There are two tests that I can think of that might be known as eccentricity in a profile projector, and since I don't know which one you mean, I'll address both. If you mean something entirely different, please describe it and I'll let you know.
MAGNIFICATION ECCENTRICITY:
The most common method that I have seen used for eccentricity is through the use of either a calibrated gage pin or gage ball. Magnification accuracy is checked in the center of the screen, using the glass scale. The stage of the projecter is then moved to place the ball or pin at the edges of the screen in each of the four quadrants (right, left, top, and bottom, or upper right, lower right, upper left, and lower left - I've seen both methods employed). The magnification error found at the center position is compared to the error at all other positions. Make sure you check both X and Y axes for magnification in each point.
If the error is excessive, but linear, the error can be adjusted out. For example, right is low, middle is nominal, left is high. By readjusting the angle of the internal mirror (it's no fun, trust me) you can remove this error. If the error is excessive and nonlinear, you just bought yourself a three thousand dollar mirror, because your's is warped.
COMPASS ECCENTRICITY:
This one requires very specialized tooling to check. Get a nail (a roofing nail works nicely because of the broad head) and make sure the point is very sharp. Position the nail in the center of the screen, so that the point is exactly in the center. Slowly rotate the compass (screen) and ensure that the point stays in the center. If there is any deviation from center, then find the worst deviation, measure its magnitude using the x-y stage, rotate the compass 180°, and measure the deviation directly opposite. Take the mathematical difference, divide by two, and you have your compass eccentricity.
To adjust compass eccentricity you have to loosen the three or four screws that retain the compass and reposition the screen. On some models the retaining screws merely hold an offset screen retainer, so you would loosen the screws and rotate the retainers to set the proper position. Be careful not to rotate the screen in the frame, as this will alter your zero compass reading.
That's it for now,
Ryan