1881.] On the 48 Co-ordinates of a Cubic Curve in Space. 301 



January 13, 1881. 



THE PRESIDENT in the Chair. 



The Presents received were laid on the table, and thanks ordered for 

 them. 



The Hon. Sir George Jessel, Master of the Rolls, was admitted into 

 the Society. 



The Right Hon. William Ewart Gladstone, whose certificate had 

 been suspended as required by the Statutes, was balloted for and 

 elected a Fellow of the Society. 



The following Papers were read : — 



I. " On the 48 Co-ordinates of a Cubic Curve in Space." By 

 William Spottiswoode, President R.S. Received De- 

 cember 29, 1880. 



(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), 

 co-ordinates of a conic in space, corresponding, so far as correspond- 

 ence subsists, with the six co-ordinates of a straight line in space. 

 And in the same papers I have established the identical relations 

 between these co-ordiuates, whereby the number of independent 

 quantities is reduced to eight, as it should be. In both cases, viz., 

 the straight line and the cubic, the co-ordinates 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 con- 

 dition 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 con- 

 veniently be replaced by forty-eight others, which are henceforward 

 considered as the co-ordinates of the cubic curve in space. The rela- 

 tion between the forty and the forty-eight co-ordinates is as follows : — 



