DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
333 
variables x, y, z, u, v, w and t, and the same expressions continue to be their defi- 
nitions when they become the variable elements of the disturbed motion. 
Now the general formulae for the variations of the set of elements here chosen 
have already been given in art. 55 ; for it is evident that the process in the present 
case would merely be a repetition, for each planet, of the process there employed. 
Here however we are to use in every case the disturbing function H given in art. 78 ; 
but if we observe the effect of the marks of exemption^ it will be evident that, for the 
planet m, the only effective part of O is 
UQ{zv—yw)-\-ai{xw — zu)-\-'oo^{yu—xv) — mR ; 
and similarly the effective part of Cl for any other planet will be given by suffixing 
the corresponding number of accents to x, y, z, u, v, w, m, R. 
Now if we put Ao, Aj, for the terras multiplied respectively by cua, ^y,, in the 
above expression, we have, by the definitions of the elements, 
(Ao-f-Ai-|-A2)^=m>/ e^) 
A 2 = —ms/ jW/a(l — e^). cos /, A,=m\/ |M.a(l — e^) .cos v sin < 
Ao= — ms / 1 —e ^) . sin v sin /, 
so that the disturbing function, so far as m is concerned, becomes 
— m{K-{-\/ — sin v sin cos v sin /-j -^2 cos/)} (94.) 
Consequently, since the expressions in art. 55 were obtained by taking — mR for the 
disturbing function, we have merely to add to them the additional terms derived 
from the part added to R in (94.). Performing the differentiations, and omitting 
afterwards the symbols of exemption over oOf,, Oi, which cease to be of any use, we 
obtain, after obvious reductions, the following simple results: if &c. represent 
those parts of the differential coefficients of a, e, &c., with respect to t, which depend 
upon the motion of the axes of coordinates, then 
B(e) Bot 
dt dt 
t . . . 
tan ^(&/, cos v—Mo sin v) — 
— {ct)„ cos sin vj 
y 
— cot I {co^ sin V — cos i') 
j 
(95.) 
where it is evident that we may write s instead of if) (see art. 55.). 
* Not their expressions in terms of x, y, z, x\ y', z' ; for though these are equivalent in the undisturbed equa- 
tions, they are not here equivalent in the disturbed equations, and therefore the general theory, which assumes 
the former mode of expression, is not here applicable to the latter. 
2 Y 2 
