ORBITS OF LARGE INCLINATIONS AND ECCENTRICITIES. 593 
— 0-016 sin ( 49) — 0-095 cos (°° 49') 
—0°071 sin (u+4g’) — 0°093 cos (u+ 49’) 
+ 0°022 sin (—3u+5g') —0:005 cos (—3u+ 59’) 
—0°037 sin (—2u+5g') +0°013 cos (—2u+4+ 54’) 
—0°054 sin (—u +59) + 0°024 cos (—u + 59’) 
+ 0°033 sin ( 5 g') — 0°013 cos ( 5g’) 
+ 0°038 sin (wu + 549’) — 0°019 cos (u+ 59’) 
This is the result for the disturbances of Encke’s comet by 
Saturn, and it is the first result of the kind. 
On counting the arguments of the above disturbances in lon- 
gitude, we find them to be forty-six in number, and in the dis- 
turbances of the logarithm of the radius-vector and of the lati- 
tude there are rather fewer. These disturbances consist of pre- 
cisely the same number of terms as of arguments, when, by a 
known transformation, we unite into one each pair of the fore- 
going terms. Under the coefficients of the disturbances of lon- 
gitude there are, if we do not include the secular variations, or 
the pair of terms multiplied by the time, only fourteen argu- 
ments whose coefficients are greater than 1", fifteen whose coef- 
ficients lie between 1" and 0!-1, and fifteen whose coefficients are 
less than 0-1. In the disturbances of the logarithm of the 
radius-vector is found very nearly the same proportion, and, in 
the disturbances of the latitude, all the coefficients, including the 
' two above mentioned, are less than 1”. 
The author here gives a comparison between the foregoing 
absolute disturbances and some relative disturbances computed 
by Encke by means of mechanical quadratures. This compari- 
son, for want of room, it is thought proper to omit here, since 
it will very shortly be published. 
In explanation of the method by which the preceding result 
has been arrived at, the author first treats of the expansion of 
the quantity: unity divided by the ratio of the mutual distance 
of the comet and planet, arranged according to powers of the 
ratio of the radii. The expansions are, as is known, as follow :— 
1 r s - 
Die Dlx 
1 a "= 
where A is the distance, 7 and 7” the fa veh 
Ee 5 U,=2H-4, U;= 7-48; &e. 
