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Referred to the asymptotes as axes the general equation becomes xy 2 obviously the axes are oblique in the general hyperbola and rectangular in the rectangular hyperbola.

00If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x, 2, z 2, yz, zx, xy from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails.

00Now f = (xy 1 - x i y) (xy 2 - x 2 y) ...

00(II.) Similarly in (I.), writing for c l, c 2 the cogredicnt pair -y2, +y1, we obtain axb5-a5bx=(ab)(xy)..

00Every symbolic product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors (xy), (xz), (yz),...

00Since, If F = An, 4) = By, 1 = I (Df A4) Of A?) Ab A"'^1Bz 1=, (F, Mn Ax I Ax 2 Axe Ax1) J The First Transvectant Differs But By A Numerical Factor From The Jacobian Or Functional Determinant, Of The Two Forms. We Can Find An Expression For The First Transvectant Of (F, ï¿½) 1 Over Another Form Cp. For (M N)(F,4)), =Nf.4Y Mfy.4), And F,4, F 5.4)= (Axby A Y B X) A X B X 1= (Xy)(F,4))1; (F,Ct)1=F5.D' 7,(Xy)(F4)1.

00For instance, x+y = x+xy and xy = x(x+y) are reciprocal.

00In 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.

00- xy.

00When the plane zx is not a plane of symmetry, we have to consider the terms in xy, 2 y, and y 3 .

00Denoting them by x, y, so that AB is axis of y and a perpendicular through A the axis of x, and rationalizing (26), we have 2 ax 2 - V 2 Xy 2 - V 2 aAy = o, which represents a hyperbola with vertices at 0 and A.

00If the direction of motion makes an angle 0' with Ox, tan B' = d0 !dam _ ?xy 2 = tan 20, 0 =-10', (9) dy/ y and the velocity is Ua2/r2.

00For in a rigid body, rotating about Oz with angular velocity the circulation round a curve in the plane xy is x ds yds) ds = times twice the area.

00Kirchhoff's expressions for X, Y, Z, the coordinates of the centre of the body, FX=y 1 cos xY--y 2 cos yY-{-y 3 cos zY, (18) FY = -y l cos xX -Hy2 cos yX+y 3 cos zX, (Ig) G=y 1 cos xZ+y 2 cos yZ+y 3 cos zZ, (20) (21) F(X+Yi) = Fy3-Gx3+i /) X 3epi.

00in the general equation of the second degree the co-efficients of x 2 and y 2 are equal, and of xy zero.

00Venn, in his Symbolic Logic, proposes the four forms, xy = o, xy = o, xy>o, xy> o (where y means " not-y "), but only as alternative to the ordinary forms. Bradley says that " ` S-P is real' attributes S-P, directly or indirectly, to the ultimate reality," and agrees with Brentano that " ` is ' never stands for anything but ` exists ' "; while Bosanquet, who follows Bradley, goes so far as to define a categorical judgment as " that which affirms the existence of its subject, or, in other words, asserts a fact."

00In quantitative judgments we may think x = y, or, as Boolero oses x = v ° p p y = - ° y, or, as Jevons proposes, x = xy, or, as Venn proposes, x which is not y=o; and equational symbolic logic is useful whenever we think in this quantitative way.

00He had now the following expression for the product of any two directed lines: xx' - yy - zz' +i(yx'+ xy')+ j(xz' '+zx') +ij(yz' - zy').

00There is, however, considerable evidence in support of the view that Greek va representing the sound arising from Ky, xy, Ty, By was pronounced as sh (s), while representing gy, dy was pronounced in some districts zh (z).4 On an inscription of Halicarnassus, a town which stood in ancient Carian territory, the sound of vv in `AXoKapvaao-Ewv is represented by T, as it is also in the Carian name Panyassis (IIavvfiTcos, geni tive), though the ordinary is also found in the same inscription.

00The three conditions of equilibrium arc therefore ~(X) = o, ~(Y) = o, ~(xY yX) = o.

