64, 65.]
§9. Laws of Optical Imagery
81
The Cardinal Points of a Compound Optical System, Composed of Two
Systems with Axes in Same Straight Line
The two component systems are designated in Fig. 38 by A and
B. The focal points and principal points of system A are designated
by P*, p// and a>, a>>, respectively; and the focal points and principal
points of system B are designated by 7r>, 7v and a>, a>>, respectively.
Let d denote the distance of the first principal point of system A from
the second principal point of system B; this interval being taken as
positive when, as in Fig. 38, the point a> lies beyond the point a>>\
The focal lengths of the first system are a'p>=f> and a-p" =/", and
A B
F!g. 38.
those of the second system are a>T>=(p> and a-7t" = Evidently,
the first focal point (w) of system B is the image of the first focal
point (L) of the compound system as formed by system A. Hence, a
ray which in the first medium crosses the axis at t> will, after traversing
system A, cross the axis at %> and emerge finally in the last medium
along a straight line parallel to the axis; because by definition this
must be the case with a ray that crosses the axis originally at the first
focal point of the compound system. Since a-?T"=d — , we have:
a.t, = £r*J)J1......(11a)
d-ip—f"
Similarly, the second focal point (t") of the compound system is the
image of the second focal point (p>>) of system A in system B. Hence
{d-f"), r>> are the principal points of the compound system. If s
is the image of r> in system A, and if r>> is the image of s in system B,
then r>> is the image of r> in the compound system. This is one of the
conditions that these points must satisfy; the other condition being
that object and image in the principal planes must be equal in every
way. Suppose, therefore, that an object of height ß> is set up at the