This is accomplished by shortening the radii of curvature of all the inner zones of the surface, leaving the outermost zone unchanged. There are two distinct methods of accomplishing this. First, by the use of full-sized polishing tools, the rosin surfaces of which are cut away in such a manner as to give a large excess of polishing surface near the central parts of the tool. Second, by the use of small polishing and figuring tools, worked chiefly upon the central parts of the mirror, and less and less upon the zones towards the edge."

The details of second method have already been given in the extract from Dr. Draper; but Prof. Ritchie further notes, that the squares around the edges of the small tools should be trimmed semicircular, as before described, in order to soften the action of the edge."

"Parabolizing with Full-size Tools. - The rosin surface can be trimmed in a variety of ways to give a great excess of action on the central parts of the mirror. The figure (scale 1 in. to 1 ft.) shows one of the best forms of tool for this purpose, six leaf. The small crosses indicate the centers from which the sides of the leaves are struck. The leaves are rosin, coated with wax. The grooves are made as shown, alternate leaves being alike, thus balancing the tool surface. The form of the edges of the rosin-covered areas can be altered, and thus the amount of action on any zone can be in some measure controlled. Length of stroke and amount of side throw are also very important factors in controlling the figure of the mirror. Tools of this kind serve admirably in parabolizing mirrors up to 36 or 40 in. in diameter, when an angular aperture is not very great."

"Small local tools, of the six-leaf form, are also excellent for polishing out zones."

"The use of an eye-piece in this test (one of Prof. Ritchey's spiral tests) is important, because it shows how fatal to good definition is even very slight convexity or concavity of a plane mirror, when used in oblique positions."

"The reflecting telescope defines well only at or near the optical axis; hence the mirrors must remain in perfect adjustment with reference to each other and to the eyepiece."

"Testing a Paraboloid on a Star. - With this method the mirror surface, as seen with the knife edge test, presents the same general appearance as in testing in conjunction with a large plane mirror. In testing on a star it is seldom indeed that the atmospheric conditions are sufficiently fine to allow any except the larger errors of surface to be seen."

And so on. In fact, I think that the two memoirs taken together, are most complete.