 • The next case is that in which u is a quadratic function of x, i.e.

• (6) The least admissible value of p is that which makes the roots equal of this quadratic in µ, and then ICI s ec 0,, u= z - p (7) the roots would be imaginary for a value of p smaller than given by Cip 2 - 4(c 2 -c i)c2C 2 u 2 =o, (8) p2 = 4(c 2 -c l)cl C2.

• (6) The least admissible value of p is that which makes the roots equal of this quadratic in µ, and then ICI s ec 0,, u= z - p (7) the roots would be imaginary for a value of p smaller than given by Cip 2 - 4(c 2 -c i)c2C 2 u 2 =o, (8) p2 = 4(c 2 -c l)cl C2.

• The transformation to the normal form, by the solution of a cubic and a quadratic, therefore, supplies a solution of the quartic. If (Xï¿½) is the modulus of the transformation by which a2 is reduced to 3 the normal form, i becomes (X /2) 4 i, and j, (Ap) 3 j; hence?

• 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.

• 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.

• 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.

• 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.

• Similarly, if a form in k variables be expressible as a quadratic function of k -1, linear functions X1, X2, ...

• Why, I can do long, complicated quadratic equations in my head quite easily, and it is great fun!

• In general the Boolian system, of the general n i °, is coincident with the simultaneous system of the n i °' and the quadratic x 2 +2 cos w xy+y2.

• the quartic to a quadratic. The new variables y1= 0 are the linear factors of 0.

• There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x .

• form rhombic or quadratic crystals.

• 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 =..

• form rhombic or quadratic crystals.

• To express the function aoa2 - _ which is the discriminant of the binary quadratic aoxi -+-2a1x2x2-+a2x2 = ai =1, 1, in a symbolic form we have 2(aoa 2 -ai) =aoa2 +aGa2 -2 a1 ï¿½ al = a;b4 -}-alb?

• The simultaneous system of two quadratic forms ai, ay, say f and 0, consists of six forms, viz.

• -, reduce s x2ax1 -x10x2 to the form j Oz ON 2 1 1 j 2 i The Binary Quintic.-The complete system consists of 23 forms, of which the simplest are f =a:; the Hessian H = (f, f') 2 = (ab) 2axbz; the quadratic covariant i= (f, f) 4 = (ab) 4axbx; and the nonic co variant T = (f, (f', f") 2) 1 = (f, H) 1 = (aH) azHi = (ab) 2 (ca) axbycy; the remaining 19 are expressible as transvectants of compounds of these four.

• (viii.) The quadratic equation is the equation of two expressions, monomial or multinomial, none of the terms involving any power of x except x and x 2 .

• But supposing them determined for the motion of a body through a liquid, the kinetic energy T of the system, liquid and body, is expressible as a quadratic function of the components U, V, W, P, Q, R.

• He was also the author of important papers in which he extended to complex quadratic forms many of Gauss's investigations relating to real quadratic forms. After 1864 he devoted himself chiefly to elliptic functions, and numerous papers on this subject were published by him in the Proc. Lond.

• It forms quadratic prisms, having a violet reflex and insoluble in boiling hydrochloric acid.

• 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).

• 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.

• For the substitution rr xl =A 11 +1 2 12, 52=A21+ï¿½2E2, of modulus A1 ï¿½i = (Alï¿½.2-A2ï¿½1) = (AM), A 2 ï¿½2 the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A lï¿½l +ai(A1/.22+A2ï¿½1) +7,2X2/22, A2 = aoï¿½l +2a1ï¿½1/ï¿½2 +a 2ï¿½2 ï¿½ We pass to the symbolic forms a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/ by writing for ao, al, a2 the symbols ai, a 1 a 2, a?

• The expression (ab) 4 properly appertains to a quartic; for a quadratic it may also be written (ab) 2 (cd) 2, and would denote the square of the discriminant to a factor pres.

• For the cubic (ab) 2 axbx is a covariant because each symbol a, b occurs three times; we can first of all find its real expression as a simultaneous covariant of two cubics, and then, by supposing the two cubics to merge into identity, find the expression of the quadratic covariant, of the single cubic, commonly known as the Hessian.

• (ab)(ac)bxcx = - (ab)(bc)axcx = 2(ab)c x {(ac)bx-(bc)axi = 1(ab)2ci; so that the covariant of the quadratic on the left is half the product of the quadratic itself and its only invariant.

• For the quadratic it is the discriminant (ab) 2 and for ax2 the cubic the quadratic covariant (ab) 2 axbx.

• Calling the discriminate D, the solution of the quadratic as =o is given by the formula a: = o (a0+a12_x2 (a0x+aix2 If the form a 2 be written as the product of its linear factors p.a., the discriminant takes the form -2(pq) 2.

• the two quadratic forms f, 4); the two discriminants (f, f')2,(0,4')2, and the first and second transvectants of f upon 4, (f,, >) 1 and (f, 402, which may be written (aa)a x a x and (aa) 2 .

• The .sextic covariant t is seen to be factorizable into three quadratic factors 4 = x 1 x 2, =x 2 1 - 1 - 2 2, 4) - x, which are such that the three mutual second transvectants vanish identically; they are for this reason termed conjugate quadratic factors.

