# Equation Sentence Examples

This

**equation**does not give us the value of the unknown factor but gives us a ratio between two unknowns.Again, the

**equation**[2N, 0] =-18500 cal.He'd asked her if she'd take herself out of the

**equation**before she hurt Gabriel.She was the last to lose hope, and it was being forced to see how out of place she was in Gabriel's

**equation**that finally broke her resolve.You care about the other person in the

**equation**.The anomaly AFQ of Q at any moment is called the mean anomaly, and the angle QFP by which the true anomaly exceeds it at that moment is the

**equation**of the centre.Let us now consider an

**equation**of the form U =a sin (nt+Lo).The cartesian

**equation**, when A is the origin and AB = 2a, is y 2 (2a - x) =x 8; the polar**equation**is r= 2a sin 0 tan 0.By the action of oxidizing agents such as nitric acid, iodine solution, &c., arsenious acid is readily converted into arsenic acid, in the latter case the reaction proceeding according to the

**equation**H3AsO3 +I2 + H2O = H3AsO4 + 2HI.Analytically it is defined by an

**equation**of the second degree of which the highest terms represent two imaginary lines.AdvertisementThe

**equation**to the tangent at 0 is x cos 0/a+y sin 0/b = 1, and to the normal ax/cos 0 - by/sin 0=a2 - b'.This is the

**equation**to a parabola, and is equivalent to the empirical formula of Avenarius, with this difference, that in Tait's formula the constants have all a simple and direct interpretation in relation to the theory.The quantity of heat liberated by convection as the current flows from hot to cold is represented in the

**equation**by dP=d(pT).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.She didn't factor Gabriel's transition into Death into the

**equation**.AdvertisementIf M is the central mass, n the angular velocity, and a the distance, the balance of the two forces is expressed by the

**equation**an' =.1111a2, whence a 3 n 2 = M, a constant.The periodic time varying inversely as n, this

**equation**expresses Kepler's third law.In the language of algebra putting m l, m2, m 3, &c. for the masses of the bodies, r1.2 r1.3 r2.3, &c. for their mutual distances apart; vi, v 2, v 3, &c., for the velocities with which they are moving at any moment; these quantities will continually satisfy the

**equation**orbit, appear as arbitrary constants, introduced by the process of integration.This fact is fully expressed by the equations (4) where we have constants on one side of the

**equation**equal to functions of the variables on the other.If we replace y' in

**equation**(4) by the value given by (3), we obtain tan w"/ y i/f2"=V, (5) the magnification of the complete microscope.AdvertisementIf this value of y be inserted in

**equation**(5), we obtain the magnification number of the compound microscope N =tan w"/ tan w =Ol/f i 'f 2 ' =Vl.The sine-condition is in contrast to the tangent-condition, which must be regarded as the point-by-point representation of the whole object-space in the image-space (see Lens), and according therefore the

**equation**n tan u/tan u' = C must exist.This involves the expenditure of a quantity of work W, the amount in any particular case being found by the

**equation**W = Q2 - Q I, where W is the work, expressed by its equivalent in British thermal units; Q2 the quantity of heat, also in B.Ther.U., given out at the higher temperature T2; and Q i the heat taken in at the lower temperature T1.While Dean was no closer to finding the identity of the bones or the person or persons trying to stop him from doing so, the introduction of Jennifer Radisson into the

**equation**was, to him, a welcome addition.Because Freud deliberately refrains from both these popular options, he has always come under fire from both sides of the

**equation**.AdvertisementAs an example, consider the

**equation**for the velocity v of an object that undergoes an acceleration a for a time t.The Wiener-Hopf

**equation**is derived as the optimum receive filter, illustrated with applications including adaptive filtering.Consider the general quadratic

**equation**ax 2 + bx + c = 0 where a 0.On the other side of the

**equation**, there is a massive backlog of sites waiting to be reviewed.Section 2 starts by looking at the direction field associated with a first-order differential

**equation**.The Numerov method [2] is used to solve the differential

**equation**.In fact, it obeys a differential

**equation**known as the Schrodinger**equation**(or a generalization thereof ).The UM uses a full compressible

**equation**set, no turbulence closure (numerical diffusion only) and a hybrid terrain following co-ordinate system.For example, where Einstein's

**equation**predicts gravitational time dilation the alternative makes clocks tick more slowly due to a gravitational mass increase.For large problems round- off errors may accumulate during the solution of the matrix eigenvalue

**equation**.For the educated, idealistic middle class, including many Northern environmentalists, the

**equation**is self-evident.In electromagnetic terms this can be explained using the diffusion

**equation**.You can show dynamic equilibrium in an

**equation**for a reaction by the use of special arrows.Approximation of the Beta function The asymptotic expansion for the T score,

**equation**31, requires the value of.The original fractal, this is a pictorial representation of a complex

**equation**to describe systems with a hierarchy of repeating patterns.With man-made global warming out of the

**equation**, mankind's ' consumption ' in biosphere terms is entirely sustainable.Even in those days I always gravitated toward the audio side of the

**equation**.In this

**equation**, H is the Hamiltonian operator associated with the classical Hamiltonian operator associated with the classical Hamiltonian function.Equations must be typeset, preferably by

**Equation**Editor in MS Word, not handwritten.In 1876, Heaviside took account of self inductance, obtaining the

**equation**of damped harmonic motion known as Heaviside's**Equation**of Telegraphy.Given the volume velocity, the PC also calculates the input impedance according to

**equation**2.7.Obtained by substituting the peak inductor current into the familiar inductor energy

**equation**.Writing down the

**equation**of a straight line given a diagram with a parallel line and the y intercept.This

**equation**means that matter and energy can be made interchangeable.It is a delicate interplay or

**equation**of forces and of natures acting one upon another.The light scattering irradiance for spheres is calculated using the

**equation**, 1.0 where is the scattering amplitude function given by [2] .Recently we have been adding some kibble into the

**equation**bones remain at 60% .On the other hand, Serbian nationalists challenged the

**equation**of the experience of the Serbs during the war.To linearise the

**equation**we take the natural logarithm.**Equation**display is automatically optimized for the resolution of the user's screen.Using the

**equation**l p = h. If the momentum of the electrons is zero, then the wavelength they have is infinite.Some sensitivity coefficients can be calculated by taking partial derivatives of the

**equation**defining the hardness value.Smith had to take coercion out of the

**equation**and let the horse discover the pleasure of speed.Experimental results drawn from the literature for each pure compound were fitted with a four-term polynomial

**equation**in reduced temperature.This simple procedure gives the first few quotients of, that we listed above in

**equation**12.Neither must we forget to add rainbow trout to the

**equation**.The issue of slaughter house spread whilst being highly relevant is only part of the

**equation**for sheep.The

**equation**might have sounded simple but, as O'Leary has constantly reminded us, they were ' only ' playing Barcelona.On later levels, enemy spaceships harry you, and the need to defend yourself enters the

