C. 8. Hastings—Color Correction of Double Objectives. 173 
easier, even if indirect, method; to a still greater degree if we 
wish to consider the far more important marginal rays. 
There are several evident methods of finding indirectly the 
constants of the objective, but the most practicable one has 
seemed to me the following: 
We have as data a, § and 7 in the formula 
n'=atfpnt+yn?; 
also F, the required focal length of the objective. Then set 
1 1 ; 
PF = (Ns6r4— 1) A+ ( 5p14— 1) B 
0 
A=—B (B+ 2y seu). 
These last two equations yield the values of A and B. In the 
general equation (1’) substitute these values of A, B, a, 8, 7 
with some known value of n other than ngj4, e. g., I have gen- 
erally used the value ng. Its solution will give the value of 
the focal length for light of the refrangibility C. Let this value 
of P be substituted in turn in equation (1’) and solve with re- 
Spect ton. We shall thus obtain two values of n, one of which 
1S mg and the other a value greater than 754 which we may call 
n., and if we substitute this in the equation connecting n and 
n’ we shall be able to derive the corresponding value n’,. This 
18 all that the consideration of infinitely thin lenses can do for 
us, but from it we have found that the rays of refrangibility 
denoted by © and ¢ should have a common focus, and we have 
also the values 
Uy 
VL, nt ¢ 
r 
Tse14 1 seis 
nN. n, 
Having given, the thickness of the lenses and the distance 
between them, and having*assumed radii proper to correct 
spherical aberration, we calculate the courses of three marginal 
Tays of refrangibilities defined above, and of one central ray of 
refrangibility 5614. By successive trials and modifications of 
three of the radii we can meet the three conditions, that the 
Marginal and central rays 5614 unite in a point on the axis, 
that this point is ata distance F from the second principal point 
of the system, and that the marginal rays C and c unite 
another point on the axis. The remaining arbitrary constants, 
viz: the thicknesses, distance and fourth radius, may be utilized 
for the purpose of satisfying other conditions; but the consid- 
eration of such conditions is at present foreign to our purpose. 
