192 W. Harkness—Color Correction of Achromatic Telescopes. 
Substituting for the quantity within the brackets, its value 
from (18); and replacing D and E by their equivalents from (6) 
Bak paws we (A,d, - A,d, + A,O,)(7, -1,') + (A,¢, +A i a A,¢,)(7,"-7,') (21) 
Substituting the values of A, and A,+A, from (12) and (16), 
and putting 
P =b.c,—b.c, (22) 
we obtain the important expression 
. a2 N+nP 
Af, = MSL 1 5 oes (23) 
If a star is viewed through a carefully focused achromatic 
telescope, and if the surface in the focus of the eye-piece is 
designated as the focal plane: then, of the infinite number of 
images which equation (7) shows will be formed, some will be 
situated before, and some behind the focal plane, but only one 
will coincide exactly with it. The cones of rays which form 
the images situated before and behind the focal plane will 
necessarily have a sensible diameter at their intersection with 
that plane, and their combined effect will be to produce a 
ringe of colored light around the image of the star, as seen 
through the eye-piece. This fringe is the secondary spectrum, 
and its magnitude, for light of any given wave length, wil 
evidently depend upon the value of df. Hence, to destroy 
the secondary spectrum, 4f, must be made equal to zero. 
Equation (23) shows that this will be the case for a triple 
objective when 
N+n”P=—0 (24) 
or for a double objective when 
N=6 (25) 
As yet no materials have been discovered whose physical 
properties are such as to satisfy these conditions. We there- 
fore proceed to investigate what form an objective constructed 
of any given materials must have in order to render the sec- 
ondary spectrum a minimum. ; 
Substituting in (23) the value of A, from (17), we find 
Any 2 2 (N - nP) 6 
AF =F". —¥;) (L + nM) (2 ) 
In the right hand member of this equation, n is the only 
quantity which depends upon the form of the objective. Con- 
_ sidering it as variable, and differentiating, we obtain 
d(4p) ice a toy 
MP) — f(y, =) (i +nM* (27) 
To make 4f a minimum, such a yalue must be attributed to 
n as will reduce the right hand member of (27) to zero. This 
