W, Harkness—Color Correction of Achromatic Telescopes. 191 
most energetically upon silver bromo-iodide must all be brought 
as nearly as possible to the same focus. This condition will 
evidently be fulfilled when the rays in question have the min- 
imum focal distance; or in other words, when they satisfy 
equation (9). Thus it appears that this equation determines 
the correction of the objective, and for that reason it will be 
called the achromatic equation, and the particular value of 7 
which satisfies it will be designated as 7, 
To find the relative values of A,, A,, A,, in a corrected 
objective, we substitute in (9) the values of D and E from (6). 
The resulting expression for the middle lens is 
pate A, (3, + 2¢,¥,) 422 A, (4, aA 2¢,y,') 
ka @, £22.) (2) 
which shows that this lens must be of the opposite kind from 
the other two,—that is, if the first and third lenses are convex, 
the middle one must be concave; or vice versa. ; 
_ To find the equivalent focal distance of the whole combina- 
tion for the ray 4,, (9) gives 
| ‘ D=— 2Ey,’ (13) 
| Substituting this in (7) 
% Se Ci 14 
Fi — C a Fy: Py ( ) 
Replacing C and E by their values from (6) 
: 1 
= 7 ‘15 
I A(a,— 1%. —1) +A,(a,—- Vo, —1) +A,(a,-¢,7, —1) ( ) 
Substituting the value of A, from (12), and putting 
a= —* 
16) 
i= (4,— 1) (6,+2¢, “ At 1) (b,+2¢,7," +7, (6 ¢,—6,¢,) 
M= (4, = 1) (anna Se WE 1) TEL, +Y¥,. (b,¢, ee 6.£,) 
% _ We obtain finally 
b, + 2¢,7," (17) 
{ ee A,(L + nM) 
The ordinate of the focal curve for the ray A,, is the difference 
tween the focal lengths of the objective for the rays A, and 
4 To find it we have 
a FA (CHD y+ 8y,.)—(04Dy,)+ By, )=DUI-1.)* (P11) (18) 
But ys ] J. = (19) 
—_— —_ = 
1 
ke AOD 
and putting f—~,=J/, this becomes 
1 1 
A. =S545 ~ zt (20) 
1 
