162 
Transactions of the Society . 
Doubling these radii, we have r" = * 830 and s" = — • 822. 
Our lens is therefore constructed, and the values of the radii are as 
follows (fig. 17). 
r" = + *830 r = — -822 r' = + *822 
F = 1-0. 
s" = - *822 s = + -822 s' = - -830 
Example 7. — It is required to construct a Steinheil triple of the 
same glass. Let the radii be r" s", r s, r’ s' as in fig. 18. a = * 2065 
and b = — *4112 as above. 
Then taking the posterior half of the triple we have, by Case 3 
(fig. 16), 
s — — a = — *2065 ; and s' = 7 ^ ; b s = *8491 ; 
b — s 
l - s = - -2047; s' = --ty-L = - -415. 
Doubling these radii we have s = — *413 and s' = — *830. 
The lens is therefore constructed, and the radii * are (fig. 18) 
r" = + *830 r = + *413 r' = - -413 
F = 1-0. 
s" = + -413 l = - -413 s' = - *830 
It is now necessary to point out that in all the above examples no 
account has been taken of the thickness of the lenses ; therefore the 
foci of these achromatic combinations will be only approximately 
correct ; therefore the quantity F, which in these cases is shorter than 
the true focus, may be termed the Nominal Focus of the combination. 
We will next investigate the method of ascertaining the True Focus 
of a combination ; but first rules for determining the thickness of a lens 
must be given. 
There are several methods by which the thickness of a lens may 
be determined. The first and most usual is to draw the lens out to 
scale and measure it ; the second is easily performed by the help of 
Barlow’s tables in which are given the squares and square roots of all 
numbers from 1 to 10,000. Let t be the thickness, d the diameter of 
/ dF 
a plano-convex lens, and r its radius, then t = r — . 
If the lens is biconvex it will be necessary to treat it as consisting 
* The radii r” in Example 6 and s' in Example 7 might easily be found with the 
assistance of a reciprocal table, as in the previous examples. Thus in Example 6 we 
have c = 4 = 4*843; e = 4- = — 2*432; then 4 - = c + e = 2*411 ; t" — *415, as 
a s r 
before. 
For the Steinheil in Example 7 we have d = ^ = — 2*432; e = - = — 4*843; 
b s 
then 4 = e — d = — 2*411; s' = — *115. 
s 
