Thus, for refined purposes, it cannot be assumed with any certainty
that the instantaneous scale-value of the heliometer is known, or that
it is a function of the temperature. Of course, for many purposes,
mean conditions may be adopted and mean scale-values be found which
are applicable with considerable precision to small angles or to
comparatively crude observations of large distances; but the highest
refinement is lost unless means are provided for determining the
scale-value for each observer at each epoch of observation.
In determinations of stellar or solar parallax, comparison stars,
symmetrically situated with respect to the object whose parallax is
sought, should be employed, in which case the instantaneous
scale-value may be regarded as an unknown quantity which can be
derived in the process of the computation of the results. Examples of
this mode of procedure will be found, in the case of stellar parallax
in the _Mem. R.A.S._ vol. xlviii. pp. 1-194, and in the _Annals of the
Cape Observatory_, vol. viii. parts 1 and 2; and in the case of
planetary parallax in the _Mem. R.A.S._ vol. xlvi. pp. 1-171, and in
the _Annals of the Cape Observatory_, vol. vi. In other operations,
such as the triangulation of large groups of stars, it is necessary to
select a pair of standard stars, if possible near the middle of the
group, and to determine the scale-value by measures of this standard
distance at frequent intervals during the night (see _Annals of the
Cape Observatory_, vol. vi. pp. 3-224). In other cases, such as the
measurement of the mutual distances and position angles of the
satellites of Jupiter, for derivation of the elements of the orbits of
the satellites and the mass of Jupiter, reference must also be made to
measures of standard stars whose relative distance and position angle
is accurately determined by independent methods (see _Annals of the
Cape Observatory_, vol. xii. part 2).
[Illustration: FIG. 17.]
Gill introduced a powerful auxiliary to the accuracy of heliometer
measures in the shape of a reversing prism placed in front of the
eye-piece, between the latter and the observer's eye. If measures are
made by placing the image of a star in the centre of the disk of a
planet, the observer may have a tendency to do so systematically in
error from some acquired habit or from natural astigmatism of the eye.
But by rotating the prism 90 deg. the image is presented entirely
reversed to the eye, so that in the mean of measures made in two such
positions personal error is eliminated. Similarly the prism may be
used for the study and elimination of personal errors depending on the
angle made by a double star with the vertical. The best plan of
mounting such a prism has been found to be the following. l^1, l^2
(fig. 17) are the eye lens and field lens respectively of a Merz
positive eye-piece. In this construction the lenses are much closer
together and the diaphragm for the eye is much farther from the lenses
than in Ramsden's eye-piece. The prism p is fitted accurately into
brass slides (care has to be taken in the construction to place the
prism so that an object in the centre of the field will so remain when
the eye-piece is rotated in its adapter). There is a collar, clamped
by the screw at S, which is so adjusted that the eye-piece is in focus
when pushed home, in its adapter, to this collar. The prism and
eye-piece are then rotated together in the adapter.
_The Double Image Micrometer._--Thomas Clausen in 1841 (_Ast. Nach._
No. 414) proposed a form of micrometer consisting of a divided plate
of parallel glass placed within the cone of rays from the object-glass
at right angles to the telescope axis. One-half of this plane remains
fixed, the other half is movable. When the inclination of the movable
half with respect to the axis of the telescope is changed by rotation
about an axis at right angles to the plane of division, two images are
produced. The amount of separation is very small, and depends on the
thickness of the glass, the index of refraction and the focal length
of the telescope. Angelo Secchi (_Comptes rendus_, xli., 1855, p. 906)
gives an account of some experiments with a similar micrometer; and
Ignarjio Porro (_Comptes rendus_, xli. p. 1058) claims the original
invention and construction of such a micrometer in 1842. Clausen,
however, has undoubted priority. Helmholtz in his "Ophthalmometer" has
employed Clausen's principle, but arranges the plates so that both
move symmetrically in opposite directions with respect to the
telescope axis. Should Clausen's micrometer be employed as an
astronomical instrument, it would be well to adopt the improvement of
Helmholtz.