**equation**.But if you don't understand the

**equation**then you should probably try a spatula instead!Their

**equation**used the standard deviation of the stock price and the risk-free interest rate to provide a value.This assertion is never directly stated because that would reveal the absurdity of the

**equation**.The sum of the twin fractions must be 1.0 Twin Data stored by CRYSTALS For a twinned crystal the following

**equation**holds.The above gear ratio

**equation**can be used to calculate the angular velocity of the large gear from the small gear.The whole picture changed when the Ira violence was taken out of the

**equation**.Numerical Equations These are based on the diffusion

**equation**in one dimension, where is the eddy viscosity.Thus by

**equation**(18) of § II of the article Diffraction Of Light, the secondary disturbance is expressed by D' - D n 2 Tsin sin (nt - kr) D 47rb2 r _ D' - D irTsin O sin (nt - kr) (3)1 The preceding investigation is based upon the assumption that in passing from one medium to another the rigidity of the aether does not change.Analytically, it is defined by an

**equation**of the second degree, of which the highest terms have real roots (see Conic Section).The reaction as before is completely expressed by the chemical

**equation**Zn+H2S04 =ZnSO 4 H+ 2, the initial and final systems being exactly the same as in the first case; yet the amount of heat generated by the action is much smaller, a quantity of the intrinsic energy having been converted into electrical energy.Berthelot's notation defines both initial and final systems by giving the chemical

**equation**for the reaction considered, the thermal effect being appended, and the state of the various substances being affixed to their formulae after brackets.To the right-hand member of the

**equation**he then adds the number expressing the thermal effect of the reaction, heat-evolution being as before counted positive, and heat-absorption negative.Ostwald has made the further proposal that the formulae of solids should be printed in heavy type (or within square brackets), of liquids (solutions, &c.) in ordinary type, and of gases in italics (or within curved brackets), so that the physical state of the substances 'might be indicated by the

**equation**itself.The hydrocarbon C20H42, for example, might be resolved into C5H12+C15H30, or CEH14+C14H28, or C7H16 +C13H26, &c., the general

**equation**of the decomposition being C„1-1 27, ± 2 (paraffin) =G_rH2(, - P)+2 (paraffin)+C P H 2 n (olefine).Succinosuccinic ester behaves both as a ketone and as a phenol, thereby exhibiting desmotropy; assuming the ketone formula as indicating the constitution, then in Baeyer's

**equation**we have a migration of a hydrogen atom, whereas to bring Ladenburg's formula into line, an oxygen atom must migrate.In the article Condensation Of Gases (see also Molecule) it is shown that the characteristic

**equation**of gases and liquids is conveniently expressed in the form (p+a/v 2) (v - b) = RT.For the subjects of this general heading see the articles ALGEBRA; ALGEBRAIC FORMS; ARITHMETIC; COMBINATORIAL ANALYSIS; DETERMINANTS;

**EQUATION**; FRACTION, CONTINUED; INTERPOLATION; LOGARITHMS; MAGIC SQUARE; PROBABILITY.In Bode's Jahrbuch (1776-1780) he discusses nutation, aberration of light, Saturn's rings and comets; in the Nova acta Helvetica (1787) he has a long paper "Sur le son des corps elastiques," in Bernoulli and Hindenburg's Magazin (1787-1788) he treats of the roots of

**equation**and of parallel lines; and in Hindenburg's Archiv (1798-1799) he writes on optics and perspective.The five processes of deduction then reduce to four, which may be described as (i.) subtraction, (ii.) division, (iii.) (a) taking a root, (iii.) (b) taking logarithms. It will be found that these (and particularly the first three) cover practically all the processes legitimately adopted in the elementary theory of the solution of equations; other processes being sometimes liable to introduce roots which do not satisfy the original

**equation**.But, if we transform the

**equation**into 4X-12s.Either of these is a statement of fact with regard to a particular quantity; it is usually called an

**equation**, but sometimes a conditional**equation**, the term "**equation**" being then extended to cover (i.) and (ii.).The quadratic

**equation**x 2 +b 2 =o, for instance, has no real root; but we may treat the roots as being +b-' - I, and - b 1, 1 - 1, if -J - i is treated as something which obeys the laws of arithmetic and emerges into reality under the condition 1 1 - I.This latter class he discussed so assiduously that they are often known as Diophantine problems, and the methods of resolving them as the Diophantine analysis (see

**Equation**, Indeterminate).The first difficulty to be overcome was the algebraical solution of cubic equations, the " irreducible case" (see

**Equation**).If the

**equation**of this line, referred to new coordinate axes in the plane area, is written xcos a+y sin a - h=o, (3) R = f f p(h - x cos a - y sin a)dxdy, (4) zR= f fpx(h - xcos a - y sin a)dxdy, (5) yR = f f py(h - x cos a - y sin a)dxdy.In the case of a steady motion of homogeneous liquid symmetrical about Ox, where 0 is advancing with velocity U, the

**equation**(5) of § 34 p/p +V +Zq'2-f(,P') = constant becomes transformed into P +V + 2- dy + 2U 2 -f(t +2Uy 2) = constant, = 1,t+4Uy2, subject to the condition, from (4) § 34, Y -2 V = - f ' (Y', y 2 2 +2Uy2).The Poisson

**equation**cannot, however, be applied in the above form to a region which is partly within and partly without an electrified conductor, because then the electric force undergoes a sudden change in value from zero to a finite value, in passing outwards through the bounding surface of the conductor.He had previously published some medical and botanical dissertations, besides his Exercitationes quaedam Mathematicae, containing a solution of the differential

**equation**proposed by Riccati and now known by his name.In that year Adriaan van Roomen gave out as a problem to all mathematicians an

**equation**of the 45 th degree, which, being recognized by Vieta as depending on the**equation**between sin 4 and sin 43/45, was resolved by him at once, all the twenty-three positive roots of which the said**equation**was capable being given at the same time (see Trigonometry).If the distance between the contour lines is h and the length of the individual contour lines 1, the sum of their lengths / (1), and A the area of the surface under investigation, then the mean angle of slope is obtained from the

**equation**h?A "cubic

**equation**" is one in which the highest power of the unknown is the cube (see**Equation**); similarly, a "cubic curve" has an**equation**containing no term of a power higher than the third, the powers of a compound term being added together.But v/V =u/U from

**equation**(2) and w =Eu/U from**equation**(3) Then 2wv/V = ZEu 2 /U 2 = 2 pu t from**equation**(6) Then in the whole wave the potential energy equals the kinetic energy and the total energy in a complete wave in a column 1 sq.At any point x from the abutment, the bending moment is M = Zwx(l - x), an

**equation**to a parabola.With unit load in the position shown, the load at D' is (p-n)/p, and that at E' is n/p. The moment of the load at C 'is' -n)lp. This is the

**equation**to the dotted line RS (fig.Starting with the exact equations of motion in a resisting medium, (43) d2t cos i = ds, d 2 y d 44 dt2 = -r sin i-g= -rds-g, and eliminating r, (45) dt - - cos z, or the