00Taking this as the plane xy, with the axis of x drawn horizontally, and that of y vertically upwards, we have X=o, Y= mg; so that d2x ~

00Ketone hydrolysis,- CH3ï¿½COï¿½C(XY)ï¿½C02C2H5-jCH3ï¿½COï¿½CH(XY)+C2HSOH+C02; Acid hydrolysis:- CH 3 ï¿½COï¿½C(XY) C02C2H5--)CH3.

00C02H+C 2 [[Hsoh+Ch(Xy)ï¿½Cooh]]; (where X and Y =alkyl groups).

00The cartesian equation, if A be taken as origin and AB (= 2a) for the axis of x, is xy 2 =4a2(2a - x).

00the order of the curve is equal to that of the term or terms of highest order in the co-ordinates (x, y) conjointly in the equation of the curve; for instance, xy - 1= o is a curve of the second order.

00The theorem of the m intersections has been stated in regard to an arbitrary line; in fact, for particular lines the resultant equation may be or appear to be of an order less than m; for instance, taking m= 2, if the hyperbola xy - 1= o be cut by the line y=0, the resultant equation in x is Ox- 1 = o, and there is apparently only the intersection (x 110, y =0); but the theorem is, in fact, true for every line whatever: a curve of the order in meets every line whatever in precisely m points.

00Stating the theorem in regard to a conic, we have a real point P (called the pole) and a real line XY (called the polar), the line joining the two (real or imaginary) points of contact of the (real or imaginary) tangents drawn from the point to the conic; and the theorem is that when the point describes a line the line passes through a point, this line and point being polar and pole to each other.

00The most simple example is in the two systems of equations x': y': z' = yz: zx: xy and x: z'x': x'y'; where yz =0, zx =0, xy = o are conics (pairs of lines) having three common intersections, and where obviously either system of equations leads to the other system.

00By taking this plane, which is that of the orbit in which the planet performs its revolution, as the plane of xy, we have only two co-ordinates to consider.

00In all such cases, there is no component to plan convexity, any curvature being entirely orthogonal to the xy plane.

00In some South American field mice and the Scandinavian wood lemming, XY females are commonplace.

00Rather than columns running along a 2D x axis, a city scape consists of square columns at xy locations.

00For the simple case of a tensile stress sigma x combined with a shear stress tau xy.

00For the simple case of a tensile stress sigma x combined with a shear stress tau xy.

00So far as (7) is concerned the alternative supposition that AD vanishes would answer equally well, if we suppose the vibrations to be executed in the plane of polarization; but let us now revert to (5), which gives w 3 = _ PAN y z - = + PAN xy _ PAN z 2 - x2 8 N r 2 N r2' W 2 + N r2 (8) from 0 along which there is no scattered light, - two along the axis According to these equations there would be, in all, six directions of y normal to the original ray, and four (y z = =x) at angles of 45Ã‚° with that ray.

00Referred to the asymptotes as axes the general equation becomes xy 2 obviously the axes are oblique in the general hyperbola and rectangular in the rectangular hyperbola.

00If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x, 2, z 2, yz, zx, xy from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails.

00Now f = (xy 1 - x i y) (xy 2 - x 2 y) ...

00(II.) Similarly in (I.), writing for c l, c 2 the cogredicnt pair -y2, +y1, we obtain axb5-a5bx=(ab)(xy)..

00Every symbolic product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors (xy), (xz), (yz),...

00Since, If F = An, 4) = By, 1 = I (Df A4) Of A?) Ab A"'^1Bz 1=, (F, Mn Ax I Ax 2 Axe Ax1) J The First Transvectant Differs But By A Numerical Factor From The Jacobian Or Functional Determinant, Of The Two Forms. We Can Find An Expression For The First Transvectant Of (F, Ã¯¿½) 1 Over Another Form Cp. For (M N)(F,4)), =Nf.4Y Mfy.4), And F,4, F 5.4)= (Axby A Y B X) A X B X 1= (Xy)(F,4))1; (F,Ct)1=F5.D' 7,(Xy)(F4)1.

