Algebraical Development of the Elliptic Perturbative Function. 303 
But before going on, a further important simplification can be intro- 
duced. With Newcomb, we write — 
{Ast. Papers, v., p. 27, &c.) 
in which the operator — 
n:;:? = n«;°xn°::;:. 
Further, as also discovered by Newcomb, the operator can be 
found from IT"- very simply. Hence the real crux is to find expressions 
for IT/. The primary operator in any term will be n;^, the secondary 
n;;''"', the tertiary 11^+% &c., &c. The expresssions to II) in terms of i and 
D are given by Newcomb. The operators with negative suffices are, as 
already mentioned, at least secondary. 
On account of its complexity, it is almost impossible to give a general 
and useful formula for IT]' in terms of the D operators. Such a formula 
starts thus — 
(-i)y(7^.y)n"=+ D« 
/^.. n(n-^) . \^ 
-^^(7 - l^n + 2n^ + 4/) 
n 
+ ^(3u3 _ 227i^ + 4:5n - 26) 
&c., &c. 
In which — 
n-j 
12 3 2 
f(n.j,) = 2"7z\ T ^T^^^ r—^ and f(n.O) = 2«7i! 
•^^ ^ n.n -l.n-2...n+j + ^2 ^ ' 
The extent given will merely serve to 7^ = 2, although it will always serve 
as a check on the coefficients of the three highest D operators in any value 
of n;. 
The primary terms are the more important as the succeeding terms 
are factored by powers of C^, &c. In the Variation of Elements " 
method of computing perturbations, the secondary terms become the 
leading terms in several of the coefficients, but in the final expressions for 
the perturbations^, of the longitude or mean anomaly the primary terms 
reassert their predominance. Thus in the theory of the motion of the 
earth we find the following perturbations of the longitude : — 