**equation**obtained, as in (18), by resolving normally in the trajectory, but di now denoting the increment of i in the increment of time dt.When The Solar

**Equation**Occurs, The Epacts Are Diminished By Unity; When The Lunar**Equation**Occurs, The Epacts Are Augmented By Unity; And When Both Equations Occur Together, As In 1800, 2100, 2700, &C., They Compensate Each Other, And The Epacts Are Not Changed.The phase rule combined with the latent heat

**equation**enables us to trace the general phenomena of equilibrium in solutions, and to elucidate and classify cases even of great complexity.The

**equation**of available energy (see Energetics) A=U+ TdA/dT may be applied to this problem.More convenient forms in terms of a single parameter are deduced by substituting x' =am t, y' = aam (for on eliminating in between these relations the

**equation**to the parabola is obtained).The envelope of this last

**equation**is 27ay 2 =4(x-2a) 3, which shows that the evolute of a parabola is a semi-cubical parabola (see below Higher Orders).The cartesian

**equation**to a parabola which touches the coordinate axes is 1 / ax+'1 / by= i, and the polar**equation**when the focus is the pole and the axis the initial line is r cos 2 6/2 = a.The

**equation**to a parabola in triangular co-ordinates is generally derived by expressing the condition that the line at infinity is a tangent in the**equation**to the general conic. For example, in trilinear co-ordinates, the**equation**to the general conic circumscribing the triangle of reference is 113y+mya+naf3=o; for this to be a parabola the line as + b/ + cy = o must be a tangent.If we remove the bar BD, and apply two equal and opposite forces S at B and D, the

**equation**is W.I(2lcosO) + 2S .1(1 sin 8)=o, A where 2 is the length of a side of the rhombus, and 8 its inclination to the vertical.Comparative Motion of Two Pistons.If there be but two pistons, whose areas are af and af, and their velocities Vf and vI, their comparative motion is expressed by the

**equation**V2/Vf = aia/2; (2)It is obvious from this

**equation**that, for an axis of rotation parallel to 0 traversing C, the centre of percussion is at the point where the perpendicular OG meets 0.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.In astronomy, the term is used in connexion with the Ptolemaic theory to denote the angular distance on the epicycle of a planet from the true apogee of the epicycle; and the "

**equation**to the argument" is the angle subtended at the earth by the distance of a planet from the centre of the epicycle.Solve a quadratic

**equation**using factors Errors Identify sources of errors.Now rearrange the first

**equation**to get distance = speed Ã— time.A second approach is to use the regression

**equation**Chart Wizard.The program is designed so as to reconstruct the polynomial in two variables represented by this

**equation**.Have signed the the

**equation**and pentagon salaries and literary and philosophical.We show, however, that asymmetric states can be moving solutions of the partial differential

**equation**.This is not true in the case of g, which would lead to the need to solve an implicit

**equation**.But if you do n't understand the

**equation**then you should probably try a spatula instead !The next step taken was to subtract the first

**equation**from the second and 3 times the first from the third.She joined the folk supergroup

**Equation**in 1995 as a replacement for another singing sensation, Kate Rusby.The perturbed wave-function reaching the telescope aperture for this case is given by setting in

**Equation**1.8.Investigating the uncertainties in the Simple Mass Balance

**equation**for acidity critical loads for terrestrial ecosystems.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.The potential due to a thin magnet at a point whose distance from the two poles respectively is r and r' is V =m(l/r=l/r') (8) When V is constant, this

**equation**represents an equipotential surface.If r and r' make angles 0 and 0 with the axis, it is easily shown that the

**equation**to a line of force is cos 0 - cos B'= constant.From the

**equation**K=(µ - I)/47r, it follows that the magnetic susceptibility of a vacuum (where µ = I) is o, that of a diamagnetic substance (where, u I) is positive.**Equation**(44) shows that as a first approximation.But though a formula of this type has no physical significance, and cannot be accepted as an

**equation**to the actual curve of W and B, it is, nevertheless, the case that by making the index e =1.6, and assigning a suitable value to r t, a formula may be obtained giving an approximation to the truth which is sufficiently close for the ordinary purposes of electrical engineers, especially when the limiting value of B is neither very great nor very small.The

**equation**F = B 2 /87r is often said to express " Maxwell's law of magnetic traction " (Maxwell, Electricity and Magnetism,, §§ 642-646).The prize was again awarded to Lagrange; and he earned the same distinction with essays on the problem of three bodies in 1772, on the secular

**equation**of the moon in 1774, and in 1778 on the theory of cometary perturbations.In algebra he discovered the method of approximating to the real roots of an

**equation**by means of continued fractions, and imagined a general process of solving algebraical equations of every degree.The method indeed fails for equations of an order above the fourth, because it then involves the solution of an

**equation**of higher dimensions than they proposed.His development of the

**equation**x m +- px = q in an infinite series was extended by Leonhard Euler, and particularly by Joseph Louis Lagrange.The one is a problem of interpolation, the other a step towards the solution of an

**equation**in finite differences.He also showed that every

**equation**of an even degree must have at least one real quadratic factor, reduced the solution of linear differential equations to definite integrals, and furnished an elegant method by which the linear partial differential**equation**of the second order might be solved.He was also the first to consider the difficult problems involved in equations of mixed differences, and to prove that an

**equation**in finite differences of the first degree and the second order might always be converted into a continued fraction.Equations with Fractional Coefficients.-As an example of a special form of

**equation**we may take zx+ 3x = Io.If the operator 12d X is omitted, the statement is really an

**equation**, giving Is.The second class of cases comprises equations involving two unknowns; here we have to deal with two graphs, and the solution of the

**equation**is the determination of their common ordinates.Thus, if we have an

**equation**P=Q, where P and Q are numbers involving fractions, we can clear of fractions, not by multiplying P and Q by a number m, but by applying the equal multiples P and Q to a number m as unit.An

**equation**of the form ax=b, where a and b do not contain x, is the standard form of simple**equation**.We therefore represent them by separate symbols, in the same way that we represent the unknown quantity in an

**equation**by a symbol.Graphical representation shows that there are two solutions, and that an

**equation**X2= 9a2 may be taken to be satisfied not only by X=3a but also by X= -3a.The consideration of cases where two roots are equal belongs to the theory of equations (see

**Equation**).Thus, to solve the

**equation**ax e +bx+c = o, we consider, not merely the value of x for which ax 2 +bx+c is o, but the value of ax e +bx+c for every possible value of x.Simultaneous equations in two unknowns x and y may be treated in the same way, except that each

**equation**gives a functional relation between x and y.The full title is ilm al jebr wa'l-mugabala, which contains the ideas of restitution and comparison, or opposition and comparison, or resolution and

**equation**, jebr being derived from the verb jabara, to reunite, and mugabala, from gabala, to make equal.The particular problem - a heap (hau) and its seventh makes 19 - is solved as we should now solve a simple

