Neiucomh Operators used in Development of Perturhative Function. 339 
In other words, although we can with Herr v. Zeipel write symbohcally 
(putting q = 2) — 
8n= = ( - 2D + 4^)^; + ( - D + i)n° 
we cannot write — 
8n-VA, - ( - 2D + 4i)n:a'A,.+, + ( - D + 2)II>'A^+, 
although we can write — 
m\a'k; = ( - 2D + 3 - H)U\a'k^ + illla'k^ 
as given in my paper above referred to. 
In a methodical application to the theory of the motion of any planet, 
we require n°a'A^, then \\\a'K^, and so on, and once in possession of these, 
(i) (i) 
we proceed to lila' &c. This is the great advantage of the recurrence 
formulae which keep to n^+"", in which the suffix i remains the same. It 
(0 
would be very nice to have expressions in which n;^+"" depended on 
(i+i) 
numbers inferior to i+j, but so far no general expression which would 
facilitate numerical work has been offered. A type of such an expres- 
sion is — 
4n^ = 4n^ - (i + 2) (3^; + 5)6t'A,+, + i{^i - l)a'k\ 
+ (0 
in which IIo for 2 + 2 is built up from the value for i plus simple multiples 
of a'k^ and a' K-^,. 
Johannesburg, 
Fehruary 21, 1913. 
