The Quarterly Journal of Pure and Applied Mathematics, Volume 11J.W. Parker, 1871 - Mathematics |
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Page 16
... variable parameter . The equation of the sphere therefore is ( x − að3w ) 2 + ( y − 2a0w ) 2 + z2 — k2w2 = 0 , - - and the ring is the envelope of this sphere . The reciprocal of the sphere is k2 ( X2 + Y2 + Z3 ) - ( a0 X + 2a0 Y + W ) ...
... variable parameter . The equation of the sphere therefore is ( x − að3w ) 2 + ( y − 2a0w ) 2 + z2 — k2w2 = 0 , - - and the ring is the envelope of this sphere . The reciprocal of the sphere is k2 ( X2 + Y2 + Z3 ) - ( a0 X + 2a0 Y + W ) ...
Page 17
... variable sphere is ( x - aw cos ) 2 + ( y − bw sin 0 ) 2 + za — k3w2 = 0 . The reciprocal of this is - - k2 ( X2 + Y2 + Z ) - ( aX cos + bY sin @ + W ) 2 = 0 . viz . writing this under the form aX cose + by sine + W + k √ ( X ' + Y2 + ...
... variable sphere is ( x - aw cos ) 2 + ( y − bw sin 0 ) 2 + za — k3w2 = 0 . The reciprocal of this is - - k2 ( X2 + Y2 + Z ) - ( aX cos + bY sin @ + W ) 2 = 0 . viz . writing this under the form aX cose + by sine + W + k √ ( X ' + Y2 + ...
Page 18
... variable sphere is - ( x − aw ) 2 + ( y — Bw ) 2 + z2 − k3w2 = 0 , - - where ( a , B ) vary subject to the condition have therefore a2 B2 + a2 b2 = 1. We аго х -aw - λ - = 0 , and thence aw = βιο = a * βω -βιο - λ = 0 , b2 a2x x -aw ...
... variable sphere is - ( x − aw ) 2 + ( y — Bw ) 2 + z2 − k3w2 = 0 , - - where ( a , B ) vary subject to the condition have therefore a2 B2 + a2 b2 = 1. We аго х -aw - λ - = 0 , and thence aw = βιο = a * βω -βιο - λ = 0 , b2 a2x x -aw ...
Page 20
... variable sphere which generates the ring . Centro - surface of a paraboloid . X * Y For the paraboloid + – 2ZW = 0 , it may be shewn a b that the centro - surface is the envelope of the quadric ax2 by2 ( a + 0 ) 2 + ( b + 0 ) 2 ̄ - 2zw ...
... variable sphere which generates the ring . Centro - surface of a paraboloid . X * Y For the paraboloid + – 2ZW = 0 , it may be shewn a b that the centro - surface is the envelope of the quadric ax2 by2 ( a + 0 ) 2 + ( b + 0 ) 2 ̄ - 2zw ...
Page 22
... variable plane is the inverse of the point on the ellipsoid . The quartic surface has the nodal conic W = 0 , X2 + Y + Z2 = 0 ; and also the node X = 0 , Y = 0 , Z = 0 ; there is consequently in the order of the reciprocal surface a ...
... variable plane is the inverse of the point on the ellipsoid . The quartic surface has the nodal conic W = 0 , X2 + Y + Z2 = 0 ; and also the node X = 0 , Y = 0 , Z = 0 ; there is consequently in the order of the reciprocal surface a ...
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a+b+c action af-g angle angular velocities axes body Cayley centre circle coefficient columns common tangent concyclic cone confocal conic conicoids conjugate coordinates corresponding cubic curvature cusps denote differential dt dt elements ellipsoid envelope equal equation Euler's equations evectant evolute expression fixed point formulæ given curve greatest value Hence inflexion instantaneous axis integral intersection line IJ log mv magic square nodal normal obtain pairs parallel curve perpendicular points of contact pole quadric quartic surface radius reciprocal respectively shewn solenoid solution stationary tangent suppose tangent planes theorem torse triads unlike signs vertex w₁ whence WILLIAM WALTON zero square
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