**equation**; but Ahmes varies his methods in other similar problems. This discovery carries the invention of algebra back to about 1700 B.C., if not earlier.A notable improvement on the ideas of Diophantus is to be found in the fact that the Hindus recognized the existence of two roots of a quadratic

**equation**, but the negative roots were considered to be inadequate, since no interpretation could be found for them.Although Pell had nothing to do with the solution, posterity has termed the

**equation**Pell's**Equation**, or Problem, when more rightly it should be the Hindu Problem, in recognition of the mathematical attainments of the Brahmans.Archimedes' problem of dividing a sphere by a plane into two segments having a prescribed ratio,was first expressed as a cubic

**equation**by Al Mahani, and the first solution was given by Abu Gafar al Hazin.The determination of the side of a regular heptagon which can be inscribed or circumscribed to a given circle was reduced to a more complicated

**equation**which was first successfully resolved by Abul Gud.Although the foundations of the geometrical resolution of cubic equations are to be ascribed to the Greeks (for Eutocius assigns to Menaechmus two methods of solving the

**equation**x 3 = a and x 3 = 2a 3), yet the subsequent development by the Arabs must be regarded as one of their most important achievements.An imperfect solution of the

**equation**x 3 +-- px 2 was discovered by Nicholas Tartalea (Tartaglia) in 1530, and his pride in this achievement led him into conflict with Floridas, who proclaimed his own knowledge of the form resolved by Ferro.This contest over, Tartalea redoubled his attempts to generalize his methods, and by 1541 he possessed the means for solving any form of cubic

**equation**.He also discovered how to sum the powers of the roots of an

**equation**.If the angular interval between the components of a double star were equal to twice that expressed in

**equation**(15) above, the central disks of the diffraction patterns would be just in contact.If we suppose that the force impressed upon the element of mass D dx dy dz is DZ dx dy dz, being everywhere parallel to the axis of Z, the only change required in our equations (I), (2) is the addition of the term Z to the second member of the third

**equation**(2).The cartesian

**equation**referred to the axis and directrix is y=c cosh (x/c) or y = Zc(e x / c +e x / c); other forms are s = c sinh (x/c) and y 2 =c 2 -1-s 2, being the arc measured from the vertex; the intrinsic**equation**is s = c tan The radius of curvature and normal are each equal to c sec t '.The first

**equation**to be established is the**equation**of continuity, which expresses the fact that the increase of matter within a fixed surface is due to the flow of fluid across the surface into its interior.Generally if S denotes any closed surface, fixed in the fluid, M the mass of the fluid inside it at any time t, and 0 the angle which the outward-drawn normal makes with the velocity q at that point, dM/dt = rate of increase of fluid inside the surface, (I) =flux across the surface into the interior _ - f f pq cos OdS, the integral

**equation**of continuity.The time rate of increase of momentum of the fluid inside S is )dxdydz; (5) and (5) is the sum of (I), (2), (3), (4), so that /if (dpu+dpu2+dpuv +dpuw_ +d p j d xdyd z = o, (b)` dt dx dy dz dx / leading to the differential

**equation**of motion dpu dpu 2 dpuv dpuv _ X_ (7) dt + dx + dy + dz with two similar equations.These equations may be simplified slightly, using the

**equation**of continuity (5) § for dpu dpu 2 dpuv dpuw dt dx + dy + dz =p Cat +uax+vay+waz?A bounding surface is such that there is no flow of fluid across it, as expressed by

**equation**(6).**Equation**(3) is called Bernoulli's**equation**, and may be interpreted as the balance-sheet of the energy which enters and leaves a given tube of flow.If homogeneous liquid is drawn off from a vessel so large that the motion at the free surface at a distance may be neglected, then Bernoulli's

**equation**may be written H = PIP--z - F4 2 / 2g = P/ p +h, (8) where P denotes the atmospheric pressure and h the height of the free surface, a fundamental**equation**in hydraulics; a return has been made here to the gravitation unit of hydrostatics, and Oz is taken vertically upward.The polar

**equation**of the cross-section being rI cos 19 =al, or r + x = 2a, (3) the conditions are satisfied by = Ur sin g -2Uairi sin IB = 2Uri sin 10(14 cos 18a'), (4) 1J/ =2Uairi sin IO = -U1/ [2a(r-x)], (5) w =-2Uaiz1, (6) and the resistance of the liquid is 2lrpaV2/2g.Similarly, with the function (19) (2n+ I) 3 ch (2n+ I) ITrb/a' (2) Changing to polar coordinates, x =r cos 0, y = r sin 0, the

**equation**(2) becomes, with cos 0 =µ, r'd + (I -µ 2)-d µ = 2 ?-r3 sin 0, (8) of which a solution, when = o, is = (Ar'+) _(Ari_1+) y2,, ?For instance, with n = I in

**equation**(9), the relative stream function is obtained for a sphere of radius a, by making it, y' =1y+2Uy 2 = 2U(r 2 -a 3 /r) sin?Employing the

**equation**of continuity when the liquid is homogeneous, 2 (cly - d z)?The continuity is secured if the liquid between two ellipsoids X and X 11 moving with the velocity U and 15 1 of

**equation**(II), is squeezed out or sucked in across the plane x=o at a rate equal to the integral flow of the velocity I across the annular area a l.I +W a W a), ' (k) 4 (I I) I+ w- R For a shot in air the ratio W'/W is so small that the square may be neglected, and formula (II) can be replaced for practical purpose in artillery by tan26= n2 = W i (0 - a) (k ð)7()4, (12) if then we can calculate /3, a, or (3-a for the external shape of the shot, this

**equation**will give the value of 6 and n required for stability of flight in the air.The cartesian

**equation**is y=a cos /2a.His first notable work was a proof of the impossibility of solving the quintic

**equation**by radicals.Hence the density v is given by 47rabc (x2/a4+y2/b4-I-z2/c4), and the potential at the centre of the ellipsoid, and therefore its potential as a whole is given by the expression, adS Q dS V f r 47rabc r' (x2/a4-I-y2/b4+z2/c4) Accordingly the capacity C of the ellipsoid is given by the

**equation**1 I J dS C 47rabc Y (x 2 +y 2 + z2) V (x2/a4+y2/b4+z2/c4) (5) It has been shown by Professor Chrystal that the above integral may also be presented in the form,' foo C 2 J o J { (a2 + X) (b +X) (c 2 + X) } (6).It may be looked upon as an

**equation**to determine p when V is given or vice versa.In the next place apply the surface characteristic

**equation**to any point on a charged conductor at which the surface density is a.The application of the first law leads immediately to the

**equation**, II=E - E,+W, .This

**equation**is generally true for any series of transformations, provided that we regard H and W as representing the algebraic sums of all the quantities of heat supplied to, and of work done by the body, heat taken from the body or work done on the body being reckoned negative in the summation.He therefore employed the corresponding expression for a cycle of infinitesimal range dt at the temperature t in which the work dW obtainable from a quantity of heat H would be represented by the

