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. 231
- (i + p 2)2 - (i + p2)2
also Q = -, Q2-;
%q, q,.
and making equal to nothing the coefficients of i(o, and
o)., we find
f (i ~ =}, {(1 + --;
i +,, i 2)
that is, at the extremities, the radii of curvature are each = o;
therefore these points are cusps; therefore the curve is a
complete cycloid.
18. In all the examples above, we have supposed, as
is commonly the case, that V does not involve the limits
of the integral. But it may happen, that V will involve
the values of x, y, p, &c. at the limits. In that case we
must recur to the investigation in (8); instead of giving
V' the value which it has there, we must put
dV dV
V' = - + î +-y,+ &c.
dx, ' dy,
+ M3x + N~y + P3p + &c.
By going through the same operation as is there performed,
we find 2u
(y p ) d (QI)
+ (YP A P),,- dx + &c.}
- (y- pîv) p - d(Q)
dx - &C
+ &c.
d, (P) d2 (Q)
+J (d-7- dy, - y + &c.).