any lot, the results are taken as exactly indicating current values.

Values have greatly increased, and the fur possessing good qualities as to colour and durability will doubtless always be in good request.

This class of skin is the most expensive fur in the world, reckoning values by a square foot unit.

The values given in the column headed " cooled " are those found by Mitchell with one end of the bar cooled.

On some occasions, owing to the sudden melting of a surface layer of ice and snow, a large quantity of cold water, percolating rapidly, gave for a short time values of the diffusivity as high as.

The values obtained in this way for waves having a period of one second and a wave-length of half an inch agreed very well with those obtained in the same cast-iron by Angstrom's method (see below), with waves having a period of i hour and a length of 30 in.

An additional difficulty arises in the case of observations made with long mercury thermometers buried in vertical holes, that the correction for the expansion of the liquid in the long stems is uncertain, and that the holes may serve as channels for percolation, and thus lead to exceptionally high values.

In any case results deduced from the annual wave must be expected to vary in different years according to the distribution of the rainfall, as the values represent averages depending chiefly on the diffusion of heat by percolating water.

The extreme divergence of the resulting values of the diffusivity, including eight independent series of measurements on different days, was less than I %.

This being inconvenient for comparison with sun-spots, use was made of his monthly values to obtain corresponding data for years commencing with January.

Even smaller mean values have been found for the angle between the auroral and magnetic " zeniths " - as the two directions have been called - e.g.

Westman, gives the values 4276.4 and 3913.5.

The values recognized at the court would naturally be recognized in noble families generally, and Philip chose Aristotle to be the educator of his son.

The values honoured by the rulers of the world must naturally impress themselves upon the subject multitudes.

At the court a limited recognition might be given, as fashion veered, to the values prevalent in the Hellenistic world.

At Assuan the mean value for the year is only 38%, that for the summer being 29%, and for the winter 51%; while for Wadi Haifa the mean is 32%, and 20% and 42% are the mean values for summer and winter respectively.

Yet, while the values of the signs were for the most part well ascertained, and the meanings of most works fixed with some degree of accuracy, few grammatical rules had as yet been established, the varieties of the language at different periods had not been.

Although a picture sign may at times have embarrassed the skilled native reader by offering a choice of fixed values or functions, it was never intended to convey merely an idea, so as to leave to him the task of putting the idea into his own words.

The history of some of the alphabetic signs is still very obscure but a sufficient number of them have been explained to make it nearly certain that the values of all were obtained on the same principles.i Some of the ancient words from which the phonetic values were derived probably fell very early into disuse, and may, never be discoverable in the texts that have come down to us.

When finely rendered they are of great value to the student investigating the origins of their values.

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.

Such measurements of fraternal correlation in the lower animal as Pearson and his pupils have at present made give values very close to 2.

Variation provides the material for selection, and although opinions may differ as to the nature of that material, the modes by which it comes into existence and their relative values and permanences, there is an increasingly wide consensus of opinion that all such material has to pass through the sieve of natural selection and that the sifted products form new varieties and species, and new adaptations.

In 1900 about nine-tenths of the total land area was inclosed in farms; the value of farm property ($2,004,316,897) was greater than that of any other state; as regards the total value of farm products in 1899 Illinois was surpassed only by Iowa; in the value of crops Illinois led all the states, and the values of property and of products were respectively 35.6% and 87.1% greater than at the end of the preceding decade.

Commerce.-A summary of import and export values of trade in the Persian Gulf, excluding Mohammerah and Basra, is appended.

Bushire, Hanjam, Bahrein, Abadan and Basra Summary showing Import and Export Values of Trade in the Persian Gulf (excluding Iraq and Arabistan) in two pre-war years and in the latest post-war year available.

If such variations of density exist, they may introduce some uncertainty in the absolute values of results obtained with the ice calorimeter, and may account for some of the discrepancies above enumerated.

The Values Deduced In This Manner For The Equivalent Agreed As Closely As Could Be Expected Considering The Impossibility Of Regulating The External Condition Of Temperature And Moisture With Any Certainty In An Engine Room.

The Heat Loss Hdo Is Determined And Eliminated By Varying The Flow Of Liquid And The Electric Current Simultaneously, In Such A Manner As To Secure Approximately The Same Rise Of Temperature For Two Or More Widely Different Values Of The Flow Of Liquid.

The Absolute Value Of The Specific Heat Deduced Necessarily Depends On The Absolute Values Of The Electrical Standards Employed In The Investigation.

But For The Determination Of Relative Values Of Specific Heats In Terms Of A Standard Liquid, Or Of The Variations Of Specific Heat Of A Liquid, The Method Depends Only On The Constancy Of The Standards, Which Can Be Readily And Accurately Tested.

It Appears Probable That His Values For Higher Temperatures May Be Adopted With This Reduction, Which Is Further Confirmed By The Results Of Reynolds And Moorby, And By Those Of Liidin.

The following table of values, calculated from these formulae, is taken from the Brit.

This quantity is now, as the result of further experiments, added to the values of h, and also represented in the formula for the specific heat itself by the cubic term.