• If 4) = rx.sx, the Y2 =1 normal form of a:, can be shown to be given by (rs) 4 .a x 4 = (ar) 4s: 6 (ar) 2 (as) 2rxsy -I- (as) 4rx; 4) is any one of the conjugate quadratic factors of t, so that, in determining rx, sx from J z+k 1 f =o, k 1 is any root of the resolvent.

• There are four invariants (i, i')2; (13, H)6; (f2, 151c.; (f t, 17)14 four linear forms (f, i 2) 4; (f, i 3) 5; (i 4, T) 8; (2 5, T)9 three quadratic forms i; (H, i 2)4; (H, 23)5 three cubic forms (f, i)2; (f, i 2) 3; (13, T)6 two quartic forms (H, i) 2; (H, 12)3.

• Further, it is convenient to have before us two other quadratic covariants, viz.

• The system of four forms, of which two are linear and two quadratic, has been investigated by Perrin (S.

• For example, take the ternary quadratic (aixl+a2x2+a3x3) 2 =a2x, or in real form axi +bx2+cx3+2fx 2 x 3+ 2gx 3 x 1 +2hx i x 2.

• denominator factors, that the complete system of the quadratic is composed of the form itself of degree order I, 2 shown by az 2, and of the Hessian of degree order 2, o shown by a2.

• Similarly, For A Linear And A Quadratic, P= I, Q= 2, And The Reduced Form Is Found To Be 1 A2B2Z2 1 Az.

• 1 A2B' Where The Denominator Factors Indicate The Forms Themselves, Their Jacobian, The Invariant Of The Quadratic And Their Resultant; Connected, As Shown By The Numerator, By A Syzygy Of Degreesorder (2, 2; 2).

• Of the quadratic axe+2bxy+cy2, he discovered the two invariants ac-b 2, a-2b cos w+c, and it may be verified that, if the transformed of the quadratic be AX2=2BXY+CY2, sin w 2 AC -B 2 =) (ac-b2), sin w A-2B cos w'+C = (sin w'1 2(a - 2bcosw+c).

• Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms.

• (iii.) In solving a quadratic equation by the method of ï¿½ 38 (viii.) we may be led to a result which is apparently absurd.

• 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.

• 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.

• A notable improvement on the ideas of Diophantus is to be found in the fact that the Hindus recognized the existence of two roots of a quadratic equation, but the negative roots were considered to be inadequate, since no interpretation could be found for them.

• 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.

• These frozen metals in general form compact masses consisting of aggregates of crystals belonging to the regular or rhombic or (more rarely) the quadratic system.

• The crystals belong to the following systems: regular system - silver, gold, palladium, mercury, copper, iron, lead; quadratic system - tin, potassium; rhombic system - antimony, bismuth, tellurium, zinc, magnesium.

• Regular crystals expand equally in all directions; rhombic and quadratic expand differently in different directions.

• Put S2 1 =12 cos 4, 12 2 = -12 sin 4, d4 d52 1 dS22 Y a2+c2 122 7Ti = 71 22 CL2- c2(121+5221)J, a2 +c2 do a2+c2 + 4c2 z dt a'-c2 (a2+,c2)2 M+2c2(a2-c2 N-{-a2+c2 2 Ý_a 2 +c 2 (' 4c2 .?"d za 2 -c 2 c2)2 2'J Z M+ -c2) which, as Z is a quadratic function of i 2, are non-elliptic so also for; G, where =co cos, G, 7 7 = - sin 4.

• Conversely, if the kinetic energy T is expressed as a quadratic function of x, x x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.

• The ordinary hydrated variety forms quadratic crystals and behaves as a strong base.

• It follows from §§ 48 and 51 that, if V is a solid figure extending from a plane K to a parallel plane L, and if the area of every cross-section parallel to these planes is a quadratic function of the distance of the section from a fixed plane parallel to them, Simpson's formula may be applied to find the volume of the solid.

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

• By drawing Ac and Ad parallel to BC and BD, so as to meet the plane through CD in c and d, and producing QP and RS to meet Ac and Ad in q and r, we see that the area of Pqrs is (x/h - x 2 /h 2) X area of cCDd; this also is a quadratic function of x.

• In the case, therefore, of any solid whose cross-section at distance x from one end is a quadratic function of x, the position of the crosssection through the centroid is to be found by determining the position of the centre of gravity of particles of masses proportional to So, S2, and 4S 1, placed at the extremities and the middle of a line drawn from one end of the solid to the other.

• His mathematical writings, which account for some forty entries in the Royal Society's catalogue of scientific papers, cover a wide range of subjects, such" s the theory of probabilities, quadratic forms, theory of integrals, gearings, the construction of geographical maps, &c. He also published a Traite de la theorie des nombres.

• It crystallizes in quadratic prisms.

• To Legendre is due the theorem known as the law of quadratic reciprocity, the most important general result in the science of numbers which has been discovered since the time of P. de Fermat, and which was called by Gauss the " gem of arithmetic."

• It may be obtained crystallized in the quadratic system by melting in a sealed tube containing hydrogen, allowed to cool partially, and then pouring off the still liquid portion by inverting the tube.

• Now these integrations are quite intractable, even for a very simple mathematical assumption of the function f(v), say the quadratic or cubic law, f(v) = v 2 /k or v3/k.

• It may be obtained crystallized in quadratic octahedra of a greenish-blue colour, by melting in a sealed tube containing an inert gas, and inverting the tube when the metal has partially solidified.

