We give the construction of a partial geometry with parameters s = 4, t = 17, σ = 2. We also obtain two new strongly regular graphs.

A (finite) partial geometry with parameters s, t and α is a 1 - (v, s + 1, t + 1) design (for which we speak of lines rather than blocks), satisfying the following two conditions.

(i) Any two distinct lines have at most one point in common;

(ii) for any non-incident point-line pair (x, L) the number of lines containing x and intersecting L equals α.

A partial geometry is called proper if 1 < α < min{s, t} (this means that the geometry is not equivalent to a combinatorial object for which another name is more common). Partial geometries were introduced by Bose [2], At that time no example of a proper one was known. In the meantime some construction methods for proper partial geometries have been found, see [15], [13], [10], [3], [5], [14], [7]. Only one of the known ones has α = 2, viz. the sporadic one of van Lint and Schrijver [10]. Here we construct a second proper partial geometry with α = 2, which is (up till now) sporadic too.

The point graph of a partial geometry is the graph whose vertices are the points, two vertices being adjacent whenever the two corresponding points lie on one line. We need to quote some results. The first one is well-known (see [2]) and easily verified.