THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
to develop the expression as far as the squares of i and^, we may at once drop out all 
those terms in these expressions, in which k occurs raised to a higher power than the 
second. This at once relieves us of the sidereal and fast semi-diurnal terms, the fast 
diurnal and the true fortnightly term. We are, however, left with one part of 
Z /3 )(jj—ZF), which is independent of the moon’s longitude and of the earths 
rotation ; this part represents the permanent increase of ellipticity of the eai'th, due 
to Diana’s attraction, and to that part of the tidal action which depends on the 
longitude of the nodes, in which the tides are assumed to have their equilibrium value. 
I shall refer to it as the permanent tide. 
Then as before, it will be convenient to consider the constituent parts of the dis¬ 
turbing function separately, and to indicate the several parts of W by suffixes as in 
§ 5 and elsewhere; as above explained, we need only consider W I( W l5 W 2 , and W 0 . 
Semi-diurnal term. 
From (37) we have 
W, /-=|[F lCT W £ '- 9 >- 3f - + FX®'*e- 2(fl, - #)+2f ‘] 
To the indices of these exponentials we must add ±2(y— x '), and for 6 write e—\p, 
and for &, e — xjj'. 
Then by (128) 
~V> 
Hence 
-¥ 2 -h' 2 -¥'~-h''~) cos r 2 (x-x') + -(e'-e)-2(f-T//)-2f 1 ] 
-ij cos[2( x - X 0 + 2( e '-e)-2(f-^) + (W-^)-2f 1 ] 
— i'f cos[2( x — x / ) + 2(e' — e) — 2(i// — i//) — (N'— i//) — 2fJ} . . 
(129) 
Slow diurnal term. 
From (38) we have 
W L / TT = W Ve 2,(3 '- ,5) ^ + srW Ve~ 2 < 6 ' 
To the indices of the exponentials we must add i( X — X ) ; sF, cv' ’’ may be obviously 
put equal to unity, and by (128) 
KK 
' = i[H ' + _p 
] 
