THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
837 
By the definitions (29) 
X= 
’e(l —e 2 )’ 
Mi, Y= 
c(l — e 2 )' 
M 2 , Z: 
c(l — e 2 ) -1 * 
Now let 
<t>(a) = 
"c(l — e 2 )' 
COS (2^-J-a), q>‘(a) = 
c(l—e 8 )' 
COS a, B: 
M, 
"c(l—e 2 )' 
Then 
X» -Y*=P*S( - 2 X ) + 2P®g^(2 X ) + (?M 2 x) 
2 XY = The same when x+i 77 is substituted for x 
YZ=-P3 W _ x)+ p^ ( p2_ Wx)+ p^ (x) 
XZ = The same when x~i n i s substituted for x 
i(X 2 +Y 2 -2Z 3 )=i(P i -P 3 ^d-g i )R+2P 3 g 2 <b(0) 
(264) 
(265) 
Hence all the terms of the five X-Y-Z functions belong to one of the three types 
<3>, % or R. 
The equation to the ellipse described by the satellite Diana is 
c( 1—e 2 ) 
= 1 +e cos (6—ttt) 
. (266) 
Hence 
Pt= 1 +fe 2 +3e(l -f-^e 2 ) cos (0—tx)-\- fe 2 cos 2(6— uT)+ie 3 cos 3 (6—zs) 
<|>(a) = R cos (2#-f a) = (1 fi-fe 2 ) cos (2 6-\-a) 
+ fe(l+ie 2 )[coS (3d-f-a — ttt) fi-COS (d-j-a-j-TTr)] 
+|e 2 [cos (4 9 -\- ol —2?rr)+ cos (a-j-2cr)] 
+ g-e 3 [cos (50-f-a — 3ar)+ COS (6— a — 3rrr)] 
1(267) 
and 4 / '(a) = R cos a. 
Now by the theory of elliptic motion, 6 the true longitude may be expressed in 
terms of LR + e and ttt, in a series of ascending powers of e the eccentricity. Hence 
(a), R, and (a) may be expressed as the sum of a number of cosines of angles of 
the form l(flt-\-e)-\-mzs-\-na, and in using these functions we shall require to make a 
either a multiple of x or zero, or to differ from a multiple of X by a constant. 
Therefore the X-Y-Z functions are expressible as the sums of a number of sines or 
cosines of angles of the form l(flt-\-e) -\-mzs-\-nx- 
Now x increases uniformly with the time (being equal to nt -\-a constant) ; hence, if 
MDCCCLXXX. 5 P 
