598 HANSEN ON THE PERTURBATIONS OF BODIES IN 
question the three components of the disturbing force, of which 
one is parallel to the major axis and another to the minor axis 
of the comet’s orbit, and the remaining one is perpendicular to 
the plane of the orbit. These are, as is known, the differential 
coefficients (a ), (Z,) , and (=) » where 2 and y have 
the same signification as above. It is clear from the foregoing — 
that their expansions have the same form as that of the quantity 
©. The differential through whose integration we must get the 
disturbances of the coordinates of the disturbing body, be those 
coordinates of what kind soever, we can always reduce to the 
following form :— 
dQ dQ dQ 
(ae) +2 (G,) +8 Gaz) 
where P, Q, and R are functions of the elliptic elements and of 
the coordinates of the disturbed body. Since now with respect to 
these functions, which are also interminable, we must avoid series 
proceeding according to the powers and products of the eccen- 
tricity and the inclination of the orbit of the comet, these functions 
P, Q, and R, in the problem before us, must be whole and rational 
functions of sin uw and cos wu, as also those which arise in the dis- 
turbing function. But since the nature of P, Q, and R depends on 
the choice of coordinates, the choice is by no means a matter of 
indifference. The examination of the different known expres- 
sions for the differentials of the coordinates shows that we can- 
not choose the ¢rue longitude and the radius-vector as the coor- 
dinates in the problem before us, since for these the functions 
P, Q, and R are not whole and rational functions of sin w and 
cos wu. If on the contrary we so arrange the disturbances that 
they must be added to the mean longitude, and to the computed 
elliptic value of the logarithm of the radius-vector, by help of 
the disturbed mean longitude, or, which is the same thing, by 
help of the disturbed true anomaly, then have the functions P, 
Q, and R the property required. The author has, in his treatise 
on perturbations, made these disturbances dependent on an ex- 
pression which he has designated by T. If then we take this 
and substitute in it the above differential coefficients of Q, the 
eccentric anomaly u, and the analogous quantity depending on 
(to be denoted by »), it is reduced to the following :— 
