DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
113 
Introducing these expressions in the value of O (art. 41.), we find 
sm 2 jcos2(<p-I); (49.) 
and when <p — l is expressed in terms of the elements and t ^see equations (R.), art. 44, 
in which is now to be written for this becomes, finally, 
n = -?(^-g)sinV.cos2[f-(|(i+g)-i)icosi.(<+r)]. . . (Cl.) 
4 7. The above expression for Cl does not contain the elements i, a ; hence, when 
there are no disturbing forces, we shall have (cos/)' = 0, a' = 0, or i and a are con- 
stant ; also 
7 , dk d£l dk d£l 
dh dr d cos j d[ 3 5 
an expression which is easily found to vanish identically ^see the values of j~-- 
in art. 44, observing to put for x) • Thus k is also constant ; and the “ prin- 
cipal plane” is still the “invariable plane,” as we know a priori. 
48. If we now suppose the attraction of another body to be introduced as a disturb- 
ing force, we shall have to take for the disturbing function 
o=n-p, 
where Cl is the same as above, and P is the potential of one body upon the other, ex- 
pressed as a function of the elements and the time*. And it follows from the remarks 
of the last article, that the variation in the position of the principal plane depends 
wholly upon P, and not upon Cl. 
I shall here conclude this part of the subject, as it would be beyond the scope of 
this essay to enter into the details of any of the various problems which might be 
taken in illustration of the theory, such as those which relate to precession and nuta- 
tion, or to the motion of the moon about its centre of gravity. The investigations of 
this section have been introduced, because the results, so far as they go, appeared 
interesting in themselves, and afforded a remarkable example of the application of 
the general method. 
P.S. Since the last sheets of this essay were in type, I have seen for the first time 
two papers by Professor Brioschi, in Tortolini’s Annali for August and October 
1853, of which the titles are “Sulla variazione delle costanti arbitrarie nei problemi 
della Dinamica,” and “ Intorno ad un teorema de Meccanica.” I have not had an 
opportunity of examining them sufficiently to judge how far any of the preceding 
investigations may have been anticipated in them. 
June 7- 
The variables which determine the position of the disturbing body are supposed to be given explicit func- 
tions of t. 
MDCCCLIV. 
Q 
