# Quadratic Sentence Examples

- 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. - 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. - 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 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 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 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. - 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, ... - 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. - The Binary
**Quadratic**.-The complete system consists of the form itself, ax, and the discriminant, which is the second transvectant of the form upon itself, viz.: (f, f') 2 = (ab) 2; or, in real coefficients, 2(a 0 a 2 a 2 1). - 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. - 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. - 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. - 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. - Of the Memoir he discusses bi-ternary quantics, and in particular those which are lineo-linear, quadrato-linear, cubo-linear, quadrato-
**quadratic**, cubo-cubic, and the system of two lineo-linear quantics. - 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 °, is coincident with the simultaneous system of the n i °' 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. - 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 . - (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. - Form rhombic or
**quadratic**crystals. - 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 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. - 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 =.. - 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. - 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. - 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? - 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 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. - It crystallizes in
**quadratic**prisms and has a bitter taste. - Why, I can do long, complicated
**quadratic**equations in my head quite easily, and it is great fun!