The Collected Mathematical Papers of Arthur Cayley volume 12

Cover The Collected Mathematical Papers of Arthur Cayley volume 12
The Collected Mathematical Papers of Arthur Cayley volume 12
Arthur Cayley
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[838 The two cubics meet in the 5 points 0=(0, oo, 1, (a, w 2 ); < = (0, oo, 1, w 2, , o> 2, 1), (1, ,, 1), where &> is an imaginary cube root of unity.
I was anxious to work out this result not for its own sake, but as an illustration of a point which first arises as to octics : a unicursal octic curve is not in general situate on any surface lower than a quartic surface ; if on
...the octic we take at pleasure 33 points, we may through these draw two quartic surfaces U=0, V= 0, each of which will entirely contain the curve : the two surfaces meet in a second octic also unicursal, and the two curves intersect in 34 points, which are points of contact of the two quartic surfaces. We cannot, by adjoining to the given octic any curve of an inferior to its OAvn, obtain a complete intersection of two surfaces : the most simple case is when we adjoin to the given octic as above another unicursal octic, and thus obtain a complete intersection of two quartic surfaces.

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