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.

INFLUENCE OF A DISTANT BODY CALCULATED. 197
Then PN= MV- MT= ysin - x cosO;
PQ = PV+ T = ycos + sin0.
Substituting these values, the moment of the force on the
particle at P
-= -- Sm. { (2 - y) sin 0 cos 0 + y (cos2 0 - sin2 0)},
22. We must now find the sums of the expressions
x2ym, y2%m, xySm, for every particle of the spheroid.
Suppose the spheroid divided into slices by planes parallel
to the plane of xy; let two of these be at the distances
z and z + ÎSz respectively; z being measured perpendicular
to the plane of xy, and bz being small; and let the included slice be divided into prisms, by planes parallel to
y; two of these being at the distances o and x +
from the plane of yz; and take a portion of this prism, included between the co-ordinates y and y + $y. The volume
of this portion = Sx. y. z; and if k be the density, the
expression wy3m becomes, for this portion, Sz. x.. keyy.
If p be the sum of xy$m for the prism,
-p = Sz. S. kay,
dy
or p = z..e.-Y;
taking this between the limits
y =- /(a _ 2), and y= + V/(a2- t,2 - 2),
a a
(since the equation to the surface of the spheroid is
x2 + z2 y2
a+ 2- 1),
p = 0.
Hence, the sum of xyÎm for the whole spheroid, is = o.
23. For the sum of a2rm it appears in the same manner, that if w be the sum for the prism,
dw
dwz, ê. kx2. or w =.z.. v.ke2 y;
dy