• The three commonest means are the arithmetical, geometrical, and harmonic; of less importance are the contraharmonical, arithmetico-geometrical, and quadratic.

• The quadratic mean of n quantities is the square root of the arithmetical mean of their squares.

• In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian parabola; a 2 x = y 3 is the cubic parabola, a 3 x = y4 is the biquadratic parabola; semi parabolas have the general equation ax n-1 = yn, thus ax e = y 3 is the semicubical parabola and ax 3 = y 4 the semibiquadratic parabola.

• In his Treatise of Algebra (1685) he distinctly proposes to construct the imaginary roots of a quadratic equation by going out of the line on which the roots, if real, would have been constructed.

• Of the phosphotungstic acids the most important is phosphoduodecitungstic acid, H 3 PW, 2040 nH 2 O, obtained in quadratic pyramids by crystallizing mixed solutions of orthophosphoric and metatungstic acids.

• Silicotungstic acid is obtained as quadratic pyramids from its mercurous salt which is prepared from mercurous nitrate and the salt formed on boiling gelatinous silicic acid with a polytungstate of an alkali metal.

• We proceed to the theory of the plane, axial and polar quadratic moments of the system.

• be the perpendicular distances of the particles from any fixed plane, the sum ~(mh2) is the quadratic moment with respect to the plane.

• /n be the perpendicular distances from any given axis, the sum ~(mp2) is the quadratic moment with respect to the axis; it is also called the moment of inertia about the axis.

• r,~ be the distances from a fixed point, the sum ~(mr2) is the quadratic moment with respect to that point (or pole).

• If we divide any of the above quadratic moments by the total mass ~(m), the result is called the mean square of the distances of the particles from the respective plane, axis or pole.

• If we take rectangular axes through any point 0, the quadratic moments with respect to the co-ordinate planes are I,, = Z(mxi), I,,= Z(my1), I, = ~(mz2); (9) those with respect to the co-ordinate axes are Ii,, = ~lm(y~+z2)~, I,, = ~tm(z2+x2)l, I,, ~tm(x2+y1)j; (10) whilst the polar quadratic moment with respect to 0 is 10 = ~tm(x2+y2+z1)}.

• If 4(x, y, z) be any homogeneous quadratic function of x, y, z, we have ~lm4o(x, Y~ 1)1 = ~}m~~+E, 3+i, ~+i)}

• Another type of quadratic moment is supplied by the deviationmoments, or products of inertia of a distribution of matter.

• The quadratic moment,s with respect to different planes through a fixed point 0 are related to one another as follows.

• If the co-ordinate axes coincide with the principal axes of this quadric, we shall have ~(myz) =0, ~(mzx) =0, Z(mxy) = 0~ (24) and if we write ~(mx) = Ma, ~(my1) = Mb, ~(mz) =Mc2, (25) where M=~(m), the quadratic moment becomes M(aiX2+bI,s2+ cv), or Mp, where p is the distance of the origin from that tangent plane of the ellipsoid ~-,+~1+~,=I, (26)

• Evidently the quadratic moment for a variable plane through 0 will have a stationary value when, and only when, the plane coincides with a principal plane of (26).

• If a, b, c be the semi-axes of the Binets ellipsoid of G, the quadratic moment with respect to the plane Xx + ~iy + vz =0 will be M(aX + bu + c2vi), and that with respect to a parallel plane ?.x+uy+vz=P (29)

• (30) Hence the planes of constant quadratic moment Mk2 will envelop the quadric a2+b2+c2~~ (31)

• The distance between the planes of and of will be of the second order of small quantities, and the quadratic moments with respect to of and co will therefore be equal, to the first order.

• Since the quadratic moments with respect to w and of are equal, it follows that w is a plane 01 stationary quadratic moment at P, and therefore a principal plane of inertia at P. In other words, the principal axes 01 inertia at P are the normals to the three confocals of the systen (3,~) which pass through P. Moreover if x, y, z be the co-ordinates of P, (33) is an equation to find the corresponding values of 0; and if Of, 02, 03 be the roots we find Oi+O2+81r1a2\$-7, (35)

• The directions of these axes are determined by the property (24), and therefore coincide with those of the principal axes of inertia at 0, as already defined in connection with the theory of plane quadratic moments.

• If we replace the mass of each particle by its moment, as thus found, we can in like manner obtain the quadratic moment of the system with respect to the line.

• The quadratic moment of the first particle will then be represented by twice the area FIG.

• The quadratic moment of the whole system is therefore represented by twice the area AHEDCBA.

• Since a quadratic moment is essentrally positive, the various areas are to taken positive in all cases.

• If some of the particles lie on one side of p and some on the other, the quadratic moment of each set may be found, and the results added.

• 60, where the total quadratic FIG.

• (38) It is easily proved that the roots of this quadratic in or1 are always real, and that they are moreover both positive unless w1 lies between pi and qf.

• The ratio B/A is determined in each case by either of the equations (37); hence each root of the quadratic gives a solution of the type (36), with two arbitrary constants A, ~.

• If the quadratic (38) has a negative root, the trigonometrical functions in (36) are to be replaced by real exponentials, and the position x=o, y=o is unstable.

• This has two soltitions of the type x+iy=ae~(~+r), where a, s are arbitrary, and ~ is a root of the quadratic (C +Moa)o2 (Cna/c)~+Moga/c = o.

• Thus T is expressed as a homogeneous quadratic function of the quantities ~i, q2,.

