Finite linear spaces with two consecutive line degrees

Let L be a non-trivial finite linear space in which every line has n or n+1 points. We describe L completely subject to the following restrictions on n and on the number of points p: p≤n2+n-1 and n≥3; n2+n+2≤p≤n2+2 n-1 and n≥3; p=n2+2 n and n≥4; p=n2+2 n+2 and n≥3; p=n2+2 n+3 and n≥4. © 1983 D. Reidel Publishing Company.