_Double-Image Micrometers with Divided Lenses._--Various micrometers
have been invented besides the heliometer for measuring by double
image. Ramsden's dioptric micrometer consists of a divided lens placed
in the conjugate focus of the innermost lens of the erecting eye-tube
of a terrestrial telescope. The inventor claimed that it would
supersede the heliometer, but it has never done anything for
astronomy. Dollond claims the independent invention and first
construction of a similar instrument (Pearson's _Practical Astronomy_,
ii. 182). Of these and kindred instruments only two types have proved
of practical value. G. B. Amici of Modena (_Mem. Soc. Ital._ xvii.,
1815, pp. 344-359) describes a micrometer in which a negative lens is
introduced between the eye-piece and the object-glass. This lens is
divided and mounted like a heliometer object-glass; the separation of
the lenses produces the required double image, and is measured by a
screw. W. R. Dawes very successfully used this micrometer in
conjunction with a filar micrometer, and found that the precision of
the measures was in this way greatly increased (_Monthly Notices_,
vol. xviii. p. 58, and _Mem. R.A.S._ vol. xxxv. p. 147).
In the improved form[19] of Airy's divided eye-glass micrometer (_Mem.
R.A.S._ vol. xv. pp. 199-209) the rays from the object-glass pass
successively through lenses as follows:
+-----------------------------------+---------------+---------------+
| Lens. | Distance from | Focal Length. |
| | next Lens. | |
+-----------------------------------+---------------+---------------+
| a. An equiconvex lens | p | arbitrary = p |
| b. " " | 2 | 5 |
| c. Plano-convex, convex towards b | 1(3/4) | 1 |
| d. Plano-convex, convex towards c | " | 1 |
+-----------------------------------+---------------+---------------+
The lens b is divided, and one of the segments is moved by a
micrometer screw. The magnifying power is varied by changing the lens
a for another in which p has a different value. The magnifying power
of the eye-piece is that of a single lens of focus = 4/5p.
In 1850 J. B. Valz pointed out that the other optical conditions could
be equally satisfied if the divided lens were made concave instead of
convex, with the advantage of giving a larger field of view (_Monthly
Notices_, vol. x. p. 160).
The last improvement on this instrument is mentioned in the _Report_
of the R.A.S. council, February 1865. It consists in the introduction
by Simms of a fifth lens, but no satisfactory description has ever
appeared. There is only one practical published investigation of
Airy's micrometer that is worthy of mention, viz. that of F. Kaiser
(_Annalen der Sternwarte in Leiden_, iii. 111-274). The reader is
referred to that paper for an exhaustive history and discussion of the
instrument.[20] It is somewhat surprising that, after Kaiser's
investigations, observers should continue, as many have done, to
discuss their observations with this instrument as if the screw-value
were constant for all angles.
Steinheil (_Journal savant de Munich_, Feb. 28, 1843) describes a
"heliometre-oculaire" which he made for the great Pulkowa refractor,
the result of consultations between himself and the elder Struve. It
is essentially the same in principle as Amici's micrometer, except
that the divided lens is an achromatic positive instead of a negative
lens. Struve (_Description de l'Observatoire Central de Pulkowa_, pp.
196, 197) adds a few remarks to Steinheil's description, in which he
states that the images have not all desirable precision--a fault
perhaps inevitable in all micrometers with divided lenses, and which
is probably in this case aggravated by the fact that the rays falling
upon the divided lens have considerable convergence. He, however,
successfully employed the instrument in measuring double stars, so
close as 1" or 2", and using a power of 300 diameters, with results
that agreed satisfactorily amongst themselves and with those obtained
with the filar micrometer. If Struve had employed a properly
proportioned double circular diaphragm, fixed symmetrically with the
axis of the telescope in front of the divided lens and turning with
the micrometer, it is probable that his report on the instrument would
have been still more favourable. This particular instrument has
historical interest, having led Struve to some of those criticisms of
the Pulkowa heliometer which ultimately bore such valuable fruit (see
_ante_).
