216 
MR. IVORY ON THE THEORY 
d x . d x , 
d 3, “f* ~7~ d 6 -f- 
d a 1 '< * ‘ 
d e 
dx 
d <37 
d VS 
= { 
dr- , . dr , dr 1 
da + Te de + d^ dr * j 
d a 
and, in like manner. 
-T- d a + ^ d e + dvs — Q 
da 1 de 'dm ’ 
dz 
d a 
, d z . t d z 1 ^ 
Cl cl *t“ “j 6 “p i Cl tsj zz 0. 
1 a e ' dvr 
Further, we have, 
dN 
«™ +%di 
and, if the expression on the right side of this formula be computed, it will be 
found equal to 
{sin (v — P) di — cos (v — P) sin i dN} X sin N sin i ; 
and, by substituting the values of sin (v — P) and cos (v — P), the same quan- 
tity may be thus written, 
{sin (X — N) d . tan i — cos (X — N) tan i </N} X 
sin N sin i cos i 
V 1 + S' 
which expression is equal to zero in consequence of what was shown in § 4. 
Wherefore we have, 
dx 7 _ T , d x , . „ 
+ &i dt — 0; 
and similarly, 
(11) 
d z 7 _ T . d z , . 
TH d ^ +il dl = 0 - J 
It follows from what has been said that the expressions of dx, dy , dz contain 
d v only, and are independent of the differentials of the five elements, a, e, vs, 
N, ?, which destroy one another and disappear. And further, if in x, y, z we 
substitute for v, its value in terms of the mean motion and the mean anomaly, 
viz. 
V = £ + £ — <P, 
