THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
727 
the radius vector, longitude and latitude. In the present problem we have to deter¬ 
mine the perturbation of a satellite under the influence of the tides raised by itself 
and by another satellite. Where the tides are raised by the satellite itself, the 
elements of that satellite’s orbit of course enter in the disturbing function in expressing 
the state of tidal distortion of the planet, but they also enter as expressing the position 
of the satellite. It is clear that, in effecting the differentiations above referred to, we 
must only regard the elements of the orbit as entering in the disturbing function in 
the latter sense. Hence it follows that even although there may be only one satellite, 
yet in the evaluation of the disturbing function we must suppose that there are two 
satellites, viz.: one a tide-raising satellite and another a disturbed satellite. 
In this place, where the planet is called the earth, the tide-raising satellite may he 
conveniently called Diana, and the satellite whose motion is disturbed may be called 
the moon. After the formation of the differential equations Diana may be made 
identical with the moon or with the sun at will, or the analysis may be made appli¬ 
cable to a planet with any number of satellites. 
As above stated, unaccented symbols will be taken to apply to Diana, and accented 
symbols to the moon. 
The first step, then, is to find the tidal distortion due to Diana. 
Let M be the projection of Diana on the celestial sphere concentric with the earth, 
and P the projection of any point in the earth. 
Let p£, prj, p 'C, be the rectangular coordinates of P and rM ls rM a , rM 3 the rectangular 
coordinates of Diana referred to axes A, B, C fixed in the earth. 
Then since p, r are radii vectores, £, rj, C, and M x , Mo, M 3 are direction-cosines. 
The tide-generating potential Y (of the second degree of harmonics, which will be 
alone considered) at P is given by 
V=f^V(cos=PM-|) 
according to the usual theory. 
Now 
cos PM = ^M x + 77 M 0 + £M 3 
and 
fc2_ T1 2 iyr 2_IV/r 2 
cos 2 PM-J= 2 ^M 1 M,+ 2 m^ -Ly-^+2,£M,M 3 +2^M 1 M 3 
3 f + Y-2£ 3 Mp + Mv —2M 3 2 
+ 2 3 3 
Also by previous definition, r=fp,w/c 3 ; so that 
3 pm t 
2 r 3 (1— e 2 ) 3 
c(l — e 2 )" 
Now let 
X= 
'e(l —e 2 )' 
M v Y= 
"c(l — e 2 )" 
Mo, Z= 
c(l — e 2 )" 
M 8 . . . . (29) 
5 A 2 
