# How to use *Equation* in a sentence

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**.AdvertisementThe 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.By his mode of regarding a liquid as a material system characterized by the unshackled mobility of its minutest parts, the separation between the mechanics of matter in different forms of aggregation finally disappeared, and the fundamental

**equation**of forces was for the first time extended to hydrostatics and hydrodynamics.'Modified Joule-Thomson

**Equation**.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.AdvertisementIf 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).AdvertisementThe 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.AdvertisementHe 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.AdvertisementThus, 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.Diverging parabolas are cubic curves given by the

**equation**y 2 = 3 -f-bx 2 -cx+d.Newton discussed the five forms which arise from the relations of the roots of the cubic

**equation**.If two roots are imaginary the

**equation**is y 2 =(x 2 +a 2) (x - b) and the curve resembles the parabolic branch, as in the preceding case.His well-known correction of Laplace's partial differential

**equation**for the potential was first published in the Bulletin de la societe philomatique (1813).If Hdo Is The Radiation Loss In Watts We Have The

**Equation**, Ec=Jsqdo Hdo.Given The Specific Heat As A Function Of The Temperature, Its Variation With Pressure May Be Determined From The Characteristic

**Equation**Of The Gas.It follows also that the normal judgment is not an

**equation**.In his Treatise of Algebra (1685) he distinctly proposes to construct the imaginary roots of a quadratic

**equation**by going out of the line on which the roots, if real, would have been constructed.He has given by means of it a simple proof of the existence of n roots, and no more, in every rational algebraic

**equation**of the nth order with real coefficients.For this

**equation**merely states that m turnings of a line through successive equal angles, in one plane, give the same result as a single turning through m times the common angle.For the rise in the boiling-point, we have by Clapeyron's

**equation**, dp/do = L/ov, nearly, neglecting the volume of the liquid as compared with that of the vapour v.These cases are really included in the

**equation**if we substitute the proper values of n or m.We may observe that the

**equation**(51) is accurately true for an ideal vapour, for which pv = (S-s)0, provided that the total heat is defined as equal to the change of the function (E+pv) between the given limits.Assuming an

**equation**of the form log (p/760) =a log (0/373), their results give a = S/R =4.305, or S=0.474, which agrees very perfectly with Regnault's value.Perry (Steam Engine, p. 580), assuming a characteristic

**equation**similar to Zeuner's (which makes v a linear function of the temperature at constant pressure, and S independent of the pressure), calculates S as a function of the temperature to satisfy Regnault's formula (10) for the total heat.From a different point of view,

**equation**(12) may be applied to determine the specific heat of steam in terms of the rate of variation of the total heat.The more logical method of procedure is to determine the specific heat independently of the total heat, and then to deduce the variations of total heat by

**equation**(52).In order to correct this

**equation**for the deviations of the vapour from the ideal state at higher temperatures and pressures, the simplest method is to assume a modified**equation**of the Joule-Thomson type (Thermodynamics,**equation**(17)), which has been shown to represent satisfactorily the behaviour of other gases and vapours at moderate pressures.Employing this type of

**equation**, all the thermodynamical properties of the substance may conveniently be expressed in terms of the diminution of volume c due to the formation of compound or coaggregated molecules, (v - b) =RO/p - co(Oo/O) n =V - c. .The rate of increase of the total heat, instead of being constant for saturated steam as in Regnault's formula, is given by the

**equation**dH/d0 =S(1 - Qdp/d0).This relation cannot be directly integrated, so as to obtain the

**equation**for the saturation-pressure, unless L and v - w are known as functions of 0.By assuming suitable forms of the characteristic

**equation**to represent the variations of the specific volume within certain limits of pressure and temperature, we may therefore with propriety deduce equations to represent the saturation-pressure, which will certainly be thermodynamically consistent, and will probably give correct numerical results within the assigned limits.The approximate

**equation**of Rankine (23) begins to be I or 2% in error at the boiling-point under atmospheric pressure, owing to the coaggregation of the molecules of the vapour and the variation of the specific heat of the liquid.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.The effect of variation of the specific heat is more important, but is nearly eliminated by the form of the

