PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE RODIES, 159 
and 
Pep {tg tt csne} Laas cs 
— {aes aoe tap (as) Py Cae OO 
where 
Par a= aay a 
Also, the equation in x can be written 
az 4 dz ‘ 
Oe 4 2 — (148)Q0 +(148)a%, 
dv? dv 
where Z, is a certain function of the positions of the planets. 
Now let us consider the form in which Gyldén wishes to express the 
solution of these equations. 
The differential equations will finally be solved by means of sums of 
periodic terms whose arguments are linear functions of v; these terms 
may be classified in the following way :— 
Firstly, there will be terms which vanish altogether when the dis- 
turbing mass is put equal to zero ; these are called coordinated terms, and 
correspond to the ‘ periodic inequalities’ of the classical planetary theory. 
Secondly, there will be terms which, when the disturbing mass is put 
equal to zero, do not vanish, but coalesce with the terms which represent 
the Keplerian elliptic motion of the disturbed planet round the sun. 
These terms involve the disturbing mass in their arguments, but not in 
their coefficients ; they are called elementary terms, and correspond to the 
‘secular inequalities’ of the classical planetary theory. Terms will also 
occur in the course of Gyldén’s process which involve the disturbing mass 
in the denominator of their coefficients, and so would become infinite if 
the disturbing mass were put equal to zero; these are called hyper- 
elementary, and it is shown that they do not appear in the final result. 
And, lastly, we have already seen that when the periods of two planets 
are nearly commensurable, certain terms of long period rise to importance ; 
these are called the semi-elementary or characteristic terms for the planet 
under discussion. 
The quantity p, as already defined, will be composed of both elementary 
and coordinated terms. Let (p) denote the elementary terms, and let R 
denote the coordinated terms, so 
p=(0) +R. 
Jt will appear that p is of the form 
ea) 
(p)=« cos {@! ie c)v —T} + poe cos {1 ee c,)v— Let 5) 
n=1 
where « (called the diastemmatic modulus) and T (called the longitude of 
the absolute perihelion at the origin of time) are two of the six constants of 
integration of the problem, «, and T, are functions of the constants of 
integration, and ¢ and ¢, are small constant quantities of the order of the 
perturbing forces, 
