This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1897 Excerpt: ... Sextic Curve with 9 Nodes. 110. A sextic curve contains 27 constants; and the number of conditions to be satisfied in order that a given point may be a node is = 3. Hence it would at first sight appear that the curve could be found so as to have 9 given nodes; this would be 9x3, = 27 conditions, or the curve would be completely determinate. But observe that through the 9 given points we have a determinate cubic curve U = 0; we have therefore U1 = 0 a sextic curve, and the only sextic curve with the 9 given nodes; that is, there is not in a proper sense any sextic curve with the 9 given nodes. The number of given nodes is thus = 8 at most. 111. The sextic curve with 8 given nodes should contain 27--3.8 = 3 constants. We may through the 8 given points draw the two cubics P--0, Q = 0; and we have then (a, b, c$P, Q) = 0, a bicubic, or improper sextic curve having the 8 nodes, and also a ninth node, viz., the remaining point of intersection of the two cubic curves, or say the remaining point of the ennead. Hence if V = 0 be any particular sextic curve having the 8 given nodes, we have (a, b, c$P, Q)J + (9V =0 a proper sextic curve having the 8 given nodes; and this, as containing the right number (= 3) of constants, will be the general sextic curve having the 8 given nodes. 112. There will be a ninth node if 0 = 0; viz., the curve is then (a, b, c$P, Q)s = 0, a bicubic, or improper sextic curve, having for nodes the 9 points of the ennead. Observe that the ninth node is here a point completely and uniquely determined by means of the given 8 nodes. Moreover the number of constants is =2, so that we have here a general (improper) solution of the question of finding a sextic curve with 9 nodes, 8 of them given. 113. But if is not = 0, then the ninth ...
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