288 Transactions of the Boyal Society of South Africa. 
of this the rigorous integrals given by Jacobi in his celebrated paper on 
the " Elimination of the Nodes " are closely approximated to.* 
If we adopt the notation for Jupiter — 
I = inclination of orbit to the invariable plane, 
e = eccentricity of orbit and S0 = e, 
w = mass, 
a = ratio of Jupiter's mean distance to Saturn's, 
and use accented symbols for the same quantities when applied to Saturn, 
and take the constant of attraction as unity, we have (see Charlier, 
''Mechanik des Himmels " formulae of 14," p. 274 of vol. i.) — 
/3 VaC0St-/3,C</),S^,=:O (A) 
/3 JaG(pCi-\-i3^C(p,Ci^ = c„ 
in which — 
^ ml(l-{-m) 
' l + ?7i' ' ' 1 + m + mi' 
and C and S are written for cosine and sine, and Ci is a constant. 
Taking — 
1047-3, 1/?;^, = 3501-6, 
LVa = 9-8682770-10, 
and dividing the equation (A) by [3 J a, we get — 
C</)St - qC(p,Si, = 0 L5 = 9-6076936-10 
C0Ct + ^C0,Ct, = c = 0-4052225, 
in which both q and c are absolute constants. 
Dr. G. W. Hill found by a comparison between the places given by 
his theory of Jupiter and Saturn and those actually observed that the 
reciprocals of the masses were 1047-38 and 3502-2 respectively; these 
values would change the value of Lq to 9-6076314-10, so that one must 
not lay too much stress on the value of this constant. 
If we square and add the two equations, we get — 
C^(p + 2qC(pG(t>,G{i + ti) + 2^0^01 = c\ 
* The not too great inequality of the three masses is also essential, and exists in the 
case of these three bodies — the integrals would furnish no useful information in the case 
of Jupiter, Sun, and Jupiter's 8th satellite, because of the smallness of the mass of JVIII, 
compared with Jupiter and the Sun. 
