# Equations Sentence Examples

Now by the theory of symmetric functions, any symmetric functions of the mn values which satisfy the two

**equations**, can be expressed in terms of the coefficient of those**equations**.Hence, finally, the resultant is expressed in terms of the coefficients of the three

**equations**, and since it is at once seen to be of degree mn in the coefficient of the third equation, by symmetry it must be of degrees np and pm in the coefficients of the first and second**equations**respectively.The general theory of the resultant of k homogeneous

**equations**in k variables presents no further difficulties when viewed in this manner.Cayley, however, has shown that, whatever be the degrees of the three

**equations**, it is possible to represent the resultant as the quotient of two determinants (Salmon, l.c. p. 89).By solving k linear

**equations**we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical.Schlafli 1 this equation may be directly formed and exhibited as the resultant of two given

**equations**, and an arbitrary linear non-homogeneous equation in two variables.By solving the

**equations**of transformation we obtain rE1 = a22x1 - a12x1, r = - a21x1 + allx2, aua12 where r = I = anon-anon; a21 a22 r is termed the determinant of substitution or modulus of transformation; we assure x 1, x 2 to be independents, so that r must differ from zero.From the three

**equations**ax = alxl+ a2x2, b.= blxl+b2x2, cx = clxi+c2x2, we find by eliminating x, and x 2 the relation a x (bc)+b x (ca) +c x (ab) =0.These

**equations**can be arrived at in many ways; the method here given is due to Gordan.To determine them notice that R = (a6) and then (f, a 5) 5 = - R 5 (k1 +k2+k3) (f, a 4 5) 5 = - 5R5 (m 1 k 1+ m 2 k 2+ m 3 k 3), (f, a352) 5 = -10R5 (m21ke +m2k2+m3k3) three

**equations**for determining k 1, k2, k3.AdvertisementHe proves, by means of the six linear partial differential

**equations**satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the contragredient line variables-it is completely determinate if its leading coefficient be known.It has been shown above that a covariant, in general, satisfies four partial differential

**equations**.In order to obtain the seminvari ants we would write down the (w; 0, n) terms each associated with a literal coefficient; if we now operate with 52 we obtain a linear function of (w - I; 8, n) products, for the vanishing of which the literal coefficients must satisfy (w-I; 0, n) linear

**equations**; hence (w; 8, n)-(w-I; 0, n) of these coefficients may be assumed arbitrarily, and the number of linearly independent solutions of 52=o, of the given degree and weight, is precisely (w; 8, n) - (w - I; 0, n).Solving the equation by the Ordinary Theory Of Linear Partial Differential

**Equations**, We Obtain P Q 1 Independent Solutions, Of Which P Appertain To S2Au = 0, Q To 12 B U =0; The Remaining One Is Ab =Aobl A 1 Bo, The Leading Coefficient Of The Jacobian Of The Two Forms. This Constitutes An Algebraically Complete System, And, In Terms Of Its Members, All Seminvariants Can Be Rationally Expressed.Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general

**equations**of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.Advertisement**Equations**(33) and (34) show that when, as is generally the case with ferromagnetic substances, the value of is considerable, the resultant magnetic force is only a small fraction of the external force, while the numerical value of the induction is approximately three times that of the external force, and nearly independent of the permeability.If these

**equations**could be assumed to hold when H is indefinitely small, it would follow that has a finite initial value, from which there would be no appreciable deviation in fields so weak that bH was negligibly small in comparison with a.He appears to have attended Dirichlet's lectures on theory of numbers, theory of definite integrals, and partial differential

**equations**, and Jacobi's on analytical mechanics and higher algebra.By considering only the particles of air found in a right line, he reduced the problem of the propagation of sound to the solution of the same partial differential

**equations**that include the motions of vibrating strings, and demonstrated the insufficiency of the methods employed by both his great contemporaries in dealing with the latter subject.Its scope may be briefly described as the reduction of the theory of mechanics to certain general formulae, from the simple development of which should be derived the

**equations**necessary for the solution of each separate problem.AdvertisementLaplace owned that he had despaired of effecting the integration of the differential

**equations**relative to secular inequalities until Lagrange showed him the way.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.To Lagrange, perhaps more than to any other, the theory of differential

**equations**is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.Instead of following the motion of each individual part of a material system, he showed that, if we determine its configuration by a sufficient number of variables, whose number is that of the degrees of freedom to move (there being as many

**equations**as the system has degrees of freedom), the kinetic and potential energies of the system can be expressed in terms of these, and the differential**equations**of motion thence deduced by simple differentiation.AdvertisementBesides the separate works already named are Resolution des

**equations**numeriques (1798, 2nd ed., 1808, 3rd ed., 1826), and Lecons sur le calcul des fonctions (1805, 2nd ed., 1806), designed as a commentary and supplement to the first part of the Theorie des fonctions.The declared aim of the author 1 was to offer a complete solution of the great mechanical problem presented by the solar system, and to bring theory to coincide so closely with observation that empirical

**equations**should no longer find a place in astronomical tables.Biot relates that, when he himself was beginning his career, Laplace introduced him at the Institute for the purpose of explaining his supposed discovery of

**equations**of mixed differences, and afterwards showed him, under a strict pledge of secrecy, the papers, then yellow with age, in which he had long before obtained the same results.The device known as the method of least squares, for reducing numerous

**equations**of condition to the number of unknown quantities to be determined, had been adopted as a practically convenient rule by Gauss and Legendre; but Laplace first treated it as a problem in probabilities, and proved by an intricate and difficult course of reasoning that it was also the most advantageous, the mean of the probabilities of error in the determination of the elements being thereby reduced to a minimum.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.The earliest algebra consists in the solution of

**equations**.For many centuries algebra was confined almost entirely to the solution of

**equations**; one of the most important steps being the enunciation by Diophantus of Alexandria of the laws governing the use of the minus sign.The treatment of

**equations**of the second and higher degrees introduces imaginary and complex numbers, the theory of which is a special subject.When we are familiar with the treatment of quantities by

**equations**, we may ignore the units and deal solely with numbers; and (ii.) (a) and (ii.) (b) may then, by the commutative law for multiplication, be regarded as identical.**Equations**with Fractional Coefficients.-As an example of a special form of equation we may take zx+ 3x = Io.Generally, we may say that algebraic reasoning in reference to

**equations**consists in the alteration of the form of a statement rather than in the deduction of a new statement; i.e.In the first class come

