M. A. AINSLEY

So much has been written on the subject of figuring a parabolic speculum, and so well has the matter been explained, that it may seem rather a waste of space for me to try and repeat what has been so often said before; but as I said in my first letter, I am writing for beginners, and I will ask the experts to bear with me if I tell them things they already know much better than I do.

Before beginning the practical figuring of a mirror, it is very necessary to have a clear idea of the effect of various curves upon the image formed by the mirror, and a little trouble taken in mastering the theory will render the practice much racier.

Speaking generally, the Tcurve produced by polishing the fine-ground mirror (and I use the word "curve" as practically signifying the same thing as " surface " in this connection)falls into classes, which I shall call A, B, and C.

In class A, the curvature is greatest at the edge, and decreases regularly to the center of the mirror, where the curve is flattest. This is known as the " oblate spheroid ", Fig. 1.

Class B consists of the sphere, in which, of course the curvature is the same all over, Fig. 2.

Class C contains the ellipse, parabola and hyperbola - in all of which the curvature is greatest at the cen ter and least at the edge. I shall call these Ct, C2, and C3. Fig. 3.

Now let us consider the action of these curves upon a pencil of parallel rays, such as we get from a star.

The effect of classes A and B is to bring the rays falling upon the outside zone of the mirror to a focus of the central rays. This is also the case C1, Fig 4.

C2 brings all the rays to the same focal point. This is what we want: Fig. 5. C2 brings the outside rays to a focus further from the mirror than that of the inside rays, the effect being the exact opposite of that of A, B and Cl, Fig 6.

Now the only way, practically speaking, of obtaining a pencil of parallel rays is to utilize the rays from a star; and if we were confined to testing the mirror in the telescope on a star, a good deal of time would be lost waiting for a suitable occasion; so it is necessary to find some test which can always be applied. Before passing on, however, it is well to say something as to the appearance of the image of a star, in the telescope, as given by the various classes of surface.

In the case of C2 (the parabola) if the imve is fo- cussed as carefully as possible, and the eye piece is then pushed in or pulled out, the image expands into a circular patch of light which is uniformly bright, and presents the same appearance inside and outside the focus, Fig. 7a.

In the case of A,B and C, the image inside toe focus will have a dark center, while outside the ceyter will be brighter than the rest of the circle, Fig. 76./ C3, the hyperbola, gives exactly the opposite effect, the patch of light having a bright center inside the focus, and a dark center outside, Fig. 7c. It will thus be seen that it is possible to judge of the correctness of the curve of a mirror by actual testing a star in a telescope; but, as I said, the speculum worker does not, as a rule, care to wait a fortnight for the chance of getting a view of a star, as is sometimes necessary.

The practical method adopted is to make use of an artificial star, formed by a pinhole in a plate of metal, and placed at a curvature of the mirror. Being at the center of the curvature, the image of the pinhole will coincide with the pinhole itself; so it is necessary to move the pinhole a little to one side in order to view the image. This does not practically affect the results except with mirrors of abnormally short focal length.

The action of the various classes of curve, however, is somewhat different from the former case, where the star was at an infinite distance, and the incident rays of light consequently parallel. In the present case they are divergent from the center of curvature, and the difference between the condition of the two cases must be carefully noted.

Upon the new conditions, class A brings the outer rays to a focus nearer to the mirror than the inner, Fig. 4.

Class B brings all the rays to the same focus, Fig. 5.

Classes C1, C2, C3, all bring the outer rays to a focus further from the mirror than the central rays, Pig. 6.

Again, if the image be examined with an eyepiece, as in a telescope, class A gives a bright center outside and a dark center inside the focus. Fig. 76.

Class B, gives the same appearance inside and outside. Fig. 7a.

And class Ci, C2, C3, gives a dark center outside and a bright inside the focus, as in Fig. 1c.

It will thus be seen that, viewed with the eyepiece, C1, C2, C3, gives the same appearance, differing only in degree, and it thus becomes necessary to have some means of determining with certainty when the parabola C2 is obtained.

If the eye be brought close up to the image of the pinhole so as to receive the whole pencil of rays reflected by the mirror, the whole mirror will be seen illuminated, and if a screen of metal be brought across the pencil of rays in the neighborhood of the image of the pinhole, it will cut off the light and apparently darken the surface of the mirrors seen by the eye. Its action, however, will be different, according as it is between the mirror and the image or beyond it.

Suppose the screen is always moved across from to left; then if it is within the focus, i. e., nearer to the mirror than the image of the pinhole, it will be seen from Fig. 8 that it will darken the right-hand side of the mirror first; if it is exactly at the focus, the mirror being supposed spherical, the mirror will darken evenly all over, while if outside the focus, the shadow will appear to move from left to right, or in a direction opposite to the motion of the screen, Fig. 9. Thus, if the shadow moves the same way as the screen, the screen is known to be inside the focus; if the opposite way, the screen is outside the focus; while, if the screen is exactly at the focus, the mirror will darken uniformly and with very great rapidity as the screen is moved across.

This gives us a very accurate means of placing the screen exactly at the focus of the mirror for rays diverging from the center of curvature; and what is true of the whole mirror is, of course, true of any part of it; so that, if the mirror is divided up into zones, and if all the mirror except the zone under examination is stopped out by means of a screen placed over the mirror, it is possible, by observing the point at which the zone darkens uniformly, to place the screen with very great accuracy at the focal point, for divergent rays, of any given zone. Thus the divergence in focus for different zones can be easily measured.

Before proceeding, however, to the actual measurement of the focal point for the different zones, it is as well to make an examination of the three classes, A,. B, or C, it belongs to. As before said in the case of class A, the outside rays will come to a focus nearer to-the mirror than the inside rays; consequently, if the screen is placed as near as possible to the image of the pinhole, so that the mirror darkens as uniformly as possible as the screen is brought across, the screen will be inside the focus for the central rays, and outside the focus for the marginal rays. Thus the shadow will advance across the mirror from left to right for the center, and from left to right for the margin, the screen being always carried across from right to left. The appearance of the mirror is shown in Fig. 10. In the case of the sphere class B, the darkening is uniform all over; while with class C, since the screen is now inside the focus for marginal rays and outside for central rays, the shadow will advance from left to right for the center, and from right to left for the outside, the appearance being exactly the opposite to that " for class A, Fig 11. C1, C2, and C3 all give the same appearance as regards this test, and it is absolutely necessary to submit the matter to exact measurement, as there is no other reliable way of deciding when C2, the parabola, is exactly attained.

The method then, is to divide the surface of the mirror into zones by means of card screens placed against it, and to determine, by means of a screen brought across the image of a pinhole, what the exact position of the focus of any zone is.

In practice it is only necessary to measure the position of the screen for the central 2 in., and for the outside inch or so. An examination of the mirror as a whole will show whether the curve is regular, or whether there are any rings; though if the polisher is carefully made, as before explained, rings ought not to appear.

In my next chapter I hope to give the formula for determining when the parabola is attained and the practical details of the testing.