Ramsden (_Phil. Trans._ vol. xix. p. 419) suggested the division of
the small speculum of a Cassegrain telescope and the production of
double image by micrometric rotation of the semispecula in the plane
passing through their axis. Brewster (_Ency. Brit._ 8th ed. vol. xiv.
p. 749) proposed a plan on a like principle, by dividing the plane
mirror of a Newtonian telescope. Again, in an ocular heliometer by
Steinheil double image is similarly produced by a divided prism of
total reflection placed in parallel rays. But practically these last
three methods are failures. In the last the field is full of false
light, and it is not possible to give sufficiently minute and steady
separation to the images; and there are of necessity a collimator, two
prisms of total reflection, and a small telescope through which the
rays must pass; consequently there is great loss of light.
_Micrometers Depending on Double Refraction._--To the Abbe Rochon
(_Jour. de phys._ liii., 1801, pp. 169-198) is due the happy idea of
applying the two images formed by double refraction to the
construction of a micrometer. He fell upon a most ingenious plan of
doubling the amount of double refraction of a prism by using two
prisms of rock-crystal, so cut out of the solid as to give each the
same quantity of double refraction, and yet to double the quantity in
the effect produced. The combination so formed is known as Rochon's
prism. Such a prism he placed between the object-glass and eye-piece
of a telescope. The separation of the images increases as the prism is
approached to the object-glass, and diminishes as it is approached
towards the eye-piece.
D. F. J. Arago (_Comptes rendus_, xxiv., 1847, pp. 400-402) found that
in Rochon's micrometer, when the prism was approached close to the
eye-piece for the measurement of very small angles, the smallest
imperfections in the crystal or its surfaces were inconveniently
magnified. He therefore selected for any particular measurement such a
Rochon prism as when fixed between the eye and the eye-piece (i.e.
where a sunshade is usually placed) would, combined with the normal
eye-piece employed, bring the images about to be measured nearly in
contact. He then altered the magnifying power by sliding the field
lens of the eye-piece (which was fitted with a slipping tube for the
purpose) along the eye-tube, till the images were brought into
contact. By a scale attached to the sliding tube the magnifying power
of the eye-piece was deduced, and this combined with the angle of the
prism employed gave the angle measured. If p" is the refracting angle
of the prism, and n the magnifying power of the eye-piece, then p"/n
will be the distance observed. Arago made many measures of the
diameters of the planets with such a micrometer.
[Illustration: FIG. 18.]
[Illustration: FIG. 19.]
Dollond (_Phil. Trans._, 1821, pp. 101-103) describes a double-image
micrometer of his own invention, in which a sphere of rock-crystal is
substituted for the eye-lens of an ordinary eye-piece. In this
instrument (figs. 18, 19) a is the sphere, placed in half-holes on the
axis bb, so that when its principal axis is parallel to the axis of
the telescope it gives only one image of the object. In a direction
perpendicular to that axis it must be so placed that when it is moved
by rotation of the axis bb the separation of the images shall be
parallel to that motion. The angle of rotation is measured on the
graduated circle C. The angle between the objects measured is = r sin
2[theta], where r is a constant to be determined for each magnifying
power employed,[21] and [theta] the angle through which the sphere has
been turned from zero (i.e. from coincidence of its principal axis
with that of the telescope). The maximum separation is consequently at
45 deg. from zero. The measures can be made on both sides of zero for
eliminating index error. There are considerable difficulties of
construction, but these have been successfully overcome by Dollond;
and in the hands of Dawes (_Mem. R.A.S._ xxxv. p. 144 seq.) such
instruments have done valuable service. They are liable to the
objection that their employment is limited to the measurement of very
small angles, viz. 13" or 14" when the magnifying power is 100, and
varying inversely as the power. Yet the beautiful images which these
micrometers give permit the measurement of very difficult objects as a
check on measures with the parallel-wire micrometer.
On the theory of the heliometer and its use consult Bessel,
_Astronomische Untersuchungen_, vol. i.; Hansen, _Ausfuhrliche Methode
mit dem Fraunhoferschen Heliometer anzustellen_ (Gotha, 1827);
Chauvenet, _Spherical and Practical Astronomy_, vol. ii. (Philadelphia
and London, 1876); Seeliger, _Theorie des Heliometers_ (Leipzig,
1877); Lindsay and Gill, _Dunecht Publications_, vol. ii. (Dunecht,
for private circulation, 1877); Gill, _Mem. R.A.S._ vol. xlvi. pp.
1-172, and references mentioned in the text. (D. Gi.)