138 REPORT—1899, 
§ V. Gyldén’s Theory of Absolute Orbits. 
In 1881 Hugo Gyldén, Director of the Observatory at Stockholm, began 
the publication of a new method for calculating the motions of the 
heavenly bodies. The method has been made of practical importance by 
its application, in the hands of Gyldén’s pupils, to the minor planets, and 
is of theoretical interest from the fact that (as in Delaunay’s, Newcomb’s, 
and Lindstedt’s memoirs) the time appears only in the arguments of 
periodic terms. In this report it seems best to give, first of all, a general 
account of the method, and then briefly to notice the series of memoirs in 
which Gyldén and his pupils have developed it. 
Consider, then, a system consisting of the sun and two planets. For 
convenience one of these will be spoken of as the distwrbed and the other 
as the disturbing planet. At any instant the motion of the disturbed 
planet is taking place in a certain (moving) plane, which passes through 
the sun; this we can call the plane of its instantaneous orbit ; in this 
plane we take (as an axis from which to measure angles) a line which 
moves with the plane in such a way that the surface formed by the 
motion of the line always cuts the plane orthogonally. The angle between 
this line and the radius vector to the planet can be called the planet’s 
true longitude, and denoted by v ; the radius vector from the sun to the 
planet will be denoted by +. 
The quantities 7, v are given by differential equations of the form 
af ,dv\ 4-60 aa S M_wy cQ 
di (" a )aMae Bush ap TN ge (i aaah? 
where © (which is called the disturbing function) is supposed to be 
expressed in terms of 7, v, and the quantities which define the moving 
plane and the position of the disturbing planet, and where M is the sum 
of the masses of the sun and the disturbed body. . 
Let the perpendicular distance of the disturbed body from some fixed 
plane be zr. Then the third differential equation of the disturbed body’s 
motion may be written in the form 
2 > 
aa + Mzr=a function of the positions of the planets. 
Now introduce new variables p, n, 8 connected with the old variables 
by the relations 
_2(1—n?) pd? — \Ma(1— n?)}} 
hos seer dt hee ary 
where @ is a constant called the protometer, as yet undetermined ; as there 
are only two conditions here to determine the three quantities p, n, S we 
can impose another condition later. 
The equations for » and v now transform into 
oA DB og431_lb an? 
1+Sdv — ‘eae 5 serra dv 
