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.

He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the contragredient line variables-it is completely determinate if its leading coefficient be known.

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.

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.

Laplace 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.

Besides 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 u1

For 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 Göttingen 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.