268
DR. ROYSTON-PIGOTT.
achromatism is obtained by the sacrifice of perfect aplanatism.
When the spherical aberration is nearly destroyed, in all the
glasses I have been able to examine during the last ten
years, then the colour is increased or achromatism is destroyed ; in a word, the most perfect definition of close points
is seen with colour; the colour and their definition vanish
together. It is an exceedingly difficult task to demonstrate
this on paper without apparatus in operation.
.One other cause of imperfection in high-cla3s objectives is,
the slight deviation of the centres of the various lenses from
the axial line. A brilliant point of light, imaged upon the
stage in the manner already described, and viewed by an
objective thus damaged by imperfect centering, displays an
immense quantity of intersecting circles, precisely resembling
the back of an engine-turned watch; and the greater the
number of the uncentered lenses the more complicated becomes
the pattern.
{To be continued.)
On the DIFFERENCES between the NOMINAL and SOLAR
FOCAL LENGTH of ENGLISH OBJECT-GLASSES. By Dr.
RoYSTON-PlGOTT, M.A., &C, &C.
OUR English opticians have been somewhat hardly accused
of pretending, for trade purposes, that the focal length of
their object-glasses is misrepresented, and that they are really
of a much deeper character than the name they bear.
Fortunately, with very little calculation, the honesty of
these nominal foci can be readily established. I have shown
(' Phil. Transactions,' p. 594, vol. ii, 1870) that if
d be the distance between an object and its image formed
by an equivalent lens,
m the number of times it is magnified,
/ the focal length of the lens for parallel rays or solar focal •
length; then
/=_*
;
if m be very large, —is so small that it may be neglected, and
then
o_
J
d
~ iw + 2"
But when m is small, ^ must be retained. For those
unacquainted with this kind of notation, it may be stated in
i
FOCAL LENGTH OF ENGLISH OBJECT-GLASSES.
269
plain English thus:—The solar focal length, when a small
lens is used and the image is greatly magnified, by dividing
the distance between the object and image by the number of
times the object is magnified, increased by 2.
Example.—Suppose a screen is 100 inches from an object,
which is found by measurement to be magnified upon the
screen 60 times : what is the focal length (solar) of the lens ?
/ = ~-^
nearly = 100 -^ 62 = 1/6129.
If greater accuracy is required, the divisor must be increased by - , or —.
J
m
60
Thus, '
,_
1 0 0 1 0 0
J—
1
The first approximation is true to three places of decimals.
The converse property is also easily ascertained when the
image is considerably enlarged. Suppose the focal length is
known, and it is required to find the magnifying power at a
distance (d), then
m=
-f-2.
(f Phil. Transact., p. 594, 1870.)
Example; f = 2 inches: what will be the magnifying
power at 180 inches' distance of screen from object ?
-i en
m = ^ - 2 = 9 0 - 2 = 88 nearly.
If it be required more accurately, then the quadratic
equation
At
*
must be solved ; or,
or,
m% + 2m + 1 = ^ r ;
/
- (j - 2 ) m =
- 1;
The focal length of a very small single lens is rather difficult of measurement, as the refractive index as well as the
curvatures or radii of surfaces require to be accurately known.
1
The positive sign must be taken, and the distance between object and
image cannot be less than four times the focal length, and then m has a
minimum value.
270
DR. ROYSTON-PIGOTT.
If, however, a small central aperture be formed by a stop,
and any convenient distance be taken between the object and
image (micrometrie lines are the best to measure), the focal
length can be easily found by the formula above quoted.
Example.—I have a Pritchard doublet, magnifying 180
times at 8 inches : what is the focal length ?
d
_ _ 8
8
&_
J ~ » + » ~ 180 + 2 ~ 182 ~ 91*
Having illustrated the use of this formula in as simple a
manner as practicable, it is now requisite to settle upon what
principle nominal focal lengths are to be ascertained. Evidently the first thing to be done is to agree upon the standard
of reference. No better, I take it, can be adopted than a
uniform distance between the object and magnified image of
ten inches. This image may be viewed by using the microscrope as a miniature-enlarging camera, be received on ground
glass, or it may be viewed as a virtual image formed within
the stop of the eye-piece, the field glass being removed,
either received there upon a glass micrometer or upon thin
oiled paper graduated finely or carefully measured by the
camera lucida.'
