VI. On the forty-eight coordinates of a cubic curve in space

Abstract

In a note published in the Report of the British Association for 1878 (Dublin), and in a fuller paper in the Transactions of the London Mathematical Society, 1879 (vol. x., No. 152), I have given the forms of the eighteen, or the twenty-one (as there explained), coordinates of a conic in space, corresponding, so far as correspondence subsists, with the six coordinates of a straight line in space; and in the same papers I have established the identical relations between these coordinates, whereby the number of independent quantities is reduced to eight, as it should be. In both cases, viz.: the straight line and the conic, the coordinates are to be obtained by eliminating the variables in turn from the two equations representing the line or the conic, and are in fact the coefficients of the equations resulting from the eliminations. In the present paper I have followed the same procedure for the case of a cubic curve in space. Such a curve may, as is well known, be regarded as the intersection of two quadric surfaces having a generating line in common; and the result of the elimination of any one of the variables from two quadric equations satisfying this condition is of the third degree. The number of coefficients so arising is 4 X 10 = 40; but I have found that these forty quantities may very conveniently be replaced by forty-eight others, which are henceforward considered as the coordinates of the cubic curve in space. The relation between the forty and the forty-eight coordinates is as follows: on examining the equations resulting from the eliminations of the variables, it turns out that they can be rationally transformed into expressions such as UP′—U′P = 0, where U and U′ are quadrics, and P and P′ linear functions of the variables remaining after the eliminations. The forty-eight coordinates then consist of the twenty-four coefficients of the four functions of the form U (say the U coordinates) together with the twenty-four coefficients of the functions of the form U′ (say the U′ coordinates), arising from the four eliminations respectively; viz.: 4 X 6 + 4 x 6 = 48. And it will be found that the coefficients of the forms P, P′, are already comprised among those of U, U′; so that they do not add to the previous total of forty-eight.