THEOREM. - If two planes are perspective, then, ?f the one plane be turned about the axis through, any angle, especially if the one plane be turned till it coincides with the other, the t?ro planes will remain, perspective ; corresponding lines w-ill still meet 021 a line called the axis., and the lines joining corresponding points will still pass throu.gh a common centre S situated in the plane.

Whilst the one plane is turned this point S ?rill 2120re in a circle whose centre lies in, the plane 7r, which is 1.-ept fixed, and whose plane is perpendicular to the axis.

The last part will be proved presently. As the plane .71-' may be turned about the axis in one or the opposite sense, there will be two perspective positions possible when the planes coincide.

Let (fig. 2) 7i, r' be the planes intersecting in the axis s whilst S is the centre of projection. To project a point A in 7r we join A to S and see where this line cuts 77.". This gives the point A'. But if we draw through S any line. parallel to 7r, then this line will cut 7r' in some point l', and if all lines through S be drawn which are parallel to 7r these will form a plane parallel to Ir which will cut the plane Tr' ill a line i' parallel to the axis s. If we say that a line parallel to a plane cuts the latter at an infinite distance, we may say that all points at an infinite distance in 7r are projected into points Which lie in a straight line i', and conversely all points in the line are projected to an infinite distance in 7r, whilst all other points are projected to finite points. We say therefore that all points in the plane 7r at an infinite distance may be considered as lying in a straight line, because their projections lie in a line. Thus we are again led to consider points at infinity in a plane as lying in a line (comp. G. §§ 2-4).

Similarly there is a line j in Ir which is projected to infinity in ir' ; this projection shall be denoted by j' so that i and j' are lines at infinity.

S will lie between them in such a position that I'S =TJ and l'T=Sif, If l'S=SJ, the point S will lie on the axis.

It follows that any one of the four points S, T, J, l' is completely determined by- the other three : if the axis, the centre, and one of the lines i' or j are given the other is determined ; the three lines s, j determine the centre ; the centre and the lines j deter-mine the axis.

We shall now suppose that the two projective planes 7r, 7r' are perspective and have been made to coincide.