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orthogonal

Such a determinant is of importance in the theory of orthogonal substitution.

20We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the Zn(n-I) quantities b are clearly arbitrary.

10The corresponding expression for two orthogonal cylinders will be With a 2 = co, these reduce to / y /, = Uy (I ra 2 p22 +-C24)..

10Clerk Maxwell, who showed amongst other things that a reciprocal can always be drawn to any figure which is the orthogonal projection of a plane-faced polyhedron.

10If we project both polyhedra orthogonally on a plane perpendicular to the axis of the paraboloid, we obtain two figures which are reciprocal, except that corresponding lines are orthogonal instead of parallel.

10The inconvenience of orthogonal illumination, which certainly gives better results, is avoided in the coaxial apparatus.

10The corresponding expression for two orthogonal cylinders will be With a 2 = co, these reduce to / y /, = Uy (I ra 2 p22 +-C24)..

10that there exists a point such that the tangents from this point to the four spheres are equal, and that with this point as centre, and the length of the tangent as radius, a sphere may be described which cuts, the four spheres at right angles; this "orthotomic" sphere corresponds to the orthogonal circle of a system of circles.

11a, /3 be the orthogonal projections of A, B on AB, we have AaB~=ABaf3=AB(I cos~,) =4AB.~2,

11We can eliminate the quantities S l, E2, ï¿½ï¿½ï¿½ In and obtain n relations AbXi = (2B 11 - Ab)'ï¿½k1 +2B21x2+2B31x3+ï¿½ï¿½ï¿½, AbX2 = 2B12x1+ (2B22 - Ab) x2 +2B32x3+..., and from these another equivalent set Abx1 = (2B11 - X1 +2B12X2+2B13X3+ï¿½ï¿½ï¿½, Abx2 = 2B21X1+(2B22 - Ab)X2+2B23X3+ï¿½ï¿½ï¿½, and now writing 2Bii - Ab 2Bik - aii, Ob = aik, Ob we have a transformation which is orthogonal, because EX 2 = Ex2 and the elements aii, a ik are functions of the 2n(n- I) independent quantities b.

00Similarly, for the order 3, we take 1 v Ab= -v 1 A =1 +x2 + 1, 2 + ï¿½ - A 1 and the adjoint is 1+A v +Aï¿½ -ï¿½ +Av -v +Aï¿½ 1+11 2 A +/-tv pt+AvA +ï¿½v 1 +1,2 leading to the orthogonal substitution Abx1 = (1 +A 2 - / 22 - v 2) X l +2(v+Aï¿½)X2 +2(/1 +Av)X3 1bx2 = 2(Aï¿½ - v)Xl+(1 +ï¿½2 - A2 - v2)X2 / +2(Fiv+A)X3 Abx3 = 2(Av +ï¿½)X1 +2(/lv-A)X2+(1+v2-A2- (12)X3.

00Orthogonal System.-In particular, if we consider the transformation from one pair of rectangular axes to another pair of rectangular axes we obtain an orthogonal system which we will now briefly inquire into.

00This is called the direct orthogonal substitution, because the sense of rotation from the axis of X i to the axis of X, is the same as that from that of x i to that of x 2.

00If the senses of rotation be opposite we have the skew orthogonal substitution x1 =cos0Xi+sinOX2r x 2 = sin °Xicos OX2r of modulus -1.

00It may denote a simultaneous orthogonal invariant of forms of orders n i, n2, n3,...; degree 0 of the covariant in the coefficients.

00Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms.

00The curves 0 = constant and 4, = constant form an orthogonal system; and the interchange of 0 and 4, will give a new state of uniplanar motion, in which the velocity at every point is turned through a right angle without alteration of magnitude.

00Thus for m =2, the spheres are orthogonal, and it can be verified that a13 a2 3 aY3 i f /' = ZU (I - 13 - 7.2 3 + 3) ' (8) where a l, a2, a =a l a 2 /J (a 1 2 +a 2 2) is the radius of the spheres and their circle of intersection, and r 1, r 2, r the distances of a point from their centres.

00The orthogonal projection of a section of this surface by a plane containing one of the perpendiculars and inclined to the axis is the quadratrix.

00We may describe, through all the points in an electric field which have the same potential, surfaces called equipotential surfaces, and these will be everywhere perpendicular or orthogonal to the lines of electric force.

00Now if the atoms are regarded as points or spherical bodies oscillating about positions of equilibrium, the value of n+3 is precisely six, for we can express the energy of the atom in the form (9 2 v a 2 v a2v E = z(mu 2 +mv 2 +mw 2 +x 2 ax2 + y2ay2-fz2az2), where V is the potential and x, y, z are the displacements of the atom referred to a certain set of orthogonal axes.

00In the case of non-intersecting circles, it is seen that the minimum circles of the coaxal system are a pair of points I and I', where the orthogonal circle to the system intersects the line of centres; these points are named the " limiti,ng points."

00Regarded as a statement concerning the orthogonal projections -~ ~ -~ -~

00In the first place, each of these figures may be conceived as an orthogonal projection of a closed plane-faced polyhedron.

