Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

806 UND1UATORY THEORY OE OPTÎIC.
Xab 2- ( _r à + b
or cos - (v t - ab r).
+6 b b Xa'
If c be the radius of the hole, this is to be taken from
r = o to r - c, and its value is
2Xtab 7ir a +b " 2r a+b 2. sn - (vt-b c). sin- c.
a +b ' 4ab X. 4ab
The intensity of illumination is consequently
4Xa_ 2. 2/2r b )
(a++ b)'s * C
(a+b'2 SlntX 4ab '
On referring to (71) it will be seen that this expression is
nearly similar to the expression for the intensity of the
reflected light in Newton's rings, considering the denominator in (71) as constant, and making
c2(a + b)
2ab
and consequently the colours are nearly the same for the
same values of V. To obtain tle colours corresponding to
those of the inner rings, we must have V as small as possiI 1
ble, or - + - must be as small as possible, and therefore b
a b
must be as great as possible. On diminishing b, V increases.
Consequently if we first place the screen at a very great
distance and then bring it nearer to the aperture, the series
of colours at the center will be the same as those found on
tracing Newton's rings outwards: but as we cannot make
- - 0, we cannot have all the orders beginning from
a b
the central black. This agrees with observation. For any
other point of the screen, the intensity can be found only
by the general method of (73)*.
* The reader will find little trouble in applying the same principles to the
demonstration of the following phenomena:
The shadow of a long and narrow parallelogram consists of several bands,
the central band being white, and the others coloured, and separated by darker
bands.
The central point of the shadow of a very small circle is white, and its
brightness is sensibly the same as if the circle did not obstruct the light.