THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
815 
Reducing these to numbers with j= 5° 9', i= 23° 28'. we have 
Si=ll*-86.* 
Hence, if the earth were homogeneous, at the present time we should have S j as the 
inclination of the proper plane of the lunar orbit to the ecliptic, and Si as the amplitude 
of the 19-yearly nutation, These are very small angles, and therefore initially the 
method of Part II. was applicable. 
* The formulas here used for the amplitude of the 19-yearly nutation and for the inclination of the 
lunar proper plane to the ecliptic differ so much from those given by other writers that it will be well to 
prove their identity. 
Laplace (‘Mec. Cel.,’ liv. vii., chap. 2) gives as the inclination of the proper plane to the ecliptic 
a.p —2" a-0 
9 — 1 
—s- sin X cos X 
Here a.p is the earth’s ellipticity, and is my t; o0 is the ratio of equatorial centrifugal force to gravity, 
and is my v?ajg, it is therefore -fe when the earth is homogeneous. 
Thus his a.p —4a0= my -ft. His g — 1 is the ratio of the angular velocity of the nodes to that of the 
moon, and is therefore my (o — /3)//2. His I) is the earth's mean radius, and is my a. His a is the moon’s 
mean distance, and is my c. His X is the obliquity, and is my i. Thus his formula is 4 —, sin i cos i 
in my notation. 
i — P c 2 
Now my t=3/iw/2c 3 , and &a?=C/M. 
Therefore the formula becomes 
But by (5) Cf2cl/iMm=k. 
Therefore it becomes 
2 
T t 
a. — ft 
(Sic) 
fiMm 
sin 2 i 
sin 2 i 
Now by (115) and (112), when |f=l, a=/cTCcos) cos 2 j. 
Therefore in my notation Laplace’s result for the inclination of the lunar proper plane to the ecliptic is 
1 
2 
a sin 2i 
- ,.sec j 
a — ft COS 2 j 
This agrees with the result (260) in the text, from which the amount of oscillation of the lunar orbit 
was computed, save as to the sec j. Since j is small the discrepancy is slight, and I believe my form to 
be the more accurate. 
Laplace states that the inclination is 20"’023 (centesimal) if the earth be heterogeneous, and 41”'470 
(centesimal) if homogeneous. Since 41'' - 470 (centes.) = 13"'44, this result agrees very closely with mine. 
The difference of Laplace’s data explains the discrepancy. 
If it be desired to apply my formula to the heterogeneous earth we must take J- of my h, because the 
•f of the formula (6) for s will be replaced by v nearly. Also t, which is -f kwi must be replaced by the 
precessional constant, which is ’003272. Hence my previous result in the text must be multiplied by 
f- of 232 x’003272 or ’6326. This factor reduces the 13''T3 of the text to 8"’31. Laplace’s result 
(20"’023 centes.) is 6"’49. Hence there is a small discrepancy in the results; but it must be remembered 
that Laplace’s value of the actual ellipticity (1/334 instead of 1/295) of the earth was considerably in 
error. The more correct result is I think 8"’31. The amount of this inequality was found by Bukg and 
5 m 2 
