20 Transacticns of the Royal Society of South Africa. 
It appears, however, that an extension can be made. It will be proved 
that if we possess a development of the disturbing function in which one 
of the eccentricities is supposed to be vanishingly small (in fact zero, but 
with a place of perihelion), we can include the effect of the second eccen- 
tricity by a series of simple operations. Thus, starting with a development 
with e — 0 and = we can arrive at the complete development for 
e and e^. 
Newcorab writes any term of the development as follows : 
eie/a'-4j,jii)C(i'9' - ig + y)r 
in which — 
k li k 
-Pjj: = Uj, + e'^nrf, + elUj'J, + terms factored by e\ e-e\, e\ etc. 
In applying the new method it is a matter of choice which eccentricity is 
assumed to be zero. If each in turn is taken as zero and the two partial 
developments completed, then an independent check is furnished. G-enerally 
as no,j' is larger than it will be better to assume that e, the eccentricity 
of the inner planet, is zero. 
Implicitly the method not only computes the development, but also its 
derivatives ; but as these are actually wanted, it is so much to the good. 
If we suppose that is zero, we will indicate the fact by replacing the 
upright P used by Newcomb by a slanting P ; if e is zero, we must add a 
dash so as to distinguish between— 
Po,o = no.o + eSn^'O + e^n^;S + etc., 
and — 
Po,o = no,o + e^nS;^ e\n% + etc. 
Assuming now that is zero, any term in the development takes 
this form — 
+ Pj_2 C[7y+ (- ^ -i-i)sf + 0" - - H- 1K+ w)] 
+ p_ -2(- i - i)C[iy+ (- i -\-j)g + 0 + - '^)^ 
+ P,--4 {i - 2) C[ig-\- (- i+j)g + {i - 2)K- w) + 2{w'+ w)] 
+ P-j-^{- i - 2) C[ig'-r <i-i^j)g + 0' t 2Xtv' ^ w) - 2(w' -f w)] 
+ etc., etc., etc. ^ 
We cannot write down the general expression for Pj, but we know it 
contains a series of terms — 
n,. 4- e^Yo + e^njfo + etc. 
* Leaded C indicates " cosine." 