**equations**in a single unknown; here the function which is equated to zero is the Y whose values for different values of X are traced, and the solution of the equation is the determination of the points where the ordinates of the graph are zero.The 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.We then obtain a set of

**equations**, and by means of these**equations**we establish the required result by a process known as mathematical induction.The first step consists in the functional treatment of

**equations**.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.Complex numbers are conveniently treated in connexion not only with the theory of

**equations**but also with analytical trigonometry, which suggests the graphic representation of a+b,l - by a line of length (a 2 +b 2)i drawn in a direction different from that of the line along which real numbers are represented.The

**equations**q'+x = q and y+q' = q are satisfied by the same quaternion, which is denoted by q - q'.On the other hand, the

**equations**q'x = q and yq' = q have, in general, different solutions.The principle underlying this expression is probably to be found in the fact that it measured the limits of their attainments in algebra, for they were unable to solve

**equations**of a higher degree than the quadratic or square.He then proceeds to discuss various artifices for the simplification of

**equations**, giving methods which are still in common use.In the body of the work he displays considerable ingenuity in reducing his problems to simple

**equations**, which admit either of direct solution, or fall into the class known as indeterminate**equations**.Ganessa, an eminent astronomer, mathematician and scholiast of Bhaskara, quotes this work and makes separate mention of the cuttaca (" pulveriser "), a device for effecting the solution of indeterminate

**equations**.Henry Thomas Colebrooke, one of the earliest modern investigators of Hindu science, presumes that the treatise of Aryabhatta extended to determinate quadratic

**equations**, indeterminate**equations**of the first degree, and probably of the second.Moritz Cantor suspects the influence of Diophantine methods, more particularly in the Hindu solutions of indeterminate

**equations**, where certain technical terms are, in all probability, of Greek origin.It is also supposed that they anticipated discoveries of the solutions of higher

**equations**.Great advances were made in the study of indeterminate

**equations**, a branch of analysis in which Diophantus excelled.In this they were completely successful, for they obtained general solutions for the

**equations**ax by = c, xy = ax+by+c (since rediscovered by Leonhard Euler) and cy 2 = ax e + b.He follows the methods of Diophantus; his work on indeterminate

**equations**has no resemblance to the Indian methods, and contains nothing that cannot be gathered from Diophantus.He solved quadratic

**equations**both geometrically and algebraically, and also**equations**of the form x 2 "+ax n +b=o; he also proved certain relations between the sum of the first n natural numbers, and the sums of their squares and cubes.Cubic

**equations**were solved geometrically by determining the intersections of conic sections.The method of solving

**equations**geometrically was considerably developed by Omar Khayyam of Khorassan, who flourished in the 1 r th century.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.The Greeks had succeeded in solving an isolated example; the Arabs accomplished the general solution of numerical

**equations**.Cardan or Cardano, who was at that time writing his great work, the Ars Magna, could not restrain the temptation of crowning his treatise with such important discoveries, and in 1 545 he broke his oath and gave to the world Tartalea's rules for solving cubic

**equations**.Cubic

**equations**having been solved, biquadratics soon followed suit.In this work, which is one of the most valuable contributions to the literature of algebra, Cardan shows that he was familiar with both real positive and negative roots of

**equations**whether rational or irrational, but of imaginary roots he was quite ignorant, and he admits his inability to resolve the so-called lation of Arabic manuscripts.Fundamental theorems in the theory of

**equations**are to be found in the same work.It includes the properties of numbers; extraction of roots of arithmetical and algebraical quantities, solutions of simple and quadratic

**equations**, and a fairly complete account of surds.He denotes quantities by the letters of the alphabet, retaining the vowels for the unknown and the consonants for the knowns; he introduced the vinculum and among others the terms coefficient, affirmative, negative, pure and adfected

**equations**.He improved the methods for solving

**equations**, and devised geometrical constructions with the aid of the conic sections.He possessed clear ideas of indices and the generation of powers, of the negative roots of

**equations**and their geometrical interpretation, and was the first to use the term imaginary roots.His principal discovery is concerned with

**equations**, which he showed to be derived from the continued multiplication of as many simple factors as the highest power of the unknown, and he was thus enabled to deduce relations between the coefficients and various functions of the roots.It is traversed by dark lines whose

**equations**are E=mfA/a, n= mfA/b.The first of these

**equations**is the condition for the formation of dark bands, and the second marks their situation, which is the same as that determined by the imperfect theory.Then the first of the

**equations**of motion may be put under the form dt ?According to this notation, the three

**equations**of motion are dt2 = b2v2E + (a2 - b2) d.s dt =b2v2rj+(a2 - b2) dy d2 CIF - b2p2+(a2_b2)dz It is to be observed that denotes the dilatation of volume of the element situated at (x, y, z).In the limiting case in which the medium is regarded as absolutely incompressible S vanishes; but, in order that

**equations**(2) may preserve their generality, we must suppose a at the same time to become infinite, and replace a 2 3 by a new function of the co-ordinates.These

**equations**simplify very much in their application to plane waves.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).His mathematical researches were also concerned with the theory of

**equations**, but the question as to his priority on several points has been keenly discussed.After his death Navier completed and published Fourier's unfinished work, Analyse des

**equations**indeterminees (1831), which contains much original matter.He considered, at every instant, the actual motion of a stratum as composed of a motion which it had in the preceding instant and of a motion which it had lost; and the laws of equilibrium between the motions lost furnished him with

**equations**representing the motion of the fluid.It remained a desideratum to express by

**equations**the motion of a particle of the fluid in any assigned direction.These

**equations**were found by d'Alembert from two principles - that a rectangular canal, taken in a mass of fluid in equilibrium, is itself in equilibrium, and that a portion of the fluid, in passing from one place to another, preserves the same volume when the fluid is incompressible, or dilates itself according to a given law when the fluid is elastic. His ingenious method, published in 1752, in his Essai sur la resistance des fluides, was brought to perfection in his Opuscules mathematiques, and was adopted by Leonhard Euler.But by Green's transformation f flpdS = f f PPdxdydz, (2) thus leading to the differential relation at every point = dy dp The three

**equations**of equilibrium obtained by taking moments round the axes are then found to be satisfied identically.These

**equations**can be made to represent the state of convective equilibrium of the atmosphere, depending on the gas-equation p = pk =RA (6) where 0 denotes the absolute temperature; and then d9 d p R dz - dz (p) n+ 1' so that the temperature-gradient deldz is constant, as in convective equilibrium in (I I).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?As a rule these

