The Quarterly Journal of Pure and Applied Mathematics, Volume 11J.W. Parker, 1871 - Mathematics |
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Page 15
... regard to the constants ; this can of course be done without difficulty , and in many cases it would be analytically advantageous to make the change . I take throughout ( X , Y , Z , W ) for the coordinates of a point on the quartic ...
... regard to the constants ; this can of course be done without difficulty , and in many cases it would be analytically advantageous to make the change . I take throughout ( X , Y , Z , W ) for the coordinates of a point on the quartic ...
Page 16
... regard to 0 , we have X { W + k √ ( X2 + Y2 + Z3 ) } − a Y2 = 0 , or , what is the same thing , ( a Y2 - XW ) 2 - k2X2 ( X2 + Y2 + Z3 ) = 0 , for the equation of the quartic surface . This has the line X = 0 , Y = 0 for a tacnodal ...
... regard to 0 , we have X { W + k √ ( X2 + Y2 + Z3 ) } − a Y2 = 0 , or , what is the same thing , ( a Y2 - XW ) 2 - k2X2 ( X2 + Y2 + Z3 ) = 0 , for the equation of the quartic surface . This has the line X = 0 , Y = 0 for a tacnodal ...
Page 17
... regard to 0 , the equation of the reciprocal surface is viz . this is - a2X2 + b2 Y2 = { W + k √ { X2 + Y2 + Z ' ) } " , - ( a * — k2 ) X2 + ( b2 — k3 ) Y2 – k3 Z2 — W2 = 2kW √ ( X3 + Y2 + Z3 ) , or { ( a * —k3 ) X2 + ( b2 — k2 ) Ya ...
... regard to 0 , the equation of the reciprocal surface is viz . this is - a2X2 + b2 Y2 = { W + k √ { X2 + Y2 + Z ' ) } " , - ( a * — k2 ) X2 + ( b2 — k3 ) Y2 – k3 Z2 — W2 = 2kW √ ( X3 + Y2 + Z3 ) , or { ( a * —k3 ) X2 + ( b2 — k2 ) Ya ...
Page 19
... regard to λ . Hence , writing this last equation in the form w3 ( a2 + λ ) ( b2 + λ ) λ − ( b2 + λ ) λx23 − ( a2 + λ ) λy2 - − ( a2 + λ ) ( b * + λ ) ( ≈2 — k2 w2 ) = 0 , - we have to equate to zero the discriminant of this cubic ...
... regard to λ . Hence , writing this last equation in the form w3 ( a2 + λ ) ( b2 + λ ) λ − ( b2 + λ ) λx23 − ( a2 + λ ) λy2 - − ( a2 + λ ) ( b * + λ ) ( ≈2 — k2 w2 ) = 0 , - we have to equate to zero the discriminant of this cubic ...
Page 22
... = 0 , * Y2 . · - W2 in regard to the variable parameter 6 , viz . the equation is X Y2 + b2 Z2 + —- ) ( a2X2 + b2Y2 + c2Z1_W1 ) — ( X3 + Y2 + Z1 ) 2 = 0 , ( see Salmon , p . 144 ) . It 22 On the Quartic Surfaces ( * X U , V , W ) 2 = 0 .
... = 0 , * Y2 . · - W2 in regard to the variable parameter 6 , viz . the equation is X Y2 + b2 Z2 + —- ) ( a2X2 + b2Y2 + c2Z1_W1 ) — ( X3 + Y2 + Z1 ) 2 = 0 , ( see Salmon , p . 144 ) . It 22 On the Quartic Surfaces ( * X U , V , W ) 2 = 0 .
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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 formulę 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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