306 
PROFESSOR DONKIN ON THE 
If we put W=Z+n, so that CL may be called the ^'disturbing function,” the above 
forinulse become 
On the first of these forms see the note to art. 38. With respect to the form (69.), 
if we put for {q, c„} its equivalent — [c^, cj, or [c„, cj (see Theorem V. art. 49.), we 
obtain the well-known expression 
The difference between this last form and (69.) consists in this ; that in the latter the 
coefficients [c„, cj are obtained from the expressions for Cj, Cj, &c. in terms of the 
variables whereas in (69.) the coefficients {q, c„} are similarly obtained from the 
expressions for c^, &c. in terms of the normal elements a^, &c., b^, he * ; and when a 
normal solution of the undisturbed problem has been obtained, the latter process will 
generally be found much more convenient than the former, since the elements c„ &c. 
will usually be much simpler functions of the normal elements than of the variables. 
55. In illustration of this, it will be worth while to deduce the expressions for the 
variations of the ordinary elliptic elements of a planet’s orbit from those of the normal 
elements given in art. 30. 
Let a and e be the semiaxis major and excentricity, / the inclination of the orbit to 
a fixed ecliptic, v the longitude of the node, ?zr the longitude of the perihelion, nt-\-{i) 
the mean longitude of the planet ; longitudes being reckoned in the plane of the 
ecliptic (from a fixed origin) as far as the node, and then on the plane of the orbit. As 
usual, n stands for Also let ?z^+(g) = j 
so that ^ ={z)' -\-tn' . 
If, then, we call the six normal elements aj, eta. 
“sj /Sa, /Ba, we have (see art. 30.) 
mu. 
2a 
ft-”-**'. 
n 
a2=/w\/ ^a{ 1 — e^). 
(3^ = wr — t/, 
«3 = ( 1 — e^) . cos /, 
(^3 ^ ? 
from which, conversely. 
mu, 
a =^ 7 -’ 
2«i 
>' —^3, 
«o 
cos /=— J 
«2 
