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.

EXAMPLES. 229
of the area is ultimately h.h. ( = ( +(p)
-2q q J )
must be a minimum.
Here, V -(1+p
q
and as this involves only p and q,
= Qq + Cp + C', by (11).
(A + p2)2 2 (1 + pl)2
And =-;... Cp C'.
q
e (Cp + C') q
Hence, = 2;
(1 + p2)2
integrating, C + C+ C' tan'p = 4 (v + a).
And (Cp + Cp) q
And (i^T ~2P3;
(1+ - 2)2
- C' - Cp3
integrating, - + C. tan-'p = (y + b).
Elimiinating tan- p,
C'2 - C' + p CC'. p
- + 2 {4C (v + a) - C'(y + b)}.
1 i+p
Let C=4fsin0; C'=4fcosO;
substituting,
cos 2 0 + sin 20. p
— = 2 -- sin 0 (x +a) - cos 0 (y + b).
To transform this into a more simple equation, we will
first change the origin of co-ordinates, preserving their direction; if m' and y' be measured frorn a point whose coordinates are - a and - b,
then, '= x + a, y'= y + b, p'= p