**equations**are established immediately by determining the component acceleration of the fluid particle which is passing through (x, y, z) at the instant t of time considered, and saying that the reversed acceleration or kinetic reaction, combined with the impressed force per unit of mass and pressure-gradient, will according to d'Alembert's principle form a system in equilibrium.Taking the axis of x for an instant in the normal through a point on the surface H = constant, this makes u = o, = o; and in steady motion the

**equations**reduce to dH/dv=2q-2wn = 2gco sin e, (4) where B is the angle between the stream line and vortex line; and this holds for their projection on any plane to which dv is drawn perpendicular.In the

**equations**of uniplanar motion = dx - du = dx + dy = -v 2 ?, suppose, so that in steady motion dx I +v24 ' x = ?'Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a quadratic function of xi, 'x2, x3, yi, Y2, Y3, U = d, V= dx, ' w= ax dT Q_ dT dT dy 1 dy2 dy The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical

**equations**follow X = dt x2 dy +x3 d Y = ..., Z ..., (3) = dt1 -y2?y - '2dx3+x3 ' M =..When no external force acts, the case which we shall consider, there are three integrals of the

**equations**of motion (i.) T =constant, x 2 +x 2 +x 2 =F 2, a constant, (iii.) x1y1 +x2y2+x3y3 =n = GF, a constant; and the dynamical**equations**in (3) express the fact that x, x, xs.Bryan, in which the analytical

**equations**of motion are deduced of a perforated solid in liquid, from considerations purely hydrodynamical.The same

**equations**apply to the case of fusion of a solid, if L is the latest heat of fusion, and v', s', v", s" the specific volumes and specific heats of the solid and liquid respectively.If we put dH=o in

**equations**(8), we obtain the relations between dv and do, or dp and do, under the condition of no heat-supply, i.e.The energy E and the total heat F are functions of the temperature only, by

**equations**(9) and (I I), and their variations take the form dE = sdO, d F = Sd0.The specific heats are independent of the pressure or density by

**equations**(to) and (12).The expression for the change of intrinsic energy E between any given limits poOo to po is readily found by substituting these values of the specific heats in

**equations**(II) or (13), and integrating between the given limits.As far as the order to which he carried the approximations - which, however, were based on a simplifying hypothesis that the molecules influenced each other through mutual repulsions inversely as the fifth power of their distance apart--the result was that the

**equations**of motion of the gas, considered as subject to viscous and thermal stresses, could be satisfied by a state of equilibrium under a modified internal pressure equal in all directions.In later memoirs Reynolds followed up this subject by proceeding to establish definitions of the velocity and the momentum and the energy at an element of volume of the molecular medium, with the precision necessary in order that the dynamical

**equations**of the medium in bulk, based in the usual manner on these quantities alone, without directly considering thermal stresses, shall be strictly valid - a discussion in which the relation of ordinary molar mechanics to the more complete molecular theory is involved.On the other hand, Vieta was well skilled in most modern artifices, aiming at a simplification of

**equations**by the substitution of new quantities having a certain connexion with the primitive unknown quantities.During the three centuries that have elapsed between Vieta's day and our own several changes of opinion have taken place on this subject, till the principle has at last proved so far victorious that modern mathematicians like to make homogeneous such

**equations**as are not so from the beginning, in order to get values of a symmetrical shape.He conceived methods for the general resolution of

**equations**of the second, third and fourth degrees different from those of Ferro and Ferrari, with which, however, it is difficult to believe him to have been unacquainted.He devised an approximate numerical solution of

**equations**of the second and third degrees, wherein Leonardo of Pisa must have preceded him, but by a method every vestige of which is completely lost.If the system is supposed to obey the conservation of energy and to move solely under its own internal forces, the changes in the co-ordinates and momenta can be found from the Hamiltonian

**equations**aE aE qr = 49 - 1 57., gr where q r denotes dg r ldt, &c., and E is the total energy expressed as a function of pi, qi,.If 2mu 2 denote the mean value of 2mu 2 averaged over the s molecules of the first kind,

**equations**(3) may be written in the form Z mu g = 2 mv 2 = 2 mw 2 = 2x,0 2 1 =.These

**equations**express the " law of equipartition of energy," commonly spoken of as the Maxwell-Boltzmann Law.When the interval between a flash and a report is measured, the personal

**equations**for the two arrivals are, in all probability, different, that for the flash being most likely less than that for the sound.In this experiment the personal

**equations**of the observers were determined and allowed for.In these

**equations**4 is to have its + or - value according to the case considered.From these

**equations**a value of W e can be obtained.Another powerful reason for taking the aether to be stationary is afforded by the character of the

**equations**of electrodynamics; they are all of linear type, and superposition of effects is possible.Now the kinetics of a medium in which the parts can have finite relative motions will lead to

**equations**which are not linear - as, for example, those of hydrodynamics - and the phenomena will be far more complexly involved.The analytical

**equations**which represent the propagation of light in free aether, and also in aether modified by the presence of matter, were originally developed on the analogy of the**equations**of propagation of elastic effects in solid media.MacCullagh's hands the correct

**equations**were derived from a single energy formula by the principle of least action; and while the validity of this dynamical method was maintained, it was frankly admitted that no mechanical analogy was forthcoming.When Clerk Maxwell pointed out the way to the common origin of optical and electrical phenomena, these

**equations**naturally came to repose on an electric basis, the connexion having been first definitely exhibited by FitzGerald in 1878; and according as the independent variable was one or other of the vectors which represent electric force, magnetic force or electric polarity, they took the form appropriate to one or other of the elastic theories above mentioned.These

**equations**determine all the phenomena.If v varies with respect to locality, or if there is a velocity of convection (p,q,r) variable with respect to direction and position, and analytical expression of the relation (ii) assumes a more complex form; we thus derive the most general

**equations**of electrodynamic propagation for matter treated as continuous, anyhow distributed and moving in any manner.He published a memoir on the integration of partial differential

**equations**and a few others, which have not been noticed above, but they relate to subjects with which his name is not especially associated.The

**equations**of motion are now, the co-ordinates x and y being measured in feet, 2 (26) - -rr- - C, dt2 dty - g' * These numbers are taken from a part omitted here of the abridged ballistic table.Replacing then the angle i on the right-hand side of

