114 
SUMMARY OF CURRENT RESEARCHES RELATING TO 
where a is the distance between the lenses, f x and / 2 their focal lengths, 
io 1 and w. 2 the widths between their principal points. 
From these formulae it follows that the resultant focal length of a 
combination of two positive lenses when separated by an interval is 
always greater than the reciprocal of the sum of their reciprocals, and 
increases as the lenses are separated. The resultant width between the 
principal points decreases from the value w x -)- w. 2 dowm to nothing, and 
then increases negatively, becoming infinite when a = /l -f- / 2 , and 
again becoming positive and finite when a exceeds that value. 
If the optical combination is not achromatic, then the positions both 
of principal planes and of principal foci will be, in general, different for 
lights of different colours. 
There exists yet another pair of points and planes, having special 
properties that should be noted. These are the points situated on the 
axis beyond the principal foci at distances respectively equal to the 
true focal length on either side. They are marked S t and S 2 in fig. 13, 
being conjugate and sym- 
metrically situated. Toepler, 
who first called attention to 
these two points, called them 
by the not very apt name 
of negative principal points 
(negative Hauptpunkte). I 
call them the two symmetric 
points, and the planes through 
them the symmetric planes. They possess the very useful properties that 
any object in one has an image of equal size, inverted, in the other, and 
that any ray which crosses one of them at any distance from the axis 
will, after traversing the lens, cross the other symmetric plane at an 
exactly equal distance from the axis, but on the other side of the latter. 
I have now cleared the way for discussing the methods that have 
been hitherto used or suggested for measuring the focal properties of 
lenses. Unfortunately, most of the ordinary methods of focometry 
which are in accepted use are based on the assumption that the lens 
may be treated as of negligible thickness, and in some others one has 
to make assumptions beforehand as to the probable width between the 
principal planes. This is easy enough in the case of a single lens, but 
for compound lenses the principal planes come in most unexpected 
positions, at an unknown width apart. After I have enumerated the 
methods of focometry, and briefly described each, I will describe a new 
method, and an instrument for carrying it out. 
Methods of Focometry. 
The methods of focometry may he classified under six general heads ; 
and under these may be grouped the several varieties adopted by different 
authorities. 
I. Methods of Direct Focal Search. — (a) Objective Methods. — The 
classic simple method of ascertaining the principal focus of a lens or 
optical combination, applicable only to positive lenses ; consisting in 
sending rays from any very distant object through the lens, and search- 
Fig. 13. 
D 
Hr"""- 
Fr S/ 

s, if 
H 1 
Q. 