**equation**.If we proceed instead by the method of integrating the

**equation**H -h =6(v-w)dp/d6, we observe that the expression above given results from the integration of the terms -dh/R0 2 +w(dp/d9)/R9, which were omitted in (25).It is interesting to remark that the simple result found in

**equation**(25) (according to which the effect of the deviation of the vapour from the ideal state is represented by the addition of the term (c-b)/V to the expression for log p) is independent of the assumption that c varies inversely as the n th power of 9, and is true generally provided that c-b is a function of the temperature only and is independent of the pressure.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.Hence in trilinear co-ordinates, with ABC as fundamental triangle, its

**equation**is Pa+Q/1+R7=o.Every additional constraint introduces an additional

**equation**of the type (10) and reduces the number of degrees of freedom by one.Each such reaction consists of two equal and oppositeforces, both of which may contribute to the

**equation**of virtual work.Again, the normal pressure between two surfaces disappears from the

**equation**, provided the displacements be such that one of these surfaces merely slides relatively to the other.The former

**equation**expresses that the horizontal tension is constant.This is the intrinsic

**equation**of the curve.Since the projection of a vector- sum is the sum of the projections of the several vectors, the

**equation**(2) gives if x be the projection of 0G.When referred to its principal axes, the

**equation**of the quadric takes the form Axi+By2+Czi=M.If all the masses lie in a plane (1=0) we have, in the notation of (25), c2 = o, and therefore A = Mb, B = Ma, C = M (a +b), so that the

**equation**of the momental ellipsoid takes the form b2x2+a y2+(a2+b2) z1=s4.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 integrate the

**equation**(I) with respect to I between the limits 1, 1 we obtain mu_mu=fXdI.The small oscillations of a simple pendulum in a vertical plane also come under

**equation**(5).In the case of a repulsive force varying as the distance from the origin, the

**equation**of motion is of the type the solution of which is x=Ae+Be, (10)This applies to the inverted pendulum, with u =g/l, but the

**equation**(9) is then only approximate, and the solution therefore only serves to represent the initial stages of a motion in the neighborhood of the position of unstable equilibrium.If the point of suspension have an imposed simple vibration f = a cos at in a horizontal line, the

**equation**of small motion of the bob is mx= mg-l-,The product 4muf is called the kinetic energy of the particle, and the

**equation**.The

**equation**(21) may now be written 3/4 muii+Vi=3/4muoi+Vo, (23)In any general dynamical

**equation**the dimensions of each term in the fundamental units must be the same, for a change of units would otherwise alter the various terms in different ratios.Again, the time of falling from a distance a into a given centre of force varying inversely as the square of the distance will depend only on a and on the constant u of

**equation**(15).Eliminating t we have the

**equation**of the path, viz.If in this we put r= I/u, and eliminate t by means of (15), we obtain the general differential

**equation**of central orbits, viz.This is recognized as the polar

**equation**of a conic referred to the focus, the half latus-rectum being hf/u.If P=pr, its polar

**equation**is rcosme=a, (27)This may be compared with the

**equation**of rectilinear motion of a particle, viz.This coincides with the

**equation**of motion of a simple pendulum [E 13 (15)] of length 1, provided 1= hf/h.The

**equation**of motion is of the type I=K0, (6)The

**equation**of energy for a rigid body has already been stated (in effect) as a corollary from fundamental assumptions.The

**equation**of the latter, referred to its principal axes, being as in II (41), the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or X, u, v.Take, for example, the case of a sphere rolling on a plane; and let the axes Ox, Oy be drawn through the centre parallel to the plane, so that the

**equation**of the latter is 1=cf.By these and the similar transformations relating to y and z the

**equation**(6) takes the form d IOT\ aTThis

**equation**expresses that the kinetic energy is increasing at a rate equal to that at which work is being done by the forces.In the case of a conservative system free from extraneous force it becomes the

**equation**of energy (T+V) =0, or T+V=const., (20)It may be shown algebraically that under theseconditions the n roots of the above

