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.

198' PRECESSION AND NUTATION.
which, taken between the limits = -c /(a2 -2 - z2), gives
a
w = 2 k S z.. c -2 /C(a- `. _ z').
a
This is ultimately the increment of the sum of x2~m for
the slice, produced by giving to x the increment ix; calling this sum v,
dv c
2dv kz= c.. x.2 2(a2 - z - m2),
dx a
and v = 2kzfx c fxe /(a2- z2 - 2),
z being considered constant in the integration.
The integral is
kszc. { - 2 -2-2 _ 42) + x (a _ w _ $2)
a 42 4
(a2-z)2 x )
t+ (a-_- )sin-'. 1/(2 2)
' 4 yV'(a- 2)}'
The limits of x are the least and greatest values of x in
the slice; that is, the values given by the equation to the
surface, upon making y= O; they are, therefore, f I/(a2 z2);
and
c (a2 - 2)2
v = k.-..'r.
a 4
Now, if u be the sum of xswm for the whole spheroid, v is
ultimately the increment of u, arising from giving to x the
increment Îz; hence,
du c 7- c
k.. (a2 - 2)2 k. - X (a, - 2aZ + );
dz a 4 a 4
7 e7r ( 4 2 2 3 X.*. =. -,, a4, _ - ar 23 a + -);
a 4 3 5
taking this between the limits z = a,
c 4a5 47r
u = k -.r. — = -kac x a 2
a 15 15