• Since T is a homogeneous quadratic function.

• The quadratic expression for T is essentially positive, and the same holds with regard to V in virtue of the assumed stability.

• The frequency equation is therefore (o2g/a)(if2g/b) ~ (12) The roots of this quadratic in rf are easily seen to be real and positive.

• where the denominator stands for the same homogeneou~ quadratic function of the qs that T is for the is.

• The value of such a fraction is the positive root of a quadratic equation whose coefficients are real and of which one root is negative.

• Since the fraction is infinite it cannot be commensurable and therefore its value is a quadratic surd number.

• Conversely every positive quadratic surd number, when expressed as a simple continued fraction, will give rise to a recurring fraction.

• The second case illustrates a feature of the recurring continued fraction which represents a complete quadratic surd.

• Muir, The Expression of a Quadratic Surd as a Continued Fraction (Glasgow, 1874).

• A saturated solution of the hydroxide deposits on cooling a hydrated form Ba(OH) 2.8H 2 0, as colourless quadratic prisms, which on exposure to air lose seven molecules of water of crystallization.

• Silver fluoride, AgF, is obtained as quadratic octahedra, with one molecule of water, by dissolving the oxide or carbonate in hydrofluoric acid.

• That for the conversion of a fraction into decimals (giving the complete period for all the prime numbers up to 997) is a specimen of the extraordinary love which Gauss had for long arithmetical calculations; and the amount of work gone through in the construction of the table of the number of the classes of binary quadratic forms must also have been tremendous.

• This concept is extended to algebra: since a line, surface and solid are represented by linear, quadratic and cubic equations, and are of one, two and three dimensions; a biquadratic equation has its highest terms of four dimensions, and, in general, an equation in any number of variables which has the greatest sum of the indices of any term equal to n is said to have n dimensions.

• Figure 6.6 - Residual maps for four quadratic approximations.

• Consider the general quadratic equation ax 2 + bx + c = 0 where a 0.

• BABY My baby is only 1 month old and can already solve quadratic equations, but has the awkward habit of eating crayons.

• A quadratic average stress failure criterion is suggested to predict delamination and the interlayer at which it occurs.

• The approximations behind the quadratic residue diffuser 's design have been tested.

• He solved problems such as pairs of simultaneous quadratic equations.

• An appropriate rescaling casts the system in a normal form, which is universal for models supporting ess through quadratic nonlinearities.

• The arbitrariness of the choice was underlined later, when the teacher introduced a quadratic graph.

• Newton strategies additionally require the second partial derivatives, thus building a quadratic internal model.

• Reminder: How to find the roots of quadratic polynomials (pdf ).

• polynomial regression, the quadratic component of the ordinate scale was significant and positive.

• include state-specific quadratic in the model price for a policy internet address http.

• Factorizing quadratics An essential skill in many applications is the ability to factorize quadratic expressions.

• If you'd like to see how to find the Julia set of the simplest quadratic, have a look at this example.

• One of the questions asked him to show that a particular quadratic had ' real and distinct roots ' .

• quadratic equation using factors Errors Identify sources of errors.

• quadratic interpolation to be used.

• quadratic spline takes longer to render than a linear spline.

• quadratic polynomials (pdf ).

• quadratic residue diffuser 's design have been tested.

• context quadratic time prominent role in these assets were in canada should.

• Thus, the Mesopotamians knew how to solve quadratic equations 4000 years ago, using essentially the same method that we use today.

• factorizing quadratics There is no simple method of factorizing a quadratic expression, but with a little practice it becomes easier.

• Use this & Gauss ' law of quadratic reciprocity, to show that 75 is a primitive root modulo 65537.

• regularizecular, the quadratic regularizing term discourages sudden changes in surface normal direction across a surface.

• short-run dynamics, assuming quadratic adjustment costs, cannot be rejected by the data.

• The quadratic spline takes longer to render than a linear spline.

• Knowing the maximum or minimum values and where the graph hits the x-axis (solving the quadratic ).

• Some of the most important results of his discoveries were communicated to the Royal Society in two memoirs upon "Systems of Linear Indeterminate Equations and Congruences" and upon the "Orders and Genera of Ternary Quadratic Forms" (Phil.

• He was also the author of important papers in which he extended to complex quadratic forms many of Gauss's investigations relating to real quadratic forms. After 1864 he devoted himself chiefly to elliptic functions, and numerous papers on this subject were published by him in the Proc. Lond.

• It forms quadratic prisms, having a violet reflex and insoluble in boiling hydrochloric acid.

• Tannery thinks that the solution of a complete quadratic promised by Diophantus himself (I.

• Among the great variety of problems solved are problems leading to determinate equations of the first degree in one, two, three or four variables, to determinate quadratic equations, and to indeterminate equations of the first degree in one or more variables, which are, however, transformed into determinate equations by arbitrarily assuming a value for one of the required numbers, Diophantus being always satisfied with a rational, even if fractional, result and not requiring a solution in integers.

• 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.

• 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).

• 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.

• Similarly, if a form in k variables be expressible as a quadratic function of k -1, linear functions X1, X2, ...

• To express the function aoa2 - _ which is the discriminant of the binary quadratic aoxi -+-2a1x2x2-+a2x2 = ai =1, 1, in a symbolic form we have 2(aoa 2 -ai) =aoa2 +aGa2 -2 a1 Ã¯¿½ al = a;b4 -}-alb?

