10 
REPORT— 1847. 
merical aubstitutioDS are also extremely laborioua, in coQsequecce of the mulbtDiie 
of tenxia which have to be considered. 
As the disturbing function, and others which require to be integrated, are fiuQj 
exhibited by M. Hansen in terms of two variables, such that direct integratusu 
itnpossible, M. Hansen han recourse to the iutegration ;;arpor/ie», in 'wbicb ead 
term by integration gives rise to a series of other terms, the nature of which ucoo* 
plicated. 
The method which 1 propose diflers from that su^csted by if. Hansen la ewy 
particular. Instead of attempting a literal development, 1 insert the numcricalwloo 
of the elliptic constants in the earliest possible stage: by this means the 
which expresses the mutual distance of tfie planet*, is explicitly a fnnrtioB rf uw 
and cosines of various angles with numerical coefficients. When f<f', I dewp 
in terms of the eccentric anomaly of m, after having obtained expressioni far tbe 
eo'ordtnatee of m' in terms of the eccentric anomaly of m. Such expression* *** 
easy to obtain, and are very convergent. It will be recollected that before I etiits- 
vou^ to develop the disturbing function in the lunar theory in terms of theffitss 
motions of the sun aud moon, tlie invariable practice had been (seeM^aoiqueCelfstE, 
vol. iii, p. 189 ) to express the co-ordinates of the sun in terms of the tree loogitwi* 
of the muon ; but the equation which connects the eccentric rmomalies of Wobodifi 
is far simpler than that which connects the true anomalies, or c' and c, and tiere- 
fore the conversion which 1 employ U made with greater facility. The 
under the radical sign in R may thus be considered as a function, of which the go** 
ral term can be represented by 
cos V n / 
a being a numerical quantity and omitting the consUnte which accompany v 
-V. The development of this quantity to the power —|or — |, may be faciliWed 
by the use of tables, which give the numerical coefficients in the development of 
{i — ^cosa^ {1 — y/cos&c. Such tables have been calculated faf 
l»y Mr. Farley. 
\\ hen r 2 »-r', that is, when the planet disturbed is superior to the disturbing 
1 am not able to suggest any other course than to develop in terms of the tru* ^ 
roaly of the disturbed planet, and the mean anomaly of the disturbing planeb M 
integrate porparris*. I have obtained the Jaw of the coefficients in the [r 
rraulta m this process, and they arc highly convergent. I am confident fhah®* 
prticesaea which 1 have attempted to describe, the perturbations of plaoete 
10 orbits, eccentric and inclined, may be calculated and may be exhibited io t»bl^ 
giving their values for an indefinite period, if required. If these methods 
M<*aiit8gu« which I ascribe to them, I hope the time U not distant when iw 
turbations of Fallas and of some of the comets may be reduced to a tabular luna> 
but the labour requireil will iw very considerable. ’ 
Although iny meibmU are specially adapted to the determioation of 
tMuuns of bodies moving in eoccutric orbits which cannot be dm'eloped in 
the mean motions, yet they embrace also the case of a planet moving *0 V 
nearly circular; and it is easy to show in what manner the labour is iocreaswi , 
the greater eccentricity. If the reciprocal of the radical which expresses the mstun 
distance of the planets be called 
tile chief difficulty arises in developing f If the numerical valncs oftbe 
elliptic constants are introduced. ^ ^ ‘ ^ 
1 cos cos = {(1 +/>)(H-P-|-y)|{l+P-f?}'' 
