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By J.W.P Hirschfeld

This booklet is the second one variation of the 3rd and final quantity of a treatise on projective areas over a finite box, sometimes called Galois geometries. This quantity completes the trilogy constructed from airplane case (first quantity) and 3 dimensions (second volume).

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Define w = 2g − k − n + 2. 8) in the non-singular case, when k = −1. 27. 28. (i) The character of Wn = Πk Qt is the same as for the base Qt . (ii) When g = k = n, then w = 2. So it is consistent to write Πn = Πn H−1 and include the whole space Πn as the quadric V(0). This becomes relevant when sections of a quadric by a subspace are considered. 29. A quadric Wn = Πn−t−1 Qt of character w has projective index g = n − 12 (t + 3 − w). 30. If Πm ⊂ Qn and Πn−m−1 is the tangent space of Πm , then Πn−m−1 ∩ Qn = Πm Qn−2m−2 , where Qn−2m−2 has the same character as Qn .

Let K be a quadratic set with a sub-generator Π and a point P of K such that P Π is not a sub-generator. Then (i) the union ΠP of the lines of K through P and a point Q of Π is a sub-generator; (ii) ΠP ∩ Π = TP ∩ Π and this subspace is a hyperplane in Π and in ΠP ; (iii) dim ΠP = dim Π; (iv) if Π is a generator of K, so is ΠP . Proof. First, ΠP = P Π ∩ TP . So ΠP is a subspace and ΠP ∩ Π = TP ∩ Π. 80, where K = (ΠP ∩Π)∪{P } in the notation used there. As ΠP cannot contain Π, so ΠP ∩ Π is a hyperplane of Π since TP is a hyperplane of Σ.

Since G(Qn ) acts transitively on Qn , there exists S1 (1) (3) (2) in G(Qn ) for which Wm S1 = Wm meets Wm . Then application of the preceding (3) (2) argument gives an element S2 of G(Qn ) with Wm S2 = Wm . Hence S1 S2 is the (1) (2) required element of G(Qn ) taking Wm to Wm . Since induction was used, the small cases have still to be considered. First assume that m = t = 0. Then the section is a point off the quadric Qn with n odd. 46. So let n ≥ 3. Assume that P1 and P2 are points off Qn and let α1 and α2 be their polar hyperplanes.

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