• For the substitution rr xl =A 11 +1 2 12, 52=A21+Ã¯¿½2E2, of modulus A1 Ã¯¿½i = (AlÃ¯¿½.2-A2Ã¯¿½1) = (AM), A 2 Ã¯¿½2 the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A lÃ¯¿½l +ai(A1/.22+A2Ã¯¿½1) +7,2X2/22, A2 = aoÃ¯¿½l +2a1Ã¯¿½1/Ã¯¿½2 +a 2Ã¯¿½2 Ã¯¿½ We pass to the symbolic forms a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/ by writing for ao, al, a2 the symbols ai, a 1 a 2, a?

• The expression (ab) 4 properly appertains to a quartic; for a quadratic it may also be written (ab) 2 (cd) 2, and would denote the square of the discriminant to a factor pres.

• For the cubic (ab) 2 axbx is a covariant because each symbol a, b occurs three times; we can first of all find its real expression as a simultaneous covariant of two cubics, and then, by supposing the two cubics to merge into identity, find the expression of the quadratic covariant, of the single cubic, commonly known as the Hessian.

• (ab)(ac)bxcx = - (ab)(bc)axcx = 2(ab)c x {(ac)bx-(bc)axi = 1(ab)2ci; so that the covariant of the quadratic on the left is half the product of the quadratic itself and its only invariant.

• For the quadratic it is the discriminant (ab) 2 and for ax2 the cubic the quadratic covariant (ab) 2 axbx.

• Calling the discriminate D, the solution of the quadratic as =o is given by the formula a: = o (a0+a12_x2 (a0x+aix2 If the form a 2 be written as the product of its linear factors p.a., the discriminant takes the form -2(pq) 2.

• The simultaneous system of two quadratic forms ai, ay, say f and 0, consists of six forms, viz.

• the two quadratic forms f, 4); the two discriminants (f, f')2,(0,4')2, and the first and second transvectants of f upon 4, (f,, >) 1 and (f, 402, which may be written (aa)a x a x and (aa) 2 .

• The fundamental system connected with n quadratic forms consists of (i.) the n forms themselves f i, f2,Ã¯¿½Ã¯¿½ fn, (ii.) the (2) functional determinants (f i, f k) 1, (iii.) the (n 2 1) in variants (f l, fk) 2, (iv.) the (3) forms (f i, (f k, f ni)) 2, each such form remaining unaltered for any permutations of i, k, m.

• The .sextic covariant t is seen to be factorizable into three quadratic factors 4 = x 1 x 2, =x 2 1 - 1 - 2 2, 4) - x, which are such that the three mutual second transvectants vanish identically; they are for this reason termed conjugate quadratic factors.

• of f=0, :and notices that they become identical on substituting 0 for k, and -f for X; hence, if k1, k2, k 3 be the roots of the resolvent -21 2 = (o + k if) (A + k 2f)(o + k 3f); and now, if all the roots of f be different, so also are those of the resolvent, since the latter, and f, have practically the same discriminant; consequently each of the three factors, of -21 2, must be perfect squares and taking the square root 1 t = -' (1)Ã¯¿½x4; and it can be shown that 0, x, 1P are the three conjugate quadratic factors of t above mentioned.

• the quartic to a quadratic. The new variables y1= 0 are the linear factors of 0.

• If 4) = rx.sx, the Y2 =1 normal form of a:, can be shown to be given by (rs) 4 .a x 4 = (ar) 4s: 6 (ar) 2 (as) 2rxsy -I- (as) 4rx; 4) is any one of the conjugate quadratic factors of t, so that, in determining rx, sx from J z+k 1 f =o, k 1 is any root of the resolvent.

• The transformation to the normal form, by the solution of a cubic and a quadratic, therefore, supplies a solution of the quartic. If (XÃ¯¿½) is the modulus of the transformation by which a2 is reduced to 3 the normal form, i becomes (X /2) 4 i, and j, (Ap) 3 j; hence?

• -, reduce s x2ax1 -x10x2 to the form j Oz ON 2 1 1 j 2 i The Binary Quintic.-The complete system consists of 23 forms, of which the simplest are f =a:; the Hessian H = (f, f') 2 = (ab) 2axbz; the quadratic covariant i= (f, f) 4 = (ab) 4axbx; and the nonic co variant T = (f, (f', f") 2) 1 = (f, H) 1 = (aH) azHi = (ab) 2 (ca) axbycy; the remaining 19 are expressible as transvectants of compounds of these four.

• There are four invariants (i, i')2; (13, H)6; (f2, 151c.; (f t, 17)14 four linear forms (f, i 2) 4; (f, i 3) 5; (i 4, T) 8; (2 5, T)9 three quadratic forms i; (H, i 2)4; (H, 23)5 three cubic forms (f, i)2; (f, i 2) 3; (13, T)6 two quartic forms (H, i) 2; (H, 12)3.

• Further, it is convenient to have before us two other quadratic covariants, viz.

• The system of the quadratic and cubic, consisting of 15 forms, and that of two cubics, consisting of 26 forms, were obtained by Salmon and Clebsch; that of the cubic and quartic we owe to Sigmund Gundelfinger (Programm Stuttgart, 186 9, 1 -43); that of the quadratic and quintic to Winter (Programm Darmstadt, 1880); that of the quadratic and sextic to von Gall (Programm Lemgo, 3873); that of two quartics to Gordan (Math.

