( 626 ) 
$ 6. When astronomers try to obtain in the Theory of the distur¬ 
bances of the movements of the planets by the application of the 
method of Lagrange expansions in series for the coordinates of the 
planets or the elements of their orbits, then terms may appear with 
abnormally large coefficients in consequence of small divisors, ori¬ 
ginating from the integration. This takes place when between the 
inverse values of the periods of revolution of some planets a linear 
relation with integer coefficients is almost fulfilled. Besides some 
other properties the terms are also distinguished according to their 
class, by which is meant: 
where a represents the exponent of [i (a small quantity indicating 
the order of greatness of the disturbing function), m the exponent 
of t, m' the exponent of the small divisor, as they appear in the 
coefficient of the term indicated. Now it is the terms of the lowest 
class which we have to take into consideration if we wish to make the 
expansions in series to hold for a long space of time. By Delaunay 
a method is indicated to determine the terms of the lowest class It 
consists principally in omitting all terms of short period (period 
comparable to the periods of the revolution of the planets) in 
the disturbing function and retaining the most important of the 
others. (Comp. e. g. H. Poincar£, Lemons de mecanique celeste, vol I 
page 341). 
The problem under discussion has much resemblance with the one 
mentioned from the theory of disturbances. In the preceding 4 in 
omitting some terms in R we have imitated what is done in the 
theory of disturbances. 
ft is easy to see that the terms omitted have really no influence 
on the first approximation, when we consider the terms which appear 
e. g. m «, by introduction of such a term. 
Osculating curves. 
* 7 In § 5 we have found that the movement of the horizontal 
projection of the material point might be represented by: 
V«, 
y ~ ^ + *»,&); 
