ASTRONOMY: A. 0. LEUSCHNER 
69 
and solution, as these involve mathematical detail. I can touch here only 
on a few points which must serve to illustrate the general character of the 
work. But first of all it would seem necessary briefly to review what objects 
are to be served in determining perturbations. 
When a planet moves solely under the central attraction of the Sun it 
describes an ellipse defined in space by six elements or constants, of which 
for our present purposes three are of prime importance, namely the mean 
daily motion, or briefly, mean motion, which is the average angular velocity 
about the Sun, or 360° divided by the period of revolution, in days; the eccen- 
tricity of the ellipse; and the inclination of the plane of its motion to the 
ecliptic. The six elements at any instant depend on the three coordinates 
defining the position of the planet at the instant and on the three components 
of velocity. When the constant elements are ascertained and the body 
continues to move solely under the Sun's attraction, the position and 
velocity of the planet may be obtained from them at any other time and vice 
versa. Such positions and velocities are called undisturbed positions and 
velocities. But if to the Sun's attraction is added that of one or more major 
planets, like Jupiter, then the velocity is changed and the planet will depart 
from its undisturbed ellipse. These departures are called perturbations. 
From a disturbed position and velocity at any instant elements may again 
be determined for that instant. They represent disturbed elements, as com- 
pared with the former. Thus different elliptic elements may be computed 
for different instants from the corresponding positions and velocities attained 
under the attraction of the Sun and major planets. These are called osculat- 
ing elements. Any one set may be adopted as undisturbed and the differ- 
ences of the others from that set represent the perturbations in the elements. 
Whether the perturbations are expressed in the elements or in the coordi- 
nates, they are determined by the integration of differential equaticns, the 
solution of which in the present state of astronomical science involves expan- 
sion in trigonometrical series. The integration introduces divisors which 
become very small, when the mean motion of the minor planet is in a com- 
mensurable ratio to that of a disturbing planet. Therefore since Jupiter's 
mean motion is 299'' or nearly 300,'' such series fail for instance for a planet 
with mean motion nearly 300" or 600" or 900". Since Hansen's method, to 
the application of which our work was originally to be restricted, is based on 
series of the character referred to, it was bound to fail for planets with a 
commensurable ratio. This was our actual experience with the first planet 
undertaken. 
This difficulty, however, is purely a mathematical one and has been over- 
come by Bohlin in his ''Gruppenweise Berechnung der Stoerungen" for the 
group 1/3 by the introduction of the simple expedient of using the exponential 
for the trigonometrical form in the series. In fact he has published tables 
for the exact commensurability 1/3 in which all of the elements appear ex- 
plicitly in the coefficients. The series referred to progress according to 