This Would Indicate That Rowland'S Corrected Values Should, If Anything, Be Lowered.

The Mean Specific Heat, Over Any Range Of Temperature, May Be Obtained By Integrating The Formulae Between The Limits Required, Or By Taking The Difference Of The Corresponding Values Of The Total Heat H, And Dividing By The Range Of Temperature.

The Values Of The Total Heat Observed By Regnault, As Reduced By Shaw, Also Show A Very Fair Agreement, Considering The Uncertainty Of The Units.

Since Much Smaller Values Are Found For More Complex Molecules, We May Suppose That, In These Cases, The Energy Of Rotation Of A Polyatomic Molecule May Be Greater Than Its Energy Of Translation, Or Else That Heat Is Expended In Splitting Up Molecular Aggregates, And Increasing Energy Of Vibration.

This Gives A Series Of Ratios 5/3, 7/5, 9/7, Ii/9, &C., For I, 2, 3, 4, &C., Atoms In The Molecule, Values Which Fall Within The Limits Of Experimental Error In Many Cases.

Its specific gravity varies according to the method employed for its preparation, the extreme values being 8.279 and 9.25.

There are postage stamps of four values.

The give and take of thought had by a swift transformation of values come by something more than its own.

Argand had been led to deny that such an expression as i 2 could be expressed in the form A+Bi, - although, as is well known, Euler showed that one of its values is a real quantity, the exponential function of --7112.

These cases are really included in the equation if we substitute the proper values of n or m.

The values thus found agreed in the main with Raoult's law for dilute solutions (see Solutions).

Taking the difference between the values of H for any two temperatures 1 " Latent Heat of Steam," Phil.

These observations give values for S ranging from 0.30 to o 46, with a mean value 0.3778.

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.

Employing the values of S calculated from dH/d0 = 0.305, he found that the product SQ was independent of both pressure and temperature for the range of his experiments.

Values calculated from these formulae are given in the table below.

The values of the saturation-pressure have been ver y accurately determined for the majority of stable substances, and a large number of empirical formulae have been proposed to represent the relation between pressure and temperature.

The values so found are necessarily erroneous if formula (io) for the total heat is wrong.

He proceeds to calculate from this expression the difference of vapour-pressures of ice and water in the immediate neighbourhood of the melting-point, but does not observe that the vapour-pressures themselves may be more accurately calculated for a considerable interval of temperature by means of formula (23), by substituting the appropriate values of the latent heats and specific heats.

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

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.

It is easy, however, to correct the formula for these deviations, and to make it thermodynamically consistent with the characteristic equation (13) by substituting the appropriate values of (v-w) and L =H -h from equations (13) and (is) in formula (21) before integrating.

But in order to deduce the values of c by the Joule-Thomson method, it is necessary to assume an empirical formula, and the type c=co(6019) n is chosen as being the simplest.

The justification of this assumption lies in the fact that the values of c found in this manner, when substituted in equation (25) for the saturation-pressure, give correct results for p within the probable limits of error of Regnault's experiments.

The latent heat L (formula 9) is found by subtracting from H (equation 15) the values of the heat of the liquid h given in the same article.

The values of the specific heat in the next column are calculated for a constant pressure equal to that of saturation by formula (16) to illustrate the increase of the specific heat with rise of pressure.

The values of the saturation-pressure given in the last column are calculated by formula (25), which agrees with Regnault's observations better than his own empirical formulae.

The value of products in 1900 ($79.6 millions) was 44% of the total farm values ($181.4 millions).

As a result of these improvements land and timber values have markedly risen, and great impetus has been given to traffic on the rivers, which carry a large part of the cotton, lumber, coal, stone, hay and miscellaneous freights of the state.

It is obvious that, both in its values and in its measurements, the survey's reckoning is very crude.

While thus the Greek philosophy furnished the dialectic and the mould for the characteristic Christian teaching, the doctrine of the Trinity preserved religious values.

The historic and religious values of the doctrine of the Trinity may be illustrated by way of contrast.

In the Mahayana gnosticism was triumphant, and the historic values of Gautama's teaching and personality are lost.

A magnetograph is an instrument for continuously recording the values of the magnetic elements, the three universally chosen being the declination, the horizontal component and the vertical component.

In each case the magnetograph only records the variation of the element, the absolute values being determined by making observations in the neighbourhood with the unifilar magnetometer and inclinometer.

The scale values of the records given by the horizontal and vertical force magnetographs are determined by deflecting the respective needles, either by means of a magnet placed at a known distance or by passing an electric current through circular coils of large diameter surrounding the instruments.

Of the foreign-born 4504 were ' Statistics for 1890 represent the value of all manufactures; those for 1900 (from this point) and 1905 show values under the factory system, excluding neighbourhood industries and hand trades.

For if the line in question be the axis of y, the first process gives us the values of mx, and the second the value of 2(mx.x) or Z(mx2).

From this point of view the equation is a mere truism, its real importance resting on the fact that by attributing suitable values to the masses in, and by making simple assumptions as to the value of X in each case, we are able to frame adequate representations of whole classes of phenomena as they actually occur.