**equation**dW =HF'(t)dt, where F'(t) is the derived function of F(t), or dF(t)/dt, and represents the work obtainable per unit of heat per degree fall of temperature at a temperature t.Applying the above

**equation**to a gas obeying the law pv=RT, for which the work done in isothermal expansion from a volume i to a volume r is W=RT loger, whence dW=R log e rdt, he deduced the expression for the heat absorbed by a gas in isothermal expansion H=R log er/F'(t).The

**equation**to these lines in terms of v and 0 is obtained by integrating dE=sd0+(Odp/de - p)dv = o .The

**equation**to the lines of constant total heat is found in terms of p and 0 by putting dF=o and integrating (it).It has the characteristic

**equation**pv=Re, and obeys Boyle's law at all temperatures.The heat absorbed in isothermal expansion from vo to v at a temperature 0 is equal to the work done by

**equation**(8) (since d0 =o, and 0(dp/d0)dv =pdv), and both are given by the expression RO log e (v/vo).If we also assume that they are constant with respect to temperature (which does not necessarily follow from the characteristic

**equation**, but is generally assumed, and appears from Regnault's experiments to be approximately the case for simple gases), the expressions for the change of energy or total heat from 00 to 0 may be written E - Eo = s(0 - 0 0), F - Fo = S(0-00).In thiscase the ratio of the specific heats is constant as well as the difference, and the adiabatic

**equation**takes the simple form, pv v = constant, which is at once obtained by integrating the**equation**for the adiabatic elasticity, - v(dp/dv) =yp.The specific heats may be any function of the temperature consistently with the characteristic

**equation**provided that their difference is constant.If we assume that s is a linear function of 0, s= so(I +aO), the adiabatic

**equation**takes the form, s 0 log e OW +aso(0 - Oo) +R loge(v/vo) =oBut this procedure in itself is not sufficient, because, although it would be highly probable that a gas obeying Boyle's law at all temperatures was practically an ideal gas, it is evident that Boyle's law would be satisfied by any substance having the characteristic

**equation**pv = f (0), where f (0) is any arbitrary function of 0, and that the scale of temperatures given by such a substance would not necessarily coincide with the absolute scale.This gives by

**equation**(9) the condition Odp/d0 =p, which is satisfied by any substance possessing the characteristic**equation**p/0=f(v), where f(v) is any arbitrary function of v.We have therefore, by

**equation**, (11), Sd0 = (Odv/d0 - v) d p,.The characteristic

**equation**of the fluid must then be of the form v/0=f(p), where f(p) is any arbitrary function of p. If the fluid is a gas also obeying Boyle's law, pv = f (0), then it must be an ideal gas.Putting d0/dp=A/0 2 in

**equation**(15), and integrating on the assumption that the small variations of S could be neglected over the range of the experiment, they found a solution of the type, v/0 =f(p) - SA /30 3, in which f(p) is an arbitrary function of p. Assuming that the gas should approximate indefinitely to the ideal state pv = R0 at high temperatures, they put f(p)=Rip, which gives a characteristic**equation**of the form v= Re/p - SA /30 2 .Neglecting small terms of the second order, the

**equation**may then be written in the form v - b=RO/p - co(Oo/O)=V - c,..The introduction of the covolume, b, into the

**equation**is required in order to enable it to represent the behaviour of hydrogen and other gases at high temperatures and pressures according to the experiments of Amagat.The value of the co-aggregation volume, c, at any temperature, assuming

**equation**(17), may be found by observing the deviations from Boyle's law and by experiments on the Joule-Thomson effect.The value of the angular coefficient d(pv)/dp is evidently (b - c), which expresses the defect of the actual volume v from the ideal volume Re/p. Differentiating

**equation**(17) at constant pressure to find dv/do, and observing that dcldO= - nc/O, we find by substitution in (is) the following simple expression for the cooling effect do/dp in terms of c and b, Sdo/dp= (n+I)c - b..The advantage of this type of

**equation**is that c is a function of the temperature only.Other favourite types' of

**equation**for approximate work are (I) p=RO/v±f(v), which makes p a linear function of 0 at constant volume, as in van der Waal's**equation**; (2) v=RO/p+f(p), which makes v a linear function of 0 at constant pressure.In the modified Joule-Thomson

**equation**(17), both c and n have simple theoretical interpretations, and it is possible to express the thermodynamical properties of the substance in terms of them by means of reasonably simple formulae.This may be interpreted as the

**equation**of the border curve giving the relation between p and 0, but is more easily obtained by considering the equilibrium at constant pressure instead of constant volume.The direct integration of this

**equation**requires that L and v" - v' should be known as functions of p and 0, and cannot generally be performed.As an example of one of the few cases where a complete solution is possible, we may take the comparatively simple case

**equation**(17), already considered, which is approximately true for the majority of vapours at moderate pressures.In the case of ferrous sulphate, prepared by dissolving iron in dilute sulphuric acid, the reaction follows the

**equation**AuCl 3 +3FeS04 = FeC13-I-Fe2(S04)3+Au.It may be that the study of such sums, which he found in the works of Diophantus, prompted him to lay it down as a principle that quantities occurring in an

**equation**ought to be homogeneous, all of them lines, or surfaces, or solids, or supersolidsan**equation**between mere numbers being inadmissible.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.The volume u may be determined by repeating the experiment when only air is in the cup. In this case v =o, and the

**equation**becomes (u --al l) (h - k') =uh, whence u = al' (h - k l) /k'.The maintenance of the conditions of steadiness implied in

**equation**(I) depends upon the constancy of F, and therefore of the coefficient of friction µ between the rubbing surfaces.Thus the pressure is given by the

**equation**(p+a/v 2) (v - b) =RNT, which is known as Van der Waals's**equation**.This

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

**equation**of energy is dQ=dE+pdv, (17) expressing that the total energy dQ is used partly in increasing the internal energy of the gas, and partly in expanding the gas against the pressure p. If we take p = RNT/v from**equation**(14) and substitute for E from**equation**(16), this last**equation**becomes dQ 2 (n +3)RNdT +RNTdv (18) which may be taken as the general**equation**of calorimetry, for a gas which accurately obeys**equation**(14).If the volume of the gas is kept constant, we put dv=o in

**equation**(18) and dQ = JC0NmdT, where C v is the specific Specific heat of the gas at constant volume and J is the mechanical equivalent of heat.He also showed that the roots of a cubic

**equation**can be derived by means of the infinitesimal calculus.This involves that its activity cannot be truly conceived of as included in an antecedent, as an effect in a cause or one term of an

**equation**in the other.The

**equation**holds, more firmly than ever; dogma = the contents of That seems to be what is meant.The Diophantine analysis was a favourite subject with Pell; he lectured on it at Amsterdam; and he is now best remembered for the indeterminate

