22 
Travsactions of the Royal Society of South Africa. 
Into these coefficients, and until they are insensible, we introduce the 
powers of e by operating firstly with = i( — 16 + D + D^), then with 
JTIo' and so on. , , - 
If these results are added together, the effect is to change P into P. 
The next coefficient will arise from the term — 
efPo,2(3)C[5^' -Sg + S(v/ - w) -i- 0(tv' + w)] 
1 
+ cr2Po.o(4)C[59' - 3^ + 4(w' -w) + l(w' +, w)] 
-t- etc. 
In this case the succession of operators is ellj -\- e^Di -r e^ul-\- etc., and 
so on. 
Similarly the term starting with — 
P„.i(l)C[5/-y+ («'-«,)], 
when operated on with eU\i + e^D^i + etc., 
will give the coefficient P_ 1.4(1) etc., etc. 
In practice each term PC[5^/' — ig + 6] will be brought to the form — 
Po,n'CeCW- ig] - PoySeS[bg' - ig] 
1 
and all the coefficients PC9, PC6\ etc., summed so that we have finally — 
2Po,.'C(9C[5/ - ig - ]2Po,.'S^S[5/ - ig]. 
Then the operator which starts with e''Ulf> + e« + 2j-j« + 2,o _^ ^^^^ acting 
on this sum will yield us— 
2P„.,,C0C[5^' -(i- n)g] - 2P,„,S0S[5^' - (i - n)g], 
which is complete in P,i 
In thus making a complete expansion of the perturbing function and its 
derivatives, the saving of labour will be great. In Newcomb's and Le 
Yerrier's expansions each coefficient requires a separate and sometimes very 
lengthy computation, whereas, as just demonstrated, blocks of coefficients 
are computed in one process. The machinery by which the other eccentricity 
is introduced is uniform throughout and simple. Moreover, it is more 
inclusive. In algebraical expansions of the perturbing function, the com- 
puter must decide at what limit he will stop. In Le Yerrier's theory of 
Jupiter and Saturn, he adopted roughly the 4th and 5th powers of the 
eccentricities. In other words, he was satisfied with such an expansion as 
The new method starting with — 
p n^'^ 4- p3n0,3 , ^5 [-10,5 
which is the same order of approximation, will at the first operation 
introduce — 
