Algebraical Developnent of the Elliptic Perturhative Function. 305 
'Chessin — 
n5=6n^ + 2{i - - {ii - 2)n^- - - ^\{2i- i)n^ - 3|^(5. - i)n: - ^i 
Chessin's formula has two advantages; firstly, it is much shorter;, 
secondly, it throws the computation of the secondary terms on to those of 
the first. In ordinary course the primary terms must be computed first, 
and it is fitting that they should form the basis for the secondary and 
higher terms. But neither Newcomb nor Chessin gave a general 
expression for the operators, and, in point of fact, it requires a very 
lengthy calculation to check, far less to extend, Newcomb's tables. 
Let us suppose that we are dealing with the primary operator !!« ; 
Newcomb's formula starts thus — 
10321920n^ = 256i8-8960^7 + 130592^6_l023120i5 + 462807^4 + &c. ' 
plus terms in D, D^... to D^. 
The figures bear no obvious relation to each other, and have to be 
taken on trust ; besides this, the computation for a particular value of i is 
-exceedingly arduous. The length of the expression for 11^, &c., is 
.appalling ; it is true that the use of the Chessin formulae would shorten 
the calculation, but Chessin does not give the constants his method 
requires for operators beyond Ul and Chessin's constants for Wl are, 
however, not difiicult to find, as they are connected with certain Besselian 
series for e used in Hansen's planetary theory. 
Newcomb gives, on p. 13 of vol. v. of the Astronom. Papers of the 
American Ephemeris, the general recurrent formula by which he builds 
.up n"; it is — 
2(7i + i)nS;-(A;;^ + /<D)n;: 
+{k':^+h:j))uiz\ 
+ 
+ {KX\}i + KX\^)ui 
in which the factors h and h depend upon the developments of the 
■equation of the centre and the logarithm of the radius vector in terms 
of the mean anomaly. The succession of the values of the A;'s given by 
Newcomb (p. 22 loc. cit.) are — 
o 5 13 103 1097 1223 47273 556403 
'2' 4' 24' 192' 160' 4608' 40320 
What the next fraction will be is by no means clear, but it will be seen 
immediately that the values are really not wanted, as a transformation at 
once discloses the values of the successive terms. Let us write out the 
Newcomb values for U\, HI in the following form : — 