**equation**ax 2 +1 = y 2, which is known by his name.The statement that the ordinate u of a trapezette is a function of the abscissa x, or that u=f(x), must be distinguished from u =f(x) as the

**equation**to the top of the trapezette.Combining this with the first

**equation**, we obtain the values of P, Q, R, ..Although Hippocrates could not determine the proportionals, his statement of the problem in this form was a great advance, for it was perceived that the problem of trisecting an angle was reducible to a similar form which, in the language of algebraic geometry, is to solve geometrically a cubic

**equation**.If we omitted it we should have to assume this, and

**equation**(6) would give us the velocity of propagation if the assumption were justified.In the momentum

**equation**(4) we may now omit X and it becomes 0.+P(U - u) 2 =poU2.This is hardly to be explained by

**equation**(I I), for at the very front of the disturbance u =o and the velocity should be normal.The receiving apparatus had what we may term a personal

**equation**, for the break of contact could only take place when the membrane travelled some finite distance, exceedingly small no doubt, from the contact-piece.In some experiments in which contact was made instead of broken, Regnault determined the personal

**equation**of the apparatus.The temperature of the air traversed and its humidity were observed, and the result was finally corrected to the velocity in dry air at o C. by means of

**equation**(ro).For the superposition of these trains will give a stationary wave between A H A (16) Y which is an

**equation**characteristic of simple harmonic motion.Putting A /M =n 2 the

**equation**becomes x+n 2 x=o, whence x =A sin nt, and the period is 27r/n.Repre - senting it by -P sin pt, the

**equation**of motion is now 2 -M sin pt=o.We may represent the displacement due to one of the trains by y l =a sin 2 i (24) where x is measured as in

**equation**(16) from an ascending node as A in fig.If th' maximum pressure change is determined, the amplitude is given by

**equation**(20), viz.But inasmuch as the successive orders are proportional to A X 2 A 3, or µµ 2 µ 3, and X and µ are small, they are of rapidly decreasing importance, and it is not certain that any beyond those in

**equation**(35) correspond to our actual sensations.In analytical geometry, the

**equation**to the sphere takes the forms x 2 +y 2 +z 2 =a 2, and r=a, the first applying to rectangular Cartesian co-ordinates, the second to polar, the origin being in both cases at the centre of the sphere.Two spheres intersect in a plane, and the

**equation**to a system of spheres which intersect in a common circle is x 2 + y 2 + z 2 +2Ax -fD = o, in which A varies from sphere to sphere, and D is constant for all the spheres, the plane yz being the plane of intersection, and the axis of x the line of centres.The value of R, the tension at any point at a distance x from the vertex, is obtained from the

**equation**R 2 = H2 +V2 = w2x4 /4Y 2 +w2x2, or, 2.The most striking addition which is here made to previous researches consists in the treatment of a planet supposed entirely fluid; the general

**equation**for the form of a stratum is given for the first time and discussed.The first

**equation**leads, as before, to t=C{T (V)-T(v)}, (29) x=C{S(V)-S(v)}.This is the consummation towards which events had been steadily moving - not at first consciously, for it was some time before the tendencies at work were consciously realized - but ending at last in the complete

**equation**of Old Testament and New, and in the bracketing together of both as the first and second volumes of a single Bible.This relation between x, a, rn, may be expressed also by the

**equation**x= log m.The logarithmic function is most naturally introduced into analysis by the

**equation**log x= x ?This

**equation**defines log x for positive values of x; if o the formula ceases to have any meaning.A relation which is of historical interest connects the logarithmic function with the quadrature of the hyperbola, for, by considering the

**equation**of the hyperbola in the form xy=const., it is evident that the area included between the arc of a hyperbola, its nearest asymptote, and two ordinates drawn parallel to the other asymptote from points on the first asymptote distant a and b from their point of intersection, is proportional to log bla.Other formulae which are deducible from this

**equation**are given in the portion of this article relating to the calculation of logarithms.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.If a= logg = - log (1-10)' 81 (1) c = 10g 80 = log 1 +sr, 126 (8) e =10g1 o = log 1 +1000 then log 2=7a-2b+3c, log 3=IIa-3b+5c, log 5=16a-4b +7c, and log 7 =2(39a - IOb+17c - d) or=19a-4b+8c +e, and we have the

**equation**of condition, a-2b+c=d+2e.In a region where there is no absorption, we have = o and therefore g=o, and we have only one

**equation**, namely, x2_ A2m' which is identical with Sellmeier's result.The polar

**equation**is r=a+b cos 0, where 2a= length of the rod, and b= diameter of the circle.Deducing from the figures of production since 1859 an

**equation**of increase, one finds that in each nine years as much oil has been produced as in all preceding years together, and in recent years the factor of increase has been higher.The

**equation**x 2 +y 2 =o denotes a pair of perpendicular imaginary lines; it follows, therefore, that circles always intersect in two imaginary points at infinity along these lines, and since the terms x 2 +y 2 occur in the**equation**of every circle, it is seen that all circles pass through two fixed points at infinity.Since the

**equation**to a circle of zero radius is x 2 +y 2 =o, i.e.The general

**equation**to the circle in trilinear co-ordinates is readily deduced from the fact that the circle is the only curve which intersects the line infinity in the circular points.The line la+ma+ny is the radical axis, and since as+43 c-y =o is the line infinity, it is obvious that

**equation**(I) represents a conic passing through the circular points, i.e.If we compare (I) with the general

**equation**of the second degree ua2 + v/ 32 +wy2 this**equation**to represent a circle we must have - kabc =vc 2 +wb 2 -2u'bc=wa g +uc 2 - 2v'ca = ub 2 +va - 2w'ab.Analytically, the Cartesian

**equation**to a coaxal system can be written in the form x 2 + y 2 + tax k 2 = o, where a varies from member to member, while k is a constant.In 1873 Charles Hermite proved that the base of the Napierian logarithms cannot be a root of a rational algebraical

**equation**of any degree.3 To prove the same proposition regarding 7r is to prove that a Euclidean construction for circle-quadrature is impossible.For in such a construction every point of the figure is obtained by the intersection of two straight lines, a straight line and a circle, or two circles; and as this implies that, when a unit of length is introduced, numbers employed, and the problem transformed into one of algebraic geometry, the equations to be solved can only be of the first or second degree, it follows that the

**equation**to which we must be finally led is a rational**equation**of even degree.The generality of treatment is indeed remarkable; he gives as the fundamental property of all the conics the equivalent of the Cartesian

**equation**referred to oblique axes (consisting of a diameter and the tangent at its extremity) obtained by cutting an oblique circular cone in any manner, and the axes appear only as a particular case after he has shown that the property of the conic can be expressed in the same form with reference to any new diameter and the tangent at its extremity.Apollonius' genius takes its highest flight in Book v., where he treats of normals as minimum and maximum straight lines drawn from given points to the curve (independently of tangent properties), discusses how many normals can be drawn from particular points, finds their feet by construction, and gives propositions determining the centre of curvature at any point and leading at once to the Cartesian

