THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 149 
To have close agreement it is necessary that all sensible terms in the expansion of 
3 
a of @ . . . . 
u (“) and wo?(“) be retained. In the expressions for these functions given a large 
number of terms and some groups of terms have been omitted as they produce no 
terms in the final results of sufficient magnitude to be retained. 
In transforming a series it will be convenient to have the values of the J functions 
on a separate slip of paper, so that by folding the slip vertically we can form the pro- 
ducts at once without writing the separate factors. 
: ° a of @ 2 ° 
The numerical expressions for u(“) and wa?(“) being known, we need next to 
have those designated by (#7) and (J), which represent the action of the disturbing 
body on the Sun. 
To find (#7) we use two methods to serve as checks. We have first 
(HT) = s[hyiyy’ + Wd8] cos (g — 9) — 3 [ys + Uys] sin (g — 9) 
+ alhyy — hoy) cos (—~g — 9) — aly. — ty,0;] sin (— 9 — 9’) 
+ ly! cos(— yf) — Br sin (=) 
+ 2[hyry! + 2'8:8:] cos (gy — 29) 9 — LM.’ + Uys8s] sin (gy — 29’) 
4. 2[hyiy.! — W’8,5:] cos (—g — 29) — 2[ ldo’ — Uy,6,/] sin (— g — 29') 
+ 2hyvy cos (— 29’) — 217d, sin (— 2¢’) 
+ $[hyys! + W0,8.] cos (g — 3g) — $[ldys + U10,'] sin (g — 39’) 
+ ete. 
where 
We Rk b= deh 
(1) (3) () (3) 
y2 = 4 [Jo — J, ] On = I [So oF J ] 
(4) 
by 2) e) 
ye = 4[Ja —Jn] & = 44a + Ya I, 
and similar expressions for 7,', 51’, 7’, 6.’, ete.; noting that y) = — de. 
