134. REPORT—1899, 
on the lines of Delaunay’s lunar theory ; Tisserand extended Lindstedt’s 
theorem, and in 1887 Lindstedt! showed how this extension could be 
derived from his own original paper. An imperfection in Lindstedt’s 
first paper was removed by Poincaré? in 1886, who, by an ingenious 
application of Green’s theorem, proved that only one secular term appears 
in each of Lindstedt’s approximations, and that this can always be 
removed. 
In 1889 Poincaré* gave a fresh method of derivation for Lindstedt’s 
series, by transforming the Gyldén-Lindstedt differential equation into a 
Hamiltonian system of the fourth order, replacing this by the corre- 
sponding Hamilton-Jacobi partial differential equation, and solving the 
latter by a series proceeding in ascending powers of a small parameter, 
the coefticients being trigonometric series of two arguments. Poincaré 
observes, however, that in the problem of three bodies this method will 
not apply if Kepler’s ellipse is taken as the first approximation, and 
consequently another intermediary orbit must be used. 
The number of independent arguments required in order to express 
the coordinates in the problem of 7 bodies, without having the time out- 
side trigonometric functions, was shown by Harzer* in 1889 to be 
3(n—1). 
The question of the convergence of sums of periodic terms, such as are 
obtained in Lindstedt’s expansions, had now become a matter of prime 
importance, Poincaré? in 1882-4 showed, firstly, that if such a series is 
absolutely convergent for certain values of the time, it is so for all values 
of the time ; and, secondly, that a function cannot be represented by two 
different absolutely convergent series of this kind. Further, a function 
represented by such a series can assume indefinitely great values if the 
convergence is not uniform. In a further note ° in 1885, he showed that 
this can happen in two ways: either the function may steadily increase, 
or its value may oscillate, the amplitude of the oscillations increasing 
indefinitely. Bruns‘ in 1884 further discussed the series of Dynamical 
Astronomy : as these are usually obtained by the integration of trigono- 
metric series, it follows that the coefficients of those terms whose periods 
are very long will be affected by very small divisors. Bruns shows that 
this causes the series to fluctuate between convergence and divergence, 
when the constants, on which the coefficients of the time in the arguments 
depend, are altered in value by small amounts. 
Features of special interest present themselves in the planetary theory 
when the periods of two planets are very nearly commensurate with each 
other. In this case some of the inequalities of long period rise to im- 
portance; thus, in the theory of Jupiter and Saturn an inequality with 
a period of 900 years has a large coefficient; the grandeur of this 
coefficient is due to the fact that its denominator contains a factor 
1 ¢Ueber ein Theorem des Herrn Tisserand aus der St6rungstheorie,’ Acta Math. 
x. pp. 381-4. 
2 «Sur une méthode de M. Lindstedt, Bull. Astr. iii. pp. 57-61. 
3 ‘Sur les séries de M. Lindstedt,’ C. R. eviii. pp. 21-4. 
4 ‘Ueber die Argumente des Problems der m-K6rper,’ Astr. Nach. cxx. pp. 
193-218. 
5 ‘Sur les séries trigonométriques,’ C.R. xcv. pp. 766-8, xcvii. pp. 1471-3; ‘Sur la 
convergence des séries trigonométriques,’ Bull. Astr. i. pp. 319-27. 
6 «Sur les séries trigonométriques,’ C. #. ci. p. 131. 
7 *Bemerkungen zur Theorie der allgemeinen Stérungen,’ Ast. Nach. cix. pp. 
215-22 
