132 
Transactions of the Society. 
will have all its radii alike, and a triple of the other kind (fig. 18), 
yiz. two flint menisci enclosing an equi-convex, will have the outer 
curves double the radii of the contact ones. 
We will now pass on to the main point of this paper, which will 
treat of dispersion formulae. A dispersion formula has for its object 
the completion of the dispersion curve when only a portion of it is 
given. If the dispersion curve were a conic section, say a parabola or 
a hyperbola, the problem would be an easy one ; but because the curve 
is of no known mathematical form, it is one of exceptional difficulty. 
Dispersion formulae may be used in two ways, either for interpolation 
or extrapolation. Interpolation means that the refractive index of 
the line sought lies between those given, and extrapolation that it is 
beyond them. For an example of interpolation, let B, C, and H be 
given, find E ; and for extrapolation, B, C, and D being given, 
find H. 
Extrapolation is the true test of the accuracy of a dispersion for- 
mula ; for some dispersion formulae will work well enough when used 
for interpolation, but altogether break down when used for extrapola- 
tion. The problem has been attacked by many eminent mathema- 
ticians, but as yet no satisfactory solution has been found. 
One of the best known dispersion formulae, owing to its being 
quoted in nearly every text-book, is that of Cauchy, but it is so 
inaccurate as to be quite useless for practical purposes ; moreover it 
is very laborious to work out. It is as follows : Let /r be a known 
refractive index, and X its corresponding wave-length, then 
** - “ + j? + X* + X* 
bed 
' il = “ + V + V + \? ; 
bed 
^2 = « + 7-3 + w + n ’* 
V 
V 
V 
Here we have four known refractive indices corresponding to four 
given wave-lengths, consequently we can determine the value of the 
four unknown quantities «, h, c and d. When once a> b , c and d have 
been determined for any particular kind of glass, any other value of 
yu for any given wave-length X x can be easily found, thus 
■ bed 
,„ = „ + _ + - + — . 
The accuracy of any dispersion formula may be tested for any 
particular glass by comparing the interpolated or extrapolated values 
with those actually measured by a spectrometer. This has been 
done, and an error has often been found in Cauchy’s formula in the 
third decimal place. 
