In a similar way by considering the intersections of the incident and emergent portions of rays which pass through F' in the image space we determine the second unit surface situated in the image space. Clearly these two surfaces have rotational symmetry about the axis and are conjugate to one another, that is, the one surface is the image of the other, and any ray striking the first unit surface in the point H will follow a path in the image space passing through H' in the second unit surface where HH' is parallel to the axis. Now let PHH'F'P' and PFKK'P' be two rays meeting in P and P', the former being parallel to the axis in the object space and the latter in the image space. Let these two parallel portions be at equal distances from the axis and on opposite sides of it. The image extending from P' to the axis is of equal height to an object lying between P and the axis, and is inverted. P and P' therefore trace out conjugate surfaces cor responding to the transverse magnification — 1. F and F' are the mid-points of PK and Hi P' and the new surfaces are there fore precisely equal to the corresponding unit surfaces but face opposite ways. Now let phh'F' pi be another ray parallel to the axis in the object space meeting the unit surfaces in lz and h' and the negative unit surfaces in p and p'. From symmetry Ph and pH intersect in a point f situated in the plane through F normal to the axis of the system, and from the congruent triangles h'F'P' , H'F' p', P' and Hip' are parallel. In other words the normal plane through F is conjugate to the surface at infinity in the image space, and similarly the normal plane through F' is conjugate to the infinitely distant surface in the object space.
By taking a pair of rays whose distances from the axis are in any assigned ratio we can construct the conjugate surfaces for this magnification.
It is a simple matter to show that the object space surfaces are all similar and similarly situated about F, and the image space surfaces also similar and similarly situated about F'. Since we have taken the ratio of the distances of corresponding points from the axis as the measure of the magnification, any corre sponding secondary elements of length (that is elements normal to the plane through the axis of symmetry) in the image and object surfaces are in this ratio. Now consider two parallel incident rays inclined to the axis, not intersecting it but situated symmetrically with respect to it, the separation between them being small. They determine on every constant magnification object surface a sec ondary element of unvarying length. In the image space these rays intersect in a point ff in the focal plane through F'. The lengths of the secondary elements intercepted on the constant magnifica tion surfaces in the image space are therefore proportional to the distances of the points of intersection from f'. In other words these surfaces must be similarly situated with respect to any point f' in the focal plane.
It follows that all the constant magnification surfaces are planes normal to the axis, and that the magnification in every such plane is uniform in all directions. All the properties of the system may therefore be related to the points in which these planes meet the axis of symmetry. With the aid of rays passing through F and F' (fig. 4) we readily prove, if U and U' are the N' F' P'