**equations**(54) - (56) by some mean value, t, we introduce Siacci's pseudovelocity u defined by (59) u = q sec, t, so that u is a quasi-component parallel to the mean direction of the tangent, say the direction of the chord of the arc.The

**equations**(66) - (71) are Siacci's, slightly modified by General Mayevski; and now in the numerical applications to high angle fire we can still employ the ballistic table for direct fire.Pfaff's researches bore chiefly on the theory of series, to which he applied the methods of the so-called combinatorial school of German mathematicians, and on the solution of differential

**equations**.The principal properties of logarithms are given by the

**equations**log (mn) = log m --Flogs n, loga(m/n) = toga m -logo.The

**equations**finally arrived at are DX2(A2_ 2) (x2_ A2m)2+g2A2 ' DgA3 (A A l m) 2 +g 2 A2 ' where is the wave-length in free ether of light whose refractive index is n, and A m the wave-length of light of the same period as the electron, is a coefficient of absorption, and D and g are constants.The

**equations**to the chord, tangent and normal are readily derived by the ordinary methods.Obviously these

**equations**show that the curves intersect in four points, two of which lie on the intersection of the line, 2 (g - g')x +2 (f - f')y+c - c'=o, the radical axis, with the circles, and the other two where the lines x2+y2= (x+iy) (x - iy) =o (where i = - - I) intersect the circles.The

**equations**to such circles may be expressed in the form x 2 +y 2 = a 2, x 2 +y 2 = /3 2 .These

**equations**show that the circles touch where they intersect the lines x 2 +y 2 = o, i.e.In various systems of triangular co-ordinates the

**equations**to circles specially related to the triangle of reference assume comparatively simple forms; consequently they provide elegant algebraical demonstrations of properties concerning a triangle and the circles intimately associated with its geometry.The corresponding

**equations**in areal co-ordinates are readily derived by substituting x/a, ylb, z/c for a, 1 3, y respectively in the trilinear**equations**.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.To Solve The

**Equations**28 X Io =19 Y 2, Or Y =X 9198, Let M= 9 19 8, We Have Thenx=2 M 172 8 9 Let M9 8 =M'; Then M= 9 M' 8; Hence X=18 M' 16 M'=19 M' 16 (I).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.In Consequence Of The Solar And Lunar

**Equations**, It Is Evident That The Epact Or Moon'S Age At The Beginning Of The Year, Must, In The Course Of Centuries, Have All Different Values From One To Thirty Inclusive, Corresponding To The Days In A Full Lunar Month.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.It is found to be very markedly superior to the other

**equations**.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.The two laws are best understood by putting the

**equations**in the form given them by Rydberg.According to him, the following

**equations**represent the connexion between the lines of the three related series.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.**Equations**of this form have received a striking observational verification in so far as they predict a tail or root towards which the lines ultimately tend when s is increased indefinitely.In this he gave

**equations**resulting from the hypothesis that the magnetism of a ship is partly due to the permanent magnetism of hard iron and partly to the transient induced magnetism of soft iron; that the latter is proportional to the intensity of the inducing force, and that the length of the needle is infinitesimally small compared to the distance of the surrounding iron.To determine these variables we may form

**equations**between the chemical potentials of the different components - quantities which are functions of the variables to be determined.These two

**equations**involve the third relation µ2 =As, which therefore is not an independent equation.Hence with three phases we can form two independent

**equations**for each component.With r phases we can form r - I

**equations**for each component, and with n components and r phases we obtain n (r - 1)**equations**.Now by elementary algebra we know that if the number of independent

**equations**be equal to the number of unknown quantities all the unknown quantities can be determined, and can possess each one value only.This condition is represented in the algebraic theory when we have one more unknown quantity than the number of

**equations**; i.e.Similarly if we have F more unknowns than we have

**equations**to determine them, we must fix arbitrarily F coordinates before we fix the state of the whole system.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.His primary aim has been declared to be the advancement and elaboration of the theory of differential

**equations**, and it was with this end in view that he developed his theory of transformation groups, set forth in his Theorie der Transf ormationsgruppen (3 vols., Leipzig, 1888-1893), a work of wide range and great originality, by which probably his name is best known.Since alite is a solid solution and, although an individual mineral, is not a chemical unit, the proportion of tricalcium silicate to tricalcium aluminate in a given specimen of alite will vary; but, whatever the proportions, each of these substances will react in its characteristic manner according to the

**equations**given above.John Wallis utilized the intersections of this curve with a right line to solve cubic

**equations**, and Edmund Halley solved sextic**equations**with the aid of a circle.At Woolwich he remained until 1870, and although he was not a great success as an elementary teacher, that period of his life was very rich in mathematical work, which included remarkable advances in the theory of the partition of numbers and further contributions to that of invariants, together with an important research which yielded a proof, hitherto lacking, of Newton's rule for the discovery of imaginary roots for algebraical

**equations**up to and including the fifth degree.Lotze's mistake is the same as that of Hamilton about the quantification of the predicate, and that of those symbolists who held that reasoning ought always to exhaust all alternatives by

**equations**.Erdmann propose new moods of syllogism with convertible premises, containing definitions and

**equations**.Now, there is no doubt that, especially in mathematical

**equations**, universal conclusions are obtainable from convertible premises expressed in these ways.All M is P. Proceeding from one order to the other, by converting one of the premises, and substituting the conclusion as premise for the other premise, so as to deduce the latter as conclusion, is what he calls circular inference; and he remarked that the process is fallacious unless it contains propositions which are convertible, as in mathematical

**equations**.Even in Hamilton's earlier work it was shown that all such questions were reducible to the solution of linear

**equations**in quaternions; and he proved that this, in turn, depended on the determination of a certain operator, which could be represented for purposes of calculation by a single symbol.The vapourpressure

**equations**are seldom known with sufficient accuracy, and the ionization data are incomplete.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.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.On the other hand, several of Wimmer's

**equations**are undoubtedly forced.The

**equations**(12) are npw replaced FIG.For the conditions of equilibrium of the forces on each pin furnish vi

**equations**, viz, two for each point, which are linear in respect of the stresses and the extraneous forces.A frame of n joints and vi 3 bars may of course fail to be rigid owing to some parts being over-stiff whilst others are deformable; in such a case it will be found that the statical

**equations**, apart from the thre identical relations imposed by the equilibrium of the extraneous forces, are not all independent but are equivalent to less thar 2,13 relations.These are the