00Work.The work done by a force acting on a particle, in any infinitely small displacement, is defined as the product of the force into the orthogonal projection of the displacement on the direction of the force; i.e.

00tion follows at once from the fact that the sum of orthogonal ~--~ ->

00If we imagine a point Q to describe a circle of radius a _________________ with the angular velocity ~, its A - 0 P orthogonal projection P on a fixed diameter AA will execute a vibration of this character.

00We have seen that a true simple-harmonic vibration may be regarded as the orthogonal projection of uniform circular motion; it was pointed out by P. G.

00in (1,3) satisfy the conjugate or orthogonal relations anaiai+aiiaiai+.

00The problem was to find the orthogonal trajectories of a series of curves represented by a single equation.

00The area of the ellipse is 7rab, where a, b are the semi-axes; this result may be deduced by regarding the ellipse as the orthogonal projection of a circle, or by means of the calculus.

00If the keyword FRACTIONAL is present, the translation is assumed to be in fractional coordinates, otherwise orthogonal angstroms.

00At a minimum this should therefore consist of observations made in the vertical direction and at 6° off-vertical in two orthogonal azimuths.

00Pass Point that lies on a local convexity that is orthogonal to a local concavity.

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

00The only restriction is that the atomic coordinates are given in Angstroms on arbitrary orthogonal axes.

00This quantity is equal to the width of the error ellipse orthogonal to the visibility vector.

00The wavelength associated with the vector resultant of these three orthogonal propagation constants is just the free space wavelength lambda.

00They note " Computational modeling has strengths orthogonal to the strengths of either traditional research discipline " (p. 7 ).

00orthogonal polynomials play an important role in the analysis.

00orthogonal axes of fine motion for the CCD mount.

00orthogonal matrices.

00orthogonal projections with each consisting of sixteen sensors.

00orthogonal coordinates.

00orthogonal transformation, Z, is applied to zero out the columns to the right of T.

00The columns T are constrained to be mutually orthogonal.

00The flags are not orthogonal, in that more restrictive flags will often make less restrictive ones redundant.

00Talking about Perl's multimethod dispatch, he pointed out that multi is " now completely orthogonal to scoping " .

00The rings have an O 120 o bridge, and are almost orthogonal to each other.

00This strongly suggests that the philosophy of intelligent tutoring is really orthogonal to the mindtool approach to learning.

00At the design stage, a design should be chosen to be as nearly orthogonal as possible to the nuisance effects.

00In both cases the solution can be made orthogonal to any number of specified functions.

00Here, orthogonal polynomials play an important role in the analysis.

00These estimators function on the principle that the noise subspace eigenvectors should be orthogonal to the signal vectors.

00Application of orthogonal wavelets to early gear damage detection.

00This theorem has been generalized for any tetrahedron; a sphere can be drawn through the four feet of the perpendiculars, and consequently through the mid-points of the lines from the vertices to the centre of the hyperboloid having these perpendiculars as generators, and through the orthogonal projections of these points on the opposite faces.

00Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Configurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geometry: applications of Differential Equations to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces.

00Such a determinant is of importance in the theory of orthogonal substitution.

00We can eliminate the quantities S l, E2, Ã¯¿½Ã¯¿½Ã¯¿½ In and obtain n relations AbXi = (2B 11 - Ab)'Ã¯¿½k1 +2B21x2+2B31x3+Ã¯¿½Ã¯¿½Ã¯¿½, AbX2 = 2B12x1+ (2B22 - Ab) x2 +2B32x3+..., and from these another equivalent set Abx1 = (2B11 - X1 +2B12X2+2B13X3+Ã¯¿½Ã¯¿½Ã¯¿½, Abx2 = 2B21X1+(2B22 - Ab)X2+2B23X3+Ã¯¿½Ã¯¿½Ã¯¿½, and now writing 2Bii - Ab 2Bik - aii, Ob = aik, Ob we have a transformation which is orthogonal, because EX 2 = Ex2 and the elements aii, a ik are functions of the 2n(n- I) independent quantities b.

00We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the Zn(n-I) quantities b are clearly arbitrary.

00Similarly, for the order 3, we take 1 v Ab= -v 1 A =1 +x2 + 1, 2 + Ã¯¿½ - A 1 and the adjoint is 1+A v +AÃ¯¿½ -Ã¯¿½ +Av -v +AÃ¯¿½ 1+11 2 A +/-tv pt+AvA +Ã¯¿½v 1 +1,2 leading to the orthogonal substitution Abx1 = (1 +A 2 - / 22 - v 2) X l +2(v+AÃ¯¿½)X2 +2(/1 +Av)X3 1bx2 = 2(AÃ¯¿½ - v)Xl+(1 +Ã¯¿½2 - A2 - v2)X2 / +2(Fiv+A)X3 Abx3 = 2(Av +Ã¯¿½)X1 +2(/lv-A)X2+(1+v2-A2- (12)X3.

00Orthogonal System.-In particular, if we consider the transformation from one pair of rectangular axes to another pair of rectangular axes we obtain an orthogonal system which we will now briefly inquire into.

