The adjustment of the mirror to reflect the imag through the tube is a long job at first. A large sheet of white paper held behind the testing apparatus to receive the image of the hole is a great help.

I assume now that the polish has became perfect all, over the glass. This is a question of time, but if the polisher is properly made, should not take more than four hours, and that the curve is either A or B - i. e., still on the safe side of the sphere. We may now bring the mirror screen into action. First of all, place the testing screen (by which, to save confusion, I shall denote the diaphragm in the eyepiece mount placed close to the eye, as described in my last) as near as possible to the image of the pinhole, and observe whether the shadows seen on occulting the image are regular. If, owing to the faulty construction of the polisher, any rings or irregularities appear, it may be necessary to return to the latter stages of the fine grinding; but such irregularities are easily perceptible in the earlier stages of polishing, and can be avoided by care in making the polisher and in adjusting the position of the central square, as before described. I have not been troubled with " rings " in my own work, and I believe they can always be completely avoided with care.

Now put in front of the screen No. 1, or if one screen only is to be used, No. 3, and gently and slowly alter the position of the tube in its Vs until the testing screen, when brought across, darkens the central part of the mirror uniformly. By keeping up an intermittent pressure with the hand on the left side of the baseboard of the testing apparatus, the mirror may be made to darken and brighten alternately, and the tube is moved very gently with the right hand until the appearance of a shadow advancing across the mirror, or rather the central portion left by the mirror screen, is lost, and the darkening is even all over. Read the scale, estimating to 1/4 mm. or 1-100 in. This should be repeated many times, the testing-screen being placed, to start with, alternately outside and inside the focus. The mean of all the reading is then taken. Now concentrate attention on the outer zone. As will be seen in Figs. 2 and 3, only two segments of this zone are visible, one on each side, and to get the best results, the testing screen, at the point where it occults the image, must be vertical - i. e., the image must be level with the center of the aperture whose edge is being used for testing. A vertical bar across the end as the tube does as well as the eyepiece diaphragm; but I found the latter more convenient. The scale is read again when the two segments are found to darken simultaneously; and, from the greater convergence of the rays from the two sides of the mirror, it is rather easier to observe the exact focus than when the central 2 1/2 in. of the mirror is observed; in any case, however, the mean of several readings should be taken.

The difference between the final means gives the actual difference between the focus of the mirror (for rays diverging from the center of curvature, of course) for the center edge of the mirror. If the focal length for the center is large, - i. e., if the first reading was greater - the mirror belongs to class A; if the focus is the same for both, the mirror is spherical, class B; and if the reading for the outer zone is greater than for the center, class C.

It is desirable, after reading off the scale for the outer zone, to make another set of observations of the the center, to make certain that the testing apparatus has not shifted. It should be heavy enough to remain still. The use of the combined mirror screen, 3, I found a great advantage, as I could remain perfectly still during the whole time, and thus risk of shaking mirror or testing apparatus was avoided.

We now know exactly what sort of curve we have got, and we now want a formula which will give the exact difference between the readings ("aberration" is the name commonly used) for the parabola, C2.

The formula giving the necessary aberration for the D2 parabola is D2/8F where D is the mean diameter of the zone, and F the focal length. It may also be written r2/R, where r = means radius of zone, and R = radius of curvature of the mirror, which is twice the focal length ( = 2 F.)

N. B. - The distance from pinhole to screen is equal to the radius of curvature. Either formula gives the same result, but I prefer the former. It will be noticed that the aberration is twice the depth of the center of the mirror below the line joining opposite middle points of the zone - e. g., if the zone used on a 9 in. mirror of 6 ft. focus is 1 in. wide, its mean diameter is 8 in., and its depth, measured from the plane of the 8 in. circle, is (8)2 / 16 X 72 = 1.18 in., while the aberration to be aimed at in the figuring would be (82) / 8 X 72 = 1.9 in. = .11 in., or 3f millimeters. It will be seen that this is but a small quantity; but it must be measured accurately, as on it depends the whole performance of the mirror.

It must be noticed, however, that this is the theoretical value. As I was told by an expert, "Here mathematics must take a back seat," and he further told me that the practical value is somewhat less. I did not find it so; but, in any case, it is safest to carry on the figuring process, (to be described later) until the aberration is about | of this, or, in the case under consideration, about .08 in. The final testing must, after all, be done in a telescope, on a star, and with a good eyepiece. I would again quote Sir J. Herschel: "That is a good form which gives a good image." If the aberration becomes too great at any time, the curve becomes C3 (the hyperbola, and it is very difficult to get back. A, B and C all mean safety, but C3 means danger. When my mirrors appeared to give perfect results in the telescope, I found, in testing with pinhole and screen that they gave the theoretical aberration. "English Mechanic."