00Put M 1 For M, N I For N, And Multiply Through By (Ab); Then { (F, C6) } = (Ab) A X 2A Y B X 1 M N I 2 (Xy), ?) 2, = (A B)Ax 1B X 2B Y L I Multiply By Cp 1 And For Y L, Y2 Write C 2, C1; Then The Right Hand Side Becomes (Ab)(Bc)Am Lbn 2Cp 1 M I C P (F?) 2 M { N2 X, Of Which The First Term, Writing C P =, ,T, Is Mn 2 A B (Ab)(Bc)Axcx 1 M 2 N 2 P 2 2222 2 2 _2 A X B X C (Bc) A C Bx M N 2 2 2 M2Ã‚°N 2 N 2 M 2 2 A X (Bc) B C P C P (Ab) A B B(Ac) Ax Cp 2 = 2 (04) 2 1 (F,0) 2.4 (F,Y') 2 Ã¯¿½?; And, If (F,4)) 1 = Km " 2, (F??) 1 1 M N S X X X Af A _Af A Ax, Ax Ax Ax1 Observing That And This, On Writing C 2, C 1 For Y 11 Y 21 Becomes (Kc) K X 'T 3C X 1= (F,0 1 ', G 1; Ã¯¿½'Ã¯¿½1(F,O) 1 M 1=1 M 2 0`,4)) 2 0, T (Fm 2.4 (0,0 2 .F ' And Thence It Appears That The First Transvectant Of (F, (P) 1 Over 4) Is Always Expressible By Means Of Forms Of Lower Degree In The Coefficients Wherever Each Of The Forms F, 0, 4, Is Of Higher Degree Than The First In X 1, X2.

00xy 2 -4z 3 +g2x 2 y+g3x 3, and also the special form axz 2 -4by 3 of the cuspidal cubic. An investigation, by non-symbolic methods, is due to F.

00For instance, x+y = x+xy and xy = x(x+y) are reciprocal.

00In 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.

00When the plane zx is not a plane of symmetry, we have to consider the terms in xy, 2 y, and y 3 .

00Denoting them by x, y, so that AB is axis of y and a perpendicular through A the axis of x, and rationalizing (26), we have 2 ax 2 - V 2 Xy 2 - V 2 aAy = o, which represents a hyperbola with vertices at 0 and A.

00If the direction of motion makes an angle 0' with Ox, tan B' = d0 !dam _ ?xy 2 = tan 20, 0 =-10', (9) dy/ y and the velocity is Ua2/r2.

00For in a rigid body, rotating about Oz with angular velocity the circulation round a curve in the plane xy is x ds yds) ds = times twice the area.

00Kirchhoff's expressions for X, Y, Z, the coordinates of the centre of the body, FX=y 1 cos xY--y 2 cos yY-{-y 3 cos zY, (18) FY = -y l cos xX -Hy2 cos yX+y 3 cos zX, (Ig) G=y 1 cos xZ+y 2 cos yZ+y 3 cos zZ, (20) (21) F(X+Yi) = Fy3-Gx3+i /) X 3epi.

00in the general equation of the second degree the co-efficients of x 2 and y 2 are equal, and of xy zero.

00Venn, in his Symbolic Logic, proposes the four forms, xy = o, xy = o, xy>o, xy> o (where y means " not-y "), but only as alternative to the ordinary forms. Bradley says that " ` S-P is real' attributes S-P, directly or indirectly, to the ultimate reality," and agrees with Brentano that " ` is ' never stands for anything but ` exists ' "; while Bosanquet, who follows Bradley, goes so far as to define a categorical judgment as " that which affirms the existence of its subject, or, in other words, asserts a fact."

00In quantitative judgments we may think x = y, or, as Boolero oses x = v Ã‚° p p y = - Ã‚° y, or, as Jevons proposes, x = xy, or, as Venn proposes, x which is not y=o; and equational symbolic logic is useful whenever we think in this quantitative way.

00He had now the following expression for the product of any two directed lines: xx' - yy - zz' +i(yx'+ xy')+ j(xz' '+zx') +ij(yz' - zy').

