Hence if a pair of lenses match up accurately for two chosen colors like those represented by the C and F lines, they will fail of mutual compensation elsewhere. Figure 60 shows the situation. Here the spectra from crown and flint glasses are brought to exactly the same extent between the C and F lines, which serve as landmarks. Clearly if two prisms or lenses are thus adjusted to the same refractions for C and F, the light passing through the combination will still be slightly colored in virtue of the differences elsewhere in the spectrum. These residual color differences produce what is known as the “secondary spectrum.” What this does in the case of an achromatic lens may be clearly seen from the figure; C and F having exactly the same refractions in the flint and crown, come to the same focus. For D, the yellow line of sodium, the flint lens refracts a shade the less, hence is not quite powerful enough to balance the crown, which therefore brings D to a focus a little shorter than C and F. On the other hand for A′ and G′, the flint refracts a bit more than the crown, overbalances it and brings these red and violet rays to a focus a little longer than the joint C and F focus. [Illustration: FIG. 60.—Irrationality of Dispersion.] The difference for D is quite small, roughly about 1/2000 of the focal length, while the red runs long by nearly three times that amount, the violet by about four. Towards the H line the difference increases rapidly and in large telescopes the actual range of focus for the various colors amounts to several inches. This difficulty cannot be avoided by any choice among ordinary pairs of glasses, which are nearly alike in the matter of secondary spectrum. In the latter part of the last century determined efforts were made to produce glasses that would give more nearly an equal run of dispersion, at first by English experimenters, and then with final success by Schott and Abbé at Jena. Both crown and flint had to be quite abnormal in composition, especially the latter, and the pair were of very nearly the same refractive index and with small difference in the quantity ν which we have seen determines the general amount of curvature. Moreover it proved to be extremely hard to get the crown quite homogeneous and it is listed by Schott with the reservation that it is not free from bubbles and striæ. Nevertheless the new glasses reduce the secondary spectrum greatly, to about ¼ of its ordinary value, in the average. It is difficult to get rid of the spherical aberration, however, from the sharp curves required and the small difference between the glasses, and it seems to be impracticable on this account to go to greater aperture than about _F_/20.