•216 
MR. IVORY ON THE THEORY 
dx . dx i dx (dr , , dr , dr 7 1 ,_ r 
and, in like manner, 
rfv 7 . dy j dz , 
•/ (/ a + j d e -j— fl® = 0, 
a a a e a ot 
, , dz , dz , 
a a + -j- e a e + ^ dm — 0. 
e? a 
Further, we have, 
dm 
d x 7T.y * d x i , f / d X . - d X \ , % . . d X , * | 
T& <A = r . j(jp °°s < + jn) <*N +-JJ- di J ; 
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 d N} 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 (a — N) d. tan i — cos (X — N) tan i dlS} X 
sin N sin i cos i 
\/l + s® ’ 
which expression is equal to zero in consequence of what was shown in § 4. 
Wherefore we have, 
dx dx . ~\ 
rt iN + di d, = 0; 
and similarly, 
d N a + d i 
di — 0 
> 
dz 7 _.. , dz , . 
cH$ dlS +^ 7 ^ = 0 . 
d i 
( 11 ) 
It follows from what has been said that the expressions of dx, dy, dz contain 
dv only, and are independent of the differentials of the five elements, a, e, vs, 
N, i, 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 = £ + $ — O, 
