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.

172 FIGURE OF TIIE EARTH,
f = - cos 0,
u
g = - (sin 0 cos w - s. sin w),
h = - (sin 0 sin w + s.cos).
u
Let F, G, H, be the forces in the directions of f, g, and
h; K and L those in the directions of k and 1; and P,
T, S, the forces parallel to EP, perpendicular to EP in
the plane of the ecliptic, and perpendicular to the ecliptic.
Then
K = G cos w + H sin w,
L = Hcos w - G sin w.
Also P = F cos 0 + K sin 0;
T = F sin - Kcos 0;
S=L.
Substituting the values of K and L,
P = F cos 0 + (G cos w + H sin w) sin 0,
T = F sin 0 - (G cos w + H sin o) cos 0,
S = H cos o - G sin w.
70. Now, by Prop. 22.
(f{ +r g'~') 12h2 -3f2 -3g;
3 (f.2_ g2+ h2)~ 5(f~+ g +h)2 )
G 4,rr q 5 (C) 12h2 - 3f - 3g2
3 4
~ 47r f +(c) Gh - 9/f - (9g2 _.1
G - (f - 7 - (c). g
(3 (2+ g'+ h2) 5(f2 + g2 + h2)'
Upon substituting these values in the expressions for P,
T, and S, the quantity multiplied by '/, (c) in each is rather
complicated, and it is, therefore, proper to take only those