The object of this paper is to prove the following:
Theorem. Every perfect septenary quadratic form assumes its minimum value at a set of 7 points with integer co-ordinates whose determinant is 1.
This is true also, as shown by Rankin , with n ≤ 6 in place of 7. The proof will be shortened considerably by using the weaker result obtained in  for n = 7, and we shall also use the following classical results, see, e.g., , for Hermite's constant γn: