Yz sentence example

yz
  • If 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.
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  • Every 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),...
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  • Two spheres intersect in a plane, and the equation to a system of spheres which intersect in a common circle is x 2 + y 2 + z 2 +2Ax -fD = o, in which A varies from sphere to sphere, and D is constant for all the spheres, the plane yz being the plane of intersection, and the axis of x the line of centres.
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  • But, assuming the distributive principle, the product of two lines appeared to give the expression xx' - yy' - zz' +i(yx' +xy')+j(xz' i j (yz' +zy').
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  • But the numerical factor appears to be yz'+zy', while it is the quantity yz' - zy' which really vanishes.
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  • For his speculations on sets had already familiarized him with the idea that multiplication might in certain cases not be commutative; so that, as the last term in the above product is made up of the two separate terms ijyz' and jizy', the term would vanish of itself when the factorlines are coplanar provided ij = - ji, for it would then assume the form ij(yz' - zy').
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  • He 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').
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  • And now a directed line in space came to be represented as ix+jy+kz, while the product of two lines is the quaternion - + yy ' +2z') +i (yz ' - zy') +j (zx' - xz') +k (x y ' - yx').
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  • Also, if 0 be the angle between them, and x", y", z" the direction-cosines of a line perpendicular to each of them, we have xx' +yy'+zz' =cos 0, yz' - zy" = x" sin 0, &c., so that the product of two unit lines is now expressed as - cos0+ (ix" +jy" +kz") sin 0.
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  • yz) =~(m.,l)+Z(m).
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  • Now if the displacement z be everywhere very small, the curvature in the planes parallel to xz and yz will be d 2 z/dx 2 and d 2 z/dy e respectively, and if T is the surface-tension the whole upward force will be d 2 z d2zl T (4x 2 + + (p - o) gz.
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  • The 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.
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  • Yz, the primary electron acceptor of the OEC, sits to the left, approximately 3.4 Angstroms away from the chloride.
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  • With the values above of u, v, w, u', v', w', the equations become of the form p x + 4 7rpAx -Fax -{-hy-}-gz =o, - p - dy+ 4?pBy + hx+ay+fz =o, P d p + TpCZ +f y + yz = o, and integrating p p 1+27rp(Ax2+By2+CZ2) +z ('ax e +ay e + yz2 2 f yz + 2gzx + 2 hx y) = const., (14) so that the surfaces of equal pressure are similar quadric surfaces, which, symmetry and dynamical considerations show, must be coaxial surfaces; and f, g, h vanish, as follows also by algebraical reduction; and 4c2 (c 2 - a2)?
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