00This is called the direct orthogonal substitution, because the sense of rotation from the axis of X i to the axis of X, is the same as that from that of x i to that of x 2.

00If the senses of rotation be opposite we have the skew orthogonal substitution x1 =cos0Xi+sinOX2r x 2 = sin Ã‚°Xicos OX2r of modulus -1.

00It may denote a simultaneous orthogonal invariant of forms of orders n i, n2, n3,...; degree 0 of the covariant in the coefficients.

00Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms.

00Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; n ap, dy, P d z, or X, Y, Z (4) are the partial differential coefficients of some function P, =fdplp, of x, y, z; so that X, Y, Z must be the partial differential coefficients of a potential -V, such that the force in any direction is the downward gradient of V; and then dP dV (5) ax + Tr=0, or P+V =constant, in which P may be called the hydrostatic head and V the head of potential.

00The curves 0 = constant and 4, = constant form an orthogonal system; and the interchange of 0 and 4, will give a new state of uniplanar motion, in which the velocity at every point is turned through a right angle without alteration of magnitude.

00Thus for m =2, the spheres are orthogonal, and it can be verified that a13 a2 3 aY3 i f /' = ZU (I - 13 - 7.2 3 + 3) ' (8) where a l, a2, a =a l a 2 /J (a 1 2 +a 2 2) is the radius of the spheres and their circle of intersection, and r 1, r 2, r the distances of a point from their centres.

00The orthogonal projection of a section of this surface by a plane containing one of the perpendiculars and inclined to the axis is the quadratrix.

00We may describe, through all the points in an electric field which have the same potential, surfaces called equipotential surfaces, and these will be everywhere perpendicular or orthogonal to the lines of electric force.

00Now if the atoms are regarded as points or spherical bodies oscillating about positions of equilibrium, the value of n+3 is precisely six, for we can express the energy of the atom in the form (9 2 v a 2 v a2v E = z(mu 2 +mv 2 +mw 2 +x 2 ax2 + y2ay2-fz2az2), where V is the potential and x, y, z are the displacements of the atom referred to a certain set of orthogonal axes.

00that there exists a point such that the tangents from this point to the four spheres are equal, and that with this point as centre, and the length of the tangent as radius, a sphere may be described which cuts, the four spheres at right angles; this "orthotomic" sphere corresponds to the orthogonal circle of a system of circles.

00In the case of non-intersecting circles, it is seen that the minimum circles of the coaxal system are a pair of points I and I', where the orthogonal circle to the system intersects the line of centres; these points are named the " limiti,ng points."

00Regarded as a statement concerning the orthogonal projections -~ ~ -~ -~

00In the first place, each of these figures may be conceived as an orthogonal projection of a closed plane-faced polyhedron.

00Clerk Maxwell, who showed amongst other things that a reciprocal can always be drawn to any figure which is the orthogonal projection of a plane-faced polyhedron.

00If we project both polyhedra orthogonally on a plane perpendicular to the axis of the paraboloid, we obtain two figures which are reciprocal, except that corresponding lines are orthogonal instead of parallel.

00Again, since the shortest distance between any two conjugate lines cuts the central axis at right angles, the orthogonal projections of two conjugate lines on a plane perpendicular to the central axis will be parallel (fig.

00Work.The work done by a force acting on a particle, in any infinitely small displacement, is defined as the product of the force into the orthogonal projection of the displacement on the direction of the force; i.e.

00a, /3 be the orthogonal projections of A, B on AB, we have AaB~=ABaf3=AB(I cos~,) =4AB.~2,

00tion follows at once from the fact that the sum of orthogonal ~--~ ->

00If we imagine a point Q to describe a circle of radius a _________________ with the angular velocity ~, its A - 0 P orthogonal projection P on a fixed diameter AA will execute a vibration of this character.

00We have seen that a true simple-harmonic vibration may be regarded as the orthogonal projection of uniform circular motion; it was pointed out by P. G.

00in (1,3) satisfy the conjugate or orthogonal relations anaiai+aiiaiai+.

00by means of the orthogonal relations (15).

00The problem was to find the orthogonal trajectories of a series of curves represented by a single equation.

00The area of the ellipse is 7rab, where a, b are the semi-axes; this result may be deduced by regarding the ellipse as the orthogonal projection of a circle, or by means of the calculus.

00The inconvenience of orthogonal illumination, which certainly gives better results, is avoided in the coaxial apparatus.

00Note the equivalence of orthogonal arrays with affine resolvable designs.

00These estimators function on the principle that the noise subspace eigenvectors should be orthogonal to the signal vectors.

00Application of orthogonal wavelets to early gear damage detection.

00Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Configurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geometry: applications of Differential Equations to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces.

01Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; n ap, dy, P d z, or X, Y, Z (4) are the partial differential coefficients of some function P, =fdplp, of x, y, z; so that X, Y, Z must be the partial differential coefficients of a potential -V, such that the force in any direction is the downward gradient of V; and then dP dV (5) ax + Tr=0, or P+V =constant, in which P may be called the hydrostatic head and V the head of potential.

01by means of the orthogonal relations (15).

01

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