**equations**of the central axis.But from the dynamical standpoint it is obvious that

**equations**which represent the facts correctly on one system of time-measurement might become seriously defective on another.The argument appears in a more demonstrative form in the theory of similar systems, or (more precisely) of the similar motion of similar systems. Thus, considering the

**equations**d2x u dix u1For its investigation we require two

**equations**; these may be obtained in a variety of forms.If the

**equations**of motion of each particle be formed separately, each such internal force will appear twice over, with opposite signs for its components, viz, as affecting the motion of each of the two particles between which it acts.For the determination of the motion it has only, been necessary to use one of the dynamical

**equations**.The remaining

**equations**serve to determine the reactions of the rotating body on its bearings.It may also be deduced from the principles of linear and angular momentum as embodied in the

**equations**(9).The resulting Z+R

**equations**are not as a rule easy of application, owing to the fact that the moments and products of inertia A, B, C, F, G, H are not constants but vary in conse- 0 quence of the changing orientation of the body with respect to the co-ordinate axes.If we eliminate P, Q, R from (22), the resulting

**equations**are integrable with respect to t; thus Moa - M0a.As a first application of the

**equations**(2) take the case of a solid constrained to rotate with constant angular velocity to about a fixed axis (1, m, n).These

**equations**are due to Euler, with whom the conception of moving axes, and the application to the problem of free rotation, originated.As in the case of (2), the

**equations**are applicable to any dynamical system whatever.These

**equations**are due to Lagrange, with whom indeed the first conception, as well as the establishment, of a general dynamical method applicable to all systems whatever appears to have originated.As a first application of Lagranges formula (II) we may form the

**equations**of motion of a particle in spherical polar co-ordinates.To apply the

**equations**(11) to the case of the top we start with the expression (15) of 22 for the kinetic energy, the simplified form (i) of 20 being for the present purpose inadmissible, since it is essential that the generalized co-ordinates employed should be competent to specify the position of every particle.For any particular root, the

**equations**(5) determine the ratios of the quantities Af, A1,.These

**equations**can be at once solved for H, H, H,.Free vibrations must of course be superposed on the forced vibrations given by (29) in order to obtain the complete solution of the dynamical

**equations**.In the case of multiple freedom, the

**equations**of small motion when modified by the introduction of terms proportional to the velocities are of the type daT -.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.The

**equations**65 and 66 are applicable to a kind of brake called a friction-strap, used to stop or moderate the velocity of machines by being tightened round a pulley.This necessitates the satisfying of five

**equations**; in other words, there are five alterations of the 3rd order, the vanishing of which produces an image of the 5th order.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 first part establishes the laws of the elasticity of a finite portion of the solid subjected to a homogeneous strain, and deduces from these laws the

**equations**of the equilibrium and motion of a body subjected to any forces and displacements.The

**equations**to the hypocycloid and its corresponding trochoidal curves are derived from the two preceding**equations**by changing the sign of b.Therefore any epicycloid or hypocycloid may be represented by the

**equations**p = A sin B+,' or p---A cos B,,G, s = A sin B11.He was professor of mathematics in the university of Deseret and wrote several books on this subject, these including Cubic and Biquadratic

**Equations**(1866).The

**equations**of surfaces of equal angular motion would be of the form r =R (i --6 cos 2 0), where e is proportional to the square of the angular motion, supposed small, and R increases as e diminishes.Instead of confining himself, as before, to the fruitless integration of three differential

**equations**of the second degree, which are furnished by mathematical principles, he reduced them to the three co-ordinates which determine the place of the moon; and he divided into classes all the inequalities of that planet, as far as they depend either on the elongation of the sun and moon, or upon the eccentricity, or the parallax, or the inclination of the lunar orbit.He investigated the problem by means of the general differential

**equations**of static equilibrium for dams of triangular and rectangular form considered as isotropic elastic solids.Jacobi and other mathematicians have developed to a great extent, and as a question of pure mathematics only, Hamilton's processes, and have thus made extensive additions to our knowledge of differential

**equations**.Of his extensive investigations into the solution (especially by numerical approximation) of certain classes of differential

**equations**which constantly occur in the treatment of physical questions, only a few items have been published, at intervals, in the Philosophical Magazine.In determining the dimensions of corresponding drums of cone pulleys it is evident that for a crossed belt the sum of the radii of each pair remains a constant, since the angle a is constant, while for an open belt a is variable and the values of the radii are then obtained by solving the

**equations**r 1 = l/ir - c(a sin a + cos a) + 2c sin a, r 2 = l/7r - c(a sin a +cos_a) - lc sin a.The old-fashioned problems about the amount of work done by particular numbers of men, women and boys, are of this kind, and really involve the solution of simultaneous

**equations**.It is to be remarked that an equation may break up; thus a quadric equation may be (ax+by+c) (a'x.+b'y+c') = o, breaking up into the two

**equations**ax+by+c = o, a'x+b'y+c' = o, viz.Each of these last

**equations**represents a curve of the first order, or right line; and the original equation represents this pair of lines, viz.The intersections of two curves are obtained by combining their

**equations**; viz.We now come to Julius Plucker; his " six

**equations**" were given in a short memoir in Crelle (1842) preceding his great work, the Theorie der algebraischen Curven (1844).In regard to the ordinary singularities, we have m, the order, n „ class, „ number of double points, Cusps, T double tangents, inflections; and this being so, Pliicker's ” six

**equations**” are n = m (m - I) -2S -3K, = 3m (m - 2) - 6S- 8K, T=Zm(m -2) (m29) - (m2 - m-6) (28-i-3K)- I -25(5-1) +65K-1114 I), m =n(n - I)-2T-3c, K= 3n (n-2) - 6r -8c, = 2n(n-2)(n29) - (n2 - n-6) (2T-{-30-1-2T(T - I) -1-6Tc -}2c (c - I).With regard to the demonstration of Pliicker's

**equations**it is to be remarked that we are not able to write down the equation in point-co-ordinates of a curve of the order m, having the given numbers 6 and of nodes and cusps.We have thus finally an expression for = m (m-2) (m2-9) - &c.; or dividing the whole by 2, we have the expression for given by the third of Pliicker's

**equations**.It is obvious that we cannot by consideration of the equation u = o in point-co-ordinates obtain the remaining three of Pliicker's

**equations**; they might be obtained in a precisely analogous manner by means of the equation v= o in line-co-ordinates,but they follow at once from the principle of duality, viz.So that, in fact, Pliicker's

