very excellent plan for testing a flat surface, which I briefly describe.  It is a well known truth that, if an artificial star is placed in the exact center of curvature of a truly spherical mirror, and an eyepiece be used to examine the image close beside the source of light, the star will be sharply defined, and will bear very high magnification.  If the eyepiece is now drawn toward the observer, the star disk begins to expand; and if the mirror be a truly spherical one, the expanded disk will be equally illuminated, except the outer edge, which usually shows two or more light and dark rings, due to diffraction, as already explained.

[Illustration:  FIG. 8.]

Now if we push the eyepiece toward the mirror the same distance on the opposite side of the true focal plane, precisely the same appearance will be noted in the expanded star disk.  If we now place our plane surface any where in the path of the rays from the great mirror, we should have identically the same phenomena repeated.  Of course it is presumed, and is necessary, that the plane mirror shall be much less in area than the spherical mirror, else the beam of light from the artificial star will be shut off, yet I may here say that any one part of a truly spherical mirror will act just as well as the whole surface, there being of course a loss of light according to the area of the mirror shut off.

This principle is illustrated in Fig. 3, where a is the spherical mirror, b the source of light, c the eyepiece as used when the plane is not interposed, d the plane introduced into the path at an angle of 45 deg. to the central beam, and e the position of eyepiece when used the with the plane.  When the plane is not in the way, the converging beam goes back to the eyepiece, c.  When the plane, d, is introduced, the beam is turned at a right angle, and if it is a perfect surface, not only does the focal plane remain exactly of the same length, but the expanded star disks, are similar on either side of the focal plane.

[Illustration:  FIG. 9.]

I might go on to elaborate this method, to show how it may be made still more exact, but as it will come under the discussion of spherical surfaces, I will leave it for the present.  Unfortunately for this process, it demands a large truly spherical surface, which is just as difficult of attainment as any form of regular surface.  We come now to an instrument that does not depend upon optical means for detecting errors of surface, namely, the spherometer, which as the name would indicate means sphere measure, but it is about as well adapted for plane as it is for spherical work, and Prof.  Harkness has been, using one for some time past in determining the errors of the plane mirrors used in the transit of Venus photographic instruments.  At the meeting of the American Association of Science in Philadelphia, there was quite a discussion as to the relative merits of the spherometer test and another form which I shall presently mention, Prof.  Harkness claiming that he could, by the use of the spherometer, detect errors bordering closely on one five-hundred-thousandth of an inch.  Some physicists express doubt on this, but Prof.  Harkness has no doubt worked with very sensitive instruments, and over very small areas at one time.