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.

312 UNDULATORY THEORY OF OPTICS.
e4
= -. The value is also considerable when 0 = 0, or = 600,
3
or = 1200, or = 1800, or = 240~, or = 3000, when it is
X2e2b/ 2 Xb. 2rre\ X b 4f 27re e
l- 1+i + 1 - cos
3Sr r' 7rre ^b J 67r4 r4 - b
This points out exactly the star-like form observed by
Sir J. lierschel (EncycL. iletrop. Light, Art. 772).
83. Let the aperture be a great number m of equal
parallelograms of length 2f and breadth e at equal distances g.
Here y' = -f, y" = +f: and the expression to be integrated is
cos (v -B + - -cos (vt -B+ x +
2rqf.2 p
= 2 sin -. sin -(vt - B + - v).
Àô X ' b'
The general integral is
Xhb. 27-qf r p
- - sin —.cos - (v t - B + o ~).
rp Xb X b
If k be the value of x corresponding to the first side of the
first parallelograin, that corresponding to the first side of
the (n + 1)th parallelogram will be k + n (e + g), and that corresponding to its last side k + n (e + g) + e. The integral
therefore for the (n + 1)th parallelogram is
Xb. 27-,rqf {f (tB+ pk pn (e+g). sin. cos vt-B+ +
7r)p hb X b b
p k pn (e + g) _g)}
- cos - vt -B+-/ p +(e
X b b b
2\b. 27rqf. rpe. 2r pk pe p(e+g)
=.sin. sin.sin - vt-B+- b + — n.
~- x6 \b À b 2b b