THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
789 
Therefore from the third 
a(a+/3)~=(/3 2 -}-ab) {(a 3 -|-ab)(/? 3 -}-ab)—ab(a+/3) 3 }z—S(/3 2 +ab)-l-2a(a-l-/3) 
dt 
and by (190) 
a(a+/3)—(/3 _ -j-ab) 7/2 -\-k^k^z— S(/3 '+ab) + 2a(a-fi/3) 
dt 2 
Similarly 
a(a-j-/3)^? —(/3'+ab) ^ -\-K x ~K^y —U(/3' + ab)-{-Ta(a+/3) 
dt 3 
b(a+/3) = (a=+ab) S +*jV>>-T( a ! +ab) + Ub( a + /3) 
(ft 2 
<& 3 
d\ 
dt 2 
b(a + /j) . —(a- + all) A +Ki"K 2 3 C— -( a ' -f-ab)-h Sb(a+/3) 
(197) 
Differentiate the first of (196) twice, using the first of (197), and wo have 
d ‘‘ = _(« 2 +ab) —(/3'+ab) ~ - K * K *z + (p+ & h+%) S-2a(«+/S) 
dt 3 
dt- 
Therefore by (190) 
z=^/3 3 +ab+|^ S—2a(a-f/3) 
Then writing (S) as a type of S, 2, U, T,— 
(S) is of the type (z)(a)(a')t-\-(a)(y)(z)-\-(a)(z) + ^ K)*+(y) ® 
Hence every term of (S) contains some small term, either (a 7 ) or (y). 
Therefore on the right-hand side of the above equation we may substitute for ( 2 ) 
the first approximation, viz.: (iq) + (%) given in (182-3). 
When this substitution is carried out, let (S x ), (S 2 ) be the parts of (S) which contain 
all terms of the speeds k x and k 2 respectively. 
Then by (191) and (193) the right-hand side in the above equation may be written 
(#CiH-#c B )(ic 3 +ot)Si — y(/Ci +a)(fc 2 H-a)(ff 1 -fi/c 3 )-'r^/fi 3 +^ S x 
+ the same with 2 and 1 interchanged. 
d i d 2 
Now let D 4 stand for the operation — -b(/c 1 3 -+-/<’ 3 3 )^+K 1 3 «:o 3 , and we have 
5 1 
+ (/< i 2 + / c 2 2 ) ^ 2 + K i K z 
MDCCCLXXX. 
