sin^2 --------- sin^2 ----------
a^2b^2 f[lambda] f[lambda]
I^2 = ------------ . ----------------- . ----------------- (3),
f^2[lambda]^2 [pi]^2a^2[xi]^2 [pi]^2b^2[eta]^2
-------------- -------------
f^2[lambda]^2 f^2[lambda]^2

as representing the distribution of light in the image of a
mathematical point when the aperture is rectangular, as is often the
case in spectroscopes.

The second and third factors of (3) being each of the form sin^2u/u^2,
we have to examine the character of this function. It vanishes when u
= m[pi], m being any whole number other than zero. When u = 0, it
takes the value unity. The maxima occur when

u = tan u, (4),

and then

sin^2u/u^2 = cos^2u (5).

To calculate the roots of (5) we may assume

u = (m + 1/2)[pi] - y = U - y,

where y is a positive quantity which is small when u is large.
Substituting this, we find cot y = U - y, whence

1 / y y- \ y^3 2y^5 17y^7
y = - ( 1 + - + --- + ... ) - --- ---- - -----.
U \ U U^2 / 3 15 315

This equation is to be solved by successive approximation. It will
readily be found that

2 13 146
u = U - y = U - U^-1 - -- U^-3 - -- U^-5 - --- U^-7 - ... (6).
3 15 105

In the first quadrant there is no root after zero, since tan u > u,
and in the second quadrant there is none because the signs of u and
tan u are opposite. The first root after zero is thus in the third
quadrant, corresponding to m = 1. Even in this case the series
converges sufficiently to give the value of the root with considerable
accuracy, while for higher values of m it is all that could be
desired. The actual values of u/[pi] (calculated in another manner by
F. M. Schwerd) are 1.4303, 2.4590, 3.4709, 4.4747, 5.4818, 6.4844, &c.

Since the maxima occur when u = (m + 1/2)[pi] nearly, the successive
values are not very different from

4 4 4
-------, ------, --------, &c.
9[pi]^2 25[pi] 49[pi]^2

The application of these results to (3) shows that the field is
brightest at the centre [xi] = 0, [eta] =