It may be assumed that the planes I' and II' are drawn where the images of the planes I and II are formed by rays near the axis by the ordinary **Gaussian** rules; and by an extension of these rules, not, however, corresponding to reality, the Gauss image point 0', with co-ordinates 'o, of the point 0 at some distance from the axis could be constructed.

If all three constants of reproduction be achromatized, then the **Gaussian** image for all distances of objects is the same for the two colours, and the system is said to be in " stable achromatism."

The **Gaussian** theory is only an approximation; monochromatic or spherical aberrations still occur, which will be different for different colours; and should they be compensated for one colour, the image of another colour would prove disturbing.

**Gaussian** logarithms are intended to facilitate the finding of the logarithms of the sum and difference of two numbers whose logarithms are known, the numbers themselves being unknown; and on this account they are frequently called addition and subtraction logarithms. The object of the table is in fact to give log (a =b) by only one entry when log a and log b are given.

The **Gaussian** theory, however, is only true so long as the angles made by all rays with the optical axis (the symmetrical axis of the system) are infinitely small, i.e.

Consequently the **Gaussian** theory only supplies a convenient method of approximating to reality; and no constructor would attempt to realize this unattainable ideal.

If, in the first place, monochromatic aberrations be neglected - in other words, the **Gaussian** theory be accepted - then every reproduction is determined by the positions of the focal planes, and the magnitude of the focal lengths, or if the focal lengths, as ordinarily happens, be equal, by three constants of reproduction.

This ray, named by Abbe a " principal ray " (not to be confused with the " principal rays " of the **Gaussian** theory), passes through the centre of the entrance pupil before the first refraction, and the centre of the exit pupil after the last refraction.