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In case F possesses a subfield F ' of order py there exists in S% at least one w-subspace .S£, that is, a finite projective w-space, on which the points contained in a line are p+l in number. The converse is also true. The object of this note is to prove the following theorem. In order to divide an S% into several Sn such that one and only one S^ contains a given pointy it is necessary and sufficient that r is a divisor of m and that m/r is relatively prime to n + 1. We first prove the necessity of the condition. From the above remark, m is evidently divisible by r. By hypothesis every point of S% is contained in one and only one 5£; we infer that ^n,o = ( ^ r ( n + 1 ) l ) / ( ^ r l ) is a divisor of iV£0 = ( £ m ( n + 1 ) l ) / ( ^ ~ l ) . Hence (m/r, n + 1) = 1 is a consequence of the following lemma.

A theorem in finite projective geometry