302 Transactioyis of the Boyal Society of South Africa. 
numerical application to the theory of the planets Jupiter and Saturn has 
been commenced, but its completion is still far off. 
The most extended tables of the algebraical development of the 
perturbative function in terms of the mean anomaUes, are due to 
Le Verrier {Les Annates, i. and x.), but a more methodical and general 
development has been given by Newcomb [Astronomical Papers of the 
American Ephemeris, v.). 
Le Verrier's development has been extended by Boquet to terms of 
the 8th order of both the eccentricities and mutual inclination, and this 
limit has also been adopted by Newcomb. The remarks which follow 
will apply more especially to Newcomb's development, but with suitable 
changes they can, if necessary, be adapted to Le Verrier's ; it is not, 
however, likely that any one who is familiar with both developments will 
prefer Le Verrier's — the use of the operator D = a ^ by Newcomb is 
alone sufficient to turn the scale. 
Every term of the development has the form='= — 
eX'Pry'COs(V,+j>+i'^') 
in which k indicates the presence of the factor o-"'" (o-^sine of semi- 
inclination). The order of this term will, be equal to n-\-n'-\-2k. 
ft 
This paper will deal with the properties of the quantities P^./, the 
expressions for which form the only difficulty in the development. As it 
o 123 
is very easy to pass from P (henceforth written P) to P, P, P, &c., 
attention will be confined to the former. 
In any given term w^e find a series of this form — 
e'e{(P;;j; + e^Pj+^-j; + e;Pj: + e^eiP;+^- + &c . ) 
(In which the suffices only may be negative and then only under certain 
conditions.) 
The first quantity Pj;jl will be called the primary term, those multiplied 
by e- or e\, the secondary terms, those by e-e\, e^ or e\ tertiary terms and so 
on. Terms in which the suffices j or j' are negative are secondary or 
higher order terms, although thsy are closely related to the primary 
terms. 
* It is assumed that the reader is conversant with the notation used by Newcomb in 
the papers above referred to. 
