ρr = 1 + (1/(ν-ν′)(ν/n - ν′/n′), and ρt = 3 + 1/(ν-ν′)(ν/n - ν′/n′)
ρ_r and ρ_t being the respective reciprocals of the radii. The surfaces are really somewhat egg shaped rather than spherical as one departs from the axis.
[13] The doublet costs about one and a half times, and the triplet more than twice the price of an ordinary achromatic of the same aperture.
[14] A very useful treatment of the aberrations of parabolic mirrors by Poor is in Ap. J. 7, 114. In this is given a table of the maximum dimension of a star disc off the axis in reflectors of various apertures. This table condenses to the closely approximate formula
a = lld/f²
where a is the aberrational diameter of the star disc, in seconds of arc, d the distance from the axis in minutes of arc, f the denominator of the F ratio (F/8 &c.) and 11, a constant. Obviously the separating power of a telescope (see Chap. X) being substantially 4.″56/D where D is the diameter of objective or mirror in inches, the separating power will be impaired when a > 4.″56/D. In the photographic case the critical quantity is not 4.″56/D, but the maximum image diameter tolerable for the purpose in hand.
[15] Instruments with a polar axis were used by Scheiner as early as 1627; by Roemer about three quarters of a century later, and previously had been employed, using sights rather than telescopes, by the Chinese; but these were far from being equatorials in the modern sense.
[16] Contributions from the Solar Obs. #23, Hale, which should be seen for details.
[17] A more precise method, depending on an actual measurement of the angle subtended by the diameter of the eyepiece diaphragm as seen through the eye end of the ocular and its comparison with the same angular diameter reckoned from the objective, is given by Schaeberle. M. N. 43, 297.
[18] The angular field a is defined by