**equations**properly understood apply to a curve with any singularities whatever.By means of Pliicker's

**equations**we may form a table - The table is arranged according to the value of in; and we have m=o, n= r, the point; m =1, n =o, the line; m=2, n=2, the conic; of m = 3, the cubic, there are three cases, the class being 6, 4 or 3, according as the curve is without singularities, or as it has 1 node or r cusp; and so of m =4, the quartic, there are ten cases, where observe that in two of them the class is = 6, - the reduction of class arising from two cusps or else from three nodes.The expression 2m(m - 2) (m - 9) for the number of double tangents of a curve of the order in was obtained by Plucker only as a consequence of his first, second, fourth and fifth

**equations**.If, however, the geometrical property requires two or more relations between the coefficients, say A = o, B = o,&c., then we must have between the new coefficients the like relations, A' = o, B' = o, &c., and the two systems of

**equations**must each of them imply the other; when this is so, the system of**equations**, A = o, B = o, &c., is said to be invariantive, but it does not follow that A, B, &c., are of necessity invariants of u.Similarly, if we have a curve U= o derived from the curve u = o in a manner independent of the particular axes of co-ordinates, then from the transformed equation u' = o deriving in like manner the curve U' = o, the two

**equations**U= o, U' = o must each of them imply the other; and when this is so, U will be a covariant of u.Zeuthen in the case of curves of any given order establishes between the characteristics pc, v, and 18 other quantities, in all 20 quantities, a set of 24

**equations**(equivalent to 2 3 independent**equations**), involving(besides the 20 quantities) other quantities relating to the various forms of the degenerate curves, which supplementary terms he determines, partially for curves of any order, but completely only for quartic curves.It is the discussion and complete enumeration of the special or degenerate forms of the curves, and of the supplementary terms to which they give rise, that the great difficulty of the question seems to consist; it would appear that the 24

**equations**are a complete system, and that (subject to a proper determination of the supplementary terms) they contain the solution of the general problem.The system has singularities, and there exist between m, r, is and the numbers of the several singularities

**equations**analogous to Pliicker's**equations**for a plane curve.It is a leading point in the theory that a curve in space cannot in general be represented by means of two

**equations**U= o, V = o; the two**equations**represent surfaces, intersecting in a curve; but there are curves which are not the complete intersection of any two surfaces; thus we have the cubic in space, or skew cubic, which is the residual intersection of two quadric surfaces which have a line in common; the**equations**U= o, V= o of the two quadric surfaces represent the cubic curve, not by itself, but together with the line.In this position he acquired a wide knowledge of Chinese religion and civilization, and especially of their mathematics, so that he was able to show that Sir George Homer's method (1819) of solving

**equations**of all orders had been known to the Chinese mathematicians of the 14th century.Considering the

**equations**ax +by +cz =d, a'x +b'y +c' z =d', a"x+b"y+cnz=d" and proceeding to solve them by the so-called method of cross multiplication, we multiply the**equations**by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = o, and the whole coefficient of z becomes = o; the factors in question are b'c" - b"c', b"c - be", bc' - b'c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value a(b'c" - b"c') +a'(b"c - bc") +a'(bc' - b'c), and this function, represented in the form a, b,c, a' b'c', a" b" c" is said to be a determinant; or, the number of elements being 32, it is called a determinant of the third order.By what precedes it appears that there exists a function of the n 2 elements, linear as regards the terms of each column (or say, for shortness, linear as to each column), and such that only the sign is altered when any two columns are interchanged; these properties completely determine the function, except as to a common factor which may multiply all the terms. If, to get rid of this arbitrary common factor, we assume that the product of the elements in the dexter diagonal has the coefficient + 1, we have a complete definition of the determinant, and it is interesting to show how from these properties, assumed for the definition of the determinant, it at once appears that the determinant is a function serving for the solution of a system of linear

**equations**.Reverting to the system of linear

**equations**written down at the beginning of this article, consider the determinant ax+by+cz - d,b,c a' x+b' y+c'z - d', b', c" a"x+b"y+c"z - d", b", c" it appears that this is viz.Thus, the Mesopotamians knew how to solve quadratic

**equations**4000 years ago, using essentially the same method that we use today.Not all single

**equations**in single unknowns may be easily rearranged to provide formulas.Using stepwise regression of a number of features of the signal stronger prediction

**equations**are possible (R = 0.95 for meat toughness ).We propose to adapt some recent results obtained for the Navier-Stokes

**equations**to the passive scalar.There are two relatively simple approaches to solve such

**equations**.Jimmy's skills are in verbalisation, not simultaneous

**equations**.This destroys the beauty of the field

**equations**, which attribute the source of curvature entirely to matter as represented by the stress-energy tensor.For a rectangular prism, we obtain upper bounds for the equilibrium energy by constructing trial configurations from local solutions of the Euler-Lagrange

**equations**.Numerical

**Equations**These are based on the diffusion equation in one dimension, where is the eddy viscosity.They are used in the

**equations**of motion to produce the output waveform, with dynamic effect of speed added.Molten anhydrous zinc chloride gives zinc (+) and chlorine (- ),

**equations**1 and 2.The

**equations**to the asymptotes are = t y/b and x = =y respectively.A long list of Boole's memoirs and detached papers, both on logical and mathematical topics, will be found in the Catalogue of Scientific Memoirs published by the Royal Society, and in the supplementary volume on Differential

**Equations**, edited by Isaac Todhunter.In the 16th and 17th chapters of the Differential

**Equations**we find, for instance, a lucid account of the general symbolic method, the bold and skilful employment of which led to Boole's chief discoveries, and of a general method in analysis, originally described in his famous memoir printed in the Philosophical Transactions for 1844.An obituary notice by his friend Auguste Chevalier appeared in the Revue encyclopedique (1832); and his collected works are published, Journal de Liouville (1846), pp. 381-444, about fifty of these pages being occupied by researches on the resolubility of algebraic

**equations**by radicals.It consisted essentially in the adoption of Delauny's final numerical expressions for longitude, latitude and parallax, with a symbolic term attached to each number, the value of which was to be determined by substitution in the

**equations**of motion.An important fact, discovered by Cayley, is that these coefficients, and also the complete covariants, satisfy certain partial differential

**equations**which suffice to determine them, and to ascertain many of their properties.It is amusing to find him speaking jubilantly of the unexpectedly large audience of eight which assembled to hear his first lecture (in 1854) on partial differential

