The President's Address. By E. M. Nelson. 
159 
negative ; similarly, the radius of the first surface of a biconcave is 
negative, while that of its second surface is positive. A meniscus, 
whether converging or diverging, when it is turned so that its surfaces 
are convex towards the left hand, will have both its radii positive ; but 
when it is turned the opposite way they will both be negative. All 
distances measured from left to right are positive, in the contrary 
direction negative.* 
The type of lens that we shall select for our general formula will 
be a biconvex doublet, having its contact curves of the same radii. 
Case 1 . — Let r, s be the radii of the positive lens, and r\ s’ those 
of the negative lens (fig. 14). Then r’ = s. 
Let /Jj be the refractive index of the glass of which 
the positive lens is composed, and y! that of the negative 
lens. 
Let W be the ratio of the radii of the exterior sur- 
faces r and s’ ; then W = > let / be the focus of the 
positive, and f that of the negative lens. 
Let a = (fju - 1) / and b = {y! — 1) /'. 
Then 
Fig. 14. 
r s' 
(W + l)ab 
s ~ W a - b~ 
as 
= r 
r = 
a +- s 
s = 
bs 
b — s 
or 
= — W r. 
Example 3. — It is required to construct an achromatic doublet 
(F = 1*0), using the same glass as above, which shall have the 
radius of the exterior surface of its negative lens twice that of the 
exterior surface of its positive lens ; thus — s’ = 2 r ; then W = 2. 
Glass No. 23 ; yu = 1*5368 ; /= *375 as above. 
Glass No. 39 ; y! = 1*6734 ; /' = — *600 as above. 
a = (p - 1)/ = *2013 ;& = (/- 1)/' = - *4040 ; 
(2 + 1 )ab ab — — *08133; 3 ab = - *244. 
S ~ 2a- b ' 2a= -4026; 2a - b = -8066. 
s = 
- *244 
" 7 MM~ 
3025 
r . 
r = 
s = 
s = 
a s 
a + s' 
- *06089 
- • 1012 
b s 
b — s' 
•1222 
- *1015 : 
as = — *06089 ; a + s = — * 1012. 
= *6017. 
bs = *1222 ; b — s = - *1015. 
= - 1*204 = - 2 r. F = 1*0. 
9 
* These conventions are the same as those employed by Sir J. Herschel. 
