PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES, 155 
5n—2n’ (where 7 and 7/ are the mean motions of Saturn and Jupiter), 
and this factor is very small, on account of the approximate commensura- 
bility of m and n’. In certain cases (called librations) the commensura- 
bility is exact ; thus a linear relation exists between the mean motions of 
three of Jupiter's satellites. 
Tisserand ! in 1887 applied Delaunay’s method of integration to dis- 
cuss the effect of approximate commensurability, showing that commen- 
surability is not inconsistent with the stability of the system. 
Bohlin 2 in 1888-9 gave a new method for treating the cases in which 
terms with small divisors appear likely to endanger the convergence, He 
considers the equation 
Os =—3iBy sin (ju), 
where the coefficients (,; are of the order of the disturbing masses and 
form an absolutely convergent series, and where the independent variable 
w is, in the applications to the planetary theory, a multiple of the time, 
ia 
If in this equation we write w= — 2, om =—p,, we have 
ae dw dp F Sarna) ps 
; apy agate Bt as —Xif3,, sin (7%¢—jw), 
which we can write 
i is pe et) edly DOM ipods Oh 
dx tp, dx cp, da of’ dx dw’ 
where 
H=}p,?—p.—/,; cos (if —jw). 
The solution of this Hamiltonian canonical system depends in the 
ordinary way on the solution of the Hamilton-Jacobi partial differential 
equation 
sV\2__8V 
This is now replaced by the equation 
Ov Non JON 
in order to mark the fact that g and the f’s are small (in the applications 
x* is the mass of the disturbing body) ; and Bohlin integrates this equa- 
tion by expanding V as a power-series in x, 
V=VotcV, te?7Vo+ Ce 
It is found that the occurrence of small divisors can be avoided in the 
series which represent the quantities V,, and hence the original difliculty 
would appear to have been removed, It is, however, possible that large 
1 ‘Sur la commensurabilité des moyens mouvements dans le systéme solaire,’ 
C. R. civ. pp. 259-65 ; Bull. Astr. iv. pp. 183-92. 
2 ‘Ueber eine neue Anniherungsmethode in der Stirungstheorie, Bihang till 
Kgl. Svenska Vet.-ak. Handlingar, xiv. No. 5; ‘Zur Frage der Convergenz der 
Reihenentwickelungen in der Stérungstheorie,’ Ast. Nach. cxxi. pp. 17-24. 