**equations**and their application to physical problems.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.The first difficulty to be overcome was the algebraical solution of cubic

**equations**, the " irreducible case" (see Equation).In his famous Geometria (1637), which is really a treatise on the algebraic representation of geometric theorems, he founded the modern theory of analytical geometry (see Geometry), and at the same time he rendered signal service to algebra, more especially in the theory of

**equations**.The stream function, y of the liquid motion set up by the passage of a solid of revolution, moving with axial velocity U, is such that y Glib = - 15 42, iI ' + Uy 2 =cons t ant, (12) over the surface of the solid; and 4, must be replaced by41' =1l.-1-1-Uy2 in the general

**equations**of steady motion above to obtain the steady relative motion of the liquid past the solid.With the values above of u, v, w, u', v', w', the

**equations**become of the form p x + 4 7rpAx -Fax -{-hy-}-gz =o, - p - dy+ 4?pBy + hx+ay+fz =o, P d p + TpCZ +f y + yz = o, and integrating p p 1+27rp(Ax2+By2+CZ2) +z ('ax e +ay e + yz2 2 f yz + 2gzx + 2 hx y) = const., (14) so that the surfaces of equal pressure are similar quadric surfaces, which, symmetry and dynamical considerations show, must be coaxial surfaces; and f, g, h vanish, as follows also by algebraical reduction; and 4c2 (c 2 - a2)?For instance, if the system is composed of a gas and a solid boundary, some of the terms in expression (2) may be supposed to represent the kinetic energy of the molecules of the boundary, so that

**equations**(7) show that in the normal state the gas has the same temperature as the boundary.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.Euler (see Differential

**Equations**).It follows from these

**equations**that the logarithm of the product of any number of quantities is equal to the sum of the logarithms of the quantities, that the logarithm of the quotient of two quantities is equal to the logarithm of the numerator diminished by the logarithm of the denominator, that the logarithm of the rth power of a quantity is equal to r times the logarithm of the quantity, and that the logarithm of the rth root of a quantity is equal to (r/r)th of the logarithm of the quantity.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.From Poisson's

**equations**Archibald Smith deduced the formulae given in the Admiralty Manual for Deviations of the Compass (1st ed., 1862), a work which has formed the basis of numerous other manuals since published in Great Britain and other countries.The

**equations**to the tangent and normal at the point x' y are yy' = 2a(x+x) and aa(y - y')+y'(x - x')=o, and may be obtained by general methods (see Geometry, Analytical, and Infinitesimal Calculus).In order to obtain at all events a qualitative representation of these it is usual to introduce into thc

**equations**frictional terms proportional to the velocities.Eliminating dS from

**equations**(is) and (14), and dividing by S, we find T = e - v??, (15) In this expression a denotes the mass of unit of area of the film, and e the energy of unit of area.In 1874 the university of GÃƒ¶ttingen granted her a degree in absentia, excusing her from the oral examination on account of the remarkable excellence of the three dissertations sent in, one of which, on the theory of partial differential

**equations**, is one of her most remarkable works.Allied to the matter just mentioned was Plucker's discovery of the six

**equations**connecting the numbers of singularities in algebraical curves (see Curve).This paper considers a linear triangular simultaneous

**equations**model with conditional quantile restrictions.The final book presents the solution of cubic and quartic

**equations**.All these measures work by setting up a set of numerical recursion

**equations**derived from the input clauses to the theorem prover.Using stepwise regression of a number of features of the signal stronger prediction

**equations**are possible (R = 0.95 for meat toughness).The results are applied to a class of retarded delay differential

**equations**.Save us from self-justifying histories and from moral

**equations**that excuse our folly.The concepts of azimuth and elevation are explored; the author includes many diagrams and simple mathematical

**equations**.The concept of instability of a basic state is first introduced using models which yield simple ordinary differential

**equations**.Graphs 2 Pages 83 to 85 Simultaneous

**equations**using graphs.Jimmy 's skills are in verbalisation, not simultaneous

**equations**.We are interested in the role that determinants play in determining new solutions to various soliton

**equations**.Some of these

**equations**are interesting from the physical point of view due to their soliton solutions.After the normal form for the

**equations**has been derived, the extent to which the**equations**may be solved analytically is covered.He escapes into the eternal verities of mathematical

**equations**as a means of avoiding the messy temporality of human life.Detailed analysis may be undertaken in a manner similar to that for the first order ordinary differential

**equations**.Molten anhydrous zinc chloride gives zinc (+) and chlorine (-),

**equations**1 and 2.The moon is over 405 kilometers from the earth, so simple mathematical

**equations**are used to translate the volume of waste into distance.If you want to be more creative, you can even search for math

**equations**!Some chat programs also let users contribute to a shared virtual whiteboard, on which they can draw sketches, jot down

**equations**, or show examples of thoughts and ideas.That's why some of the free online preschool games are focused on learning the numbers and learning basic arithmetic

**equations**.The app library for each phone also has academic uses, with apps available for math calculations, scientific

**equations**, history facts and trivia quizzes.How in the world will their child be able to ever get a job unless he memorizes those math facts and, later, algebraic

**equations**?Many believe math is learned best by learning a concept and then solving

**equations**using that concept.The more

**equations**that are solved, the more ingrained the concept can become.A crossword-style game, players must create

**equations**and place them on the board with number tiles.**Equations**receive varying point levels, and the player with the highest score at the end of the game wins.Head Full of Numbers is a fun way to create and solve math

**equations**.A deck of playing cards can also help teach preschoolers their numbers and work as visual aides for addition and subtraction

**equations**.Do you need it to be fast or able to perform complex digital drawings or

**equations**?And the algebraists or arithmeticians of the 16th century, such as Luca Pacioli (Lucas de Borgo), Geronimo or Girolamo Cardano (1501-1576), and Niccola Tartaglia (1506-1559), had used geometrical constructions to throw light on the solution of particular

**equations**.In pure algebra Descartes expounded and illustrated the general methods of solving

**equations**up to those of the fourth degree (and believed that his method could go beyond), stated the law which connects the positive and negative roots of an equation with the changes of sign in the consecutive terms, and introduced the method of indeterminate coefficients for the solution of**equations**.'The well-known Treatise on Differential

**Equations**appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work.During the last few years of his life Boole was constantly engaged in extending his researches with the object of producing a second edition of his Differential

