1 and 2 to be accurately convex and 3 accurately concave, of the same radius.  Now it is evident that 3 will exactly fit 1 and 2, and that 1 and 2 will separately fit No. 3, but when 1 and 2 are placed together, they will only touch in the center, and there is no possible way to make three plates coincide when they are alternately tested upon one another than to make perfect planes out of them.  As it is difficult to see the colors well on metal surfaces, a one-colored light is used, such as the sodium flame, which gives to the eye in our test, dark and bright bands instead of colored ones.  When these plates are worked and tested upon one another until they all present the same appearance, one may be reserved for a test plate for future use.  Here is a small test plate made by the celebrated Steinheil, and here two made by myself, and I may be pardoned in saying that I was much gratified to find the coincidence so nearly perfect that the limiting error is much less than 0.00001 of an inch.  My assistant, with but a few months’ experience, has made quite as accurate plates.  It is necessary of course to have a glass plate to test the metal plates, as the upper plate must be transparent.  So far we have been dealing with perfect surfaces.  Let us now see what shall occur in surfaces that are not plane.  Suppose we now have our perfect test plate, and it is laid on a plate that has a compound error, say depressed at center and edge and high between these points.  If this error is regular, the central bands arrange themselves as in Fig. 9.  You may now ask, how are we to know what sort of surface we have?  A ready solution is at hand.  The bands always travel in the direction of the thickest film of air, hence on lowering the eye, if the convex edge of the bands travel in the direction of the arrow, we are absolutely certain that that part of the surface being tested is convex, while if, as in the central part of the bands, the concave edges advance, we know that part is hollow or too low.  Furthermore, any small error will be rigorously detected, with astonishing clearness, and one of the grandest qualities of this test is the absence of “personal equation;” for, given a perfect test plate, it won’t lie, neither will it exaggerate.  I say, won’t lie, but I must guard this by saying that the plates must coincide absolutely in temperature, and the touch of the finger, the heat of the hand, or any disturbance whatever will vitiate the results of this lovely process; but more of that at a future time.  If our surface is plane to within a short distance of the edge, and is there overcorrected, or convex, the test shows it, as in Fig. 10.  If the whole surface is regularly convex, then concentric rings of a breadth determined by the approach to a perfect plane are seen.  If concave, a similar phenomenon is exhibited, except in the case of the convex, the broader rings are near the center, while in the concave they are nearer the edge.  In lowering