208 
MR. IVORY ON THE THEORY 
and hence, in consequence of what has been shown, 
2- k ' 2 ■ 2 • k " 2 * 2- ^ 
cos 2 1 — ; sm 2 1 = p ; tan 2 * = p. 
Let the momentary plane of the planet’s orbit, that is, the plane passing 
through r and r -f- dr, intersect the immovable plane of xy, and put N for the 
place of the ascending node: then s and s + ds will be the tangents of the 
latitudes at the distances X — N, and X + d X — N from the node : and, i being 
the angle contained between the two planes, we shall have, 
s = tan i sin ( X — N), 
dS . / -».T\ 
= tan i cos (X — N). 
By adding the squares of these equations, 
d s 3 h" 3 
S , + _ =ta „2 i=F ; 
by differentiating, making d X constant. 
^ + •) = 
WdLW - h'dh ' 
ds 
and, by substituting the values of h" d h" and h! d h!, 
dds r 2 ( d R 1 d R ds 1 
d\ 2 ‘ S h 12 ' ^ d s 1 + s 2 ' d X d X J ’ 
Since i is variable in the equations (5), it is obvious that N, or the place of 
the node, must likewise vary. By combining each of the two equations with 
the differential of the other, these results will be obtained, 
„ d . tan i . . _ _ . d N . . 
0 = — sin (X — N) — tan * cos ( X — N) 
dds d.tani , N . . . - T . 
dJ* + 5 = ~inr cos (X - N) + tan i sm (X — N); 
from which we deduce, 
d i = cos 2 i cos (X — N) . 
{& + *j 
l . dX, 
^ cos i sin (x — N) j 
sin i ' J 
[iv + *} 
>.dX; 