**Equations**much more complete than the first edition; and part of his last vacation was spent in the libraries of the Royal Society and the British Museum.Thus, 1 - x would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1 - x) (1 - y) would give us all things neither horned nor sheep. By the use of such symbols propositions could be reduced to the form of

**equations**, and the syllogistic conclusion from two premises was obtained by eliminating the middle term according to ordinary algebraic rules.He showed that the heat motion of particles, which is too small to be perceptible when these particles are large, and which cannot be observed in molecules since these themselves are too small, must be perceptible when the particles are just large enough to be visible and gave complete

**equations**which enable the masses themselves to be deduced from the motions of these particles.But the bulk of the work consists of problems leading to indeterminate

**equations**of the second degree, and these universally take the form that one or two (and never more) linear or quadratic functions of one variable x are to be made rational square numbers by finding a suitable value for x.A few problems lead to indeterminate

**equations**of the third and fourth degrees, an easy indeterminate equation of the sixth degree being also found.Since the distance of a body from the observer cannot be observed directly, but only the right ascension and declination, calling these a and 6 we conceive ideal

**equations**of the form a = f (a, b, c, e, f, g, t) and 5=0 (a, b, c, e, f, g, t), the symbols a, b,.Then by solving these

**equations**, regarding the six elements as unknown quantities, the values of the latter may be computed.Instead of the six ideal

**equations**just described we have to combine a number of**equations**of various forms containing other quantities than the elements.For Tartaglia's discovery of the solution of cubic

**equations**, and his contests with Antonio Marie Floridas, see Algebra (History).Problems in artillery occupy two out of nine books; the sixth treats of fortification; the ninth gives several examples of the solution of cubic

**equations**.His largest work,Trattato generale di numeri e misure, is a comprehensive mathematical treatise, including arithmetic, geometry, mensuration, and algebra as far as quadratic

**equations**(Venice, 1556, 1560).The combination, as it is ordinarily termed, of chlorine with hydrogen, and the displacement of iodine in potassium iodide by the action of chlorine, may be cited as examples; if these reactions are represented, as such reactions very commonly are, by

**equations**which merely express the relative weights of the bodies which enter into reaction, and of the products, thus Cl = HC1 Hydrogen.A physicist, however, does more than merely quantitatively determine specific properties of matter; he endeavours to establish mathematical laws which co-ordinate his observations, and in many cases the

**equations**expressing such laws contain functions or terms which pertain solely to the chemical composition of matter.Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic

**Equations**," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential

**Equations**," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential**equations**of applied mathematics, Dirichlet's problem; "Difference**Equations**and Functional**Equations**," with the topics recurring series, solution of**equations**of finite differences and functional**equations**.For the subjects of this heading see the articles DIFFERENTIAL

**EQUATIONS**; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS, CALCULUS OF.They teach further the solution of problems leading to

**equations**of the first and second degree, to determinate and indeterminate**equations**, not by single and double position only, but by real algebra, proved by means of geometric constructions, and including the use of letters as symbols for known numbers, the unknown quantity being called res and its square census.The difference of potential between two solutions of a substance at different concentrations can be calculated from the

**equations**used to give the diffusion constants.The results give

**equations**of the same logarithmic form as those obtained in a somewhat different manner in the theory of concentration cells described above, and have been verified by experiment.On these lines the

**equations**of concentration cells, deduced above on less hypothetical grounds, may be regained.Other " Galois " groups were defined whose substitution coefficients have fixed numerical values, and are particularly associated with the theory of

**equations**.Arithmetical groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and

**equations**, came into existence, and also particular linear and higher transformations connected with analysis and geometry.The adjoint determinant will be seen subsequently to present itself in the theory of linear

**equations**and in the theory of linear transformation.This relation implies six

**equations**.Linear

**Equations**.-It is of importance to study the application of the theory of determinants to the solution of a system of linear**equations**.For further information concerning the compatibility and independence of a system of linear

**equations**, see Gordon, Vorlesungen fiber Invariantentheorie, Bd.Resultants.-When we are given k homogeneous

**equations**in k variables or k non-homogeneous**equations**in k - i variables, the**equations**being independent, it is always possible to derive from them a single equation R = o, where in R the variables do not appear.R is a function of the coefficients which is called the " resultant " or " eliminant " of the k

**equations**, and the process by which it is obtained is termed " elimination."We cannot combine the

**equations**so as to eliminate the variables unless on the supposition that the**equations**are simultaneous, i.e.Resultant Expressible as a Determinant.-From the theory of linear

**equations**it can be gathered that the condition that p linear**equations**in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz.Assuming then 01 to have the coefficients B1, B2,...B,, and f l the coefficients A 1, A21...A,n, we may equate coefficients of like powers of x in the identity, and obtain m+n homogeneous linear

**equations**satisfied by the m+n quantities B1, 2, ...B n, A 1, A 2, ...A m.Forming the resultant of these

**equations**we evidently obtain the resultant of f and 4,.From these m+n

**equations**he eliminates the m+n powers xmE.-1, xm+n- 2,..We can prove that if the three

**equations**be satisfied by a system of values of the variable, the same system will also satisfy the Jacobian or functional determinant.Hence in all there are mn such systems. If, therefore, we have a third equation, and we substitute each system of values in it successively and form the product of the mn expressions thus formed, we obtain a function which vanishes if any one system of values, common to the first two

**equations**, also satisfies the third.Hence this product is the required resultant of the three

**equations**.The consideration of cases where two roots are equal belongs to the theory of

**equations**(see Equation).Observe that for a determinant of the n-th order, taking the decomposition to be r + (n - I), we fall back upon the

**equations**given at the commencement, in order to show the genesis of a determinant.The germ of the theory of determinants is to be found in the writings of Gottfried Wilhelm Leibnitz (1693), who incidentally discovered certain properties when reducing the eliminant of a system of linear

**equations**.Differential

**equations**which express the changes of the co-ordinates are then constructed.The process of discovering the laws of motion of the particle then consists in the integration of these

**equations**.Such

**equations**can be formed for a system of any number of bodies, but the process of integration in a rigorous form is possible only to a limited extent or in special cases.In this case the two bodies really revolve round their common centre of gravity; but a very slight modification of the

**equations**of motion reduces them to the relative motion of the planet round the sun, regarding the moving centre of the latter as the origin of co-ordinates.In the actual problems of celestial mechanics three co-ordinates necessarily enter, leading to three differential

**equations**and six**equations**of solution.