112 W. Harkness—Color Correction of Achromatic Telescopes. 
Clause V involves the assumption that the focal plane must 
be tangent to the focal curve at the point where the latter 
makes it nearest approach to the objective. No reason is 
assigned for this, and I do not believe any exists. 
Clause VI virtually asserts that the focal distance of an 
objective is a function of the power of its ocular. For all 
astronomical instruments carrying filar micrometers, the first 
business of the observer is to place the wires accurately in the 
focus of the objective. This once done, they are not again dis- 
turbed, unless to make some radical change in the instrument. 
A dozen eye-pieces may be used in the course of a single 
evening; but no matter what their power, when they are 
with the wires. But the plane of the wires is fixed; and the 
focal curve, as I have defined it, is also fixed. Consequently, 
the points of intersection of the focal plane with the focal curve 
are fixed, and the universal experience of astronomers demon- 
strates that the positions of the points 7, and y, do not vary 
with the power of the ocular. 
As Clause VII affirms the correctness of my fourth conclu- 
sion, it is only necessary to express my thanks for such an 
indorsement; but I cannot refrain from adding that, since this 
clause rests upon equations condemned by my critic, there may 
be people wicked enough to inquire how these erroneous equa- 
tions finally led to a correct result. 
In this connection it is desirable to state that some months 
ago I investigated the relations existing in achromatic objec- 
tives between aperture, focal length and secondary spectrum. 
As the admissible limit of the latter of these elements is arbi- 
trary, it is not possible to fix absolutely the relations between 
the other two; but I believe the focal distance should rarely 
be less than that given by the formula 
F = (9:04a° + 1296)? — 36 (38) 
in which F is the focal distance, in feet; and a the clear aper- 
ture in inches. For small apertures, the foci given by this 
expression are inconvenienutly short; while for large apertures, 
they considerably exceed those in general use. 
Now consider a system of infinitely thin lenses in contact ; 
and let us inquire how many lenses are needed in the system, 
to bring the greatest possible number of light-rays of different 
degrees of refrangibility to a common focus, with any give? 
law of dispersion. 
For this purpose we revert to equation (5), which may be 
written 
f= (#, = DAA, — A, + (4, — A, + be. (69) 