**equation**in r2 are all real and positive.These statements require \ some modification when two or more of the roots urn of the

**equation**(6) are equal.This leads to a determinantal

**equation**in X whose 2n roots are either real and negative, or complex with negative real parts, on the present hypothesis that the functions T, V, F are all essentially positive.The series of equations of the type (3) is then replaced by a single linear partial differential

**equation**, or by a set of two or three such equations, according to the number of dependent variables.These limit the admissible values of a-, which are in general determined by a transcendental

**equation**corresponding to the determinantal**equation**(6).Hence, forming the

**equation**of motion of a masselement, plx, we have pSx.fi=I(P.Oy/8x).When the pressure and temperature of the air can be maintained constant, this machine fulfils

**equation**(2), like the hydraulic press.The only modification required in the formulae is, that in

**equation**(26) the difference of the angular velocities should be substituted for their sum.Removing the summation signs in

**equation**(52) in order to restrict its application to two points and dividing by the common time interval during which the respective small displacements ds and ds were made, it becomes Pdsfdt = Rds/dt, that is, Pv = Rv, which shows that the force ratio is the inverse of the velocity ratio.But, p and q being respectively the perpendiculars to the lines of action of the forces, this

**equation**reduces to Pp=Rq, FIG.The result of this investigation, expressed in the form of an

**equation**between this energy and the useful work, is called by Moseley the modulus of the machine.Moseley, however, has pointed out that in most cases this

**equation**takes the much more simple form of E=(1+A)U+B, (55)The efficiency corresponding to the last

**equation**is U ILet v be the common velocity of the two pitch-circles, ri, C2, their radii; then the above

**equation**becomes /1 IBy

**equation**(72) we have E = w(v,2 vi2)/ag which, being divided by V=1/2(v,+vi), gives E/V=w(vfvi)/g; and consequently V2 in = gE/Vw (73)Let M be the moment of the unbalanced couple required to produce the deviation; ther by

**equation**57, 104, the energy exerted by this couple in tht interval dt is Macit, which, being equated to the variation of energy gives da R2W da -A very gentle heating gives decomposition approximating to the

**equation**of 22KC103=14KC10 4 +8KC1+50 2, whilst on a more rapid heating the quantities correspond more nearly to loKC10 3 = 6KC104+4KC1+ 302.Lagrange used simple continued fractions to approximate to the solutions of numerical equations; thus, if an

**equation**has a root between two integers a and a+1, put x=a+I/y and form the**equation**in y; if the**equation**in y has a root between b and b+i, put y = b + I /z, and so on.The solution in integers of the indeterminate

**equation**ax+by=c may be effected by means of continued fractions.The value of such a fraction is the positive root of a quadratic

**equation**whose coefficients are real and of which one root is negative.In the case of a recurring continued fraction which represents N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and pnr/gnr the nr th convergent, then p 2 nr - Ng2nr = (- I) nr, whence, if n is odd, integral solutions of the indeterminate

**equation**x 2 - Ny 2 = I (the so-called Pellian**equation**) can be found.If n is even, solutions of the

**equation**x 2 -Ny 2 =+1 can be found.In the Supplement to the Theory of Capillary Action, Laplace deduced the

**equation**of the surface of the fluid from the condition that the resultant force on a particle at the surface must be normal to the surface.This condition when worked out gives not only the

**equation**of the free surface in the form already established by Laplace, but the conditions of the angle of contact of this surface with the surface of a solid.In particular he maintained that the constant pressure K, which occurs in Laplace's theory, and which on that theory is very large, must be in point of fact very small, but the

**equation**of equilibrium from which he concluded this is itself defective.In both theories the

**equation**of the liquid surface is the same, involving a constant H, which can be determined only by experiment.Integrating the first term within brackets by parts, it becomes - fo de Remembering that 0(o) is a finite quantity, and that Viz = - (z), we find T = 4 7rp f a, /.(z)dz (27) When c is greater than e this is equivalent to 2H in the