**equation**of the evolute of any conic.It is in this department of criticism that the personal

**equation**has the freest play, and hence the natural adherents of either school of critics should be specially on their guard against their school's peculiar bias.An inquirer who examines the stars with a shilling telescope is not likely to make observations of value, and even a trained astronomer has to allow for his "personal

**equation**" - a point to which even a finished critic rarely attends.Now In This

**Equation**P' May Be Any Number Whatever, Provided 15 P Exceed 52.This Method Of Forming The Epacts Might Have Been Continued Indefinitely If The Julian Intercalation Had Been Followed Without Correction, And The Cycle Been Perfectly Exact; But As Neither Of These Suppositions Is True, Two Equations Or Corrections Must Be Applied, One Depending On The Error Of The Julian Year, Which Is Called The Solar

**Equation**; The Other On The Error Of The Lunar Cycle, Which Is Called The Lunar**Equation**.The Solar

**Equation**Occurs Three Times In 400 Years, Namely, In Every Secular Year Which Is Not A Leap Year; For In This Case The Omission Of The Intercalary Day Causes The New Moons To Arrive One Day Later In All The Following Months, So That The Moon'S Age At The End Of The Month Is One Day Less Than It Would Have Been If The Intercalation Had Been Made, And The Epacts Must Accordingly Be All Diminished By Unity.Thus The Epacts 11, 22, 3, 14, &C., In Consequence Of The Lunar

**Equation**, Become 12, 23, 4, 15, &C. In Order To Preserve The Uniformity Of The Calendar, The Epacts Are Changed Only At The Commencement Of A Century; The Correction Of The Error Of The Lunar Cycle Is Therefore Made At The End Of 300 Years.The Years In Which The Solar

**Equation**Occurs, Counting From The Reformation, Are 1700, 1800, 1900, 2100, 2200, 2300, 2500, &C. Those In Which The Lunar**Equation**Occurs Are 1800, 2100, 2400, 2700, 3000, 3300, 3600, 3900, After Which, 4300, 4600 And So On.In That Year The Omission Of The Intercalary Day Rendered It Necessary To Diminish The Epacts By Unity, Or To Pass To The Line C. In 1800 The Solar

**Equation**Again Occurred, In Consequence Of Which It Was Necessary To Descend One Line To Have The Epacts Diminished By Unity; But In This Year The Lunar**Equation**Also Occurred, The Anticipation Of The New Moons Having Amounted To A Day; The New Moons Accordingly Happened A Day Earlier, Which Rendered It Necessary To Take The Epacts In The Next Higher Line.When The Solar

**Equation**Occurs Alone, The Line Of Epacts Is Changed To The Next Lower In The Table; When The Lunar**Equation**Occurs Alone, The Line Is Changed To The Next Higher; When Both Equations Occur Together, No Change Takes Place.Hence In The Julian Calendar The Dominical Letter Is Given By The

**Equation**L= 7M 3 X () W This**Equation**Gives The Dominical Letter Of Any Year From The Commencement Of The Era To The Reformation.In Order To Investigate A Formula For The Epact, Let Us Make E=The True Epact Of The Given Year; J =The Julian Epact, That Is To Say, The Number The Epact Would Have Been If The Julian Year Had Been Still In Use And The Lunar Cycle Had Been Exact;, S =The Correction Depending On The Solar Year; M =The Correction Depending On The Lunar Cycle; Then The

**Equation**Of The Epact Will Be E=J S M; So That E Will Be Known When The Numbers J, S, And M Are Determined.Had The Anticipation Of The New Moons Been Taken, As It Ought To Have Been, At One Day In 308 Years Instead Of 3121, The Lunar

**Equation**Would Have Occurred Only Twelve Times In 3700 Years, Or Eleven Times Successively At The End Of 300 Years, And Then At The End Of 400.In Those Years In Which The Line Of Epacts Is Changed In The Gregorian Calendar, The Golden Numbers Are Removed To Different Days, And Of Course A New Table Is Required Whenever The Solar Or Lunar

**Equation**Occurs.Thus if a molecule were set into vibration at a specified time and oscillated according to the above

**equation**during a finite period, it would not send out homogeneous vibrations.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.Halm subsequently showed that if N may differ in different cases, the

**equation**is a considerable improvement on Rydberg's.As he takes N to be strictly the same for all elements the

**equation**has only three disposable constants A, a and b.Rydberg discovered a second relationship, which, however, involving the assumed

**equation**connecting the different lines, cannot be tested directly as long as these equations are only approximate.On the other hand the law, once shown to hold approximately, may be used to test the sufficiency of a particular form of

**equation**.As has already been mentioned, the law is only verified very roughly, if Rydberg's form of

**equation**is taken as correctly representing the series.The fact that the addition of the term introduced by Ritz not only gives a more satisfactory representation of each series, but verifies the above relationship with a much closer degree of approximation, proves that Ritz's

**equation**forms a marked step in the right direction.Hicks 1 has modified Rydberg's

**equation**in a way similar to that of Ritz as shown by (5) above.This form has the advantage that the constants of the

**equation**when applied to the spectra of the alkali metals show marked regularities.If we compare Balmer's formula with the general

**equation**of Ritz, we find that the two can be made to agree if the ordinary hydrogen spectrum is that of the side branch series and the constants a', b, c and d are all put equal to zero.'The other method starts from the observed values of the periods, and establishes a differential

**equation**from which these periods may be derived.This is done in the hope that some theoretical foundation may then be found for the

**equation**.Riecke, 3 who deduced a differential

**equation**of the 10th order.Lord Rayleigh,' who has also investigated vibrating systems giving series of lines approaching a definite limit of " root," remarks that by dynamical reasoning we are always led to equations giving the square of the period and not the period, while in the

**equation**representing spectral series the simplest results are obtained for the first power of the period.The

**equation**which expressed " Deslandres' law " was only given by its author as an approximate one.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 r =1, we obtain Pickering's

**equation**, which is the one advocated by Halm.The distance between the lines measured on the frequency scale does not, according to the

**equation**, increase indefinitely from the head downwards, but has a maximum which, in Pickering's form as written above, is reached when (s +, u) 2 = 3a.Halm,' to whom we owe a careful comparison of the above

**equation**with the observed frequencies in a great number of spectra, attached perhaps too much weight to the fact that it is capable of representing both line and band spectra.These two equations involve the third relation µ2 =As, which therefore is not an independent

**equation**.The slope of these curves is determined by the so-called "latent heat

**equation**" FIG.The phase rule combined with the latent heat

**equation**contains the whole theory of chemical and physical equilibrium.Since, in dilute solutions, the osmotic pressure has the gas value, we may apply the gas

**equation**PV=nRT =npvi to osmotic relations.In the vapour pressure

**equation**p - p' = Pa/p, we have the vapour density equal to M/v 1, where M is the molecular weight of the solvent.The relation between the equilibrium pressures P and P' for solution and solvent corresponding to the same value po of the vapour pressure is obtained by integrating the

