The Collected Mathematical Papers of Arthur Cayley volume 11

Cover The Collected Mathematical Papers of Arthur Cayley volume 11
The Collected Mathematical Papers of Arthur Cayley volume 11
Arthur Cayley
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Starting with an arbitrary line (x, y, z) in the first cone, then the reciprocal plane thereof (in regard to the absolute cone) is the plane Xx + Yy+Zz = 0, which meets the second cone in two lines, say (2) and (2 ), each of which is a line reciprocal to the line (1); and we have thus two planes (12) and (12 ), each of which envelopes, as is to be shown, the same cone q"r"X 2 + r"p"Y* +p"q"Z 2 = 0.
Suppose, in general, that we have an arbitrary line (x, y, z) and an arbitrary plane aX + /3Y+
...=Q, and that it is required to find the equation of the two planes through the line (x, y, z), and the intersections of the plane aX+/3Y+In the present case, the plane aX + @Y+yZ = is the plane xX+yY + zZ=0, which is the reciprocal of the line (x, y, z) in regard to the absolute cone, and the equation of the pair of planes is +q y 2 x n - + y* + z-) (p Xx + q Yy + r Zz) = 0, 783] ON MR WILKINSON S RECTANGULAR TRANSFORMATION.

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