• The system of four forms, of which two are linear and two quadratic, has been investigated by Perrin (S.

• For example, take the ternary quadratic (aixl+a2x2+a3x3) 2 =a2x, or in real form axi +bx2+cx3+2fx 2 x 3+ 2gx 3 x 1 +2hx i x 2.

• denominator factors, that the complete system of the quadratic is composed of the form itself of degree order I, 2 shown by az 2, and of the Hessian of degree order 2, o shown by a2.

• Similarly, For A Linear And A Quadratic, P= I, Q= 2, And The Reduced Form Is Found To Be 1 A2B2Z2 1 Az.

• 1 A2B' Where The Denominator Factors Indicate The Forms Themselves, Their Jacobian, The Invariant Of The Quadratic And Their Resultant; Connected, As Shown By The Numerator, By A Syzygy Of Degreesorder (2, 2; 2).

• Of the quadratic axe+2bxy+cy2, he discovered the two invariants ac-b 2, a-2b cos w+c, and it may be verified that, if the transformed of the quadratic be AX2=2BXY+CY2, sin w 2 AC -B 2 =) (ac-b2), sin w A-2B cos w'+C = (sin w'1 2(a - 2bcosw+c).

• In general the Boolian system, of the general n i Ã‚°, is coincident with the simultaneous system of the n i Ã‚°' and the quadratic x 2 +2 cos w xy+y2.

• Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms.

• For the quadratic aoxi +2a i x i x 2 +a 2 x, we have (i.) ax = 7/1x1+2aixix2-I-7/24, (ii.) xx=xi+xzi (ab) 2 =2(aoa2 - ai), a a = a o+712, _ (v.) (xa)ax= i'?- (a2 - ao)xix2 - aix2.

• There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x .

• 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.

• (viii.) The quadratic equation is the equation of two expressions, monomial or multinomial, none of the terms involving any power of x except x and x 2 .

• (iii.) In solving a quadratic equation by the method of Ã¯¿½ 38 (viii.) we may be led to a result which is apparently absurd.

• The quadratic equation x 2 +b 2 =o, for instance, has no real root; but we may treat the roots as being +b-' - I, and - b 1, 1 - 1, if -J - i is treated as something which obeys the laws of arithmetic and emerges into reality under the condition 1 1 - I.

• 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.

• 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.

• A notable improvement on the ideas of Diophantus is to be found in the fact that the Hindus recognized the existence of two roots of a quadratic equation, but the negative roots were considered to be inadequate, since no interpretation could be found for them.

• 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.

• These frozen metals in general form compact masses consisting of aggregates of crystals belonging to the regular or rhombic or (more rarely) the quadratic system.

• The crystals belong to the following systems: regular system - silver, gold, palladium, mercury, copper, iron, lead; quadratic system - tin, potassium; rhombic system - antimony, bismuth, tellurium, zinc, magnesium.

• Regular crystals expand equally in all directions; rhombic and quadratic expand differently in different directions.

• 1 a +') - (121+52277)2] 4 - (a2+0)2 [L M -N2 { L 2c (a c 2 c 2) ae - N az+c2 l Y2 T L + 2 z 2 M (a2+c2) (9a2 - c2) 4 J 16c4 (a2-c2) = Z, where Z is a quadratic in ?

• Put S2 1 =12 cos 4, 12 2 = -12 sin 4, d4 d52 1 dS22 Y a2+c2 122 7Ti = 71 22 CL2- c2(121+5221)J, a2 +c2 do a2+c2 + 4c2 z dt a'-c2 (a2+,c2)2 M+2c2(a2-c2 N-{-a2+c2 2 Ã_a 2 +c 2 (' 4c2 .?"d za 2 -c 2 c2)2 2'J Z M+ -c2) which, as Z is a quadratic function of i 2, are non-elliptic so also for; G, where =co cos, G, 7 7 = - sin 4.

• But supposing them determined for the motion of a body through a liquid, the kinetic energy T of the system, liquid and body, is expressible as a quadratic function of the components U, V, W, P, Q, R.

• Conversely, if the kinetic energy T is expressed as a quadratic function of x, x x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.

• 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 =..

• It crystallizes in quadratic prisms and has a bitter taste.

• The ordinary hydrated variety forms quadratic crystals and behaves as a strong base.

• The next case is that in which u is a quadratic function of x, i.e.

• It follows from §§ 48 and 51 that, if V is a solid figure extending from a plane K to a parallel plane L, and if the area of every cross-section parallel to these planes is a quadratic function of the distance of the section from a fixed plane parallel to them, Simpson's formula may be applied to find the volume of the solid.

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

• By drawing Ac and Ad parallel to BC and BD, so as to meet the plane through CD in c and d, and producing QP and RS to meet Ac and Ad in q and r, we see that the area of Pqrs is (x/h - x 2 /h 2) X area of cCDd; this also is a quadratic function of x.

• In the case, therefore, of any solid whose cross-section at distance x from one end is a quadratic function of x, the position of the crosssection through the centroid is to be found by determining the position of the centre of gravity of particles of masses proportional to So, S2, and 4S 1, placed at the extremities and the middle of a line drawn from one end of the solid to the other.

• His mathematical writings, which account for some forty entries in the Royal Society's catalogue of scientific papers, cover a wide range of subjects, such" s the theory of probabilities, quadratic forms, theory of integrals, gearings, the construction of geographical maps, &c. He also published a Traite de la theorie des nombres.

• It crystallizes in quadratic prisms.

• To Legendre is due the theorem known as the law of quadratic reciprocity, the most important general result in the science of numbers which has been discovered since the time of P. de Fermat, and which was called by Gauss the " gem of arithmetic."

• It may be obtained crystallized in the quadratic system by melting in a sealed tube containing hydrogen, allowed to cool partially, and then pouring off the still liquid portion by inverting the tube.

• The theoretical assumptions of Newton and Euler (hypotheses magis mathematicae quam naturales) of a resistance varying as some simple power of the velocity, for instance, as the square or cube of the velocity (the quadratic or cubic law), lead to results of great analytical complexity, and are useful only for provisional extrapolation at high or low velocity, pending further experiment.

• Now these integrations are quite intractable, even for a very simple mathematical assumption of the function f(v), say the quadratic or cubic law, f(v) = v 2 /k or v3/k.

• It may be obtained crystallized in quadratic octahedra of a greenish-blue colour, by melting in a sealed tube containing an inert gas, and inverting the tube when the metal has partially solidified.

• The three commonest means are the arithmetical, geometrical, and harmonic; of less importance are the contraharmonical, arithmetico-geometrical, and quadratic.

• The quadratic mean of n quantities is the square root of the arithmetical mean of their squares.

• In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian parabola; a 2 x = y 3 is the cubic parabola, a 3 x = y4 is the biquadratic parabola; semi parabolas have the general equation ax n-1 = yn, thus ax e = y 3 is the semicubical parabola and ax 3 = y 4 the semibiquadratic parabola.

• In his Treatise of Algebra (1685) he distinctly proposes to construct the imaginary roots of a quadratic equation by going out of the line on which the roots, if real, would have been constructed.

• Of the phosphotungstic acids the most important is phosphoduodecitungstic acid, H 3 PW, 2040 nH 2 O, obtained in quadratic pyramids by crystallizing mixed solutions of orthophosphoric and metatungstic acids.

• Silicotungstic acid is obtained as quadratic pyramids from its mercurous salt which is prepared from mercurous nitrate and the salt formed on boiling gelatinous silicic acid with a polytungstate of an alkali metal.

• We proceed to the theory of the plane, axial and polar quadratic moments of the system.

• be the perpendicular distances of the particles from any fixed plane, the sum ~(mh2) is the quadratic moment with respect to the plane.

• /n be the perpendicular distances from any given axis, the sum ~(mp2) is the quadratic moment with respect to the axis; it is also called the moment of inertia about the axis.

• r,~ be the distances from a fixed point, the sum ~(mr2) is the quadratic moment with respect to that point (or pole).

• If we divide any of the above quadratic moments by the total mass ~(m), the result is called the mean square of the distances of the particles from the respective plane, axis or pole.

• If we take rectangular axes through any point 0, the quadratic moments with respect to the co-ordinate planes are I,, = Z(mxi), I,,= Z(my1), I, = ~(mz2); (9) those with respect to the co-ordinate axes are Ii,, = ~lm(y~+z2)~, I,, = ~tm(z2+x2)l, I,, ~tm(x2+y1)j; (10) whilst the polar quadratic moment with respect to 0 is 10 = ~tm(x2+y2+z1)}.

• If 4(x, y, z) be any homogeneous quadratic function of x, y, z, we have ~lm4o(x, Y~ 1)1 = ~}m~~+E, 3+i, ~+i)}

• Another type of quadratic moment is supplied by the deviationmoments, or products of inertia of a distribution of matter.

• The quadratic moment,s with respect to different planes through a fixed point 0 are related to one another as follows.

• If the co-ordinate axes coincide with the principal axes of this quadric, we shall have ~(myz) =0, ~(mzx) =0, Z(mxy) = 0~ (24) and if we write ~(mx) = Ma, ~(my1) = Mb, ~(mz) =Mc2, (25) where M=~(m), the quadratic moment becomes M(aiX2+bI,s2+ cv), or Mp, where p is the distance of the origin from that tangent plane of the ellipsoid ~-,+~1+~,=I, (26)

• Evidently the quadratic moment for a variable plane through 0 will have a stationary value when, and only when, the plane coincides with a principal plane of (26).

• If a, b, c be the semi-axes of the Binets ellipsoid of G, the quadratic moment with respect to the plane Xx + ~iy + vz =0 will be M(aX + bu + c2vi), and that with respect to a parallel plane ?.x+uy+vz=P (29)

• (30) Hence the planes of constant quadratic moment Mk2 will envelop the quadric a2+b2+c2~~ (31)

• The distance between the planes of and of will be of the second order of small quantities, and the quadratic moments with respect to of and co will therefore be equal, to the first order.

• Since the quadratic moments with respect to w and of are equal, it follows that w is a plane 01 stationary quadratic moment at P, and therefore a principal plane of inertia at P. In other words, the principal axes 01 inertia at P are the normals to the three confocals of the systen (3,~) which pass through P. Moreover if x, y, z be the co-ordinates of P, (33) is an equation to find the corresponding values of 0; and if Of, 02, 03 be the roots we find Oi+O2+81r1a2\$-7, (35)

• The directions of these axes are determined by the property (24), and therefore coincide with those of the principal axes of inertia at 0, as already defined in connection with the theory of plane quadratic moments.