**equation**V'dP' = vdp between corresponding limits for solution and solvent.From this

**equation**the osmotic pressure Po required to keep a solution in equilibrium as regards its vapour and through a semi-permeable membrane with its solvent, when that solvent is under its own vapour pressure, may be calculated from the results of observations on vapour pressure of solvent and solution at ordinary low hydrostatic pressures.The slope of the temperature vapour pressure curves in the neighbourhood of the freezing point of the solvent is given by the latest heat

**equation**.The variation of gases from Boyle's law is represented in the

**equation**of Van der Waals by subtracting a constant b from the total volume to represent the effect of the volume of the molecules themselves.By an imaginary cycle of operations we may then justify the application to solutions of the latent heat

**equation**which we have already assumed as applicable.In the

**equation**dP/dT= X/T(v 2 - v 1), P is the osmotic pressure, T the absolute temperature and X the heat of solution of unit mass of the solute when dissolving to form a volume v2 - v1 of saturated solution in an osmotic cylinder.This result must hold good for any solution, but if the solution be dilute when saturated, that is, if the solubility be small, the

**equation**shows that if there be no heat effect when solid dissolves to form a saturated solution, the solubility is independent of temperature, for, in accordance with the gas laws, the osmotic pressure of a dilute solution of constant concentration is proportional to the absolute temperature.The osmotic pressure of a solution depends on the concentration, and, if we regard the difference in that pressure as the effective force driving the dissolved substance through the solution, we are able to obtain the

**equation**of diffusion in another form.By comparison with the first

**equation**we see that RT/F is equal to D, the diffusion constant.Its cartesian

**equation**is x 3 -1-y 3 =3axy.It may be traced by giving m various values in the equations x=3am/ ('1-1-m 3 ),' y=3am2 (1-1-m 3), since by eliminating m between these relations the

**equation**to the curve is obtained.Monge's memoir just referred to gives the ordinary differential

**equation**of the curves of curvature, and establishes the general theory in a very satisfactory manner; but the application to the interesting particular case of the ellipsoid was first made by him in a later paper in 1795.The most simple case is presented by the two platinum compounds PtC12(NH3)2, the platosemidiammine chloride of Peyrone, and the platosammine chloride of Jules Reiset, the first formed according to the

**equation**PtC1 4 K 2 + 2NH 3 = PtCl 2 (NH 3) 2 + 2KC1, the second according to Pt(NH 3) 4 C1 2 =PtC1 2 (NH 3) 2 +2NH 3, these compounds differing in solubility, the one dissolving in 33, the other in 160 parts of boiling water.In this

**equation**a relates to molecular attraction; and it is not improbable that in isomeric molecules, containing in sum the same amount of the same atoms, those mutual attractions are approximately the same, whereas the chief difference lies in the value of b, that is, the volume occupied by the molecule itself.In algebra it denoted the characters which represented quantities in an

**equation**.If Q is expressed in terms of this unit in

**equation**(I), it is necessary to divide by c, or to replace k on the right-hand side by the ratio k/c. This ratio determines the rate of diffusion of temperature, and is called the thermometric conductivity or, more shortly, the diffusivity.To find the conductivity, it is necessary to measure all the quantities which occur in

**equation**(I) to a similar order of accuracy.This gives an average value of the conductivity over the range, but it is better to observe the temperatures at three distances, and to assume k to be a linear function of the temperature, in which case the solution of the

**equation**is still very simple, namely, 0+Ze6 2 =a log r+b, (3) where e is the temperature-coefficient of the conductivity.If the thickness of the glass is small compared with the diameter of the tube, say one-tenth,

**equation**(1) may be applied with sufficient approximation, the area A being taken as the mean between the internal and external surfaces.We thus obtain the simple

**equation**k'(de'/dx') - k"(de"/dx") =c (area between curves)/(T - T'), (4) by means of which the average value of the diffusivity klc can be found for any convenient interval of time, at different seasons of the year, in different states of the soil.To illustrate the main features of the calculation, we may suppose that the surface is subject to a simple-harmonic cycle of temperature variation, so that the temperature at any time t is given by an

**equation**of the form 0 - 0 0 = Asin 27rnt= A sin 27rt/T, (5) where 0 0 is the mean temperature of the surface, A the amplitude of the cycle, n the frequency, and T the period.The wave at a depth x is represented analytically by the

**equation**0 - 0 0 = Ae mx sin (21rnt - mx).The dotted boundary curves have the

**equation**0 =omx, and show the rate of diminution of the amplitude of the temperature oscillation with depth in the metal.The

**equation**of the method is the same as that for the linear flow with the addition of a small term representing the radiation loss.We thus obtain the differential

**equation**gk(d 2 0/dx 2) =cgdo/dt+hpo, which is satisfied by terms of the type =c" sin where a 2 -b 2 = hp/qk, and ab = urnc/k.The differential

**equation**for the distribution of temperature in this case includes the majority of the methods already considered, and may be stated as follows.We thus obtain the

**equation**C 2 R 0 (i +ao)/1 =-d(gkdo/dx)/dx+hpo+gcdo/dt+sCdo/dx.If h also is zero, it becomes the

**equation**of variable flow in the soil.If do/dt = o, the

**equation**represents the corresponding cases of steady flow.In this case the solution of the

**equation**reduces to the form e =x(1 - x)C 2 Ro/2lgk.Neglecting the external heat-loss, and the variation of the thermal and electric conductivities k and k', we obtain, as before, for the difference of temperature between the centre and ends, the

**equation**O, Tho z Go = C 2 R1/8qk = ECl/8qk = E 2 k'/8k, (11) where E is the difference of electric potential between the ends.Trans., A., 1893) that this frequency may be closely represented by the curve whose

**equation**is y = O.21 122 5 x-( 332 (7.3 2 53 - x) 3.142.The simplest

**equation**to the parabola is that which is referred to its axis and the tangent at the vertex as the axes of co-ordinates, when it assumes the form y 2 = 4ax where as = semilatus rectum; this may be deduced directly from the definition.An

**equation**of similar form is obtained when the axes of co-ordinates are any diameter and the tangent at the vertex.Expressing this condition we obtain mb = 1/ nc = o as the relation which must hold between the co-efficients of the above

**equation**and the sides of the triangle of reference for the**equation**to represent a parabola.The pedal

**equation**with the focus as origin is p 2 =ar; the first positive pedal for the vertex is the cissoid and for the focus the directrix.In the geometry of plane curves, the term parabola is often used to denote the curves given by the general

**equation**a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian parabola; a 2 x = y 3 is the cubic parabola, a 3 x = y4 is the biquadratic parabola; semi parabolas have the general**equation**ax n-1 = yn, thus ax e = y 3 is the semicubical parabola and ax 3 = y 4 the semibiquadratic parabola.