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.

VARIATION 0F AN INTEGRAL. 221
then, (to the first power of Sx, &c.)
V'= V + M3x + N3y + P3p + QÎq + &c.,
d x' d. Sx
and — = 1 + -—;
dx dx
dx' d.c9x... — V = V.. d M x + NMy + PN p + QSq + &c.
dx dx
Hence, Îu, (integrating the first term by parts)
= V.x - f x. d ) f (Mc) + NSy + Pip + Qcq + &c.).
The integrations are performed with respect to x, considering
d(V)
y, p, q, &c. as functions of x; hence, for — ) we must put
dx1
dV dV dy dV dp dV dq
-+. +. +&c.
d x dy dx dp dx dq dx
= M+ Np+ Pq+ Qr+ &c.;
then, ]u = V.x + f N(Sy - p x)
+ P(3p - q3x) + Q(3q - r)) + &c.}.
9. Now, upon substituting the values found above for
sp, 2q, &c., we find
d.èy d.( x d(cy-p x)
p - qxP = - P - qx =
d.êp d. x
$q - rê, = — - q -- - rSx
dx div
d (2p-q qx) ds (3y -p x)
dx dx2
and so on. Let iy - px= Wc; then Îu =
rV +.(+P +dwo dw+&c
vcx + f (Nc + d x7 + &c.).