BANDS SEEN IN THE SPECTRUM. 
237 
line being parallel to the axis of y. The luminous line is supposed to be a narrow 
slit, through which light enters in all directions, and which is viewed in focus. Con- 
sequently each element of the line must be regarded as an independent source of 
light. Hence the illumination on the object-glass due to a portion of the line which 
subtends the small angle (3 at the distance of the object-glass varies as (3, and may 
be represented by A(3. Let the former origin O be referred to a new origin O' situated 
in the plane xy, and in the image of the line ; and let fj. q' be the ordinates of O, M 
referred to O', so that q — q’ — ri. In order that the luminous point considered in the 
last article may represent an element of the luminous line considered in the present, 
we must replace by Ad^ or -jd?i ; and in order to get the aggregate illumination 
due to the whole line, we must integrate from a large negative to a large positive 
value of n, the largeness being estimated by comparison with y- Now the angle 
changes by t when q changes by which is therefore the breadth, in the di- 
rection of y, of one of the diffraction bands which would be seen with a luminous 
point. Since I is supposed not to be extremely small, but on the contrary moderately 
large, the whole system of diffraction bands would occupy but a very small portion 
of the field of view in the direction of y, so that we may without sensible error sup- 
pose the limits of ri to be — oo and +00 . We have then 
^^d' ^ 2 xf /s.in ^ \ 2 
dA 
by taking the quantity under the circular function in place of '/] for the independent 
variable. Now it is known that the value of the last integral is tt, as will also pre- 
sently appear, and therefore we have for the intensity 1 at any point. 
2Axl[ 
sm -j- ^sm 
Ttpky 
y 
+2 
sin 
( 12 .) 
which is independent of q’, as of course it ought to be. 
13. Suppose now that instead of a line of homogeneous light we have a line of 
white light, the component parts of which have been separated, whether by refraction 
or by diffraction is immaterial, so that the different colours occupy different angular 
positions in the field of view. Let be the illumination on the object-glass due 
to a length of the line which subtends the small angle (3, and to a portion of the 
spectrum which subtends the small angle at the centre of the object-glass. In the 
axis of X take a new origin O", and let p' be the abscissae of O', M reckoned from 
O", so that p=p In order that (12.) may express the intensity at M due to an 
jg 
elementary portion of the spectrum, we must replace A by l^d'^, or -jd^ ; and in 
order to find the aggregate illumination at M, we must integrate so as to include all 
values of | which are sufficiently near to p' to contribute sensibly to the illumination 
2 I 2 
