From equation (1) f/x = x′/f′, we obtain by subtracting unity from both sides (f − x)/x = (x′ − f′)/f′, and consequently

f − x= − x= − f = γ.
f′ − x′ f′x′

(4)

From equations (1), (3) and (4), it is seen that a simple relation exists between the lateral magnification, the axial magnification and the convergence, viz. αγ = β.

Fig. 5.

In addition to the four cardinal points F, H, F′, H′, J. B. Listing, “Beiträge aus physiologischen Optik,” Göttinger Studien (1845) introduced the so-called “nodal points” (Knotenpunkte) of the system, which are the two conjugate points from which the object and image appear under the same angle. In fig. 5 let K be the nodal point from which the object y appears under the same angle as the image y′ from the other nodal point K′. Then OO1/KO = O′O′1/K′O′, or OO1/(KF + FO) = O′O′1/(K′F′+ F′O′), or OO1/(FO − FK) = O′O′1/(F′O′− F′K′). Calling the focal distances FK and F′K′, X and X′, we have y/(x − X) = y′/(x′− X′), and since y′/y = β, it follows that 1/(x − X) = β/(x′− X′). Replace x′ and X′ by the values given in equation (2), and we obtain

1= β / ( ff′− ff′) or 1 = −β xX.
x − X xX ff′

Since β = f/x = x′/f′, we have f′ = −X, f = −X′.

These equations show that to determine the nodal points, it is only necessary to measure the focal distance of the second principal focus from the first principal focus, and vice versa. In the special case when the initial and final medium is the same, as for example, a lens in air, we have f = −f′, and the nodal points coincide with the principal points of the system; we then speak of the “nodal point property of the principal points,” meaning that the object and corresponding image subtend the same angle at the principal points.

Equations Relating to the Principal Points.—It is sometimes desirable to determine the distances of an object and its image, not from the focal points, but from the principal points. Let A (see fig. 3) be the principal point distance of the object and A′ that of the image, we then have