"M. Chrétien reviewed the derivation of the aplanatic curves mathematically and gave an example of the calculation of a large mirror and its secondary. His equations are identical with those of Schwartzschild, with a slight change in the case of the secondary mirror to take care of the Cassegrain type. The plan of telescope proposed by Schwartzschild is generally considered the best possible, although it is inconvenient in arrangement and has not yet been built. In his best telescope he proposes the use of a secondary mirror with a diameter one half of that of the primary, which would intercept 25 percent of the incident light.

"Consider now Figure 4, which is adapted from Chrétien's drawing to show the theory of aplanatic mirrors: If we let: M be the distance from the vertex of the secondary mirror to the focus, F; E be the distance between the mirrors; x be the distance of point P measured from F along the axis F-V-N; x' be the distance of point P' measured from F along the axis F-V-N; y be the distance of point P measured vertically above the axis F-V-N; y' be the distance of the point P' measured vertically above the axis F-V-N; L be the diagonal distance from focus F to point P' on the secondary; R be the diagonal distance from the point P' to P on the primary; u' be the angle the ray from point P' makes with the axis at F; and, finally, let the origin of x and y be at F, the focus-then the first few terms the series of Schwartzschild's equations may be written as follows:

"For mirrors under 20 inches in diameter any further terms would not affect the movement of the knife-edge and pinhole moving together more than one or two thousandths of an inch, a distance much to small for even an expert to measure with certainty, unless conditions of temperature are rigidly controlled. At the focus the involved error would be only one half of this amount.

"The mathematically-minded amateur will at once recognize that the first terms of these equations are those of the parabola, that the added terms are quantities which deform it according to a given set of conditions. These conditions are embodied in the equations below, covering the fundamental dictum of Hamilton and the sine condition.

"If we should trace rays of oblique light making an angle of three degrees or four degrees with the parallel rays, we would find the same law holding almost exactly; remembering, of course, that the field of the aplanatic telescope is still spherical. Professor Ritchey expects to eliminate the aberration of the curved field by the use of photographic plates spherically curved to conform to the image field.

"The curves cannot be considered as having any known form, and have no name. Because of the appearance assumed by them when y is greatly increased, it has occurred to one of the authors to call them 'bows.' The appearance of the bows was clearly shown in an illustration in SCIENTIFIC AMERICAN, July, 1932, page 22.

"Both Schwartzschild and Chrétien made their calculations and built their equations on the basis of taking the final equivalent focal length as unity and all other quantities as multiples of this unit. It is as if the focal length of the telescope were one meter and all dimensions were given as fractional parts of the meter.

"In an ordinary reflecting telescope one can place a plane mirror about half way to the focus and return the light toward the primary, turning it aside to an eyepiece at the side of the tube. The plane mirror would be one half the diameter of the primary, as in the Schwartzschild telescope, and would intercept 25 percent of the incident light from the stars. If the primary were figured to the aplanatic curve described above, and the secondary slightly deformed on the concave side in order to compensate, a superior telescope would result. If such a telescope were figured to f/6, a 12-inch mirror would require a tube only three feet long. It would have the light-grasp of a ten-inch, and the resolving power of a 12-inch telescope, so far as double stars, planetary detail and nebulae were concerned.

"In principle, the calculation of an aplanatic telescope is simple. When the diameter of the primary mirror, the radius of curvature, the distance between the mirrors and the distance from the secondary to the final focal point are known, these quantities are transformed into proportional parts of the equivalent focal length by dividing each in turn by F, and substituted as such in equations (1) and (2). This gives the value of x in terms of y. Then, by assigning different values to y, the value of the sagittae for the different zones can immediately be determined.

"In order to determine the distance the knife-edge and pinhole or slit must move together at the center of curvature we must know the point where the normals at the point x, y (centers of the zones) cut the axis-the point marked N in Figure 4. Th formula for finding this distance is derived from equation (1) and can he written a follows:

"This equation is solved for the different zones evaluated above, also for the case where y is equal to zero. This gives a calculation of the radius of curvature which one might expect would check the radius of the sphere when the mirror is completely polished and before figuring starts. Usually, however, it will not check exactly. In the calculations, some decimals will be dropped, hence the calculated radius will be slightly different from the radius of the polished mirror. This difference is too small to have any influence on the calculations of the zones themselves, but it indicates the need of an adjustment in the estimation of the corrections at the point where the knife-edge and pinhole move together. The differences between this calculated radius of curvature and the various distances FN, found for each zone, will be the corrections to apply-- the distance to move the knife-edge and pinhole from the focus of the center zone.

"In the test the aplanatic mirror will appear similar to a parabolic or hyperbolic mirror, and care must be exercised throughout to maintain a smooth curve. In applying the corrections they must first be multiplied by the focal length of the telescope in inches.

"After the primary is correctly figured the Ritchey test on Cassegrainian telescopes is arranged and the secondary is figured to look perfectly flat under the pinhole and knife-edge.

"In this way an aplanatic telescope can be produced in which the star images two or three degrees from the center of the field are only very small diffraction circles or slightly elongated ellipses, entirely different from the winged, hairy shapes mentioned at the start."

This completes the discussion proper. Next month the zonal radii for three aplanatic telescopes will be given.

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