158 
Transactions of the Society. 
As these two division sums can be performed by slide rule, and the 
combined focus 85*33 found by Mr. Tamblyn- Watts’ calculator,* the 
above is a simple as well as rapid method, as the values can be found 
by inspection without having recourse to any arithmetical process 
whatever. 
Method II, which is the simpler for arithmetical calculation. We 
have the following well known two equations for finding the values of 
/and/', viz. 
111 . 1 . 1 _ 
/ / I vf V J 
Putting F = 1 * 0 and combining, we have 
Let nf = —; and n = its reciprocal $ or — ; then / = 1 — n; 
f = 1 — n' ; and F = 1*0. 
Example 2. — Using the same glasses as before, we have v = 51*2, 
and v — 32*0; then n — = * 625. 
51*2 
The reciprocal of *625 = 1*600 — n! — o.y q 5* 
Then / = 1 - *625 = *375; and f = 1 - 1*600 = - *600; 
F = 1*0, which is the same result as obtained above. 
We now have to find the radii ; but as they will depend on the 
form of lens we wish to construct, it will be better to give a general 
formula that will be suitable for every case. First, however* we must 
agree to the convention employed with regard to the signs of the 
radii. The radii of all surfaces which are convex towards the left 
hand will be considered positive, and of all those which are concave 
towards the left hand negative. Example, the radius of the first 
surface of a biconvex lens will be positive, and of its second surface 
* Journal K.M.S., 1897, p. 1. 
t In 1827 Sir J. Herschel wrote his work on Optics (published inEncy. Metrop., 
art. ‘Light’), and in 1829 Henry Coddington’s celebrated book appeared. These 
two books are still regarded as standard works, and have never been surpassed. 
Both these eminent mathematicians use a convention of signs which makes the focus 
of a converging lens positive and that of a diverging one negative, and the symbol a> 
to denote the dispersive ratio. W. N. Griffen wrote a treatise on Optics (2nd ed., 
1 842) in which he reversed this convention, by making the focus of a converging 
lens negative, &c., and w to denote the dispersive power. (The Gauss method is given 
in this book.) In Parkinson’s ‘ Optics,’ which is a reprint of Griffen’s work omitting 
Gauss’ method, this altered convention is maintained. It surely is a great pity 
to introduce such confusion of signs and symbols without obtaining any commen- 
surate advantage. In this paper the refractive index will be denoted by n, the term 
commonly adopted by English authors ; but owing to the above mentioned confusion 
with regard to the dispersive ratio , n will be used for it. 
X The reciprocals of all numbers from 1 to 10,000 are given in Barlow’s Tables 
(Spon, price 4s. 6d.), a most useful book for opticians. 