00There is, however, considerable evidence in support of the view that Greek va representing the sound arising from Ky, xy, Ty, By was pronounced as sh (s), while representing gy, dy was pronounced in some districts zh (z).4 On an inscription of Halicarnassus, a town which stood in ancient Carian territory, the sound of vv in `AXoKapvaao-Ewv is represented by T, as it is also in the Carian name Panyassis (IIavvfiTcos, geni tive), though the ordinary is also found in the same inscription.

00The three conditions of equilibrium arc therefore ~(X) = o, ~(Y) = o, ~(xY yX) = o.

00Taking this as the plane xy, with the axis of x drawn horizontally, and that of y vertically upwards, we have X=o, Y= mg; so that d2x ~

00Ketone hydrolysis,- CH3Ã¯¿½COÃ¯¿½C(XY)Ã¯¿½C02C2H5-jCH3Ã¯¿½COÃ¯¿½CH(XY)+C2HSOH+C02; Acid hydrolysis:- CH 3 Ã¯¿½COÃ¯¿½C(XY) C02C2H5--)CH3.

00C02H+C 2 [[Hsoh+Ch(Xy)Ã¯¿½Cooh]]; (where X and Y =alkyl groups).

00The cartesian equation, if A be taken as origin and AB (= 2a) for the axis of x, is xy 2 =4a2(2a - x).

00the order of the curve is equal to that of the term or terms of highest order in the co-ordinates (x, y) conjointly in the equation of the curve; for instance, xy - 1= o is a curve of the second order.

00The theorem of the m intersections has been stated in regard to an arbitrary line; in fact, for particular lines the resultant equation may be or appear to be of an order less than m; for instance, taking m= 2, if the hyperbola xy - 1= o be cut by the line y=0, the resultant equation in x is Ox- 1 = o, and there is apparently only the intersection (x 110, y =0); but the theorem is, in fact, true for every line whatever: a curve of the order in meets every line whatever in precisely m points.

001806) shown how for any given position of the summit the plane of contact is determined, or reciprocally; say the plane XY is determined when the point P is given, or reciprocally; and it is noticed that when P is situate in the interior of the surface the plane XY does not cut the surface; that is, we have a real plane XY intersecting the surface in the imaginary curve of contact of the imaginary circumscribed cone having for its summit a given real point P inside the surface.

00Stating the theorem in regard to a conic, we have a real point P (called the pole) and a real line XY (called the polar), the line joining the two (real or imaginary) points of contact of the (real or imaginary) tangents drawn from the point to the conic; and the theorem is that when the point describes a line the line passes through a point, this line and point being polar and pole to each other.

00The most simple example is in the two systems of equations x': y': z' = yz: zx: xy and x: z'x': x'y'; where yz =0, zx =0, xy = o are conics (pairs of lines) having three common intersections, and where obviously either system of equations leads to the other system.

00By taking this plane, which is that of the orbit in which the planet performs its revolution, as the plane of xy, we have only two co-ordinates to consider.

00Rather than columns running along a 2D x axis, a city scape consists of square columns at xy locations.

00For the simple case of a tensile stress sigma x combined with a shear stress tau xy.

00Females have two X chromosomes (the XX combination); males have one X and one Y chromosome (the XY combination).

00This means that rather than having the normal XY combination, the male has an XXY combination.

00Males have one X and one Y chromosome (XY).

00Since the egg has no gender, it's the sperm's job to combine either a Y or X chromosome with egg's X chromosome to produce a boy (XY) or a girl (XX).

00If the sperm that fertilizes the egg is Y, then the baby will be XY or male.

00Plec, who also worked on Scream 2 with Williamson as well as the ABC Family Drama Kyle XY.

00xy 2 -4z 3 +g2x 2 y+g3x 3, and also the special form axz 2 -4by 3 of the cuspidal cubic. An investigation, by non-symbolic methods, is due to F.

01

The word usage examples above have been gathered from various sources to reflect current and historial usage. They do not represent the opinions of YourDictionary.com.