THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
729 
Hence in the expression (30) for cr, we must put 
£=M/, rj = M/, £=M 3 ' 
Then by analogy with (29), let 
and we have 
W=~ 
X'= 
1 
V(l—e' 2 )" 
M/, Y'= 
V(l-e' 2 )- 
Mf, Z'= 
V(I — e' 2 )" 
Ms 7 
g (1—e 2 ) 3 (l—e' 2 ) 3 _ 
V-/2_V'2 ¥ 3 _9F>2 
2X'Y' ^P + 2-~- ^-^ + 2Y / Z / |32; + 2X'Z' ^ 
-hi 
2 2 
X' 2 + Y /2 -2Z /2 + |f-255 3 
(31) 
This is the required expression for the disturbing function on the moon, due to 
Diana’s tides. 
So far the investigation is general, but we now have to develop this function so as 
to make it applicable to the several problems to be considered; 
II. 
SECULAR CHANGES IN THE INCLINATION OF THE ORBIT OF A SATELLITE. 
§ 5. The perturbed satellite moves in a circular orbit inclined to a fixed plane .— 
Subdivision of the problem. 
In this case e=0, e'=0, r—c, r'=c, so that the functions X, Y, Z and X', Y', Z' 
are simply the direction cosines of Diana and the moon, referred to the axes A, B, C 
fixed in the earth. Hence X=M x , Y=M 3 , Z = M 3 , and the five formulas (24-8) give 
the functions X 3 —Y 3 , 2XY, 2YZ, 2ZX, ^—Z 3 . In order to form the functions in 
gothic letters we must express these functions as simple time-harmonics. 
The formulas (24) to (28) are equivalent to the expression of the five functions as a 
series of terms of the type A cos (ay+/30-byY+S). Now y is the angle between a 
point fixed on the equator and the autumnal equinox, and therefore (neglecting 
alterations in the diurnal rotation and the precessional motion) increases uniformly with 
the time, being equal to nt- j-a constant, which constant may be treated as zero by a 
proper choice of axes A, B, C. 
0 is the true longitude measured from the autumnal equinox, and is equal to 
— ijj, since the orbit is circular; also x/j may for the present be put equal to zero, 
without any loss of generality. 
Then if in forming the expressions for the state of tidal distortion of the earth 
we neglect the motion of the node, the five functions are expressed as a series of simple 
time-harmonics of the type A cos (unt-\-(3flt-\-C). 
