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. 237
The part depending on the variation of the limits, besides
the usual terms
(rl- P"p" a- (V,-P $/+ p y _ p y
must have the terms which express the variation of - (y+y,, 2),
that is, a(y,y, + y,,y,,).
Making the whole = o,
Cx,,- CC,+ ay,,, + {//(1 +py)
+ ay, 1- (1 +p2)} Y,.
The first two coefficients shew that C= o, and that the
solid is part of a sphere; the third and fourth require
that y,= o, y,=; or that - =o, - =, which agree with
P,, P,
the former; hence, the solid is an entire sphere.
25. Required the curve of quickest descent from one
given point to another given point, the length of the curve
being given.
Here V = - / )..+ l+ ')
V(y)
and V= Pp + C gives ' - + 1 = C /(1 + p)
the differential equation to the curve. The constants must
be determined so as to make the length of the curve equal
to the given quantity, and to make the curve pass through
both points.
26. Given the mass of a solid of revolution, required
its form, that the attraction upon a point in the axis may
be a maximum. Let the solid be divided into slices by
planes perpendicular to the axis of revolution; then, since
the attraction of a circle, whose thickness is h and radius
y, upon a point at the distance x from its center, is ultimately
27rh {1 V( + y1