THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
839 
complete products the products of terms of the like argument in the two sets of 
functions. 
Whence it follows that all the part of the disturbing function, which is here 
important, consists of terms of the form 
F(a +be fi- ce 3 + cle s -f-/e 4 '-f &c. ) 3 cos \m(e—e') + n( rx —sr') — f~\ 
or 
F (a 3 + 2 cibe + (2«c-f-//)e 2 + (2cic l + 2 6c)e 3 + (2 off 2bd +c 3 )e 4, -j- &c.) 
cos [_m(e — e — c/)—y] 
Now it is intended to develop the disturbing function rigorously with respect to 
the obliquity of the ecliptic, and as far as the fourth power of the eccentricity. 
The question therefore arises, what terms will it be necessary to retain in developing 
the X-Y-Z functions, so as to obtain the disturbing function correct to eh 
In the X-Y-Z functions (and in their constituent functions <b(a), 'P(a), R) those 
terms in which a is not zero will be said to be of the order zero ; those in which a is 
zero, but b not zero, of the first order ; those in which a=b — 0, but c not zero, of the 
second order, and so on. 
Then, by considering the typical term in the disturbing function, we have the 
following— 
Rule of approximation for the development of the X-Y-Z functions and of <k(a), 
T(a), R: develop terms of order zero to e 4 ; terms of the first order to e 3 ; terms of 
the second order to e 3 ; and drop terms of the third and fourth orders. 
To obtain further rules of approximation, and for the subsequent developments, we 
now require the following theorem. 
Expansion of cos ( k9-\-/3 ) in poivers of the eccentricity. 
6 is the true longitude of the satellite, flt-\-e the mean longitude, and rn- the 
longitude of the perigee. For the present I shall write simply ft in place of /2^-f-e. 
By the theory of elliptic motion 
12 = 9 —2e sin ( 9 —sr)-|-fe 3 (l + ^e 3 ) sin 2(9 —«r) — ^e 3 sin 3 (9 —ttt)+- 3 - 2 -e 4 sin 4(0— »r) 
If this series be inverted, it will be found that* 
0=/2-f-2e(l — |e 3 ) sin (/2 — ttt) H-^e 2 (l — ^e 2 ) sin 2(/2 — T^+yfe 3 sin 3(/2 — to-) 
+ Vij-e 1, sill 4(/2 — sr) 
* See Tait and Steele’s ‘ Dynamics,’ art. 118, or any other work on elliptic motion. 
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