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.

*3'74 UNDULATORY THEORY OF OPTICS.
164. Now the form of the brushes interrupting the rings
will be discovered by making the multiplier of sin2 o.
This gives
sin 2 = 0, or sin (20 + x)= O.
Consequently
tan (20 + 2,3) = tan 2/3, or = tan (2/ - 2a).
Now refer P to the point C bisecting AB, by rectangular
co-ordinates, x being measured in the direction CA and y
perpendicular to it; let CA = b. Then tan PAF - Y
- b
tan PBF = y: whence tan (20 + 293) = tan 2 PQA
x~+b
= tan (PBF + PAF) because PQ bisects the angle at P
2 xy
v2 b2- y2
Hence the brushes are determined by these equations
=b, tan 23, or (a2 -b - y2) tan 2/ - 2y = o;
2u _ b2 - y2
2xy
- 2 =tan (2 3 - 2a),
x2-_ b2 _ y2
or (-2 - b - y2) tan (2/ -2a) -2xy=0.
These are evidently equations to hyperbolas, of which C is
the center. As in both of them y = o when x = b, the
hyperbolas defined by both equations pass through A and
B. The position of the asymptotes will be determined by
supposing V and y very great compared with b: this gives
in the first equation
2cot, tan o -cot
t + 2cot2y - 1=0, or - +tan or -coti,
and similarly in the second
-= tan ( - a) or -cot ( - a).
o?