160 
Transactions of the Society. 
The lens has therefore been constructed, and it fulfils the required 
condition that its posterior surface has twice the radius of its anterior 
surface.* 
If it was required that the doublet should be equiconvex, then 
W = 1, and s — ^ a ^ ; the other radii being found as before, thus 
r = 
as 
; and s' = — r ; and r' = s ; and similarly with any 
other ratios of r and s'. 
We will now investigate the other cases which are 
included in our general formula. 
Case 2. — Let the negative lens have a plane ex- 
terior surface (fig. 15). Then s' = co , and W = go . 
Therefore s = h, and r = 
a s 
a + s 
; r' = s; and s' = go . 
Example 4. — We found above that h= - '4040; 
this then is the radius of the contact curve, s ; r can be found by 
putting this value for s in the formula 
•2013 x - '4040 
a s 
, a being '2013 
a + s & 
All the radii are therefore deter- 
r = ------ - = *4012. 
•2013 - '4040 
mined, and it will be observed that the crown is practically an equi- 
convex. F, the focus of the ’combination, = 1 • 0. The lens is there- 
fore constructed. 
Case 3. — Let the positive lens have a plane exterior surface (fig. 16). 
Then r — go and W = 0. Therefore s = — a; and s' = s . ; 
b - s 
r' = s as before. 
Example 5. — We found above that a = • 2013 ; the radius of the 
contact curve s is therefore - '2013; this is also the value of r ’ ; 
7 a c\a c\ u, , -4040 x - '2013 
i= - -4040; therefore s — ^ + , 2Q13 = - -4012; 
F = 10. 
* When Barlow’s Tables are at hand, the following is a simple method of finding 
the radii. Let c be the reciprocal of a, d that of b , and e that of s. 
Then - = c + e, and - = e — d. 
r s 
Thus c = 4-968, d = — 2-475, and e — — 3-306 : 
and - = 1-662; r=-6017: - = - *831 ; s' =- 1-203. 
r s' 
The reciprocal method has not been inserted in the text, because it is not so 
rapid for computing by means of a slide rule. 
It may be as well to state here that Prof. Fuller’s spiral slide rule, which reads 
to 4 significant figures, is very suitable for optical computations. If logarithms are 
employed, a capital 5-place table of numbers by Sang is published by Blackwood, 
price 6d. Chambers’ new ‘ Mathematical Tables,’ price 4s. 6d., which contains a 
7-place logarithmic table of numbers, and natural and logarithmic canons of trigono- 
metrical functions to 1' of arc, is an excellent work. 
