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. 235
from which
V- P. p, = o, V- Pp,= o.
Since V- Pp = C, these equations are satisfied by making
C= O;
(x- C') + y+ = a2:
that is, when the lengths of the ordinates AM, BN, and
the arc AB are given, the area AIMNB is a maximum if
AB be a circular arc, whose center C is in the line MN.
If AM, BN, each = o, MN is the chord of AB: and, it
appears that the curve which, with a given length, contains between its chord and its arc the greatest area, is
the semi-circle.
22. Given the length of a curve, to find its form,
that its center of gravity may be the lowest possible. Let
the length = b; then, the depth of the center of gravity
= b y /(1 + p): the length = f\/(1 + p2); hence
yo(i ^P2)
V= a </(i + p+) Y,/(1 + )
a +and, making V= Pp + C, ( l ) C,
I/(1 + P'')
whence - =bC.log {y ba + </(y + ba b-C2) + C';
the equation to the catenary. If the extreme points be not
fixed, but move on curves, it will be found that the catenary will cut the curves at right angles.
23. Given the surface of a solid of revolution, to find
its nature, that the solid content may be a maximum. Let
x be measured along the axis of revolution;
the surface = 2r y /(1 +p2); the solidity = 7rfy2;
~ V=yg +ayv/(i+p2); V=pp+Cgives ay _C-y2