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VI. On the Forty-eight Coordinates of a Cubic Curve in Space. 
By William Spottiswoode, P.R.S. 
Received December 29, 1880—Read January 13, 1881. 
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 corres¬ 
pondence 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 4X10 = 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.: 
4X6 + 4X6 = 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. 
The number of identical relations established in the present paper is 34. But it will 
be observed that the equations, UQ'— U'P = 0, are lineodinear in the U coordinates 
