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.

234 CALCULUS OF VARIATIONS.
su be = 0 for any values whatever of cVx and iy, but only
for such values as make yv =; a condition which would
by that process be entirely neglected. If, however, we make
eu + av = O (a being a constant to be determined), on the
supposition that êx and Îy have any values whatever, then,
the values of Si and Îy which make vv = 0, and no other,
will make 3u = 0. And, at the same time, an additional
constant is introduced into the equation between x and y,
which enables us to give to v the value required in the
statement of the problem. Hence, when u is to be made
a maximum or minimum, while the value of v is constant,
we must make 3 (u + av) = o, proceeding in the same manner as in the simpler cases. And, if it were required that
u should be a maximum or minimum, the values of the
functions v and w being constant, the same reasoning would
shew that we must make 3 (u + av + bw) =; and so for
any number of functions.
21. Taking the instance above, the area= y: the
length =,/( + p2);... u + av = J y + a. /(1 + p);.. r y + a,/(l + p2):
and V= Pp +C, or y + a /(1 + p2) = + C;
V(l \ p )
a dx 1 C-y
)c+p) y; dy p /{2( )2
= C'-/{a-2 _ (C- y)2};.- (-_ C')2 + (y- C2=a2,
the equation to a circular arc. If the limits be fixed, the
values of the constant must be determined so as to make
the length of the arc equal to the given length, and to
make it pass through the two given points. If the limits
be not fixed, suppose the first and last ordinates AM,
BN, fig. 4, to be given, their abscissoe being not given;
then, since Syy,= o, y,,= 0, the equation for the limits reduces itself to
( V,-,p,,) (- Pp,), o=0;