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.

DEVELOPMENT OF R. 101
It is plain that as ', increases from o till cos 2\/= - a,
'/ increases from 0 to g0~; and as xJ/ increases to 900,
\]/ increases to 180~.
By differentiating the equation tan j,' (cos2 \ + a)= sin 2 /,
we get
cos 2 xr + a d\'
--- - d - - 2 ta sin 2'. sim 2\ = 2 cos 2 g,;
cos~" / ' aI.
cos 2xJ. +, a.
or acos21. q = o (cos 2 / Cos 4' + sin l / sin 4').
cosj,'.d\..
On substituting the values of cos ^' and sin xk', this becomes
d\ 1 + a cos 2 q\/
V/(1 + 2a cos 2l + a) 2acos2 a2)
= 2 / f( - > 2) = 2 s2(1 - a 2 sin2",'),
X 1+2acos2x +a2'
~1 1 d/,'
whence d
</(1 + 2a cos 2\ + a2) 2 /(1 - a sin"j/') cd
hence C(u)= -1 (fro to
4 2a /,(l1-a2sin ') ' '
_= 2aJ3' Ç ~(î - - a4k2 ) (fror!= O to -'k'= 7r)
2 a J / (1 -asin v/) (fiom '=o to f-)
_= / i - in' - 2 from = 0 to '= —.
ii IV -a sin'r') \ 2
1i P
Let -
1 ______ __ P ____o t
et (1- - na'sm^//) si / (1 + a)2 -4asina 1' }
- 4. a, - 2/(1 - a 2)
then P= 1 + a': and ( )2= a', or a -
(1+ ')2 +v/(and thus C (O)= 1+ a
4 a JV V/(1 +2a'cos 2 /+ a2
from \k'= 0 to /'=; where a' is smaller than a. If we
proceed, forming k" from ', "' froum NV, &c. a" from