240 
Proceedings of the Royal Society 
or more divergent than the less refrangible rays, according as they 
were convergent or divergent originally. If the rays pass through 
the prism before they reach the lens, the pencils will be divergent 
at incidence, and the more refrangible will be most divergent at 
emergence. If they then fall on the lens, they will be more con- 
verged than the rest; so that by a proper arrangement all may be 
brought to their foci at approximately the same distance. If the 
violet rays come to their focus first, we must turn the base of the 
prism more towards the light, and vice versa . 
We proceed to the numerical calculation of the proper arrange- 
ment. 
To find the variation of position of the focus of light passed 
through a prism as dependent on the nature of the light. 
Let ix be the index of refraction of the prism, a its angle, <f> x and 
<£ 2 the angles of incidence and emergence, 6 l and 0 2 the angles of 
the ray within the prism with the normals to the first and second 
surfaces, 8 the difference of these angles — then by geometry 
0 X + 0. 2 = a 0 X - 0 2 = 8 
and by the law of refraction, 
sin cfi 1 = /x sin 6 X sin <£ 2 — fx sin 0 2 
. 
<f> ± is constant, being the angle of incidence for all kinds of light, 
but the other angles vary with [x , so that 
d0 x _ sin 6 X d$ 2 _ sin 6 X d<j>. 2 _ sin a 
d[x fx cos 0 X d[x /x cos 6 X dfx cos 6 X cos $ 2 
The last expression gives the dispersion, or breadth of the 
spectrum, and shows that it increases as the base of the prism is 
■ 
turned from the light. 
As the slit is parallel to the edge of the prism, we have only to 
consider the primary foci of the pencils when we wish to render the 
image distinct. 
