THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 49 
We notice that 
x) (-1) 
dE ) S = — 36, 
(0) 
0 == alk 
For all other values of 7 
(i) 
We a=)! 
If a large number of the -/ functions are needed they are computed by means of 
equations (24) to (27), as shown in the example given in Chapter V. 
If we wish to determine any of them independently we have from 
(m) _— pmjm we na i he, 28 
Ne aL Tem ) L.2m-lm+2 — 1.2.38.m2lm2m+3 = ete. |, 
fe Ge 
Fe= li-F Gta a a S| 
ie = ae pie oe ect | ) (30) 
mote D Peta 
oe be 
In these expressions we have written for @ its value 3¢. 
(m) 
Since h has all values from hk = +o to— o we find any value of J, by at- 
tributing proper values to h. 
From equations (29) we find the values of the functions cos Ze, sin ze, in terms of 
cos hg, sin hg, and the J functions just given; always noting that when 4 = 0, we 
have only for 7 = +1, — $e as the value of the function. 
We can employ equation (22) when only a few functions are needed, or as a 
check. 
A. P. §.— VOL. XIX. G. 