**equation**of Laplace.But this

**equation**is applicable only at points in the interior, where p is not varying.] [The intrinsic pressure and the surface-tension of a uniform mass are perhaps more easily found by the following process.This

**equation**gives a relation between the inclination of the curve to the horizon and the height above the level of the liquid.Hence the

**equation**of work and energy is p dV = Tds (6) 41rpr 2 dr = 8zrrdrT (7) p = 2T/r (8) This, therefore, is the excess of the pressure of the air within the bubble over that of the external air, and it is due to the action of the inner and outer surfaces of the bubble.Differentiating

**equation**9 with respect to s we obtain, after dividing by 27 as a common factor, pyds - T cos a ds + Ty s i n ad s =o...Hence dividing

**equation**io by y sin a, we find p= T(I/R1+I /R2) (14).This

**equation**, which gives the pressure in terms of the principal radii of curvature, though here proved only in the case of a surface of revolution, must be true of all surfaces.Before going further we may deduce from

**equation**9 the nature of all the figures of revolution which a liquid film can assume.When the ellipse differs infinitely little from a circle, the

**equation**of the meridian line becomes approximately y = a+c sin (x/a) where c is small.Suppose, therefore, that the

**equation**of the boundary is r =a+a cos kz, (I) where a is a small quantity, the axis of z being that of symmetry.The wave-length of the disturbance may be called A, and is connected with k by the

**equation**k= 271/A.In the case of a continuous jet, the

**equation**of continuity shows that as the jet loses velocity in ascending, it must increase in section.One form of the solution of the

**equation**, and that which is applicable to the case of a rectangular orifice, is z =C sin px sin qy.Substituting in the

**equation**we find the condition +" stable.If the wave-length is X, the

**equation**of the surface is y=b sin 2lrxx The pressure due to the surface tension T is p= - Td 2 = 4Ty.When the orifice is circular of radius a, the limiting value of a is 1 J' z, where z is the least root of the

**equation**FIG.If the current-function of the water referred to the body considered as origin is Ili, then the

**equation**of the form of the crest of a wave of velocity w, the crest of which travels along with the body, is d =w ds where ds is an element of the length of the crest.To integrate this

**equation**for a solid of given form is probably difficult, but it is easy to see that at some distance on either side of the body, where the liquid is sensibly at rest, the crest of the wave will approximate to an asymptote inclined to the path of the body at an angle whose sine is w/V, where w is the velocity of the wave and V is that of the body.Taking the case where the motion is strictly in two dimensions, we may write as the polar

**equation**of the surface at time t r =a cos nB cos pt, (4) where p is given by p2 = (n3_ n)..It may be shown that if the distance of the carried point from the centre of the rolling circle be mb, the

**equation**to the epitrochoid is x = (a+b) cos 0 - mb cos (a+b/b)0, y = (a +b) sin 9 - mb sin (a +b/b)0.If the radius of the rolling circle be one-half of the fixed circle, the hypocycloid becomes a diameter of this circle; this may be confirmed from the

**equation**to the hypocycloid.If the ratio of the radii be as I to 4, we obtain the four-cusped hypocycloid, which has the simple cartesian

**equation**x 2'3+ y 213 = a 21 '.The cartesian

**equation**was first given by Wilhelm Gottfried Leibnitz (Ada eruditorum, 1686) in the form y = (2xx 2)-1.The intrinsic P

**equation**is s =4a sin 4,, and the**equation**to the evolute is s= 4a cos 1P, which proves the evolute to be a similar cycloid placed as in fig.The radius of curvature at any point is readily deduced from the intrinsic

**equation**and has the value p=4 cos 40, and is equal to twice the normal which is 2a cos 2B.The cartesian

**equation**in terms similar to those used above is x = a6+b sin 0; y=a-b cos 0, where a is the radius of the generating circle and b the distance of the carried point from the centre of the circle.The cartesian

**equation**, referred to the fixed diameter and the tangent at B as axes may be expressed in the forms x= a6, y=a(I -cos 0) and y-a=a sin (x/afir); the latter form shows that the locus is the harmonic curve.