• If we replace the mass of each particle by its moment, as thus found, we can in like manner obtain the quadratic moment of the system with respect to the line.

• The quadratic moment of the first particle will then be represented by twice the area FIG.

• The quadratic moment of the whole system is therefore represented by twice the area AHEDCBA.

• Since a quadratic moment is essentrally positive, the various areas are to taken positive in all cases.

• If some of the particles lie on one side of p and some on the other, the quadratic moment of each set may be found, and the results added.

• 60, where the total quadratic FIG.

• (38) It is easily proved that the roots of this quadratic in or1 are always real, and that they are moreover both positive unless w1 lies between pi and qf.

• The ratio B/A is determined in each case by either of the equations (37); hence each root of the quadratic gives a solution of the type (36), with two arbitrary constants A, ~.

• If the quadratic (38) has a negative root, the trigonometrical functions in (36) are to be replaced by real exponentials, and the position x=o, y=o is unstable.

• This has two soltitions of the type x+iy=ae~(~+r), where a, s are arbitrary, and ~ is a root of the quadratic (C +Moa)o2 (Cna/c)~+Moga/c = o.

• Thus T is expressed as a homogeneous quadratic function of the quantities ~i, q2,.

• Since T is a homogeneous quadratic function.

• The quadratic expression for T is essentially positive, and the same holds with regard to V in virtue of the assumed stability.

• The frequency equation is therefore (o2g/a)(if2g/b) ~ (12) The roots of this quadratic in rf are easily seen to be real and positive.

• where the denominator stands for the same homogeneou~ quadratic function of the qs that T is for the is.

• The value of such a fraction is the positive root of a quadratic equation whose coefficients are real and of which one root is negative.

• Since the fraction is infinite it cannot be commensurable and therefore its value is a quadratic surd number.

• Conversely every positive quadratic surd number, when expressed as a simple continued fraction, will give rise to a recurring fraction.

• The second case illustrates a feature of the recurring continued fraction which represents a complete quadratic surd.

• Muir, The Expression of a Quadratic Surd as a Continued Fraction (Glasgow, 1874).

• A saturated solution of the hydroxide deposits on cooling a hydrated form Ba(OH) 2.8H 2 0, as colourless quadratic prisms, which on exposure to air lose seven molecules of water of crystallization.

• Silver fluoride, AgF, is obtained as quadratic octahedra, with one molecule of water, by dissolving the oxide or carbonate in hydrofluoric acid.

• That for the conversion of a fraction into decimals (giving the complete period for all the prime numbers up to 997) is a specimen of the extraordinary love which Gauss had for long arithmetical calculations; and the amount of work gone through in the construction of the table of the number of the classes of binary quadratic forms must also have been tremendous.

• This concept is extended to algebra: since a line, surface and solid are represented by linear, quadratic and cubic equations, and are of one, two and three dimensions; a biquadratic equation has its highest terms of four dimensions, and, in general, an equation in any number of variables which has the greatest sum of the indices of any term equal to n is said to have n dimensions.

• Include state-specific quadratic in the model price for a policy internet address http.

• Factorizing quadratics An essential skill in many applications is the ability to factorize quadratic expressions.

• If you 'd like to see how to find the Julia set of the simplest quadratic, have a look at this example.

• One of the questions asked him to show that a particular quadratic had ' real and distinct roots '.

• Solve a quadratic equation using factors Errors Identify sources of errors.

• Values of 1 or 2 will cause linear or quadratic interpolation to be used.

• The quadratic spline takes longer to render than a linear spline.

• Context quadratic time prominent role in these assets were in canada should.

• Thus, the Mesopotamians knew how to solve quadratic equations 4000 years ago, using essentially the same method that we use today.

• Factorizing Quadratics There is no simple method of factorizing a quadratic expression, but with a little practice it becomes easier.

• Use this & Gauss ' law of quadratic reciprocity, to show that 75 is a primitive root modulo 65537.

• In particular, the quadratic regularizing term discourages sudden changes in surface normal direction across a surface.

• Quadratic residue diffuser In this case, the longest wells in the diffuser were replaced by shorter, active equivalents.

• Finally, a forward-looking interpretation of the short-run dynamics, assuming quadratic adjustment costs, cannot be rejected by the data.

• Knowing the maximum or minimum values and where the graph hits the x-axis (solving the quadratic).

• Among the great variety of problems solved are problems leading to determinate equations of the first degree in one, two, three or four variables, to determinate quadratic equations, and to indeterminate equations of the first degree in one or more variables, which are, however, transformed into determinate equations by arbitrarily assuming a value for one of the required numbers, Diophantus being always satisfied with a rational, even if fractional, result and not requiring a solution in integers.

• 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.

• of f=0, :and notices that they become identical on substituting 0 for k, and -f for X; hence, if k1, k2, k 3 be the roots of the resolvent -21 2 = (o + k if) (A + k 2f)(o + k 3f); and now, if all the roots of f be different, so also are those of the resolvent, since the latter, and f, have practically the same discriminant; consequently each of the three factors, of -21 2, must be perfect squares and taking the square root 1 t = -' (1)ï¿½x4; and it can be shown that 0, x, 1P are the three conjugate quadratic factors of t above mentioned.

• It crystallizes in quadratic prisms and has a bitter taste.

• 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.