TRANSACTIONS OP THE SECTIONS. 
11 
4 4 &c. are numerical coefficients, which I here suppose ranged in th e order of 
toDomerical magnitude. 1 make 
cos v^jCOSej^ .... |1 —^jCOS •.Jasl + P + 2 ', 
iedaiiDg 8 limited number of terms in I'l-P-}-;. 
|1—jl —^5C0S«3}^»...&C., 
an be obtained at once by means of a table. But as the coefficients given by such 
liable do not readily famish, by interpolation, the values required unless it be con- 
fldinbiy extended, I take for vf,, Jf, See. the nearest value given by the table, and 
llwt the residue to form part of q. 
5«* 2flo' 
1+P=l——^-cos 
' tr r'a * 
5 is the angle included by lines drawn from the sun to »» and n»', 
iV 
Tbe terms contained in — —cos S obviously exceed greatly in magnitude those 
ff ^ A 
tcuaioed m ji*g. 
-cos 3«i cM/'ri-* sin/'; 
a 
I' Itisg the true anomaly of m', and »the eccentric anomaly of m, 
— e9+3cDsw— Vl— e^liisin v, 
il= — eC+Ccos ^ I— e*B sin i*i 
®'8.CandJ9heing constants, each necessarily leas than unity, which depend only 
inclination of the orbits and the position of their line of intersection and their 
* 1^1 ind such that when they are in the same plane 
0 • iO and 3J C. 
iM process b the same in substance, whether the orbit of m is highly eccentric 
or circular, and in the same plane with that of m' the only difference 
jJ*f that, while in the former case it may be convenient to detach four terms to 
^thequ4ntityi.^p^^. in the latter case, supposing e'tlie eccentricity of n»' to 
^Bsiderable, may conveniently contain only two factors if the deve- 
^tU made in terms of the mean anomalies, and therefore ^ I+P+j}^, &c. 
[ wi& greater facility. 
case of orbits situated in tfie same plane and not very eccentric, as in the 
U p of the older planets, if their inclination to the ecliptic be neglected 
v? may contain only one factor, 
1— /<co8 n'f— i'}, 
i^^^Joogitudes are introduced; but if the mean anomalies are retained, 
IKrnaps preferable, then there will generally be two factors, 
{l--^iCOs(£-? 0 J-{l-^ 28 inC«-l')}. 
iJ!f ? hlie quantity (1+P+Q)^ depends not only upon 
which it is desirable to retain, but also on the magnitude of 
K ““wrical coefficients. 
(H-;>)(l.fP+y)=l-q, 
{l+P}-5=Cl+P+y)^/l+3q4.|:5qa+&c J. 
'*^culation of the series 
“"•ilttto 
hbat of the series 
