Complete quadrics and collineations in S n

In this paper we shall set out the generalization, for n-dimensional space S n , of some recent results about complete quadrics and complete collineations in S 2 , S 3 and S 4 . For the results about complete conies in S 2 , originally introduced by Study [1], we refer the reader to papers by Severi ([2], [3]), van der Waerden [4], Semple [5]; for those about complete quadrics in S 3 , to Semple ([6], [7]); for the extension to S 4 to Alguneid [8]; for the general concept of complete collineations in S n , and for results in S 2 and S 3 , to Semple [9].