130 Transactions of the Society. 
Another pair, the same flint combined with a different crown : 
Flint 606 713 607 
Crown 633 705 571 ri = 1*56, 
— 27 +8 +36 
Here by judicious selection we have not only obtained an increase 
in the value of n', but have also secured a more favourable condition 
in the secondary spectrum. 
If, however, we make use of some of the newly made substances, 
we shall obtain a better result still. Take, for example, the 1st and 
the 5th in Table L, we have foci of 50*8 and 65 • 2 giving a dispersive 
ratio of 1 * 28, thus : 
Flint 615 704 571 
Crown 644 703 565 v! =1*28. 
+ 1 + 1 
+ 6 
Here the secondary spectrum has almost disappeared. 
Now, with regard to the importance of n', we saw in my last 
Address that 1 — n' =/', /' being the focus of the negative lens, 
when the focus of the combination was equal to 1 • 0. The radius of the 
contact curve, when the contact curves are identically the same, and the 
negative lens is a plano-concave, is equal to (// — 1)/' or (y! — 1) 
(1 — n'), consequently the value of n' is an index of the flatness of 
the contact curve ; it will necessarily vary also with the value of y!, but 
then the limits of the variation of y! are not large. In general there- 
fore, the greater the value of n' the flatter is the contact curve, and 
we have seen that as a rule the greater the value of n r the greater 
will be the amount of secondary spectrum present. Further, as steep 
curves are synonymous with spherical aberration, we see that aplanatism 
or flat curves are antagonistic to achromatism. Thus it would be 
easier to aplanatise a doublet made of the n' = 1 • 89 pair of glasses, 
because of the flatness of their curves, but here we have very imper- 
fect achromatic conditions; on the other hand, the n' = 1*28 pair 
have favourable achromatic conditions, but then the curves of the 
doublet will be steep, and therefore not so suitable for aplanatism. In 
some instances, when the contact curves are the same, the formula 
for aplanatism gives imaginary roots ; this means that under those con- 
ditions perfect aplanatism is impossible. 
These statements must, however, be taken with certain limitations. 
For example, in a telescope objective, where the ratio of — ^ U S — varies 
L 1 aperture 
from say 10 to 15, n' may be smallj but in lenses employed in the 
Microscope, this ratio sometimes equals 1 * 0, and then n' must be 
large. 
Before passing on, allow me to give one further illustration 
with regard to choosing a pair of glasses for an achromatic doublet. 
