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.

178 FIGURE OF THE EARTH,
75. PiRor. 32. The Earth being supposed a spheroid, to
express the length of a small arc of the meridian at any point,
in terms of the difference of latitude of its extremities.
Let APQB, (fig. 19), be the meridian passing through the
extremities P and Q of a small arc of latitude; at P and Q,
draw normals meeting in R; then, R is very nearly the center
of curvature of p, which bisects PQ. And, if D be the difference of latitude of P and Q, PRQ = D;.. PQ=D.pR.
Now pR (Whewell's Doctrine, of Limits, p. 63)
CK2 A C2. BC2 a c2
pF pF' pFd '
putting a and c for the semi-axes.
c2 a2. p L3
But pF;. pR=.
And pLN = latitude of p =;.. LN L cos L;.. CN=- pL cosl; and Np=pLsin l.
c
CN2 Np2
Substituting these in the equation - C + -- 1, we get
3 C'
pL-9 cosiI +~cîsinol '
pL'= C(dS~I 3- e~silC
and pR= a=:
(a cos1 + c sin1)'
ac2
hence PQ = D). i)
(a, cos2 l -r c' sin' 1)If the ellipticity be small, let a = c (1 + e);
D.c'(_ + e).-. PQ = l (1 + 2)col+ il = Dc( + 2e - 3ecos2),
co (1 + 2e) cos + es2 in l nl
or =D. c(l - e +-3e sin'l), nearly.