MORENO. 



ON RULED LOCI IN 71-FOLD SPACE. 



123 



we can, theoretically, solve these equations for ^ — 1 of the parameters 

 in terms of the remaining one, so that this case is the same as the previ- 

 ous one. 



The actual elimination may be avoided. Let the parameters be A, 

 fx, . . . . V. Differentiate totally all the equations, 



-7;r- rf A + ^r— C?/X + + -— d V = 



a A. d ^ d V 



^r— dX + -^.r— d/x -\- 

 d A d fj. 



d V 



From these we may eliminate the differentials, 



J, 9 A 9 A 9 A 



■ti : 



= 



This is the equation of an (n — l)-flat. The equation involves h 

 parameters but they are connected hjh — 1 equations. Two consecutive 

 (n — l)-flats of the system intersect in an {n — 2)-flat whose equations 

 are ^ = 0, B= 0. 



Three consecutive (n — l)-flats of the system intersect in the (n — 3)- 



flat, 



^ = 0, ^ = 0, C = 0, 



where is the determinant B, with A replaced by B. The equation of 

 the aS'„_i is found by eliminating the parameters between the equations 

 of tlie (n — 2)-flats and the equations connecting the parameters. The 

 equations of the other spreads are derived in a similar manner. The 

 system of related spreads is of the same character as before. 



2. Muhidl relations of connected loci. 



Let us consider more in detail these connected loci. We will use F,^ 

 to denote a i-flat of the 1-fold infinite system of ^--flats. Two consecu- 



