190 W. Harkness—COolor Correction of Achromatic Telescopes. 
combination bef Then, by a well known optical theorem, 
1 
- — (H,—1)A, — (#,—1)A, + (u,—1)A, (5) 
Substituting the values of yw, w,, “4, from equation (2), 
putting 
C= A,(a,— 1) - A,(a,— 1) sa A,(a@,—1) 
Ab, 
D=A6,+A,,+ 
(8) 
E=Ave,+ A,e, + A,e, 
and arranging the terms according to the powers of 7, we have 
1 
This equation expresses the relation between the focal dis- 
tance of the combination, and the wave length of the light. 
It shows that when white light enters an objective there will 
generally be an infinite number of foci, situated one behind 
the other, and all contained between the two values of f which 
correspond to the limiting values of 7. For our purpose, how- 
ever, it will be more convenient to consider / as the ordinate, 
and 7 as the abscissa, of a curve which we will designate as the 
focal curve. To investigate its properties, we differentiate with 
respect to fand y, and obtain 
d 
So = — 2yf"(D + Ey) () 
Putting the left hand member of this expression equal to 
zero, we fin 
D 
Ae 9 
ed (9) 
Differentiating (8) a second time 
a. 
in = 2f*y(2D + 4Ey’)? —f7(2D + 12Ey’) (10) 
Substituting the value of 7* from (9), this becomes 
aft 2 ‘ 
a= or (11) 
which shows that, so long as D remains positive, the curve is 
convex toward the objective, and the value of 7 given by equa- 
(9) corresponds to the minimum focal distance. 
An achromatic objective, or more accurately, and with greater 
Se: a corrected objective, is one in which all rays of the 
nd for which the correction is made are brought to as nearly 
as possible the same focus. For example; if an objective 1s 
corrected for visual purposes, then the rays which produce the 
greatest effect — the human eye must all be brought as 
nearly as po to the same focus; or, if the objective 1s 
corrected. 20t¢ hic purposes, then the rays which act 
