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.

140 FIGURE 0F THE EARTH
Let E (fig. 11) be the point, EOoF a canal; take O
and o, two points very near each other; draw ON, on,
perpendicular to the plane of the equator, and OM, om,
perpendicular to the axis; draw 01 perpendicular to no,
and Ok perpendicular to mo: let MO=x, NO=y, EO=s;
mo= +4 x, o = y + y, Eo=s- + s. The accelerating
force in the direction ON=P. ON P.y: the resolved part
of this in the direction of the canal
= P.. cos Oon = P. y. = ultimately P y. y.
And the accelerating force in the direction OM
= (Q- ). (ar2).;
= (Q - -~2 0O= Q -
the resolved part, in the direction of the canal, is found
ok i
by multiplying it by cos Ook, that is, by - or -, or
Co o s
ultimately by d: hence, the whole accelerating force in
ds
that direction
dy / 4 r2 dx
Py Y - + Q --.
Y ds ' T2) ds
and the pressure produced by the action of this force on
the fluid in Oo (see the Note to Art. 25) may be represented by
os. (Py dy s r s
2s. /y djy + 47r2 dx)
ds 4w2 dx
This, if we put p for the pressure, is the decrement of p,
produced on increasing s by 2s: hence, dp = - ultimately
dy 47r1 dx
Y ds Q T2 ds
P y2 4- r2 *v2
Integrating, p = C - --— Q.-.
2 T" ~T