[Illustration:  FIG. 5.]

If any error of convexity exists in the plate, the focal plane is disturbed, and the eyepiece must be moved out.  If the plate is concave, it must be moved in to obtain a sharp image.  Irregular errors in the plate or surface will produce a blurred or indistinct image, and, as in the first instance, no amount of focusing will help matters.  These methods are both good, but are not satisfactory in the highest degree, and two or three important factors bar the way to the very best results.  One is that the aberrations of the telescopes must be perfectly corrected, a very difficult matter of itself, and requiring the highest skill of the optician.  Another, the fact that the human eye will accommodate itself to small distances when setting the focus of the observing telescope.  I have frequently made experiments to find out how much this accommodation was in my own case, and found it to amount to as much as 1/40 of an inch.  This is no doubt partly the fault of the telescopes themselves, but unless the eye is rigorously educated in this work, it is apt to accommodate itself to a small amount, and will invariably do so if there is a preconceived notion or bias in the direction of the accommodation.

[Illustration:  FIG. 6.]

Talking with Prof.  C.A.  Young a few months since on this subject, he remarked that he noticed that the eye grew more exact in its demands as it grew older, in regard to the focal point.  A third and very serious objection to the second method is caused by diffraction from the edges of the holes or the slit.  Let me explain this briefly.  When light falls upon a slit, such as we have here, it is turned out of its course; as the slit has two edges, and the light that falls on either side is deflected both right and left, the rays that cross from the right side of the slit toward the left, and from the left side of the slit toward the right, produce interference of the wave lengths, and when perfect interference occurs, dark lines are seen.  You can have a very pretty illustration of this by cutting a fine slit in a card and holding it several inches from the eye, when the dark lines caused by a total extinction of the light by interference may be seen.

[Illustration:  FIG. 7.]

If now you look toward the edge of a gas or lamp flame; you will see a series of colored bands, that bring out the phenomenon of partial interference.  This experiment shows the difficulty in obtaining a perfect focus of the holes or the slit in the diaphragm, as the interference fringes are always more or less annoying.  Notwithstanding these defects of the two systems I have mentioned, in the hands of the practical workman they are productive of very good results, and very many excellent surfaces have been made by their use, and we are not justified in ignoring them, because they are the stepping stones to lead us on to better ones.  In my early work Dr. Draper suggested a