270 
[212 
212. 
A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC 
MOTION. 
[From the Memoirs of the Royal Astronomical Society, vol. XXVIL, 1859, pp. 1—29. 
Read March 9, 1858.] 
I venture to take up the problem of disturbed elliptic motion, for the sake of 
a further elaboration of the analytical theory. The points which present difficulty are 
the measurement of longitudes in the varying plane of the orbit, and (in the lunar 
theory) the determination of the position of the orbit by reference to the varying 
plane of the sun’s orbit; it is, in memoirs and works on the lunar and planetary 
theories, often difficult to discover where or how (or whether at all) account is taken of 
these variations, and the analytical mode of treatment is for the most part very imperfect. 
I must except always Hansen’s Fundamenta Nova [investigationis orbitcc vene quam 
Luna perlustrat, Gotha 1838] where the points referred to are treated in a perfectly 
rigorous manner. There is, however, a want of clearness in the form under which his 
investigations are presented; and the comprehension of them is greatly facilitated by 
Jacobi’s remarks, published under the title “Auszug zweier Schreiben des Prof. Jacobi 
an Herrn Director Hansen” (Crelle, t. xlii. pp. 12—31 (1851)). Jacobi observes 
that the integration of Hansen’s system of differential equations introduces seven 
arbitrary constants, which, in the expressions for the coordinates referred to fixed axes, 
reduce themselves to six. The seventh constant, neglecting the disturbing forces, is 
in fact a constant which determines the position in the orbit of the arbitrary origin 
from which the longitudes in orbit are reckoned. I have, in my paper “On Hansen’s 
Lunar Theory,” Quarterly Mathematical Journal, vol. i. pp. 112—125 (1855), [163], 
termed this origin “ the departure-point,” and longitudes measured from it “ departures.” 
The seventh constant may be taken to be the departure of the node. I reproduce in 
the present memoir the explanation of what is meant by the departure when the plane 
of the orbit is variable. If the problem is treated by the method of the variation of the 
elements, the seventh constant becomes, like the other elements, variable ; and we have