Let us take the example of an ordinary one-inch objective
as now made. If the tables be examined printed by different
opticians, it will be found that with a C eye-piece in general
this power is given as 100 diameters. Now, the C eye-piece
magnifies ten times, and if" all the glasses be removed from
it there will be found a certain length of tube for the body
of the microscope, where, within the stop, an image will be
found to be exactly magnified ten times. This point should
be exactly ten inches' distance from the object. In different
glasses this distance will be found to vary slightly, but in
the main it is ten inches, neither more nor less. If now, an
eye-lens of exactly one inch focal length, i. e. magnifying ten
times, be inserted within the empty eye-piece, so that the
stop shall be precisely in focus, then the magnifying power
at this standard distance will be 100.
Now, let us calculate what the focal length (/") of an
equivalent lens would be to magnify an object ten times upon
a screen placed ten inches from the object.
Now, if m be large, — may be neglected; but in the case of
the inch m only equals 10.
- n o
<*
/ really then =
2
10
FOCAL LENGTH OF ENGLISH OBJECT-GLASSES.
271
— _2_
~~ 12-1
= -826446,
so that the maker, in producing an " inch" objective capable
of forming an image one hundred times larger with a C eyepiece at 10 inches' distance, must form it of a solar focal
length of
0-826446,
or nearly -i-th of an inch less than one inch.
But as the powers increase, and the focal length diminishes, the solar focal length more and more nearly approaches the nominal.
Example 2.—" One half inch," magnifying 200 with C eyepiece at 10 inches' distance :
d
*
/ —
10
r —
i
m+2+ _
20 + 2 + j—
— 2_
~ 22-05
= 0453514,
focal length of " half inch" nominal.
If the first approximation be taken, leaving out —
-
d
/i = i T R
10
=
5 /
5
] \
22 = H ^ nearl y = I o o r T J
= 0-545454
Error, y^th of an inch.
Example 3.—" One sixteenth :" nominal power ] 600 : C
eye-piece; m = 160.
_ _d_
10
10
•ft ~ m + 2 ~ 160 + 2 — 102
= -0617284
= 0625000
T'-«th
Difference = -0017716
or, say,. -roVVotns of an inch.
A nearer approximation will be obtained by not neglecting
- or m , when
1
0
1
0
f
272
DR. ROYSTON-PIGOTT.
= -0624975
/, = -0617284
•0007691
Error, T-^i^ths, or -00077.
/ = -0624975
V^th = 0625000
Difference = -0000025
between -r^th the actual focal length and the nominal -Vth,
which amounts to an almost inappreciable quantity of
i oooooooths of ap inch. No English maker could possibly make a TT^h object-glass which magnifies 1600 times
with a C eye-piece at 10 inches more accurately than this
standard.
Example 4.—" One-eighth" nominal.
1 0 1 0
,
= 012193263
th = 0-12500000
Difference = 0-00306737
or ! 030 o ths of an inch.
But, if we take a " two-inch " or " three-inch/' then the
focal length is considerably shorter than the nominal, as the
conjugate focus is more widely separated from the solar for
parallel rays.
Example 5.—" Two-inch:" magnifying power 50 at 10
inches, and 5 without the C eye-piece.
10
d
J=
i =
i
5+2 + —
«»+2 + —
77*
5
_ 12
~ 9-2
= 1-42857.
Which is less than H inch.
Resume.
Nominal objective.
2-inch
.
1
,,
i >,
•
Solar focal length for parallel rays.
1-|- inch, less -j-^-j-ths.
1
„
i
„