If we take as rough values a=21 X,o6 feet, g=32 foot-second units, we get a velocity of 36,500 feet, or about seven miles, per second.

The motion being assumed to be small, the tensions of the two strings may be taken to have their statical values Mgb/(a+b), Mga/(a+b), where a, b are the distances of G from the two threads.

For given values of n and 0 we have two possible values of i,l provided n exceed a certain limit.

When, in any problem, the values of u, v, w, p, q, r have been determined as functions of t, it still remains to connect the moving axes with some fixed frame of reference.

It is easy ti construct in this connection a proof that the n values of a2 an all real and positive.

If in (21) we imagine that x, y, I denote infinitesimal rotations of a solid free to turn about a fixed point in a given field of force, it appears that the three normal modes consist each of a rotation about one of the three diameters aforesaid, and that the values of in are proportional to the ratios of the lengths of corresponding diameters of the two quadrics.

These limit the admissible values of a-, which are in general determined by a transcendental equation corresponding to the determinantal equation (6).

Since y must vanish at the upper end (x 1), the admissible values of a are determined by o2=gz2/41, Jo(z)=o.

The agreement between the observed and calculated values of the critical speeds was in most cases quite remarkable.

It implies a subsequent contribution, from all the parties concerned, rateably to the values of their respective interests, to make good the loss thus occasioned.

But if all are lost the burden of the expenditure ought not to remain upon the interest which at first bore it; and the proper rule seems to be that contributions must be made by all the interests which were at stake when it was made, in proportion to their then values.

This has only been possible owing to the temper of the Oriental mind which, while clinging tenaciously to its rites, values dogma only in so far as it is expressed in rites.

On assuming the directorship of the Nautical Almanac he became very strongly impressed with the diversity existing in the values of the elements and constants of astronomy adopted by different astronomers, and the injurious effect which it exercised on the precision and symmetry of much astronomical work.

Accordingly he resolved to " devote all the force which he could spare to the work of deriving improved values of the fundamental elements and embodying them in new tables of the celestial motions."

A valuable summary of a considerable part of this work, containing an account of the methods adopted, the materials employed, and the resulting values of the various quantities involved, was published in 1895, as a supplement to the American Ephemeris for 1897, entitled The Elements of the Four Inner Planets and the Fundamental Constants of Astronomy.

Kriimmel's more recent calculations (see Ocean) give values somewhat higher than those of Karstens.

Silver values ran, in the years 1880-1902, from 11.3 to 23.1 million dollars; and the quantities in the same years from 11.6 to 26.3 million ounces.

Taxation has been very erratic. From 1877 to 1893 the total assessment rose steadily from $3,453,946 to $238,722,417; it then fell at least partly owing to the depreciation in and uncertain values of mining property, and from 1894 to 1900 fluctuated between 192.2 and 216.8 million dollars; in 1901 it was raised to $465,874,288, and fluctuated in the years following; the estimated total assessment for 1907 was $365,000,000.

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

But a new agency began to engage the speculations of thinkers, the moral values of action embodied in the Deed.

Chile is a member of the International Postal Union, and has arrangements with the principal commercial nations for the exchange of postal money values.

The values of imports and exports (including bullion, specie and re-exports) in pesos of 18d.

For N to be constant for all values of w, a' tan w'/ a tan w must also be constant.

Tradesmen are less affected, because they can sell the articles which they manufacture at values which are more in proportion with the increased prices of food.

Under the influence of Cerdo, Marcion carried out his ethical dualism in the sphere of cosmology; but the fact that his system is not free from contradictions is the best proof that all along religious knowledge, and not philosophical, had the chief values in his eyes.

The values of coin and notes are expressed in multiples of the real (plural reis), a monetary unit which does not actually exist.

Taxation for ordinary municipal purposes is limited to 1 Y„ on property values, extra taxes being allowed for unusual purposes; but the city cannot be bonded without the affirmative vote of two-thirds of the electorate.

For the fiscal year1906-1907the assessed value was $375,93 2, 447, indicating the drop in values immediately after the earthquake and fire, and, by comparison with the 1910 figures, the extent of recovery.

Realty values rose ten-fold in 1849.

In1855-1856a disastrous commercial panic crippled the city; and in 1858, when at the height of the Fraser river gold-mine excitement it seemed as though Victoria, B.C., was to supplant San Francisco as the metropolis of the Pacific, realty values in the latter city dropped for a time fully a half in value.

In September 1869 the Central Pacific line was completed to Oakland, and in the next four years there was a crash in real estate values inflated during the railway speculation.

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.

If the thickness of the film is greater than 2E, there will be a stratum of thickness c-2E in the middle of the film, within which the values of p and x will be pc and In the two strata on either side of this the law, according to which p and x depend on the depth, will be the same as in a liquid mass of large dimensions.

The only cases, however, in which we have experimental values of this quantity are when the substance is either liquid and surrounded by similar liquid, or gaseous and surrounded by similar gas.

When a liquid is in thermal and dynamical equilibrium with its vapour, then if p' and x' are the values of p and x for the vapour, and po and Xo those for the liquid, x' - xo=JL - p(I/p' - I/pc),.